David Mumford
Tata Lectures
on Theta II
Birkhauser
Almost periodic solution of K-dV given by the genus 2
2
p-function D2log-& (z, St) with 92 = 10
2
10
An infinite train of fast solitons crosses an infinite
train of slower solitons (see Ch. II2a,§10,IIIb,§4).
Two slow waves appear in the pictures: Note that each
is shifted backward at every collision with a fast
wave.
David Mumford
With the collaboration of C. Musili, M. Nori,
E. Previato, M. Stillman, and H. Umemura
Tata Lectures
on Theta 11
Jacobian theta functions and
differential equations
1984
Birkhauser
Boston Basel
Stuttgart
Author:
David Mumford
Department of Mathematics
Harvard University
Cambridge, Massachusetts 02138
Library of Congress Cataloging in Publication Data
(Revised for volume 2)
Mumford, David.
Tata lectures on theta.
(Progress in mathematics ; v. 28,
)
Vol. 2 has title: Tata lectures on theta.
"Contains ... lectures given at the Tata Institute
of Fundamental Research in the period October 1978 to
March 1979" - v. 1, p. ix.
Includes bibliographical references.
Contents: 1. Introduction and motivation : theta
functions in one variable ; basic results on theta functions
in several variables - 2. Jacobian theta functions and
differential equations.
1. Functions, Theta. I. Tata Institute of Fundamental
Research. II. Title. III. Series: Progress in mathematics (Cambridge, Mass.) ; 28, etc.
QA345.M85
1982
515.9'84
82-22619
ISBN 3-7643-3109-7 (Switzerland : v. 1)
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Mumford, David:
Tata lectures on theta / David Mumford. With the
collab. of C. Musili... - Boston ; Basel ; Stuttgart :
Birkhauser
Progress in mathematics ; ...)
2. - Mumford, David: Jacobian theta functions and
differential equations
Mumford, David:
Jacobian theta functions and differential equations /
David Mumford. With the collab. of C. Musili... Boston ; Basel ; Stuttgart : Birkhauser, 1984.
(Tata lectures on theta / David Mumford ; 2) (Pro=
gress in mathematics; Vol. 43)
ISBN 3-7643-3110-0 (Basel, Stuttgart)
ISBN 0-8176-3110-0 (Boston)
NE: 2. GT
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without prior permission of
the copyright owner.
o Birkhauser Boston, Inc., 1984
ISBN 0-8176-3110-0
ISBN 3-7643-3110-0
Printed in USA
CHAPTER III
Jacobian theta functions and
Differential E uatiohs
Introduction
IIIa:
§0.
§1.
("92)
§3.
§4.
An Elementary Construction of H
ix
erelli tic Jacobians
Review of background in algebraic geometry
Divisors on hyperelliptic curves
3.1
3.12
Algebraic construction of the Jacobian of a
hyperelliptic curve
The translation-invariant vector fields
Neumann's dynamical system
3.28
3.40
3.51
Tying together the analytic Jacobian and algebraic
Jacobian
3.75
§6.
Theta characteristics and the fundamental
Vanishing Property
3.95
§7.
Frobenius' theta formula
3.106
§8.
Thomae's formula and moduli of hyperelliptic curves
3.120
§9.
Characterization of hyperelliptic period matrices
3.137
§10.
The hyperelliptic fir-function
3.155
§11.
The Korteweg-deVries dynamical system
3.177
IIib:
Fay's Trisecant Identity for Jacobian theta functions
The Prime Form E(x,y)
Fay's Trisecant Identity
3.207
§2.
§3.
Corollaries of the Identity
3.223
§4.
Applications to solutions of differential equations
3;.239
§5.
The generalized Jacobian of a singular curve and solutions
3.243
§1.
IIIc:
Resolutions of Algebraic F uations b
by Hiroshi Umemura
Bibliography
3.214
Theta Constants,
3.261
3.271
Introduction to Chapter III
In the first chapter of this book, we analyzed the classical
analytic function
(z T) =
I
e7T in2T+21T inz
of 2 variables, explained its functional equations and their
geometric significance and gave some idea of its arithmetic
applications.
In the second chapter, we indicated how 17
generalizes when the scalar z is replaced by a vector variable
z E Cg
and the scalar
T
by a gxg symmetric period matrix 0.
The geometry was more elaborate, and it led us to the concept of
abelian-varieties:
space.
complex tori embeddable in complex projective
we also saw how these functions arise naturally if we
start from a compact Riemann surface X of genus g and attempt
to construct meromorphic functions on X by the same methods used
when g = 1.
However, a very fundamental fact is that as soon as g >
the set of gxg symmetric matrices
n
4,
which arise as period
matrices of Riemann.surfaces C depends on fewer parameters than
g(g+l)/2, the number of variables in
that the
ing tori
0.
Therefore, one expects
Q's coming from Riemann surfaces C, and the correspondXX, also known as the Jacobian variety Jac(C) of C,
will have special properties.
Surprisingly, these special
properties are rather subtle.
I have given elsewhere.
(Curves and their Jacobians, Univ. of Mich. Press, 1975), a
survey of some of these special properties.
What I want to
x
explain in this chapter are some of the special function-
theoretic properties that * possesses when
Riemann surface.
0 comes from a
One of the most striking properties is that
from these special 4's one can produce solutions of many
important non-linear partial differential equations that have
arisen in applied mathematics.
For an arbitrary
0, general
considerations of functional dependence say that 4 (zSl) must
always satisfy many non-linear PDE's:
equations are not known explicitly.
interesting problem.
but if g > 4, these
Describing them is a very
But in contrast when
Sl
comes from a?
Riemann surface, and especially when the Riemann surface is hyperelliptic,
-S
low degree.
satisfies quite simple non-linear PDE's of fairly
The best known examples are the Korteweg-de Vries
(or KdV) equation and the Sine-Gordan equation in the hyperelliptic
case, and somewhat more complicated Kadomstev-Petriashvili (or KP)
equation for general Riemann surfaces.
We wish to explain these
facts in this chapter.
The structure of the chapter was dictated by a second goal,
however.
As background, let me recall that for all g > 2, the
natural projective embeddings of the general tori
X.
lie in
very high-dimensional projective space, e.g., IP3g--1) or IP45_1)
and their image in these projective spaces is given by an even
larger set of polynomials equations derived from Riemann's theta
relation.
The complexity of this set of equations has long been
a major obstacle in the theory of abelian varieties.
It forced
mathematicians, notably A. Weil, to develop the theory of these
varieties purely abstractly without the possibility of
motivating or illustrating results with explicit projective
examples of dimension greater than 1.
I was really delighted,
therefore, when I found that J. Moser's use of hyperelliptic
theta functions to solve certain non-linear ordinary differential
equations leads directly to a very simple projective model of the
corresponding tori XQ.
It turned out that the ideas behind this
model in fact go back to early work of Jacobi himself (Crelle, 32,
1846).
It therefore seemed that these elementary models, and
their applications to ODE's and PDE's are a very good introduction
to the general algebro-geometric theory of abelian varieties,
and this Chapter attempts to provide such an introduction.
In the same spirit, one can also use hyperelliptic theta
functions to solve explicitly algebraic equations of arbitrary
degree.
It was shown by Hermite and Kronecker that algebraic
equations of degree 5 can be solved by elliptic modular functions
and elliptic integrals.
H. Umemura, developing ideas of Jordan,
has, shown how a simple expression involving hyperelliptic theta
functions and hyperelliptic integrals can be used to write down
the roots of any algebraic equation.
He has kindly written up
his theory as a continuation of the exposition below.
The outline of the book is as follows.
The first part
deals entirely with hyperelliptic theta functions and hyperelliptic
jacobians:
§0
reviews the basic definitions of algebraic
geometry, making the book self-contained for
analysts without geometric background.
xii
§§1-4 present the basic projective model of hyperelliptic
jacobians and Moser's
use of this model to solve the
Neumann system of ODE's.
§5 links the present theory with that of Ch. 2, §92-3.
§§6-9 shows how this theory can be used to solve the
problem of characterizing hyperelliptic period matrices
0
among all matrices
2.
This result is new, but it
is such a natural application of the theory that we
include it here rather than in a paper.
§§10-11 discuss the theory of McKean-vanMoerbeke, which
describes "all" the differential identities satisfied
theta functions, and especially the
Matveev-Iits formula giving a solution of Kd V.
We
present the Adler-Gel'fand-Manin-et-al description of
Kd V as a completely-integrable dynamical system in the
space of pseudo-differential operators.
The second part of the chapter takes up general jacobian theta
functions (i.e., ,9
Riemann surface).
,0) for 0 the period matrix of an arbitrary
The fundamental special property that all such
.D's have is expressed by the "trisecant" identity, due to John Fay
(Theta functions on Riemann Surface, Springer Lecture Notes 352),
and the Chapter is organized around this identity:
§1 is a preliminary discussion of the "Prime form" E(x,y)
- a gadget defined on a compact Riemann surface X which
vanishes iff x = Y.
§2 presents the identity.
§§3--4 specialize the identity and derive the formulae for
solutions of the KP equation (in general) and KdV,
Sine-Gordan (in the hyperelliptic case).
§5 is only loosely related, but I felt it was a mistake
not to include a discussion of how algebraic geometry
describes and explains the soliton solutions to KdV
as limits of the theta-function solutions when g of
the 2g cycles on X are "pinched".
The third part of the chapter by Hiroshi Umemura derives the
formula mentioned above for the roots of an arbitrary algebraic
equation in terms of hyperelliptic theta functions and
hyperelliptic integrals.
There are two striking unsolved problems in this area:
the first, already mentioned, is to find the differential
identities in z satisfied by
(z,SZ) for general
second is called the "Schottky problem":
jacobians
X2
The
to characterize the
among all abelian varieties, or to characterize
the period matrices
U2
of Riemann surfaces among all
problem can be understood in many ways;
geometric properties of
of zeroes of
U.
XS
U.
The
(a) one can seek
and especially of the divisor e
,9-(Z,Q) to characterize jacobians or (b) one can
seek a set of modular forms in
comes from a Riemann surface.
U whose vanishing implies
One can also simplify the
problem by (a) seeking only a generic characterization:
conditions that define the jacobians plus possibly some other
irritating components, or (b) seeking identities involving
xiv
auxiliary variables:
is a jacobian iff
the characterization then says that
X0
3 choices of the auxiliary variables such
that the identities hold.
In any case, as this book goes to
press substantial progress is being made on this exciting
problem.
I refer the reader to forthcoming papers:
E. Arbarello, C. De Concini, on a set of equations
characterizing Riemann matrices,
T. Shiota, Soliton equations and the Schottky problem,
B. van Geemen, Siegel modular forms vanishing on the
moduli space of curves,
G. Welters, On flexes of the Kummer varieties.
The material for this book dates from lectures at the
Tata Institute of Fundamental Research (Spring 1979), Harvard
University (fall 1979) and University of Montreal (Summer 1980).
Unfortunately, my purgatory as Chairman at Harvard has delayed
their final preparation for 3 years.
I want to thank many
people for help and permissions, especially Emma Previato for
taking notes that are the basis of Ch. Iila, Mike Stillman for
taking notes that are the basis of Ch. Ilib, Gert Sabidusi for
giving permission to include the Montreal section here rather
than in their publications, and S. Ramanathan for giving permission to include the T.I.F.R. section here.
Finally, I would
like to thank Birkhauser-Boston for their continuing encouragement
and meticulous care.
3.1
§0.
Review of background in algebraic geometry.
We shall work over the complex field C.
Definition 0.1.
An affine variety is a subset X c (Cn, defined
as the set of zeroes of a prime ideal
{x E cnlf(x) = 0 for all f EP }1).
V(P) or by V(f,...;fk) if
4)
a (C[Xl,...,Xn]; X
X will sometimes be denoted by
D.
generate
A morphism between two affine varieties X,Y is a polynomial map
f: X ->Y, i.e., if (X1'...,Xn)E X, then the point f (X1,..., X) has coordinates Yi =fi(Xl, ...fi(n),
where fi E cE[Xl,.... Xn];
following this definition, we will identify
isomorphic varieties, possibly lying in different (dimensional) Cn's.
A variety is endowed with several structures:
a)
2 topologies; the "complex topology", induced as a subspace
of (Cn, with a basis for the open sets given by
{(xl,...,xn)Ilxi-ail< s, all i}, and the "Zariski topology" with
basis {(x1,...,xn)lf(x)
b)
0},
f E (E[Xl,...,Xn].
the affine ring R(X)= CE[Xl,..,Xn]/p, which can be viewed as
a subring of the ring of (C-valued functions on X since D
is the
kernel of the restriction homomorphism defined on CE-valued polynomial
functions on Cn, by the Nullstellensatz.
c)
the function field (C(X), which is the field of fractions of
R(X); the local rings
and
subvariety of X, defined by a x =
&Y,X, where x is a point, Y a
{f/gIf,g ER(X) and g(x)
01, with
maximal ideal m{ = {f/g EV I f('x) = 0 } , C5`Y,X = { f/gl f,g E R(X) ,g t 0 on y}
= R(X)Q 2) if Y = Y(q) ;
1) If a polynomial f ECE[X
is zero at every point of V then
f E3 ; this is Hilbert 's Nullnstellensatz.
2) We denote by A the localization of a domain A with respect to its
prime ideal Q,q Aq = {a/bja,b EA, b
q}.
3.2
if x E Y, R(X) c C. c
(notice:
c T(X)the structure sheaf (Y,
subsheaf of the constant sheaf UXi --- T (X) , which assigns to any
Zariski-open subset U of X the ring 0U ax = r (u,cX) ; and a dimension
given by dim X = tr.dtC(X). dim X is related to the Krull dimension of
Oy,X
(maximum length of a chain of prime ideals), by:
Q Proposition 0.2.
d)
dim X - dim Y = Krull dim. V xX
the Zariski tangent-space at x E X, which can be defined
in a number of equivalent ways:
TX,x = vector space of derivations d: R(X) - T centered at x
(i.e., satisfying the product rule d(fg) = f(x)dg+g(x)df); or
TX, x = (rft / fl2) V ,
the space of linear functions on 1v/m2 ; or
such that for all
TX,x = the space of n-tuples
0 mod e2
f f
where from a derivation d a linear function £(Xi-xi) = dXi
and an
n-tuple (xl,...,xn) with dXi = xi are obtained; this sets up the
bijection.
(xi)a/3x;
This vector is also written customarily as
i=1
Proposition O.D3
such that
3
a non-empty Zariski open subset U c X
tr.d.T T(X) = dim TX,x
for all
x E U; if
x 0 U, then
dim TX x > dim X.
O U is called the set of "smooth" points of X, X-U the "singular
locus".
It can be shown from this proposition that U (with the complex
topology) is locally homeomorphic to T,d where d is tr.deg.TCC(X).
3.3
For any
Lemma 0.4
x E X,
.3
a fundamental system of Zariski
neighborhoods U of x such that U is isomorphic to an affine variety.
In fact, for any f E R(X) such that f(x)
0, Uf = {y E x(f(y) 3g 0}
is a neighborhood of x and if RX = T[X1,...,X ] /', then Uf is isomorphic to the sub-
variety of
Tn+l
is defined by the ideal
the isomorphism is
realized by
-1
(x1,...rxn)
(xl,...,xn, .f(xl,...,xn))-
But we need a more subtle definition of morphism from an open set
to an affine variety.
Definition 0.5.
f: U
is a morphism if (equivalently):
Y
flopen
x
(1) for any
on Y,
g E R(Y), thought of as a complex-valued function
gof E r(U,O )
(2)
3 gikhk E (C[X1,---,X AI
there is a suitable k such that
is given by
such that for any (x1,---,xn) E
hk(x)
34
0, and the i-th coordinate of
gik(x1,-.-,xn)
hk(xl,...,xn
whenever
(n.b.. there may not exist a single expression
with
h--1
U
)ti=
E r(u,O'x) .,
Theorem 0.6 (Weak Zariski's Main Theorem).
hk(x)
If
0.
h(X ,--,X n
1
'
f: X - Y is
-el
an injective morphism between affine varieties of the same dimension
and
Y
is smooth, then f is an isomorphism of X with an open subset of Y.
3.4
X
QThe product of affine varieties is categorical, i.e., given
Y c Cm
(,n and
affine varieties, i) X x Y is an affine variety.
(in (Cn+m),
ii) the projections are morphisms,
variety and morphisms
unique morphism
Z -> X,
of affine varieties:
are given, then there is a
making a commutative diagram
Z --> X x Y
Definition 0.7
Z -- ;;,Y
iii) if Z is an affine
A variety in general is obtained by an atlas
X =
a S
Xa, S a finite set,
n
Xa c ( a, glued
by isomorphisms
(where Ua,s is a nonempty Zariski-o en subset of Xa), such that one
of the equivalent (se aration) conditions holds:
(1)
X is Eausdorff in the "complex topology" (a subset of ,X
being open in the complex topology if and only if its intersections
with Xa
are open for all
a's)
(2)
the graph of
(3)
for any valuation ring
a$r,, c
function field of Uf c Uas
Xa X Xa is Zariski-closed.
R c T(X) = C(Xa)
(any a, for the
coincides with that of Xa, hence
identifies E(Xa) and VX)) there is at most one point
x E X
has
3.5
such that
i.e.,
R > (Y
R D(Y
(R "dominates" 0 x, or R is "centered" at x,
and inR = rrX) .
for all affine varieties Y and morphisms
(4)
set
X
{y E YIf(y) = g(y)}
f,g: Y --.X, the
is (Zariski) closed in It.
Such an X carries:
(a'))
2 topologies (the complex and the Zariski; as with the
complex topology, a subset of X is Zariski-open if and only if its
intersection with all the Xa's is Zariski-open)
the function field Q(X); the local rings)
(c')
{f/gIf,g E R(Xa), g(x)
n
Ui
xEU
res f:
if
x E X
a
;
the structure sheaf
ox
the Zariski tangent space Tx,X =
(d')
0 f:
0}
Tx,X
if x E Xa.
X --- Y is a morphism between two varieties if the, restriction
Ua f f-l(Va) ---ov
is a morphism for all a,$'s or, equivalently,
if. for any open set U c Y and g E r (U,0 )
,
gof E r (f-lu,
)
is
satisfied.
Key example.
Projective varieties, defined by homogeneous
ideals P c cE(X0,...,Xn1, as
V(P) _
{(xo,..-,xn) E 7Pn If(x) = 0 for all f E}7 };
an atlas is given by V(). = f p E v (p) Xi (p) 4 0} .
variety X can even be defined as a set of local rings {Cr) with
the same fraction field M(X). Then the topology on X is defined as
follows - for each f ET(X), let Uf be the set of the local rings
containing f.
3.6
The product of varieties is again a variety; we take (UxV)Zf
to be a basis for the open sets in
g
XxY, where U,V are open subsets
of X,Y isomorphic to affine varieties, fi E r(U,OX), giE F(V,( ) and
(UXV)Efigl
is the set of points (x,y) E UxV
Efi (x) gi (y)
such that
0-
The product of projective varieties is again a projective variety,
for instance the map (xi,yj) i (xiyj) embeds 7Pn x3Pm
into
g, (n+1) (m+l)-1 and the image is given on the affine pieces
by the equations
(7Pn+m+nm)X
where
sij = sihskj for all i
h and j # k,
hk
sij = Xij/Xhk'
Definition 0.8.
A variety X is complete (or proper if one of
the following equivalent condition holds:
(1)
X
is compact in the complex topology
(2)
3
a surjective birational morphism f: X' --> X, X' projective
(3)
for all valuation rings R c OW, 3 x E X such that R y ax
(4)
for all varieties Y, Z c XxY closed, pr2Z
is closed in Y.
A subvariety of a variety X is an irreducible locally closed
subset Y of X; the variety structure is given by the sheaf (7Y which
assigns to any open subset V of Y the ring
r(v,OY) =
T-valued functions f on V
b x E V, 3 a neighborhood U
of x in X and a function
f1E r(U,OX) such that
f = restriction to UOV of fl
So, any open subset of X is a subvariety; but a subvariety which
is a complete variety must be closed.
3.7
Divisors and linear systems.
The theory of divisors is based on a fundamental result of Krull.
if R is a noetherian integrally closed integral domain, then
a)
minimal prime ideal, R
for all p c R,
is a discrete
valuation ring.
b)
fl
R=
4D
R
min.
P
.
prime
Thus if Ordd
= valuation attached to RP, and K is the fraction field of R,
we get an exact sequence:
1 -> R* > K* ---D
free abed.. group
on min. prime ideals
W
v1
f d>
ord
f- [P )
=
(f)
be the primes occurring positively in (f),
Let
P
, ... , }
negatively in (f), then
to
m
in R,
For all prime ideals p
Corollary 0.10.
fER
f-1 E R
neither f or f-1 are in R
"f is indeterminate at
l?
(in particular, if f is indeterminate at
minimal prime ideal).
3)
,
any i
any i,
+
hence
for
some
then 3) is not a
i,j
3.8
We will apply Krull's result to the following geometrical
situation:
Theorem 0.11:
X = J Xa
if
is a smooth variety, then RX a is
integrally closed, the minimal primes
J
in RX
one (closed) subvarieties Y of X which meet Xa
Idea of the proof:
being smooth means
are the codimension
and (RX ) P _ ay X.
for all points P E X,,, the hypothesis of
dim
/M2
= dim X = Krull-dim. ,, i.e., ( p is
'regular" (this can be taken as a definition).
One proves that a
regular local ring is integrally closed, hence C is integrally
closed.
Since, for any affine variety,
RX
a
is integrally closed.
Lemma 0,12
=
f,
PEXa 0 P
5),
RX
a
The rest of the statement follows from the:
A (closed) subvariety Y of Z is maximal
dim Y = dim Z-l.
'(This follows from (o;%), or else can be used to prove (o.2.) . )
Thus the map f {. >(f) defines a homomorphism
T (X) *
>Div X =
free abel. group
on codim. 1 subvar.
Elements of Div X are called divisors on X and 2 divisors D1,D2 are
called linearly equivalent (written
Dl='D 2)
if D1-D2 = (f), some fET(X)*.
0 The corollary 0.10 has the following geometrical meaning:
any f EQZX)*, set (f) = (f)O-(f).
for
with M0 (zero-divisor) and (.f).,
(pole-divisor) both positive divisors, and let, for any divisor
D =
5)
IniYi,
supp D = U Yi; then
e,, consider the ideal A = {zERX
since x/yE 0j,
PEX
y
a
al
x/y can be written w/z, with wERX , zERX MP, so P MP. Therefore A is not
If x/yE f,
contained in any maximal ideal, s8 A = Ra P. This means that lEA, i.e., 2ERx
a
y
a
.
3.9
fEt9'
f
4=
supp (f)
P
c c, 4
.
P / supp (f) o
supp (f) o n Supp (f).'
0 Moreover, if X is a smooth, affine variety of dimension 1 with affine
f is indeterminate At P 44 P E
ring R, then R is a Dedekind domain, so all its ideals are products
of prime ideals.
_
(f)
If fER, let:
En.Y. where Yi corresponds to the prime ideal
Vi in R.
Then:
Corollary
n
f-R
=T7 :Pii
i
We define Div+(X) to be the semi-group in Div(X) of divisors with
only positive coefficients.
We define Pic(X) as the cokernel:
T (X) *
r Div X
Pic (X)
}
0
i.e., as the obstruction to finding rational functions with given
zeroes and poles. Elements of Pic(X) are called divisor classes.
Example.
Pic (En) = Z.
In fact, any hypersurface is given by
the zeroes of a homogeneous polynomial.
D = EniYi
is defined by
The degree of a divisor
deg D = Enideg Yi where
deg Yi
is the
degree of the irreducible homogeneous polynomial defining it.
any divisor of degree zero comes from a rational function, and
degree gives an isomorphism
Pic(1Pn)
-.>2Z
Then
3.10
Suppose D is 4 positive divisor; we define the vector space
Y_(D) =
Note;
{fEu(X')*i
(f)+D > 0} u {0},
The condition (f)+D > 0 is equivalent to M. < 0 '(the
poles of f are bounded by D).
Note that f(D) is a sub-vector space
of T (X) .
Lemma 0.15.
If X is proper, dim t (D)<c
and for all f.E T(X)*
(f) = 0 if and only if fE(C'
0 In this case, we form the associated projective space 2
(s..,(D))
of one-dimensional subspaces of Y-(D) and note:
F (x(D) ) !-'
[,-1 (irD)
n Div+ (X)l _
W
W
line{a.flaek} +
Divfibre
+
through D of
(X)) Pic(X)
)divisor (f)+D
These projective spaces and their linear subspaces are the so-called
"linear systems" of divisors.
If
L c
,DI
B(L) =
1P (.,x (D))
is denoted
D J.
is a linear subspace of dimension k, set
()
Supp E, the "base locus" of L.
The fundamental construction associated to linear systems is the map
PL :
where
x i
Lv
(X-B(L))
* 1111
is the projective space of hyperplanes in L, given by
> [hyperplane in L consisting of the EEL s.t. x E Supp EJ
cpL isa morphism.
let's choose a
To prove this and to describe
cpL explicitly,
3.11
projective basis of L, i.e., k+1 points which are not contained in a
hyperplane:
E, E+(fl) ,E+(f2) ,... ,E+(fk) .
Set fo = 1; the map
x'-- (f o(x)...,fk(x))
is defined on the open set X-Supp E since the poles of fi are all
contained in Supp E;
it coincides with q7L on X-Supp E, as we
see if we let coordinates on L
x
E
Supp(E +
be
and note:
k
(
I c.f.))
i=0
c.
i=O
for x
Supp E,
f . (x) = 0.
1
hence f.L(x) = hyperplane in L with coefficients fo(x),--, fk(x)
= pt. of LO with homogeneous coordinates
---
fk (X).
3.12
Divisors on hyperelliptic curves.
§1.
Given a finite number of distinct elements a i E T, i E S, let
(t-ai) . We form the plane curve C1 defined by the equation
f (t) =
iES
f (t) .
s2 =
The polynomial s2_f(t) is irreducible, so (s2-f(t)) is a prime ideal, and C1 is
a 1-dimensional affine variety in C2. in fact, C1 is smooth.
To
prove this, we will calculate the dimension of the Zariski-tangent
space at each point, i.e., the space of solutions (s,t) E
T2
to
the equation
fl(t+e t-ai )
(s+e s)2
mod
e
2
for (s,t) E C1.
That is equivalent to the equation
2ss = t .
T (t-a
I
j ES i#j
if s
0, the solutions are all linearly dependent since
s'= 0, we get from the equation of the
s = 2s( I T (t-a )); if
j ES i34j
curve
) ;
1
hence
Ti (t-ai) = 0,
iES
0 = t.TT(a.-a.), so
0.
t = ai
for some i; thus
Thus at all points, the Zariski
tangent space is one-dimensional.
We add points at infinity by introducing a second chart:
C2 :
s2=
i T(l-ait')
iES
s' 2 = to .TT (1-ait')
iES
if #S
2k
if #S = 2k--1,
3.13
glued by the isomorphism
t' = 1
t
s
s
between the open sets t
The points at
tk
0 of C1 and t'
0 of C2.
of C1 are:
Col 1100
given by t' = 0, s' _ +1 if #S even
2
w
00
to
t' = 0 = s'
if #S odd.
On the resulting variety C we can define a morphism
n:
C
3P1.
then define
u
Let t and t' = 1/t be affine coordinates in 7P 1
by
(s,t)'+- t, on the chart Cl.
(s' , t') 6
n
>
t', on the chart C2 .
is 2:1 except over the set B of the "branch points" consisting
in the ai's, and o
in the case #S odd.
is therefore an even number 2k'in both cases.
The number of branch points
Topologically C is a surface with
k-l handles, so we say that it is of genus g = k-l;- this is called the genus of
the curve.
This is usually visualized by defining 2 continuous functions
+,,T(t) , - f t
1
for t E JP -(k"cuts") and reconstructing C by glueing the 2 open
pieces of C defined by s = + f t) and s = - f t):
k disjoint cuts on each
copy of 1PI
glueing ai's, ai's
3.14
Since C is smooth,
the af fine rings of C1 and C2 are Dedekind domainsl),
and their local rings V are discrete valuation rings,
1:
(s,t) t> (-s,t)
is an automorphism of C, that flips the sheets of the covering, hence
is an involution, with the set of orbits C/{±l}
k (o1 + 00 2
]P1.
Tr(B) is the
if #S is even
D=
if #S is odd.
2koo
is a basis for the vector space
Lemma 1.1.
(D).
1) We already know that the tangent space to the curve at each point
has the right dimension, in each of the two affine pieces; but it's also
easy to see directly that T[t,s]/(s2-II(t-a i))=R is integrally closed,
the reason being that f(t-ai) is a square-free discriminant over the
If we let a
U.F.D. cC[t].
(s,t)
of R is
to
(-s,t),
a+bs,
be the automorphism which sends
then the general element of the quotient field
with a,bE(C(t), and for all a+b.s integral over R,
(a+bs) +r; (a+b s) = 2a
and
(a+b s) a (a+b s)
=
a2-b2d are in V t) and are
integral over U[tl, which is integrally closed.
Thus 2aEC[t],
a2-db2ECC[tl, so db2ET[tl; since d is square-free and
U.F.D. we conclude b E T ['tl , hence a+bs ER.
M[t] is
3.15
The function field of X,
Proof:
II (] ,
(a (t) [
has an
involution over (C(t), that interchanges1'002, or fixes the
point
hence sends 2 (D) into itself.
,
the sum of the +1 and --1 eigenspaces of
Thus
,F_ (D) = [i(D) n T (t) ] ® [d° (D) n SC (t) ]
If
h(t)
V(D) splits into
i
.
n r(t), since it has no poles for finite values of t,
E Y (D)
then it must be a polynomial in t, h E (C[t].
the case #S even the maximum ideal of
}
On the other hand, in
= R(C2)
is
generated by t' since the equation of the curve gives
s'-l = ((s'+1)-1 (TI(l--t'ai)-1) E(t')R(C2) (t',s'_l); in the case
the max. ideal of C = R (C 2)
is generated by s', since
t' =
vOP
l
#S odd
S,2 (II (1-t' ai)) -1; thus
( t ')
or
= -v l (t) =1 (similarly v
2
(t' )= 1) , A
v (t') = -v. (t) = 2,
CX) 1 + 00 2, #S even
i.e.,
(t)
2°°
So in order for (h),
to be < D
Now consider h (t) E X (D) n s(C (t) ,
,
#S odd
we must have deg h < k.
h = sg (t) with g (t) E Q(t);
g may only have poles in C1 where s has zeroes, i.e., in the set
{Pi
...
(O, ai) } .
The order of vanishing of s at Pi is 1 and that of
(t-a1) is 2, since the max. ideal
and
(t-a1)
= S2 (
11
in_
(gyp
(t-a.)) l.
T(t
is generated by s and
1 That
prevents g (t)
E
3.16
from having a pole at Pi, because the product sg(t) would still have
a pole at Pi.
Thus the only poles of g(t) must be at
must be a polynomial
-., i.e., g(t)
in t; but now
k (°°l+°°2) #S even
#S odd
(s't k
_ (2k--1)-
hence g(t) must be constant in order to have D+(sg(t)) > 0.
This proves the lemma. Ci
Now a projective base for
since W. =
IDI
D+(s)
is
;
either k(O1+02) or 2k0
we have D+(tk) =
2=
where
0
is a branch point in the 2nd case, and where 01,02 are the
two points in the fiber over the point t = 0 of IP1 in the 1st case. I
Hence
contains D and
1DI
whose supports are disjoint,
hence 4IDI) is empty. F]
Thus explicitly, corresponding to IDI, we get
by
(s,t)
F
and (s',t')I
> (l,t,t2,-..,tk,s)
--. (t'k...... ,t',l,s')
Note that these 2 maps
do agree on the overlap:
(l,t,t2,,---tk,s)
(PIDI: C
on
on
`IC
7pk+l
Supp D = Ci,
C2.
t
This map, which is an isomorphism of C with its image, makes C
into a projective curve.`
3.17
If : 1P1
Remark.
a b
at+b
(
fi(t) = ct+d'
li"e''
P1P1
(c d)
is a linear fractional transformation
E SL(2,(D)), then it is not hard to check
that the two hyperelliptic curves whose sets of branch points are,
OB)
respectively, B and
are isomorphic.
assume that #S is always odd by sending one
So we can henceforth
branch po:int,to
Co.
Our aim is to describe a variety of divisors on C, and from
this the Jacobian variety of C; the idea of this construction is due
and appeared in
originally to Jacobi
"Uber eine neue Methode zur Integration der hyperelliptischen
Differential leichun en and fiber die rationale Form ihrer vollstandigen
algebraischen Integralgleichungen"
Crelle, 32, 1846.
of Div(C) given by all the
Let's consider the subset Divv(C)
divisors of degree
v
:
LL Divv(C),
Div(C) =
and inside the set
vE7,L
of
ositivelones
Div + ` v j j those with the following property:
f CV(
I
Div+,v
(C) =
Div+,v
(C) =
D E Div+'v (C)
I
if D =
o
i=1
Pi r
then P
and Pi 34 t (P
all i
all ij
invdl
Our basic idea is to associate to
D
E
Div+'v(C) three polynomials:
V
(a) U(t) _
(b)
fl (t-t(Pi)), monic of degree v
i=l
If the Pi's are distinct, let
Y (J
(1,a'(
(t(Pi) is the value of
at Pi)
3.18
V (t) _
IT (t-t(Pj))
s (Pi)l
i=1
V
TT (t (Pi ) -t (P
jEi
V(t) is the unique polynomial of degree <
V (t (Pi) ) = s (Pi) ,
If
P.
1<
i.
J
) )
v-1 such that
<v
has positive multiplicity in D, then we want to "approximate
the function
V-f--(t)
up to the order
mD(Pi)", and in order to do
that we let
V (t) =
the unique polynomial of degree <
v-l such that, if
mD (Pi) = ni,
)'[V(t) -
it
(t-a2)]
=0
LES
for 0 < j -< ni -1
t=t(Pt)
By construction f(t)-V(t)2 is divisible by U(t), hence
(c)
Define W (t) by:
f(t)-V(t) 2 = U (t) .W (t) .
Let's assume v < g+1; then deg V(t)2
manic in t of degree
< deg f(t) and since U(t) is
v , W is manic of degree 2g+l
-v.
Conversely, given any U,V,W such that f-V2 = UW, U and W manic,
having degrees as above, we get the divisor (U)o of
v
points on the
t-line; over each zero of U, the corresponding value of V gives a
square root of f(t), i.e., a value of s (either one of ±4f-(t) );
thus the divisor of the points on the curve so obtained isin Div'v.
0
3.19
Remark.
Given U,V,W
satisfying the above equation, then in
ct[s,t];
s2_f (t) _ (s -V (t)) (s+V (t) ) -U (t) W (t) , hence
(s2-f(t)) C (U (t) , s-V (t)),
and the bigger ideal defines a zero-dimensional subset of C, which
is in fact supp D, or a zero-dimensional subscheme, which is D.
We have now proven:
'Proposition 1.2.
Diva'
and
There is a bijection between
triples of
polynomials
U,V,W
f-V2=UW, U, W
are monic,
deg 'V < v-l, deg U = v, deg W =2g+1- v
Notice how the bijection gives us a way to introduce
coordinates into
U(t) =
Div +'y (x):
0
tv
+
V(t) =
t2g+l-v
W(t) =
let
Ulty-1
+... + Uv
Vlty-1
+... + Vv
+ W0t2g-v
+... + W29-v
be 3 polynomials with indeterminant coefficients, and expand:
2g
f - V2
UW =
I
a a(Ui,Vj,WQ)ta
a=0
Then, taking Ui,Vj,Wk as coordinates;
[the set of triples (U,V,W) as above] =
c
(r2g+l+v_
3.20
Or else, since U and V determine W whenever the division is
possible, we can write using the Euclidean algorithm:
f (t) -V (t) 2 = U (t) .
[t2g+l-v+B0
(Ui,V)
t2g_v+..
V.)tv-l+...+R
]+Rl( 1,
(U1,V)
remainder
Using only Ui,Vj as coordinates, we find:
[the set of triples (U,V,W) as above] ~
c
2v
.
The structure of affine variety is the same in both cases because the
morphisms:
(Ui), vi , k)
projection
(Ui,Vj'Bk (U. 1v
))
>
(Ui,V
and
(Ui,Vj )
=*
are inverse of one another.
On the other hand, we can parametrize Div+'v(C) by points of C,
in the following way:
we have a surjective map
Cv
now let (C9)o c CV
)
Div+'v(C)
be the Zariski open set defined as follows:
3.21
_
V
CV - (Cv)o = [ U pi1(°°) Jut
i=1
U pij ()
O<iaj<i
'A- i<j.V
where
pi:
is the i-th projection, pij: CV - ;:C2 the
CV
> C
(i, j) -th projection, r = [locus of points (P,1 P)1
closed subset of C2 given by the equations sl = -s
(si,ti,s2,t2) are coordinates.
(
t1 = t2 if
Then everything is tied together in:
a prime ideal
(C[Ui,Vj,WzI, the variety
is smooth and the composite map
,a2g) is a surjective morphism making
(orbit space for the group of permutations
CV)o ---->-> Divo`v (C) = V(a0,
V($, -.,a2g) =
iJ\
2,
The equations
Proposition 1.3.
in
1 the Zariski
,_..
I
acting on (Cv) o
Sv
1
The proof of proposition 1.3 will consist of 2 steps.
1.
In order to prove that V(aa) is smooth, let's consider a
small perturbation of the coordinates (U1,...,Uy,V1,...,Vv,WO,...,2g`v)'
Starting with any solution U,V,W to the equation f-V2 = UW
(with
prescribed degrees) we will show that the vector space of triples
U,V,W
(deg U,V <,v-l, deg W < 2g-v) such that
f- (V+EV) 2
has dimension
v
=
(U+sU) (W+cW) mod
s2
(*)
.
The dimension must be > v
since in general
k equations in
n-dimensional affine space define a closed set whose irreducible
components are varieties of dimension > n-k; which in our case means
>
(2g+l+v) - (2g+l) =
v.
3.22
On the other hand, the condition (*) is equivalent to the equation
Uw + Uw + 2VV = 0.
(*)
If we can prove that any polynomial of degree < 2g can be written
in the form TjW + UW + 2VV, then the number of linear conditions
imposed by (*) equals the dimension of the space of polynomials
in t of degree < 2g, which is 2g4 l, and we conclude that the dimension
of the space of solutions of (*) equals (2g+l+v)-(2g+l) = vV.
But
notice:
WU gives all the polynomials that are multiples
of U
2VV assumes any given values at the points t where U = 0, V
tt
UW
of
it
If
to
tt
It
These make a vector space of dimension (2g-v+1)+v
n
it
U = 0 , W ..F
= 2g+l, since
U = V = W = 0 never happens, or f_= V2+UW would have a double zero.
2.
(CV)o - V(aa)
is a morphism.
First observe that the map is a
morphism on the smaller Zariski-open set (C)) oo = (Cv)o - U pi1(6),
7
i<j
where
is the diagonal.
o c CXC
of U,V
This is because the coefficients
(hence W) were given above by an explicit formula as
rational functions in the coordinates s(Pi),t(Pi) with denominators
products of t(Pi)-t(Pj).
or P. =
I(P
(i
These denominators are zero only if Pi = Pa
j), i.e., only on
pg(/ UT").
To see that the map is a morphism elsewhere, we use Newton's
Interpolation formula.
This is expressed in
0
0
3.23
Theorem 1.4 (Newton):
U
Let
f
be a Cm
function on an open set
(resp. an analytic function on an open set
c kt
Define
U c (L).
by induction on n
f(xl,...Ixn) =
Then
f
is a em
f (xl . ' . , xn-1) -- f (x2 , ' , xn)
x1 _ xn
function (resp. an analytic function) on Un,
symmetric in its n arguments
xi,
and for all
f (x) - [f(a1)+(x-)f(a11)++(x-a1) -f (av...n ,a )] +--a f
i=l
Note that the expression in brackets is therefore the unique
polynomial V(x) of degree < n-l such that:
(
-)k [f(x)-V(x)l
0<k<
0,
Or
Ix=a
(
of a. \
equal to a
)
--
1
.
To apply this to our problem, we define by induction on n rational
functions on Cn by:
S(P1,..,Pn)
=
t (P1) -t (Pn)
As in Newton's theorem,. it is an easy calculation that
is symmetric in P1,.-,Pn.
I claim
3.24
For
n = 2, note that
s (P'P)
12
s2 (P1)-S 2 (P2)
s (P1) -s (P2)
t (P1
-t
(P1 +s (P2) ]
f (t (P1)) -f (t (P2)) 1
t (P1)-t(P2)
1
-
s(P)+s(P2)
1
polynomial in
t(P1) ,t(P2)
Thus s(P1,P2) has no poles in the open set t(P1) 76 t(P2) nor in the
open set s(P1)
-s(P2)
The union of these 2 open sets is C2-r
since t(P1) = t(P2) and s(P1)
-s(P2) implies P2 = L(P1).
s(P1,P2) has no poles on (C2)o, hence is in
Thus
r((C2)o, 0 2).
For
C
n > 3, by induction and, the expression for s(P1' 'Pn),
has poles only if t(P1) = t(Pn).
if t(P2) = t(Pn) too.
But by symmetry, it has poles only
The subset t(P1) = t(P2) = t(Pn) has
codimension 2 in (Cn)o, so
has no poles at all in (Cn)o.
Finally, by Newton's theorem, the interpolating polynomial V(t)
can be expressed by:
v-1
V(t) = s(P1)+(t-t(P1)).s(P1,P 2)+...+ TT
i=1
Thus the coefficients Vi of V(t) are polynomials in t(Pi) and
hence are functions in
that (C') 0
--
r((Cv)0, 0 )
C
V(a) is a morphism.
.
This proves
3.25
A consequence is that the set V(aa).is irreducible since (CV)o
maps onto V(a
and (Cu)o is irreducible.
To complete the proof,
use the elementary:
Lemma 1.5:
If
V c (Cn is an affine variety,
f1'...'fk E T[X1,...'Xn
are polynomials such that
V=xE
(Cn
p x EV, TV,x = ;x E
fi (x) = 0 all
I
n fi(x+ek) = 0 mod
e2,
all i}
is the prime ideal of all 2olynomials zero on V.
then
(Proof omitted).
We want to emphasize at this point the rather unorthodox use
U,V,W:
that we are making of the polynomials
we have a bijection
divisors D on C
(of
a certain type
three polynomials
(U
(t) ,V (t) ,W (t) ofa certain type
Thus
b)
these divisors D become the points of a variety for
which the coefficients of U,V,W are coordinates.
To take the coefficients of certain auxiliary polynomials as
coordinates for a new variety is quite typical of moduli
constructions, although it is usually not.so explicitly carried out.
In all of this Chapter, U,V,W will play the main role, and we will
talk of (U,V,W) as representing a point of the variety Divo'v(C).
3.26
Actually, for any smooth projective curve X,
describe Div
+,V
it's possible to
(X) as a projective variety, although not as
explicitly asinthe above construction.
we outline this without
giving details, as it will not be used later.
we use the bijection
under the action of the
Div +'v(X) - Symmv(X), the orbit space of Xv
symmetric group permuting the factors.
A)
X r
Given an embedding
>
3P n,
we have the associated
Segre embedding:
j
X'
Pa
=
>
2P
(n+l)v -1
given by:
V
(Xpa),...,Xna))EX
v
(a)
(P1,...,Pv) r--- (..., TT
Xa(W
a=l
then
(one coordinate for every map a
B)
;
j is equivariant under the action of the symmetric group on Xv
v
on the homogeneous coordinate ring of X, R =
S
Sv
acts preserving the grading; the ring of invariants R v
is finitely generated by homogeneous polynomials and
are a basis of R. in degree M, then:
such that if
Sd
"h
OM
S
R V
(r[g0r...rgN
I
Ctt mehfS w,oqree.
ivisible by M
a
3.27
C)
Via gi we have the embedding:
Symm VX C
Xv
D)
]PN
3P (n+1)V -1
C
The smoothness of Symm V X follows from local analytic
description:
SymmV(z-disc.)
{open set in a
}
biholomorphically
via
->
Celem. symm. functions of z(Pi)l.
The explicit coordinates given by prop. 1.2 are particular to
the case of hyperelliptic curves.
3.28
§2. Algebraic construction of the Jacobian of a hyperelliptic curve.
Let's recall that a hyperelliptic
equation
curve C is determined by an
s2 = f(t), where f is a polynomial of degree 2g+l; C has
one point at infinity, and (t). = 2--
(s) .
= (2g+1) --.
We shall study the structure of Pic C = {group of divisors modulo
linear equivalence}.
Since the degree of the divisor (f) of a rational function is
zero, there is a homomorphism
deg:
P
i
Definition 2.1.
The Jacobian variet
Jac C = Ker[deg: Pic C
`
of C is given by:
2Z]
we wish to endow Jac(C) with the structure of an algebraic variety.
dO-Cl
The possibility of doing this by purely algebraic constructions was discussedsby A. Weil.
In the hyperelliptic case, his construction becomes
quite explicit.
For the general case, see Serre
Step I.
that
Pi # -
[
].
Given any g+l points on the curve P1.....Tg+1, such
and P i # 1Pj if i # j, the function
gs+4(t)
tr (t-t(Pi))
i=l
where
(t(Pi)) = s(Pdeg c
¢ g,
3.29
has simple poles at all of the Pi's and no poles anywhere else, for
qhd
numerator and denominator are both zero at LPi,Athe numerator is 54 0
at Pil) . At
(2g+l) -°°.
(s)ue
(t) )co < 2g.co
g+l
(n (t-t(Pi))
so the function is -z
(2g+2) co,
i=l
Thus :
r-o.-.
C", -c
g+l
Pi
00
Q1 ., suitable Q's.v'
+
i=1
i=l
t-a
Consider also the function
1
for any numbers a,b; it gives
an equality of divisor classes:
P
a
+ tP
-
a
t-a
t-a
t-b) o
_
Qt-b)
Pb
+ t Pb
Similarly, using the function t-a, we see:
Pa + tPa = 2=
9)
degree 2 that
Let's define ]G to be the divisor class of
-
ntains
con
P+tP, all P E C.2)
I
The above remarks show that for every divisor D of degree zero
g
SPl,...,P
such that
D
P. .
Q
degree 0, D =
In fact for any D of
Q
R.
i=1
also write
- g-03.
i=1
g
S. _
--
i=1
1
GRi + IlSi
- P1Si+Si) =- JRi +Jt i-22-co;
whenever a pair R+tR occurs in
JRi+ ItSi; now we can
use the construction above in order to decrease the number of points
Or if Pi is a branch point, then the numerator vanishes to 1st order
and the denominator to 2nd order at P..
2)
th1
Also calledAfundamental "pencil" on
e h perelliptic curve; "pencil"
because the projective dimension of IP+tPI= W(d(P+i P) )
is 1. In the
affine part of the curve C1 c STS, IP+tPI is cut out on C1 by the pencil
of lines through the point at infinity of the curve.
3.30
in
R; + iSi to < g.
Therefore the map:
I:
SymmgC
Ip
`
Jac C
Ipi-g..'
i
i
is surjective.
(This in fact is true for every curve.)
Step II.
Pi, Pi
Given a divisor
ZP. if i > j,
-, Pi
i=1
then
X
a non constant rational function on C whose poles are
bounded by
IP
i.
g
Proof.
Let h be such a function; then h II (t-t(Pi)) has poles
i=l
a polynomial in the affine coordinates s,t,
only at
where
i.e., it has the form
q
and
iy
are polynomials;
wlhrcti ''
now
vco(s) = 2g+lAodd, vo,(flt)) is even, hence v,(std) #
0 = v,,, (h) = v. (fl
(pi))/
>
v,,. (sip)
so
- 2g = 1+v (V) > 1,
which is a contradiction
(t) = 0, i.e., h is a function of t only, hot = h; this
unless
implies that the poles of h are bounded by
I1Pi also, thus h cannot
have poles, i.e., is a constant.
Definition 2.2. 0 = [subset of Jac C of divisor classes of the
g-l
form
i=1
P.-(g-l)°'
].
1
Steps I and II imply that a suitable restriction of the map I is
injective:
3.31
divisors
Pi such that
i=l
z
if i `j
Pi co
" Jac C - 0
res I
SymmgC
->a
Jac C
in OL4
,by Step II, I(D) = I(D') for-D E Z implies D = D' because a function
such that D'-D = (h) would have poles only on
a constant; in particular
D =
z n 1-10 = ¢ since
0
7 P., hence be
i=l i
is the image of,
g-l
Pi + CO.
Pi-g--
Now if we represent any divisor class in Jac C--S
by Step I, then
i=1
P. is in Z, because if P
_
as
or
1
Pi = 1Pi, i.e.,
Pi + Pi = 2--,
then
Pi-q--) E
(
S.
1
By the previous section, Z is a smooth g-dimensional variety;
by translation, we will cover Jac C by affine pieces isomorphic to Z.
Step III.
defined by
Recall that B c C is the set of branch points P,
P = iP: thus 2P = L, for all P E B.
Definition 2.3.
Let T c B be a subset of even cardinality;
define
eT = (
PET
P)( LT-)
EJac C
3.32
Lemma 2.4.
a) 2eT = 0
where T1oT2=(T1UT2)-(T1fT2),
eT1+eT2 = eT16T2
b)
(symmetric. difference)
c) e T1 = e, if and only if T 1 = T2 or T1 = CT2
12
the complement of T2
,,
in
Thus,the set of the eT's forms a group isomorphic to (a/2Z2)2'.
Proof.
12P - (#T)L E 0.
2eT =
a)
PET
#T1+#T2
_
b)
eTl+eT2
occur once in
P +
I
PET1
PET P + PET P
1
P
I
the P's that
2
(
2
are those in
T1OT2; the others can be
PET2
cancelled against L's because 2P = L, and the multiple k of L in
+e
e
T1
is determined by
P) = 2k.
deg(
PET1oT2
T2
c)
eT + eCT = eg =
1 P - (g+1)L:
PEB
the function s has
a simple zero at each of the branch points except'
(S)CO = (2g+l) - -, so
0
=
-- =)- (2g+1)'
(s)
cEB
and
,
eB.
To prove the converse, it is enough to check that if T
then eT # 0.
=
(h)w _
1
or B
By replacing if necessary T by CT, we may assume
#T < g+l and, in the case
I P
PET
Sb
L =
#T = g+1,
- is in T.
eT = 0 means
Therefore there must be a function h with
I P; by putting
PET
- on the right if it occurs, we bound the
poles of h with at most g distinct branch points, none of which is
since two distinct such P's cannot be conjugate, by step II f must
be a constant and T = 95.
3.33
Lemma 2.5.* U (Jac c - 0) + era = Jac C
T
or
(1 (e+eT)
T
r
Proof.
Q
Qi
Write any D =
i=1 1
if it occurs).
distinct from the Qi's
Then
and
D+
(g--r) co) _
IQi + IRi -goo E [Jac C-0]
(because it's the image of a point in Z).
(Jac C-0) + e{R
,,,,R
1,
D E
with
i=1
(by replacing P+1P with 2-
IQ
Now choose g-r branch points
D E
Q.-r=
as =
g-r
1
if g-r is odd
}
(Jac C-0) + eJR ,,..,R
If g-r is even, then
}.
QED
g-r"00
So, we take one copy of Z for each T, and we glue them together
according to their identification as subsets of the Jacobian; we have
to see that this glueing satisfies the conditions to give the atlas
of a variety.
*
Here - in Jac C-0 is a difference of sets, but + in (Jac C-0)+eT
means translation of a set by a point using the group law on Jac C.
3.34
Step IV.
Lemma 2.6.
Given
any
{Bi}iET
1
of even cardinality, let
P
rT
1
,T 2
T1,T2
2
i=1 1
Then
c B, with
2
be the set of pairs
rT .T c ZxZ
1
{Bi}iET
,
is Zariski closed and
+e
i1Q.
=
+e
T1
T2
i
ro'ects isomorphically to
Zariski open subsets of each factor.
Proof.)
Rewrite the definition of
I'T
1
JPi' YQi
Pi +
i=1
i=1
tiQi +
Bi +
I
iET1
r
I
T
as
2
Bi
=-
iE T2
Consider the vector space V of functions whose poles are bounded by
N = 2g+#T1+#T2; as we saw .before (lemma 1.1)
polynomials in t
V
polynomials in t
+
of degree < I N
s
g-1]
of degree < {N_2_l])
is
Say f1, ... , fM a basis of V, where M = N-g+l.
Among these functions, those which have zeroes at
I Pi +
i=1
I tiQi +
i=l
I
iET1
Bi +
I
B.
for fixed
Pi, lQi, are just 3)
iET2
the elements of the ideal in
34ve. 6J ftie troducf
use the fact that a(s,t]/(s2-f(t)) is a Dedekind domain (91, Note 1)
plus Corollary 0.14.
3.35
`u(1)
(*) IEP.,EQ.
(t),s-V(1) (t)). (U(2)
e
(t),s+v(2) (t)JO
1T (s,t-a.) iT (s,t-a.
iETI
where the divisor
P.
iET2
<--->
(U
(U(2) ,V(2) ,W(2))
and
e
Note 1Lntif h E V n IEPa.EQi
then
(
Ipi, IQi) E rT
since h has
T
1' 2
exactly N zeroes, and poles only at -
.
Note also that membership in I imposes N linear conditions on a
function, so
codim I = 2g + #T1+ #T2 = N, independent of
Let RZ =(affine ring of Z)- (E[U.,V.,Wk}/(aa).
Ipi' 'Qi-
Then we get a
"universal" I
Ic
R(1) ® RZ(2) [s,t(s2-f (t) )
defined by the same formula (*) with U.1),V(1)
E R(1), U(2),V(2)E R(2)
1
1
Z
being variables.
1
1
Z
Consider
A = R(1) ® R(2) '[s,t)/(s2-f (t))+I;
A is an algebra over RZ1)®R Z2), finitely generated and integrally
dependent (I contains,a monic polynomial in t) such that for all
homomorphisms
RZ1)® R(2) ->
T
(evaluation of the coordinates) it
3.36
becomes a c-vector space of fixed dimension N.
It follows`) that A
is "locally free", i.e.,
3ha,ga E RU) ® R(2)
such that 1 =
jhaga
Va 3 e(a),...,e(a)
basis of Aha as (R(l)®R(2)
N
ha --module.
1
Z
Z
and
Now let the map
>A
V
be given by
14
fi
N--g+lu
rk (c (
)
<
defines
rT
in the open set ha # 0 of
1,2
T
ZXZ
4)This follows from the
Proposition.
If R is the affine ring of an affine variety, S a
finitely generated R-module, and
dime S ®R R/,.
is constant as fl varies among the maximal ideals of R, then S is
a locally free R-module.
E S
If m is a maximal ideal in R, let
Proof:
a basis for the vector space S
®R RM /MR m = S ®R R/tn;
by
generate S. as R. -module; we claim
they are a free set of generators. Since S is finite we can express
i)ei,
its generators as combinations
for any
generate 1
f
it follows that
Now if there were a
max. ideal
of R which doesn't contain f.
relation among the e1,...,eN, jXiei = 0, Xi E R,tt,F and say X 31 0,
let's express the X.'s as g'/f', g',f'E R. Then g'Em., since
Nakayama's lemma, e11
.. ,eN
,
11.
I
1
1
I
e1 , ... ,eN is a basis for S./MS.. There is a maximal ideal IS
that
glff'
!Z f, since R is the ring of an affine variety.
such
But if
f then al is not zero in R/t. R and dimcc St 0R R/4R < N,
n
which contradicts the assumption. (See Hartshorne, Algebraic
Geometry, Ex. 5.8, p. 125.)
g1ff'
'
3.37
(such open sets cover Z x Z because
rT
l,
Jhaga = 1).
This proves that
is Zariski-closed.
T
2
Note:
the rank of (ci7a)) is never less than N-g since
2 functions in the kernel of V -o A must be linearly dependent,
having the same zeroes and poles.
by g equations.
Therefore
r
is defined locally
This follows from:
Proposition 2.7.
Given a matrix
0
all (x) ... aln (x) al,n+l (x) ..
alm(x)
N
and (x)
where
... ahnN) an,n+l (x)
..
anm(x)
aij(x) are polynomial functions on an affine variety, let
MIJ =
det(ai.), #I = #J,
iEI
jEJ
and suppose
3
M
(O) # 0;
open neighborhood U of 0 such that for all
or n < i < mu if and only if 'irk M (x) = n-l`.
Therefore all components of
projection
rT1,T
---> Z
3
a Zariski
x E UftM(x)-0
have dimension > g.
rT FT
l
then
But each
2
is injective and
dim Z = g.
Therefore
2
r
TT
1' 2
must be irreducible and of dimension g too.
Main Theorem (0.6),
rT ,T
1
under each projection.
By Zariski's
is isomorphic to an open subset of
2
Z
3.38
This proves that rT,
1
,T 2
can be used to.glue the T1 and T2 copies
This procedure therefore constructs Jac C as a variety;, we will
of Z.
Note that it is complete, because
see later that it is projective.
there is a surjective map Cg -> Jac C, and Cg is compact in the complex
topology as C is complete.
In fact, Jac C is an abelian variety:
Definition 2.8.
An abelian variety X is a complete variety with
a commutative grou2 law such that addition X x X - X and inverse
X -b X
are morphisms.
We know that Jac(C) is a complete variety and a commutative group.
To see that the group law is a morphism we need:
Lemma 2.9.
{JPi,JQi,JRi
For all T1, T2, T3
even sets of branch points
IPi+IQi+G B + GBi = I Ri+1
i
1
1
T1
T2
i=1
T3
is Zariski closed in
Bi+(g+#T1+#T2-#T3)-}
ZxZxZ, and projects isomorphically via p12 to Zx2.
(This can be proved in the same way as Lemma 2.6, so we omit the
details.)
Proposition 2.10.
As a complex manifold, every abelian variety
X of dimension n is a complex torus Cn/L.
Proof:
We use the Lie group structure of X.
Cn = (Lie algebra of X) --- X
The exponential mapping
is a homomorphism because X is commutative;
thus, being a diffeomorphism in a neighborhood of the identity, it is open,
and since the image is connected exp is surjective.
Again by bijectivity
in a neighborhood of 0, the kernel is a discrete subgroup of Cn, and
Tn/L
the only discrete subgroups L such that
is compact are lattices.
QED.
3.39
In fact, we already showed in Chapter lI that Jac C was
a complex torus (Abel's theorem 11.2.5).
we have thus a 2nd
proof of this based on the chain of reasoning:
Jac C is a complete variety --> Jac C is an abelian
variety -> Jac C is a complex torus.
Corollary 2.11.
Every 2-torsion element of Jac C is of the
0
form eT, some T c B.
Proof:
The 2-torsion subgroup of the abstract group
Tg/L = Jft2g/ZZ2g
is 2L/L, which is isomorphic to (72/ZZ) 2g.
But
it contains the group of the eT's, which has the same order, so
they coincide.
QED.
3.40
§3.
The translation-invariant vector fields
Then a vector field D on X is given
Let X be a variety.
equivalently by:
a)
a family of tangent vectors D(x) E TX x, all x E X
such that in local charts
n
X a Ua
D (x) _
b)
rd.
c CC
ai E (C [X1, ... ,Xn ] .
ai (x). a/aXi,
a derivation D:
In fact, given D(x), f
O -> (.
E
r(u,OX), define Of by
Df (x) = D (x) (f)
.
When X is an abelian variety, then translations on X define
isomorphisms
TX'x
TX'O
for all x E X (0 = identity),
invariant vector fields.
so we may speak of translation-
It is easy to see that for all D(O)E TX'O,
there is a unique translation-invariant vector field with this value
at O.
In general, the vector fields on X form a Lie algebra under
commutators:
[D1,D2](f) = D1D2f-D2D1f.
For translation-invariant vector fields, the commutativity of X
implies that bracket is zero (see Abelian Varieties, D. "Mumford,
Oxford Univ. Press, p. 100,
3.41
The purpose of this section is to give explicit formulas for
the invariant vector fields in the chart Z in Jac C.
Let
this.
P E C
> P(e) E C
E
of P
and choose a non-zero 6P E TC P.
Our method is
Let
be analytic coordinates in a small neighborhood
with P(O) = P, so that 6P is the image of the unit tangent vector
a/ae at 0 in this coordinate.
Then we get
DP (O) E
TJac C,O
defined as the image of a/ae for the map
c-disc -----> Jac C
S
at
[divisor class P(0)-P(&)]
1
e = 0, i.e., the tangent vector to this little analytic curve
in Jac C at 0.
Note that
dp and DP(O) are determined by P only
up to a scalar.
Starting with any divisor D =
Pi - g-=) E Z, let
(
i=l
D - P(e) + P
Pi(e)
g--
and let
-
lPi (e)
Since Z is open, choosing
(D - P(e) + P) E Z.
IcI
_
(Ue (t) ,Ve (t) ,We (t)) .
small enough, we can suppose
3.42
Then
(due
dE
dW
dVs
C=O
de
'
E=O
de
/
E=0 )
E TZD
and this represents the translate of-DP(O) to
TJac C,D
Note that
Tyac C ,D'
for this to be an invariant vector field, it is possible to use
different uniformizations a ti--. P(e) for each D, so long as the
tangent vector
SP to this ma
is independent of D.
The result is:
Theorem 3.1.
For any
P E C, P 7( -, for suitable
above tangent vector is given at (U,V,W) E Z
U(t)
V(t)
t - t(P)
U(t(P))).W(t(Pw(t(P)) U (t) _ U(t(P))'U(t)
t (P)
DP:
by
V(t(P)) -U(t)--U(t(P)) V(t)
W(t) = W(t(P))V(t)-V(t(P))-W(t)
t .Note.
SP the
L*(t(P))'V(t)
Equivalently, this means we have a derivation
cr[Ui,Vj,Wk]/(a(,)O
given by
DP(Ui) = [coeff. of tg-i in V (t (P)) U (t) U (t (P)) -V (t)
DP(Vi), DP(Wi) = (coeff. of tg-i in the other expressions)
.
3.43
Note.
Corresponding to P = w , we get the vector field
U (t) = V (t)
V (t) = !I-W(t) + (t-U1+w0) U (t) ]
z
W (t) = - (t-u1+w0) v (t) ,
obtained by letting t(P) go
dP/t (P) g-l. To c keck 4 L
by
o
-
,
and replacing
6P in the Theorem
ve Calculkf e :
lim[_t(P)Lv(t)+lowerordertennsin t (P) = V (t)
]
(-t(P)+'t)
lim U (t)1 =
t(P)g-1
P->-t(P)te
lower terms}
lim
V (t)
P->00 t (p)g-1
= lim
t (P) gw(t)-(t (P)1+Wt (P )g) U (t)+ in t (P)
2
/-(t
(P))U (t)
(-t (P) +t) t (P) 9-1
lower terms
lim 2 t(P)q(t)-w0 t(P)gU(t)+Ult(P)gU(t)-t(P)gtu(t)+ in t(P)
2
-t(P)g + lower order terms in t(P)
W (t) + (t-U+W) U (t1D
Jl
lower termsJ))
lim W(t)
P->.t(P)T-1
= lim
= -(t-U
Note.
(t(?)9+U1t(P)9
DP(O) E TJac
)V(t)+ in t(P)
(-t(P)+t)t(P)g-1
1+W0).V(t).
C,O
will depend on P and on the chosen
uniformization; as P varies, we should only have g independent
vector fields.
To see this, it suffices to expand the above
expressions in powers of t(P).
As before let:
3.44
g
u (t) _
Ui
I
U0 = 1
i=O
i0IV i
V (t) =
W (t) _
i=-1
tg-i
V0 = 0
tg-i.
W
W-1 = 1;
1
then
U (t) =
0<.<3<g
0<.
i,j=0
V1Uj
t (pp-y-j-t (P) g-jtg-i
t - t (P)
V. U t (P) g-jtg-j
t (P) -i-tJ-a
so g-1'g-j
(iUj V3Ui)
i >j
+
t-t(P)
1 3
V
g>i>j>0 i
so g-i<g-j
t (P)
g-itg-i /ti-j-1
1
+ .... + t(P)3_)
ll
t (P)
gktg-R,
k+R;-.A.+j+1
1 < j+1<k,R, <i<g
_
I t(P)g-k[1..=1
k=1
tg-P,
I
i+j= (k+i) -1
ix (k, R,)
3 <min (k, !,) -1
(VU.
i -VU. )
tl t
t
(PP)']
3.45
t(P)g-itg-j-t(P)g-itg-i
2V(t) _
U.W.
O<i<g
t-t(P)
17
I (U1Wj-UjWi) t (P)
O<i, j<g
g-ltg-i (ti -j -1+.:.+ t (P) i-i
U.U.t(P)g-1tg J
IJ
-EUiUjt
7
I tg-k(
I t(P)g-k 1 k=1
I
k=1
(UiW. i-Ui W
i+j=k+k-1
i
UkUk >
i>max (k, k)
j <min (k, k) -1
-60 rVy%
(k and
k are allowed from 0 to g, but for k = O,k = 0
j(UWj-UjWi)-UkUk = UW-1-Uk, =UkW_i-Uk(resp.) = 0).
W(t) _
_
1Wivj
t(P)g-ltg-,-t(P)g 3tg-1
t - t(P)
IUiVjt(P) g-i tg-j
tg-k
t(P)g-k
1
k=1
+
(WiVj-WjVi) + UkUk
1
k=0
i+j=k+k-1
i>max (k, k)
j <min (k, k) -1
(k is allowed to be zero but for k = 0 7 I(WiVj-WjVi)+UkV = 0,
while for k = 0 we get
-
I
k=1
t(P)g-ktgVk).
(P)g`at9 7
3.46
So if
D U
k Z
_
(V.U
i --V.U
i+j=k+i-l
3
3
].
)
i>max(k,k)
j <min (k, Q) -l
DkVQ
(UiW.-UjW.) -
= ZC
same
UkURJ
as above
DkWQ
(WiV.-W.Vi) + UkVQ
I
same
as above
.
then we find
DP =
I
I t(P)q-kDk.
Proof of the theorem 3.1.
P 0 Supp
i=l
For the proof, we also assume
P1., and that neither P nor any P
a.
is a branch point.
The result will follow by continuity for all P and
IPi.
Let
Pi correspond to (U,V,W) as usual and note that as no Pi is a
i=l
branch point, U,V have no common zeroes.
We consider the function
q (s, t) = U (P).(s+V (t)) +U (t).(s (P) -V (P) )
U (t), (t--t (P) )
the denominator is zero at
is zero at
L(P .)+ t(P) so q has poles at
1
principal
q is like
Pi + Iti(Pi)+P+r,(P), but the numerator
1
part at P is 2s (P)
t-t(P))
so
q
independent of
has a zero at
P. and at P.
i=1 1
IPi.
Its
At infinity,
3.47
So the equation
q(s,t)-i
or U (t) (t-t (P)
has solutions(
i [U (P) (s+V (t))+U (t) (s (P) -V (P)) ]
Pi(e)
near
P
i=1
near P,
P(E)
and
f
Pi (E)+P (e) ]
-
111
[
Ll
Pi+P]
Unfortunately, this analytic family
=
Iq-
t
E0
eI--a P(E) of points near P
depends on the choice of U,V,W, hence on the divisor
IPi.
since the principal part of q at P is independent of
1Pi, the
tangent vector
of "jPi.
But
SP to the family E F P(e) at E = 0 is independent
In fact, this gives
zu (P) s = U (t).(t-t (P)) -
[U (P) V (t) -U (t)v (P)+U (t) s (P) ]
z
squaring both sides
U(P)2f(t) = U(t)2[t-t(P)]2-U(t).(t-t(P)).E[U(P)V(t)-U(t)V(P) +
2
+ U(t)s(P)I+ 4 [U(P)2V(t)2+2U(P)V(t)U(t) (s (P)-V (P)) +U (Q
(s (P) -V (p))
or (substituting f(t) = V(t)2+U(t)W(t) and dividing by U(t)(t-t(P)) )
U (t).(t--t (P)) -c [U (P) V (t) -U (t) V (P) +U (t) s (P) ]
+
E2 {-u(P)2w(t)+2u(P)v(t) (s (P) -V (P))+U (t) (s (P) -V (P)) 2
t - t (P)
I
3.48
or O =
[u(t) + g U(t)V(P)-U(P)V(t)
+ s2(
t-t(P)
)+...1[t-t(P)-es(P)+e2( )+...]
L
degree g in t
JJJ
degree 1 in t;
defines P(s)
Thus the 1st factor is U5(t), hence differentiating, we find:
(P) -U (P) V (t)
U (t) = U (t) V
t-t (P F
Now from the relation
f - (V+eV) 2
-2VV
=- (U+&U) (W+sW)
=
mod. E 2,
UW + WU
UV(P) -U(P)V
t -t(P)
W + WU
we have
O = V(2V - t(t(P))
U1 W +
t-t`P)
)
Therefore
2V - U(P)W
t-t P
= - a(t)U
t--t(P) = + a (t) V
W + V (P) W
or
U (P) -W
2t-t P)
- a (t) U I
W = a(t)-V - V(P)W
t-t P)
where a is a rational function
3.49
Set
a (t) =
t-W (t P)
(P)
+ a (t) .
Then
V = 1 U(P)W-W(P)U 2
t-t (P)
W
so
aU, aV
W (P) V-V (P) W
=
t-t (P)
a(t)U1
a (t)V
are polynomials in t; it follows that
a
(since U,V are relatively prime), and since deg V < g
is a polynomial
and
V = [UJP)tg-a(t)tg + (lower order terms in t)], then a(t) = U(P).
If U,V have common zeroes, the formula follows by continuity.
QED
In fact, we have something more here than an expression for the
invariant vector fields on Jac C.
s2 = f(t) vary too.
C3g+1
AM
Suppose we let the curve
We see that we have a morphism
space of all polynomials U,V,W
s.t. U monic, deg. g
,
coord. Ui,Vj,Wk
deg.Eg-1
V
W monic, deg. g+l
Tr
I
2g+l .
(space of polynomials f1
s.t. f monic, deg 2g+1
where r is defined by:
= V2 + UW.
coord. as
f(t) t2g+1+ a1 2g+...+a2g+l
3.50
The fibre of
over any f with distinct roots is the affine
7r
piece Z of the Jacobian of s2 = f(t).
Thus all Z's fit together
The formulae above define vector fields Dk,
into a fibre system.
1 < k < g, on all of T3g+1, which are tangent to the subvarieties
7r-1(f)
(and generate their tangent spaces at each point).
E D
= 0
F D
l
Dk(aa)
Thus
(because the Jacobians are commutative
2
groups, hence [Dk
= 0
each Z)
D
1
I
k
is zero on
2
To summarize we have found an explicit set.of.g commuting vector
fields on
Tag+1, with 2.g+l polynomial invariants
manifolds
Jac C - 0, where
C
aa, and integral
varies over all hyperelliptic curves.
3.51
§4.
Neumann's dynamical system.
In classical mechanics, one encounters the class of problems:
M = real 2n-dimensional manifold, with a closed non-degenerate
differential 2-form w
w = dual skew-symmetric form on TM
H =
XV
00-function on M, called the Hamiltonian.
the vector field on M defined by
w(XH,Y) = <Y,dH> for all vectors Y
H
or
<XH,a> for all 1-forms a
Recall that we define the Poisson bracket by:
{f,g} = +<Xf,dg>
= -<Xg,df>
= w(df,dg),
and that the compatibility condition
[XfFXgl
X{f,g}
holds.
Moreover, {f,g}
= 0 means that dg is perpendicular to Xf, or that g
is constant on the orbits of the integral flow of Xf.
The main
problems of classical mechanics were all to integrate various vector
fields XH.
Unfortunately, it never happens except in trivial.cases that
there exist 2n-1 functions defining
tt-1(x)
= orbits of XH.
M
1R 2n-1
such that
However, what does happen occasionally and
3.52
Cco
to date unpredictably is that there exists. a
(a) if Xi are the coordinates on ]Rn
U open in ]Rn .
{Xioh, Xioh} = 0,
(d)
-map h: M ---)U,
(b) h is submersive,
H = foh, f a Coo function on U.
it follows from
{Hi,H.} = 0
hence the fibres of
(c) h is proper, and
In this case, if Hi = Xioh,
that the
XH
commute,
are
h
n-dimensional compact submanifolds whose connected components are
orbits of
{xH
,..
YES
,XH
1
}
hence are isomorphic to,tori: 1n/lattice).
n
It can be proven that near each one of these tori M has coordinates
xi determined mod 2Z, Hi = Hi(yl,...,yn)
independent of (xl , ... , xn) ,
w=
coordinates on the torus ]Rn/2,n;
Idxi A dyi
and
(xl r,'-
, xn )
such (canonical) x,y are called
the action-angle variables.
But the orbits of H by itself are almost all dense 1-parameter
subgroups (as soon as the
's
ay
are rationally independent, for
k
instance, in action-angle coordinates); in this case the closure
of 5in1eorbit of XH is already an n-dimensional torus, and that's why
A
we cannot find any more rational continuous invariants for the flow,
wQ CA"Kef f,hj
In particular
7: H
> ]R2n-l which would give 2n-1 functions
constant on the orbit.
A Hamiltonian vector field XH with properties (a), (b) , (c), and (d) i
called a completely integrable system.
Given a completely integrable system, suppose M is the set of
real points on an algebraic variety and that
w,Hi are rational
differentials and functions without poles on M.
Then the Lori
3.53
Mc =
M.
7r-1(c)
are the real points on complex algebraic varieties
It may then happen, although this is a strong further
assumption, that the vector fields XH
still have no poles on
a compactification of M.
M c I
(Typically
of real points of an affine variety
MC, leaving plenty of room
for poles at infinity: e.g., take M = IR2,
H = x4+y4.)
and is the set
w = dx A dy,
If this does happen, then for a suitable complex-
ification of the system, each Mc will be the group of real
points on an abelian variety (or a degenerate limit which is a
group formed as an extension of (C*)k by an abelian variety).
We call such systems algebraically completely- integrable.
More
precisely:
Definition:
(M2n,w,H) is an algebraically completely integrable
system if there exists a smooth algebraic variety
structure w on
Il,
i.e. ,
h:
U
R, a co-symplectic
w E A2TR. and a morphism
in ---B U
a Zariski open subset of ci
all defined over the real field,
,
such that
a)
{Xi o h, X. o h}
b)
h
is submersive
c)
h
is proper
d)
M is a component of
RI
(d(X3 oh),d(Xi oh) ) = 0
am, the w on M is the w on
along M, and H is a
Cco-function of
Xi oh
IM,
3.54
In such a situation, it is easy to prove that the fibres
of h are abelian varieties or extensions of these by (T*)k.
These remarkable cases give us methods of describing families
of abelian varieties by dynamical systems.
Neumann discovered a remarkable example of this in:
C. Neumann, De problemate quodam mechanico, quod ad primam
integralium ultraellipticorum classem revocatur, Crelle, 56 (1859).
To describe this we start with n particles in simple harmonic
oscillation, whose position is given by
x1,...,xn.
The equations
of motion are
1
-a.x.
1 1
or equivalently a system of 1st order differential equations
xi = yi
yi = -aixi.
We assume that
a1 < a2 < < an, and we want to
constrain the position to lie on the sphere
jx2 = 1; then
Jxiyi = 0, too, and the equation of motion is given by
adding a force normal to Sn-l that keeps the particle on
Sn-1.
3.55
We get
yi
xi
(4.1)
yi = -aixi
2
xi (lakxk-- yk)
(In fact,
2
these imply: (Zxkyk) =
lyk +
:xk'(.-akxk+xk(yaQxP.
x2
2
ayk
`
yk))
+ (jxk),( aixQ- YZ
Hence if we start with a point such that
which is zero if jxk = 1.
Ixk = 1, Ixkyk = 0, these will continue to hold if we integrate
these equations.)
Let
T(Sn-1) _ (locus of points s.t.
Ixk = 1,
Ixkyk = 0), i.e., the
Sn-1.
tangent bundle to
(4.1) gives a vector field on T(Sn-l)s
D = lyk axk
jak xk ayk +
ayk
.
If we put a symplectic structure on T(Sn-1) in the usual way by
restriction of the 2-form
jdxiA dyi, then (4.1) is Hamiltonian
with
H =
(4.2)
1
2
2
{jakx k +
2
ly k}
(= potential + kinetic energy).
To check this, it is convenient to develop formulae which
express Hamiltonian flows on symplectic submanifolds in general.
Thus say
3.56
M C IR 2n
is defined by f = g = 0.
structure on IR
2n
Let
w =
and let resMw
define a symplectic
Jdxindyi
define one on M.
We assume resMw
is non-degenerate, so w gives us a splitting
_ TM ® TM
TM
is generated by the vector fields Xf,Xg, and for all functions
h on )R2n
the vector field
Xres(h)
gotten from the Hamiltonian
structure on M is the projection of Xh to TM:
Xres (h) = xh _
w(XhrXf)
w (x ,X ) }Xg
w(xh,x
\w (X f,Xg
f
g
where, as usual, on ]R 2n
Xh =
v ah
axi
Now, consider the special case
X
f
= -2 Ix i
_ v ah
axi
a
yi
a
ayi
,
f = (Ix2 - 1), g =
xg
w(Xf,Xg) = 2 Ix2 = 2
hence
a
ayi
a
i axi -
1xiyi.,
a
yi ayi
Then
3.57
X
ah
ayk
resW
+
k
a
x. ah
ayi ] axk
- xk (
ah
r
axk + xk
_
ah
ah 1
xi ax
yi ayi +yk
/
ah 1 1
ayi/
Substituting Z(Jakx2+Iyk) for h, we get (4.1.).
Following Moser, we can link these equations
follows:
with Jacobians as
let n = g+l; define a map:
T(Sg)
7r:
a
(x,y)
by
t
(c3g+l
(Ux,Y"Vx,Y'Wx,Y)
where we let
f1 (t) _
n
(t-ai)
i=l
2
Ux,y(t) =
k
Vx,Y (t)
t-a
monic,
deg=n-1 = g
k
Xkyk
fl (t). I
k
deg < n-2 = g-1
k
2
Y
Wx,y (t) = fl W. t_k
k
k
+ 1)
monic,
deg=n = g+l
and the coefficients of Ux"y,VX'y,Wx,y are taken as the coordinates
in a:
3g+1
a
ayk
3.58
Then
x 22
ky
U
_
2
xrYwxrY+Vx,Y
2
xk
R
{kr
f l(t)
l,Q t-ak
22+x22
xkYiQyk-2xkxtykyi
f (t)2
1
xkYkXP,
t-ak - k-ak t-aR
t-aQ)
t-ak
k
t-ak)(t-aQ)
k<Q
2
xk
+
y
(xkY
-x k
L k<lQ (t-ak t-aQ)
= f1 (t) 2
because the second factor has only simple poles at
1 [x2k + Qk
"singular part" t--ak
(x kyz-xQ k )2
ak-aQ
J
/
ak
(with
and is 0 at o0
we can re-expand by partial fractions
2
t-ak Lxk
fk
If we set
F
k
= x 2k
+
(xkYQ-xQ
2
f1(t)
+ Jk
ak-as
(xkyz-xQyk)2
I
ak-ak
ilk
then
f (t-ai)Fk
f2(t) =
is manic, of degree n-l,
k
and finally:
2
Ux,y x,y + vx,Y
fl, f2
so that x,y defines a point of the affine point of the Jacobian of the
algebraic curve s2 =
embedded in T3g+l by the method of §2!
3.59
The map
7:
(U,V,W) extends to a map wT on the
(x,y)
complexification T(Sg)T of T(Sg), i.e., the complex variety given by
equations
Irk = 1, lxkyk = 0, and the image if contained in the set
of complex polynomials tJ,V,W such that f1IV2+UW; or equivalently the
set of affine parts of the Jacobians of the curves s2 = f(t) for which
filf.
The situation is summarized in the diagram (4.4) below.
Lemma 4.3:
7r(C is surjective;
TrCC (x, Y) =
rr(C (x' ,y') if and only if
(.x',y') is the image of (x,y) under one of the transformations
hence
+1, which form a group of order 2g+1
(Ekxk'Ekyk), sk
(xk'yk)
is unramified outside the subvariety of the (U,V,W) such that
7r
T,
U(ak) = V(ak) = W(ak) = 0
Proof:
such that
for some k.
Given f with the property f11f, and polynomials (U,V,W)
f = UW+V2, then we make partial fraction expansions
u
ff 1
_
I
Xk
t-alt
k
uk
V
t-ak
f
k
w
f1
it follows that
uk = 0 because
I
k
vk
t-ak + 1
Irk = l because U Monic, and it follows that
ii
because
has a simple pole.
Now we
deg V < g-1, and it follows that XV i = p1
2
at each ai, UW+V2 has a zero, hence
UWZV
i
fl
can solve for (xi,yi) E T (Sg) T
Xi _ xi
2
2
vi = yi
4i = xiyi,
uniquely up to a single sign
for each i.
QED
3.60
3g+1
U Zariski-closed
f-V2=U W,all f such that f
7r
(4.4)
U
U
T (Sg)
res R
g+l
real dim. 2g
is real and V pure imaginary
g
f1(t) _ 1T(t-ai),
f2(t) =
i=l
. g
(t-bi),
U(t) = fl (t-ci)
i=1
If
1!
subset where f,U have
Creal roots as below, W
real dim. 2g
Lemma 4.5:
cx.dim. 2g
union of the Jatobian
cx.dim 2g
(U,V,W) satisfy
i=l
f1IUW+V2, and (U,V,W)
then x and y are real if and only if U,W are real, V is pure
imaginary and f(t),U(t) have real roots separated as in (4.4).
Proof:
If x and y are real, then
u (t)
._
I T (t-ai) xk 2
k
ifk
is a real polynomial and sign U(ak) _ (-1)g-k+l, so U must have a
zero in each of the intervals (ak,ak+l),
k < g.
Also U is monic
3.61
so U(t) > 0 for t >> 0.
Thus U(t) has signs like this:
+
0
at
+
c1 A5 5 a ft l
a7-
C,
0-p
Next, f-V2 = U-W and V(a) is pure imaginary, all a E ]R,
hence
f(t) is negative at all zeroes of U(t), hence f2(t) is alternately
+ and - at these zeroes.
Thus all zeroes of f(t) are real with one
zero of f1 and one zero of f2 in each interval (-co,cl) , (Cl,c2) ,---,(c
as shown in (4.4).
9-1!c g
(In all of this we have assumed the zeroes of f
are distinct, but limiting cases ai = bi and bi = bi+1 are possible.)
Conversely, if the zeroes of U and f are real, and interweave like
in the diagram, then in the partial fraction expansion above
ak > 0 and Riuk is imaginary, so the equations
x
2
2
have real solution (x,y).
QED
If we fix f with real zeroes, the curve s2 =f{¢)is a double covering
of the t-line; it has a real structure given by coordinates
S, t) .
Since
2
g+l
g
i= l
i=l
sI = - iT (t--ai) TI (t-bi), the real points
3.62
on C are givens by :
2-sheeted 1
covering map JJ
C---> JP
V
f<0
f<0
f>0
By complex conjugation
f>O
f>0
. - - - - - f>0
(s;,t) t----- (Set), we get an antiholomorphic
involution
- > Jac C
Jac C
(Jac C)IR
is defined as the set of fixed points of this involution:
it is a subgroup of Jac C which must consist in a g-dimensional real subtorus plus
Since oo is fixed under this involution, we can determine
a finite number of cosets.
the real points in Jac C-0 as follows:
(Jac C) 3R ` 0 =
Note that
{
pi -- g- Pi
I
i=1
3'
oo,P
lpj ,
if i
j, and Pi = JPiJ.
Pi = Ipi means that Ipi consists in some real points and, some
pairs of conjugate complex points. If (Io,---,lg) are the intervals in
IR where f < 0 (see diagram (4.6)), then any subset S of
whose cardanality is <g and
component
KS
S g mod 2
{0,1,---,g}
defines a connected
of (Jac. C) IR - 0, namely the set of the
3.63
Pi-goo
divisor classes
such that, if S =
fill ... is),
i=1
t (Pk) E Ii
A)
ah
, far
1
.
k. -
S
'`
Ps+l
ICs
5o Pk = Pk
01
or
ps+2
t(Ps+l),t(Ps+2) E (same TP,
)
1
Clnd
's+3
or
Ps+4
t (Ps+3) I t (Ps+4) E (same IQ )
2
etc.
Example:
21
1
12
0
the real (affine) part of the Jacobian breaks up into 4 components;
2
2
in order for
P
to be equal to
i=1 i
P., the only possibilities are:
i=1 1
P1 over 20, P2 over I1
P1 over T0, P2 over 12
K{01)
P1 over T1,
P2 over 22
K{t
K
(Pl,P2 over the same 1K) or (P2
Note that this last set
K0
is connected because we can continuously
move P+P on the complex curve C until P = P El
k
and then move the two
points independently on the real loop over Ik; KO also contains tin the limit, hence it's the connected component of the origin.
Thus we verify that when g = 2,
(Jac C), is a real 2-dimensional closed subgroup,
isomorphic to
3R2 /lattice x
(7L/2?Z).
2
.
3.64
Returning to Neumann's dynamical system, the following theorem
will show that the functions Fk(x,y) are integrals of the vector field XH give
ahaw fl.a
commuting flows on T(Sg) T and their image under v
is tangent to all
Jacobians, and gives the translation-invariant flows on them.
Theorem 4.7 (Moser-Uhlenbeck*).
{Fk,FQ}
b)
c)
= 0
the
-j
= 1,
Il
2 k=l
ckDa
k
k
= CD.,
Tr* (XH)
d)
1
xk2
kElFk =
Tr* (XF
On T(Sg)
XF
akFk = H
ck = 4/ T
c
fl
(ak-aZ)
= -2v "-- T
span a g-dimensional s ]Race, except over the
k
Zariski closed subset of triples (U,V,W) which have a common root.
Proof of a).
First one checks that on (C 2g+2, with coordinates xi,yi,
{Fk
=0
with respect to the symplectic form I dx1Adyi = w; if we let
zkZ _ xkyZ-ykxZ, then
{zkP'zij} {zkV'zij} = 0
azkfZ azi
oaxa aya
if
{k,2.}
13
aykZ Ox
!1
hence
{i,j} =
=-YQ(_xj)+(-xz)yj
{zkZ,'zkj}
=-ZQj
*
J. Moser, Various aspects of hamiltonian systems, C.I.M.E. conference
talks, Bressone, 1978; K. Uhlenbeck, Equivariant harmonic maps into
spheres, Proc. Tulane Conf. on Harmonic Maps.
3.65
2
Thus if
k.
Jk
{
ak-a.
and
k 34 k
1
1
2
2
1
1
j#k ak-aj aCa {zkj' zki} = 4 3 fk ak-a.
ak-ai zkjzki{zkj'zki}
i `k
ilk
zkk zkj
1
- q 7
jk
1
zk j
ak-aj ak-ai I zkk zk i zki
z
kj zkk z jk
k
r
1
=-4 zkk
1
i;k ak-ai aCa
i
zki zki
+
or k
+
4
;..4
zki zki
1
zk. akak i L ak -ai + 4z k k ak -ak
z
kk ilk
(ak-ai}
or k
zk7,.,,k
jqdk
1
ak -ai) - (ak-ak) (ak-ai - (aak) (ak
(ak-ak) - (ak-ai) + (ak-ai)
(ak-ai) (ak-ai (ak -ak)
zkizki
0
Finally
{Fk,FL} _
{xk2+qk,
{xk2,x2k} + {xk2,Ok} + {Ok,xk2}
=-2xk
ay 4.
k
ak-a .
+ 2xk ayk
i
3.66
2(xiyk-xkyi)(-xk)
2(xkyi-xiyk)(-xi)
= 2x
=
0
- 2x
ak-ai
k
aQ-a.k
lC
.
To conclude use the following
Lemma 4.8:
Let M2
=,N2n-2
be symplectic with basis 2-forms w, res w; let
f,g,h be functions on M such that df is nowhere zero and f = 0 on N.
Then if the Poisson brackets on M satisfy {f,g}M = {g,h}M = 0, the
Poisson bracket on N satisfies {g,h}N = 0.
Proof:
Via
w, we can split the cotangent bundle to M into the
cotangent bundle to N and its orthogonal complement
TM* _ TN* a C*
Now write
dg = d(res g) + a
dh = d(res h) + S
df =
From
w((X,y) =
a,a.Y e
y 76 0
C*
by
it follows that a =
w(S,Y) = 0
w(dg,dh) =
Proof of b).
w(d res g, d res h) =
Fk =
+
k,k
kQ
_
= 1
2
Xk
QED
{res g, res h}.
(xkyR-xQyk)2
2
xk
are
w(a,s) = 0,
linearly dependent, because C* is 2-dimensional, so
so 0 =
.
+
a -a
k Z
kZk(xkyi-xiyk)
2--!-+
1
(ak-at ai-ak
3.67
1 j1
and
2
akFk
2
2
+
akxk
2
kk
ak-ak
ak
2
kk
k?`k
a
2
akxk
+ 1 k
k
2
+ at-ak) (xkyk-xkyk) 2
(xkyk-xkyk)2
akxkk +
_
a
(ak-k ak
k
2
akxk + 412 E xk'1 yk - 2(Q xkyk)2l
LL
k
k
X2 + 2 Yyk
Proof of c):
=
k
H.
Under the Fk flow on T(Sg)(,, by the formulae for
Hamiltonian flows on a submanifold:
ark
aFk
- xk 1 p xp ayP)
pp
ayk
Since
aFk
2(xkyp--xpyk)xk +
x
x
p
p ayP
P39k
then
P
x
ak-ap
k
P54k
2(xky2-xpyk)(`xp) - 0
ak-ap
2(xkyk-xkyk)
x
k
a k -a k --- x k
=
2(xkyp-x pyk )
or
_
(-xp)
p34k
ak-a p
if k = k.
3.68
it, this vector field becomes
Under the action of
xk x 9 (x k y Q ""x Q X k )
2
Q = (xk)
= 2xz
4
P,
ak-ak
ak-aL
p#k
I
or
-4xpxk(xkYp-xpYk)
ak-ap
=
akap
pfk
U (t) = f1(t) I t
Xkpk-XZuk
f X (t)[4
ak--a
#k
_ 4
1
t-ax
SOP.-xkuk
=
1
p 34k
l
i k (t-ai)(t-ak J
Xk
4 f1(t)
uQ
t-ak k t-az
I AkV(t)/vT-UkU(t)
= - 4
But
if R=k
xpuk
aku
-4
So
if
xkuQ-x Z"k
4
U (t) =
V(t) =
t
I
k
ak
1 r (t-ak) Ak
£k
I iT (t--a2)uk
k
,34k
Ilk
xQ
t-ak R t-a2
akup-Apuk
ak-ap
l
t-ak
3.69
U (ak) _
so
fGk
(ak-aQ) . lk, V (ak)
rV
and finally
U(t)
'T -ff
Qk
(ak) U (t) -U (ak) V (t)
_
=I 4
=
t - ak
(ak-a
1-cr
- fl (ak-aR) -1)- Dak (U (t)) .
kft
The argument at the end of the proof of theorem 3.1 shows that for
any vector X on the space of (U,V,W)`s which is tangent to the fibre
UW+VZ = f at a point where U,V have no common zeroes, the equality
(P) -U (P) V (t) ,
X (U (t)) = c- U (t) Vt-t(P)
X (V (t) ) =
X(W(t) )
2. IU (P) (t
t
(P)
c a constant, implies
(P) U (t) _ U (P) U (t)
= c. IW(P)V(t)-V(P)W(t)
t-t(P)
_ U(P)V(t)
J
.
So in order to finish the proof of c r, where the constant is
c = 4/T -'TT (ak-aQ)-1, we notice that by a) the collection of
vector fields
XF
are tangent to the loci (Fi = constant, all Q.),
k
hence their images via
f = fl
k
r* are tangent to the fibres sitting over
Fk - l( (t_a,). Thus we get
Zk
= cDa
7r*XF
k
k
on the part of
the fibre where U,V are relatively prime; the result holds everywhere
by continuity.
The formula for w*(XH) is proven similarly, the calculation
being much simpler.
*Alternatively, one may calculate 1 (t) ,CV' (t) directly as we did U W.
3.70
Proof of d).
are constants d
i
Assume that at a particular triple (U,V,W), there
such that
i=1
d
i
d.D
ia
Then
= 0.
V(a.)U(t)--U(a.)V(t)
z
t -- ai 1
-
0
or
V(a )
t_a. )U (t) _ (1
(i= di
1
U(a )
d
V (t)
or
di V (ai)
(t-a
If
jai
] U (t) _
di U (ai)
V*
I[
jai
(t-aj )
V (t)
U
since deg U = g, and deg U* < g-1, this implies that U and V have
a common root a where U*(a)
0
(or at least the
multiplicity of a
as a root of u* is less than its multiplicity for U).
Likewise
g
rU(ai)W(t)-W(ai)U(t)
t _ ai
i=ldi L
- U(ai)U(t) I = 0
or
U(a
)
(I d, t-ai) W(t) =
implies
W(a.
1
t-ai
)
+
1 di U (ai) IU (t)
3.71
I
di
U(ai)jITl(t-ai)+W(t)
=
{jd.W(ai)
U*
IT
j54i
(t-ai )+Id U(ai)II(t-aiU(t)
7
JJ
W*
hence U(a) = 0, U*(a)
74
0 implies W(a) = 0 too.
Thus U,V,W have a
common root.
Corollary
.9: Almost all orbits of {XF }
(defined by
k
Fk = const., all k)
are compact real tori, isomorphic to connected
components of the real points ona 29+1--order covering of the Jacobian
of a hyperelliptic curve.
The covering that occurs here will be described analytically in §5.
Finally, Moser discovered a beautiful link between the dynamical
system T(Sn-l),{Fk}
and the problem of finding the geodesics on an
ellipsoid.
The result is so elegant that we want to reproduce it here:
x2
k
Theorem 4.10 (Moser). Let E be the ellipsoid
= 1
n..'
ak
g+l
if x,y EIRg+l satisfy
j
= 1, <x,y> = 0
then:
/the line LxrY {y+tx t E 7R
k(x,y) = 0) if and only if (
is tangent to E
\
and if this holds:
xkyk
=y-
(E
ak
x
=
L
x,Y
n E.
C ak2
k
If x (t) 'Y (t)
then
is an integral curve for the vector field
t(t) is a geodesic on E, up to reparametrization.
1
a
ak Fk'
3.72
Proof:
First we calculate out
Lak
l (x2 +
Fk(x,Y) _
a
(xkyR, -xkyk)
ak-at
k
t7k
k
xk +
aQ)
2>k ak
ak
(xkyQ-xQyk)2
1
1
ak-a91
2
xk
ak
x
- 1
2
2
akaZ
2
2
xk
YR
(1
_I
ak)
/
2
I ukvk/akr
Fk (x, Y) = B (X, X) (1 - B (y, y) ) + B (X, y) 2
Call this function F.
We calculate the flow associated to F.
_
a)
xayk!
xkyk
ak
k
or if B(u,v) is the bilinear form
a
+
k
I
\
ka
k ak
ak
ak
(xkyRTxQyk)2
z,k
xk ayk
Y
_ B(x,x) I(- 2xk)ak + 2B(x,y)
k
k
I xk(xk/ak)
k
= -2B(x,x)B(x,y) + 2B(x,y)B(x,x) = 0
a3F
b)
c)
k xk
xk
Likewise
=
2F
because F is homogeneous in x of degree 2.
I yk ax
picks out the quadratic y-terms in F, i.e.,
k
aF =
l-2B
(x, x) B (Y,Y) + 2B (x,y)2,
Yk
ayk
so
Xk 8x k
.
Yk aF
yk
2B(x,x)
3.73
d)
The flow therefore is
xk = aE
x
Y.
a kk B (x,x) + 2 ak B (x,y)
2
yk
aF
Yk
axk
+ x (2B(x,x))
k
- 2 ak(l - B(Y,Y))
k
Let E be the ellipsoid B(x,x) = 1.
minimum at
t = -B (x,y) /B (x,x) ,
Y
--
2 ka B(x,Y) + 2 xk B(x,x).
k
B(y+tx,y+tx) has a
Note that
i.e., at B (t,
if
')
=y-
X
(X,y) x.
But
B(E,) = B(y,Y) - B(x,Y)2
B x,x
So
Lxry is tangent to E if and only if B(,) = 1, which holds if
and only if l-B(y,y) + B(x,y)2/B(x,x) = 0, i.e., if and only if
F(x,y) = 0.
Now differentiating along a flow line:
B (X, y)
k
k
B(x,x)xk
B (xrY)
B(x,x ) xk
y) + 2 xk B(x,x)
+ 2 yk
Y)
- 2 xk B(
a x,x )2
Now define a function T(t) by setting
T(W '
y)
= 2B(x,x) - (BB (X,
(x,x) y
_
/B(x0`
1B(x,x)/
x
k
3.74
along this flow line.
Then
dT
dt*xk
dt
or
d9k
xk.
dT
Therefore
2
d 9k
d
dt Xk
dT dt
yk
l
xk
- B(x,Y)
B(x,x) xk
dT/dt[
2B(x,x)
dT/dt
(-2 B(x,x))
ak
.
Ek
ak
This simply says that the acceleration of
normal to the ellipsoid E, i.e., that
t(T)6 E is always
1(T) is a geodesic.
QED
3.75
Tying together the analytic Jacobian and algebraic Jacobian
§5.
So far in this Chapter, we have definetan algebraic variety
Jac C and studied its invariant flows.
In Chapter II, we associated
to any compact Riemann Surface C a complex torus Jac C.
If
is
C
hyperelliptic so that both constructions apply, they are isomorphic
by Abel's theorem.
We would now like to make this isomorphism
explicitly, i.e., express the algebraic coordinates on Jac C-S as theta functions.
To study C as we did in Chapter II, the first thing we must do
is to choose a homology basis A±,Bi.
to do this in the hyperelliptic case.
There is a traditional way
One first chooses on
a simple closed curve P through the set of branch points B.
One then chooses paths in 1P1-B as in the diagram below. Noting
7P1
= (C
U
(co) ,
that each of them circles an even number of branch points, these
paths can be lifted to the double cover C.
see Figure on next page.
On C, the paths Ai are disjoint from each other as are the paths Bi1
and Ai,Bj meet only if i = j, and then in one point so that
i(AA) = i(Bi.Bj) = 0
i(Ai,Bj) = Sij.
Thus Ai,Bj are a symplectic basis of HI(C,2Z).
To make this picture
clearly homeomorphic to the figure in Ch. II, §2, we can also add
disjoint tails to all Ai,Bi, connecting them to the base point
P1.
Widening each tail into 4 parallel paths, we can lengthen Ai,Bi to
disjoint simple closed loops AiBi' all beginning and ending at P1,
which is exactly as in §11.2.
3.76
top layer
curves on
bottom layer
(5.1)
crosscuts where
layers join
3.77
4,111
Next, on C we can describe the g-dimensional vector space of
holomorphic 1-forms:
Proposition 5.2:
dt
w = P (t)
s
Proof:
consists in the 1-forms:
r(C,& )
P a polynomial of degree < c_1,
s2 = f(t), we have
Because
2s ds =
so
P (t) dt
s
= 2P (t) ds
f' (t)
On Cl (the affine piece of C with coordinates s,t), s = 0 implies
f(t) = 0 which implies fl(t)
Thus at every
0 because f has no double roots.
P E C1,either s(p)
34
0
or
f'(t(P))
one of the above expressions it follows that
Now at
and s' =
t' =
t
t}a
w
74 0,
so using
has no poles on C1.
are coordinates.
Then
3.73
ds
- (g+l)
ds' =
tg+l
f' (t) ( 2.t g1
s-dt
tg+T
+1) s2
tg+2 )
tf' (t) - (251+2) f (t)
2tg+2
dt
s
.
dt
s
_t2g+1+(lower terms in t)
dt
s
2tg+2
Now we saw in §1 that s',t' have respectively a simple and a double
hence s' is a local coordinate near
zero at
-
and ds' is a
So the above equation
1-form with neither zero nor pole at
shows that
st = (-2(t') g-l + higher order terms in
i.e.,
dt
s
Thus if deg P < g-1,
has a zero of order 2g-2 at
P(t) has pole at - of order < 2g-2 and w has no poles at all.
Thus we have found a g-dimensional space of 1-forms without poles
and as dim F(C,21) = g, this must be all of them.
(We could also
and show
start with an arbitrary rational 1-form n =
directly that if n
has no poles, then
= 0,
(t) = (polyn. of deg < g-1)/f(t).)
The next step is to choose a normalized basis
w. _
P. (t) dt
s
QED
3.79
of holomorphic 1-forms such that
A.
I
The period matrix of the curve C is then
Wi
and the analytic Jacobian is by definition:
(lattice 2Zg + S2.7Z g) .
By means of the indefinite abelian integrals we have holomorphic maps
I
>
Og/LS2
P.
(P 1,
mod LP
,Pk)
.
Abel's theorem (I1, 52) states that these induce an isomorphism
Jac C
>
k
if we map a divisor class
Cg/Ln
k
Pi
--
Qi
to
1(P1,...,Pk) -
Taking k = g, we compare this with our algebraic
description of an affine piece of Jac C:
3.80
I
Cg
a*
CTg/LQ
U
Z c CCZg
(Cg) o
11
11
open set of
PiO°°,Pi01P
variety of polyn. U,V
if i 0 .7 g f)
J
such that UIf_V2
We have seen that z is the open piece Jac C-0
)
of the Jacobian,
g-l
where
0 = (locus of divisor classes
P.-(g-l).=).
i=1 1
Our goal
now is to prove:
Theorem 5.3:
1)
There are
E
ZZ,g
such that for all
P I...FP
1
lg ] (z. St) = 0
z
2)
Thus
Jac C-0
g-1
g-1
i-1
z E Ego,
E C such that1
i,W
mod L
can be described analytically as
and algebraically as the above variety Z,
whose coordinates are the coefficients of U(t),V(t).
Thus
the coefficients of U(t),V(t) are meromorphic functions on
CTg/LQ
with poles where''
S
= 0.
3.81
3)
For all branch points akE B, there are r1' (k)
and a constant ck such that for all divisors
D =
in
p.
(k) E 12Z9
9
(Cg)., if
UD(t) _
i=l
(t-t(Pi)) is the
corresponding polynomial, then
+q
'LS +n
k
UD(ak) = ck.
2
P .
9
4.
\iI1 JAW)
CO
pi->.
jd° l (
I
\i l
f )
CO
I
This determines the coefficients of
functions on
U(t)
as meromor hic
CCg/Ln.
In the course of proving this, we shall determine
d,rl(k) explicitly.
In fact, ck can also be determined, but this will not be done until
the next section.
We, first prove (1).
Riemann's Theorem,
E
2LQ
Note that (1) is exactly Corollary 3.6 of
(Ch. I1,§3), except that we assert that
and we want to compute
the fact that
z) = t4' (z) )
Q
too.
(Also, we have used
WeAdetermine
4
by arguing
backwards from some of the Corollaries of Riemann's theorem, or else
we can go back to the proof of 11.3.1 and work a little harder on
the integrals there.
We shall do this although the reader should
be warned that the details are such that it is almost impossible
not to make mistakes of sign, orientation conventions, etc.
result is that for hyperelliptic C,
The
3.82
I=
(5.4)
S11' +
mod L
"
where
_
8
1 1 ._.
2
g
1
2
E
........ 1 2) E
2
indefinite integral of
let gk be the
mod L2:
Recall the expression for Z
Proof:
gk(-) = 0
1l
22
wk on C - UA! - UBi, normalized so that
(we are extending Ai and Bi by "tails" to get a figure
homeomorphic to the one in §12.2).
P
kk
A
k
-
2
- f
Then
1
k
+
R,
1
f +gkwk
A
P1
(In the term
f
wk, the path should be taken in C-UA'-UB
from co
m
to
P
considered as the beginning of B'.)
1
f
Firstly, wQ = dgp,
2
d (gk)
.
This is
Ak
A +
1Ig2(end of Ak,+)-g2(beginning of
the beginning of
of
1t
at the end because
Wk = +1.
J
Ak
So this term is
the
3.83
P1
2
f
z[(J wk)
- (
2
rPl
Wk -1
1)
I
00
Secondly, if
and endof
has the same value at the beginning
k, then
k
A+
because
wk = 0.
So the contribution of the
J
Ak
tails on
A'+
in
is zero and we may as well integrate around Ak.
on
gk
is evaluated
AQ by paths as follows, missing all Ai'P
-,4
-R
= access road to A
3.84
here we have chosen
exactly along the cut between a2X_1,a2Q,
Ai
from a2Q to a2Q_1 and a return
so that it consists in a path az
along
i(cL).
Now
= -WQ
i*Wz
t(ak) is traversed backwards:
and
,
so
J+gkWQ
=
AR
But
J
(gk +
is constant because
gk+l*gk
J
(gk+l*gk) WQ
l*gk)WQ
aQ
d(gk+l*gk) =
2gk (a22, ) J
Wk+l*Wk = 0.
Thus
WR
aQ
a21
(since
gk (a22)
JU)k
pQ
1
Wk
2
Roop pQ.i*pz]
=2
(since
J
(Al+... +At).-BZ
0
+ {1/2
pQ-l* pi
is homologous
to (A1+...+Ai) _BQ
if k > i
if k < i
3.85
Altogether, this shows
1
if g-k+l even
0
1
'2 Q=1
k
k£
+
if g-k+l odd
.1/2
which proves (5.4).
Part (2) is just a restatement of Part (1).
Before proving (3),
we need to tie together the different descriptions we have introduced
In fact, in §2, we showed that
for the 2-torsion in Pic C.
group of divisor classes eT
(Pic C)2
(T c B, #T even, mod eT = eCT
where
err
=
IP-
#T.-
PET
We also know
(Pic C)2
=
2-torsion in
Tg/L
The link between these is given by
P
eT I
>
I (eT)
w
P T
3.86
th place
We can calculate I(eT):
.r
Lemma 5.6:
a) I(e {a
2i-I
I (e {a2i
a 2il
)
_ tt0,...,2, ...0)
a2i+l' ... ,
2g+1}
-"I" li
2
._
'
'
'GC3l)mod I
2
ith place
b) I (e
{a2i-l"'
I (e {a2i , 00} )
Proof:
1
1(2: 1 ...r2'0,
t (2,2,
1 1 ..
Ln
0)
2'2
)
ith p lace
+
2,
2
2
mod L
The path Ai in the diagram (5.1) above may be moved so
that it follows
p
from a2i to a2i_1 on one sheet of C, and then
goes back on the other sheet:
C.
N
TI
3.87
But each Cok reverses its sign when you switch sheets.
As the
direction in which Ai is traversed also changes:
w
a2i-lam
=
w
-
0*
a2i
f
w
J
00
e{a2i-l'a2i}
mod
LS2
z
a(2i
(Note:
2.J
w E L
because I(2eT) = 1(0) = 0 E Cg/LQ.)
CO
The same argument with Bi shows
= I(e
})
mod Ln
.
{a2i,...a2g+l
This proves (a).
(b) follows because of
{al,a 2}o...o{a
2a.-3,a2i-2}o{a2i,...,a2g+1} = C{a
and
{al,a2}o ..o{a2i-l'a2i}o{a2i,...,a2g+l} =
and lemma (2.4).
QED
C{a2i,o}
2i--1,Go}
3.88
.th
1
Definition 5.7*:
place
tn2i-1
2
r
2i-1
= the 2xg matrix
0
2
2
n2i-1
ilace
0...0 2 0...0
n2i = the 2xg matrix (
2.
I
nT
'nk
_
)
.2
1
2
for all
0...0
fl2i
(
)
if
2i
T c B
akET
a
k#o
Then lemma 5.6 says that
I (eT) =
(5.8)
SZn +
TIT
and the more precise version of part (3) of the theorem states that
UD (ak )
= ck
CO
To prove this, note that both sides are meromorphic functions
on Jac C with poles only on the irreducible divisor 0
.
Suppose we
prove that both sides are zero precisely on the translate of 0
* There is an unfortunate conflict here between conventions for row and column vectors.
In Ch. 2,
{n
was defined for
n',n" columns of height g.
W,n" were rows of length g (easier to write!).
put a transpose in this definition.
In the 19th century,
To make these compatible, we must
3.89
ak
by
J
w
and vanish to 2nd order there.
It follows that the ratio
CO
of the LHS and RHS is finite and non-zero on J'ac-0, hence it is
either
1) a constant, or 2) has a zero on
3) has a pole on
and no zeroes.
0
0
and no poles, or
But using the fact that a
bounded analytic function on a compact analytic space is constant,
applied to the ratio or its inverse, we see that 2) and 3) are
impossible.
Consider therefore the zeroes.
on the translate of V(9[6])
this is the translate of
ak
f
w.
0
by
sink + ?I
by I
As for the LHS, as D =
The RHS has a double zero*
(e
{k,
,
}
By our remarks above,
k*
)' i.e.,
Pi'
CO
UD(ak) = 0
for some i
Pi = ak
Pi)-ak
(
(effective divisor of
degree g-1)
1
(divisor class
Pi-ak- (g-l) -) E
P.
0
ak
W
E
(translate of 0
J
by 1
w)
Note that if '& [S] vanished to some higher order r > 2 on 0, this
would contradict Riemann's theorem: because for a general choice
of P1, ' - , pg E C,
f (P)
w+
00
order at
P = P1'...'Pg.
1w)
00
vanishes to first
3.90
To check the order of vanishing, go back to the covering:
res ir
(Cg)00
Z0
open set of Z of U,V
(such that U has g
distinct roots
g-tuples Pl,---,Pg
(
s.t.
P.P.,#t(p),
1
if j O i,# o0
The group of permutations acts freely on (Cg)
00,
so
7T
is an
unramified covering map between g-dimensional complex manifolds,
i.e., they are locally biholornorphic.
Now
i h place
(res rr
1 zeroes of
UD (ak)
}=
g
U
[Cx...x{a
i=l
k
I
The pull-back of the function UD (ak) is f (Pl, ... ,Pk) _
g
11
(t (Pi) -ak) -
i=l
But the function t-ak on C vanishes to order 2 at the point s = 0,
t = ak, i.e., at the point we are calling ak.
So f vanishes to
g
order 2 on i,,;prll(ak) as required.
QED
i=l
An interesting restatement of part (3) of the Theorem is
3.91
Corollary'5.9:
#[-Ik] (z)
The 2g+2-meromorphic functions
2
f
*[Olw
k E B, on
Jac C
span a vector space V of dimension
)
only g+1.
In the projective space 1P(V),
the individual functions
lie on a rational curve D of degree ,g and on this curve, give a
finite set projectively equivalent to B in IPI.
In this way, we
can reconstruct the hyperelliptic curve C from Jac C and i9'
Proof:
Part (3) says that, up to a translation in z, these
functions are D i--.. UD (ak) . But
UD(ak) _
So the 2g+2 function
I U?-ag-'
i=0
,
UDy = coefficients
of UD(t).
UD(ak) are all constant linear combinations
of the g+1 functions D t----> UD
(including UD which is the
constant function 1).
Taking these U? as a basis of V, the individual functions
UD(ak) have coordinates in V
(ak,ak-1,...,ak,l)
The rational curve D in the theorem is just the locus of points
in IP(VJ
whose homogeneous coordinates in V are:
some
b E C.
Thus b is a coordinate on D and the individual functions UD(ak.)
have coordinates b = ak.
restatement of Part (3).
Thus the Corollary is just a geometric
QED
3.92
In 93, we described algebraically the translation invariant
vector fields on the variety Z of polynomials U,V,W such that
on (Cg, the translation
In analytic coordinates
f-V2 = U-W.
invariant vector fields are just
c-, E.
,
Ic1 8z
We can tie these
The result is:
together too.
Let
Pro position 5.10.
Wi =
(t) =
(t)dt/s,
Then in the isomorphism
the vector field Da on Z corresponds to the vector field
- vi(a) az
and the vector field D,,
vector field
-
Proof:
ei 8z
on Z corresponds to the
1
Let D(E) _
Pi(s)
represent an integral curve
i=1
of the vector field
U E(t) =
(P
g
IT
i=l
Da
(t-ci (E) ) = UD (E)
= V E(ci (E)) .
ci(E) = t(Pi(e)),
Let
(t), and Vs (t) = VD (E) (t), so that
Then
U F- E (t) =
VE (a) UE (t) --UE (a) VE (t)
C
t
a
,
D(E)
w
The corresponding curve in (Cg-space is
J
prove
=
E=0
and we want to
3.93
Letting ci(0) = ci, we calculate
Therefore
aE (U,: (ci) ) =
in 2 ways:
UE (a) VE (ci)
(aE UE) (ci) _ _
and
aEUE(c
ci-a
(ci-ck) (- as ci (E))
II
ki
.
U
ae (ci (E) )
=
1
(ci-a)
(ci-ck)
II
k#i
Letting t=a, s=b
be the point on C over a, we recall the rational
function
U (a) . (s+V (t))+U (t) . (b-V (a) )
U(t) (t-a)
on C used in
§3, which has poles at P = (a,b) and at
Take its product with
w
and use the fact that the sum of its
7
residues at all poles is zero:
0 =
(t) dt\
+U (t) (b-V (a))
resQ (U(a). (s+VU(t))
(t) (t-a
;(a)dt l
= resP U(a) (t-a)
(
b
l
+
s
2U(a)V(c)
resP-ci)
11 (ci ck)(ci-a)
((t
-71i
(using s(Pi) = V(ci))
20.(a) + 2
W(ci
)
E=0
V(c)
.(ci) dt
V(ci))
3.94
But
C. (C)
a
r
J
wj
j (t)dt
j
g.Co
i
The proof for the vector field D.
s
E=0
0*
C=0
j (Ci)
V(ci
a
ae
is similar.
(ci(e))
E=0
QED
3.95
§6.
Theta characteristics and the fundamental
Vanishing Property
The appearance of
mysterious.
in the main theorem of '§5 looks quite
It appeared as a result of an involved evaluation of
the integrals in Riemann's derivation.
As in the Appendix to §3,
Ch. II, we would like to introduce the concept of theta characteristics
in order to give a more intrinsic formulation of (5.3) and clarify
the reason for the peculiar looking constant
It cannot be
eliminated but it can be made to look more natural in this setting.
Recall that theta characteristics on a curve C are divisor
2D = KC.
classes D such that
For hyperelliptic curves, we can
describe them as follows:
Proposition 6.1:
i)
ii)
KC
(g - l)L
=
Every theta characteristic is of the form
f7
def
IP+
(
T)L
PET
for some subset T a B with
,iii)
if and only if
fT = fT
1
set
#T ; (g+1)(mod 2).
T1 = T2 or CT2, hence the
2
I
of theta characteristics is described by:
set of subsets T c B
#T
-= (g+l) mod 2
1/
modulo
T - CT
3.94
3.96
g-l
iv)
For all such T,
such that
E C
P.
But
if and only if
#T 34 g+1, and if #T < g+l
g-1
(f T) = dim oG (
dim
Pi)
1
g+l--#T
2
(if #T > g+1, replace T by CT to compute dim t(f,V
Proof:
In the proof of (5.2), we saw that the divisor of the
differential dt/s
was just (2g-2)-, which belongs to the divisor
The pi
class (g-l)L.
This proves (i).
2fT
=
As for (ii) and (iii), note that
I 2P + (g-l-#T) L
=
(g-l) L
PET
hence fT
But all 2-torsion is representable as divisor
classes e,,, and it's immediate that:
(6.2)
fT + eS
=
fTOS
.
Since any 2 theta characteristics differ by 2-torsion, they are all
of the form fT for some T.
fT
1
=
fT
Moreover
eT
2
o
1
T
= 0
by
(6.2)
2
T1oT2 =
T1 = T2
or B
or
T1 = CT2.
3.97
Finally, to calculate 9(fT), use
I P,
(g-l+#T)c
PET
hence
_
fT)
space of fcns. f with (g-l+#T)-fold
(pole at - and zeroes at all PE T )
We assume #T < g+l, so (g-l+#T) < 2g.
poles at
pole at
are polynomials in t
).
Now functions with 2g-fold
of degree < g (s has a 2g+l-foa.d
So
polynomials in t of degree < (5-1+#T)
T
zero at all P E T
The dimension of the latter space is g+2-#T, hence (iv).
QED
Comparing Prop. 6.1 with 11.3.10, we come up with a set of
canonical isomorphisms as follows:
T= B: #Tsg+l(2)1
and 'T - CT
Tt
>
>
fT
fsymmetric translates}
I
f
2
of 0 in ,Yac C
locus of div. classes
-- p +...+P
fT
}
1
g-l-
zeroes of '$[nl
n
3.98
Thus the symmetric translates of
0
in Jac C can be described
2g/m2g.
combinatorially in 2 ways:
p E
by subsets T of B and by
Riemann's theorem tells us how to link these up.
2
The result can be
better phrased like this:
Proposition 6.2:
al,a3,
Let
U c B
be the set of g+l branch points
In the above correspondences, the.followin
,a2g+l.
objects
correspond to each other:
itself ,
(a)
if g odd
¢i
<
(g-l)00 4-;
if g even
(b)
T c B
For all
T
0
i.e.,locus of
(P1+...+Pg-l
l-(g-1)co
such that #T E (g+l)mod 2:
<
TIToU
especially:
U C
Proof:
0
(a) is a rephrasing of (5.3) part 1, except for the
first description.
For this note that
0
= (g+l)mod 2
and
f,
=
(g--l)-,
g even -=> # ({ }) = 1
= (g+l)mod 2
and
f{-}
_
(g-l)
g odd
#
3.99
nU
To check (b), build from (a) as follows:
[eSI
rtranslate
if g odd
if g even
ISoS
1:II) i
(...)+(
:iii:)(
z
2
[6+
of0 by
L
If g is odd, #U is even and one checks
1
for all S c B, #S even,
I(eS)
6
)
...
2 z
z
while if g is even #(Uo{0o}) is even and
0...0l (0.. -0
0 0...2
0)+....
2,a4,...,a2
}
g
(1 0...OJ+\1 L.
2
2 2
1
1...3.)
22
=
S
2
Letting T = S if g is even, T = So{oo} if g is odd, part (b) follows.
QED
4.
This gives the following "explanation" for
symmetric translates of
1_,q+1(2).
the
0 are - without any unnecessary choices -
naturally parametrized by the divisor classes
T c B, #T
A and 6:
Z
,. hence by subsets
The points of order 2 on Jac C are naturally
parametrized by subsets T c B, #T even.
The theta function, after
a lot of non-canonical choices, picks out a particular symmetric
i.e.,
't9 (z) = 0.
0,
(6.2) shows that in effect all these choices just
boil down to fixing. a "base point" in the set I
of odd-numbered branch points.
which is the set U
S1.
3.100
In Ch. II, §3, Appendix, we also noted that
I
came with a
natural division into even and odd subsets." We can identify this
division in the hyperelliptic case:
Proposition 6.3:
rnS
a)
e2(n5
1
(-1)#(S1ns 2)
2
b)
e* (nToU)
c)
If
for
Sic B, #Si even,
(#T-g-1)
2
for T c B, #T
(g+l) (2) ,
hence:
T c B satisfies #T = (g+l)(mod 2), fT is an even
element of
if and only if
I
odd if and only if #T
Proof:
=
#T
- (g+1)(mod 4),
(g--1) (mod 4) .
Check (a) as follows:
Note that
# (Sl n (s2Os3)) -
# (S1 n s2) + # (Sl n s3) (mod 2)
(see figure 6.4), hence #(S1n S2) mod 2
is a symmetric bilinear
2Z/22Z-valued form on the group of sUJ,sets of B.
are generators {ak ,°°},{ak ,=}
1
directly.
When S1 and S2
of this group, one checks the result
2
This proves (6.3a).
A = S1 n (S2oS3)
(6.4)
B = points occurring in
both S1ns2 and S1ns3.
3.101
To check (b), recall that
e (a+13)
e*' a)e*
=
s)
all
e2(a, ).
aR
E
I 2Z2g
#T--g-1
Let e* (T) _ (-1)
We check that
#slns2
e (ToS1QS2) e* (T)
e,(To51)e*(ToS2)
(--1
for all S1,S2,T C B, #Si even, #T = g+1(2).
(6.5)
)
This is equivalent to:
#(ToS1OS2) + #T - #ToS1 -- #ToS2 = 2#(S1ns2)(mod 4)
Proof by Venn diagram:
7
(6.6)
(+ for membership in ToS10S2 or T; - for membership in ToS1 or TOS
Thus in (6.5):
2).
3.102
LES = 2# (TfS1fS2) - 2# (S1f,S2fCT)
2# (Sill S2) (mod 4) = RHS.
Putting together part (a) and this equality, we find that
e* (ToU) /e* (r1T
T I
is a homomorphism from the group of even subsets of B to (+1).
Next check
e* (Uo{ak,co
+1
if k is odd
if k is even
while
e*(pk)
+1 if k is odd
=
-1 if k is even.
This proves (b).
(c) is a restatement of (b).
QED
Note that (6.1 iv) and (6.3b) together confirm the formula:
(-1)dim t (fT)
=
e,ti (nToU)
asserted without proof in 11.3 for all corresponding divisor classes
D with 2D = KC and theta functions &[n].
Putting together (6.1) and (6.2), we obtain the following
vezy important Corollary:
Corollary 6.7:
points B.
Let C be a hyperelliptic curve, with branch
Describing the topology_of_ C as:above, let U c B be the
(g+1) odd branch points and let 92 be its period matrix.
Then for
3.103
S a B, with #S even, let I(eS) E Jac C
all
2-division'point.
151
[nS) ( 0 , n )
Proof:
be the corresponding
Then
=0
'&(I (eS) ,n) = 0
4r` (SOU) # (g+ 1) .
Combine Car. 3.12. of Ch. II with (6.2) to find:
(fSOU
(4[nsl(0,n)'= 0)
Pl+...+Pg_l for some Pi).
Then apply 6.1 iv.
QED
The importance of this Corollary is that it provides a lot
of pairs
n',n" E
22Z2g
such that for hyperelliptic period matrices
SZ,
I
0
nE2Zg
We know (II.3.14). that for all odd
vanishes for all
n
n',n", i.e., 4tn'. n" odd, this
because in fact the series vanishes identically.
But Cor. 6.7 applies to many even
n',n" as well.
We shall see, in fact, that these identities characterize
hyperelliptic period matrices.
To get some idea of the strength
of this vanishing property, it is useful to look a) at low genus
and b) to estimate by Stirling's formula, what fraction of the
2-division points are covered by this Corollary for very large genus.
3.104
g = 2:
{S c {1,2,3,4,5,6}
=
I
#S = 1, 3, or 5}/(S ^ CS)
[the 6 odd characteristics
U
the 10 even characteristics
{1,2,3},{1,2,4}, --,{1,5,6}
[normaizing S by assuming 1 E S)
g = 3:
1 _ {S c {1,2,.--,8} #S = 0,2,4,6 or 8}/(S - CS)
[the one even characteristic S =
=
g = 4:
=
with g(fS)
.(0)l
U
[the 28 odd characteristics S = {i,j}l
U
[the 35 even characteristics S = {1,i,j,k}, J(f5)=(0)l
{S
I
c {1,2,...,10}
#S = 1,3,5,7 or 9}/(S - CS)
[the 10 even characteristics S = {i}, with dim X(fS)=2l
U
[the 120 odd theta characteristics S = {i,j,k}l
U
[the 126 even theta char. S =
£(f)= (0) l
.
3.105
Fraction of 2_ ;s.ovt
pts.wh,ck are odd
Fraction of
2-division pts. a
(so that &(a,f)=0
_,.9
where i3 (a,St)=0
all S)
S2 hyperelliptic
in dimension 2,
hyperelliptic Q's
2
6/16
6/16
3
28/64
29/64
4
120/256
130/256
5
496/1024
562/1024
large g
^- 1/2
"- (1-
2
-IT
are an open,
dense set
l
)
vg-+1
The last estimate comes from:
# ({S c B I#S=g+l}/S _ CS) -
1 (2g+2) !
2
(g+l )! 2
1[(2ge2)2«+2
2Tr (
22g (_2 __L_]
JTr
/g-+l
-
.C(9+1) 9+1
2Tr
)-2
3.106
Frobenius' theta formula
§7.
In this section we want to combine Reemann's theta formula
(11.6) with the Vanishing Property (6.7) of the last section.
An
amazing cancellation takes place and we can prove that for
hyperelliptic
SZ
,
4(z,S2) satisfies a much simpler identity
discovered in essence by Frobenius*.
applications of Frobenius' formula.
We shall make many
The first of these is to make
more explicit the link between the analytic and algebraic theory of
the Jacobian by evaluating the constants ck of Theorem 5.3.
The
second will be to give explicitly via thetas the solutions of
Neumann's dynamical system discussed in §4.
Other applications will
Because one of these is to the Theorem
be given in later sections.
characterizing hyperelliptic Q by the Vanishing Property (6.7), we
want to derive Frobenius' theta formula using only this Vanishing
and no further aspects of the hyperelliptic situation.
Therefore,
we assume we are working in the following situation:
1.
B = fixed set with 2g+2 elements
2.
U
3.
- E B-U a fixed element
4.
T j' 71 T an isomorphism:
B, a fixed subset with g+l elements
even subsets of B
(modulo
1
)
S ^- CS
2Z
2g/ 2Z2g
2
such that
a)
b)
=
ns O5'
2
1
+ fS
1
e2 (n5 rns
1
W-
n
)
2
_ (-1) #S11 52
2
Uber die constanten Factoren der Thetareihen, Crelle, 98 (1885);
see top formula, p. 249, Collected Works, vol. 11.
3.107
# (ToU) -g-l
5.
6.
St
2
e*(nT) _ (-1)
c)
Ej
satisfies
2g
n9jnT1(O,SZ)
We fix
ni E 2 ZZ
equals
n{i,.}, and also let
for all i
E
=0
B-=
no, = 0.
if
#ToU
g+l.
such that ni mod 7L2g
(This choice affects
nothing essentially.)
We shall use the notation
eS (k) = +1
-1
for all k E B, subsets
if
if
kES
k%S
S c B.
Theorem 7.1 (Generalized Frobenius' theta formula).
above situation, for all zi E Cg, 1 < i < 4
z1+z2+z3+z4 = 0, and for all
such that
ai E(p2g, 1 < i < 4, such that
al+a2+a3+a4 = 0, then
(Fch)
e
jEB
4
U
(j)
U
(j)exp(4Tritfl cfl
11
i=l
&(a+nj]
(z. ) = 0
i
or equivalently:
(F)
e
jEB
In the
3
4
)
IT
i=1
z+Pn+T1
i
J
3
=0
3.108
By (Rch), for every
Proof:
2-g
w E I 2G2g
(z4)
1
XE22Z2g/IL2g
a1+2w+a2--a3-a4
_ '1} [wl (0) l C
or
2
l (z1+z2-z3-z4) ( ...) (.... )
19'[w1 (0) = 0
4
0=
exp(4Tri
I
i=l
2Z2gtZ2g
XE
[ai+al (zi)
since
-e (a 1
Therefore,
2Trit (a'+a') 2w"
l
i,9,[a1+x1 (zl)
.
V T c B, #T even, #ToU # (g+l),
0=
ScB,#S even
(-1) #snT
4
i=1
'&[ai+nsl (zi) .
SMCS
Thus, for any coefficients cT,
(7.2)
0 =
I
I
S
I
B
#S even
mod S^+CS
T c B
#T even
# (ToU) 34g+1
c
(_l) #SnT
T
4
1 z9'[ai+nsl (zi) .
i=1
3.109
What we must do is to choose the cT's so that "most" but not
all of the terms in brackets vanish!
For this, we resort to a
combinatorial lemma:
Lemma 7.3.
For all S c B, #S even,
,{-,kj,B-{-,k},B
if S #
2 g2 if either S
0, {-,k } Or
S=B-{- k},B and gis odd
22g-2 if S - B,B-,k} and g is even.
0
#T- (q+l)
Y
.
(-l) #Sf1T =
2
TcB
ET
#T w (g+1)mod 2
We note first the following points:
Proof:
a)
for all finite non-empty sets R,
# (subsets T c R)
skt #T even
# /subsets TcRR = 2#R-1
#T odd
J
In fact, the subsets T c R
form a group under o and the even subsets
are a subgroup of index 2.
b)
for all finite sets R with at least 2 elements,
rr
(#T)
I
TTeven)
=
(#T) = (#R)
I
.2#R-2
#T odd)
In fact, the first sum here is the cardinality of the set of
pairs (i,S),
where i E R
we count this by (a).
an even subset of
and S
is an odd subset of R-{i}, and
The second sum is the same except that S is
R-{i}.
3.110
Given these facts, we can easily work out the sum of the lemma.
Note that it is invariant, up to the sign (-1)#T = (-1)g+1, under
S H CS = B-S, so we may assume
w E S.
We then have
#T1+#T2- (g+l)
#T- (g+1) . (-1) #SnT __
2:
TcB
I
2
® ET11S
T2cCS
2
00 ET
#T =(g+1)mod 2
#T
(--1)
#T1+#T2=g+l
#T
1
[#T2+(#T1-(g+1))) ]
T2cCS
#T2(g+l-#T1)
If
#CS > 2 and #S > 3, then
#T1
(-12)
#CS-2#CS-2
+ [#Tl- (g+1)12 #CS-11
coET1cS
= 2#CS-3
I
{ -ET1c
#CS+2[#T1-(g+l)]-
S
#T1 even
= 0, using b) again.
I
-ET 1cS
#T1 odd
#CS+2[#T1-(g+1)]}
3.111
If either #CS < 1 or #S < 2, we must have S = {-,k} or S = B;
in the first case we compute directly:
2#CS-3f
E
#CS + 2[#T1_(g+1)]
I
--
_
#T, odd
#T1 even
=
#CS+2[#T1-(g+1)]}
ET1CS
co ET1CS
22g-3 (2g+2 [2- (g+1) ]
- 2g - 2[1-(g+1)})
=
22g-2
;
in the second case,
#T1
_
2g+1)
(-1) #T1
=
(-l)
ET1CS
g+1
[ (2g+l) 2g+1-3 - (g+l) 22g+1-2
2
#T12g+1
= (-1)g+1
22g-2.
QED
To apply the lemma, note that
#S n (ToU)
# (snT) + # (SAU) mod 2.
=
In formula (7.2), set
# (ToU) - (g+1)
2
if
0
if not
E ToU
cT
Then by the lemma,
# (ToU) - (g+1)
0=
S CB
TC B
#S even
#T even
.
(-1)#s n (ToU), (_1)#(snrJ)
4
i=1
mod S - CS
22g-2
S={}o{k}
kEB
ti
(-1) #Snu
(zi) .
QED
(zi)
3.112
Corollary 7.4.
Let
S = ToUo{oo}, so #S even.
jET
T c B have g+2 elements and let
Then
e.(j)exp(4Tri tT1 Snj) 9 [ns+nj] (O)2.9,rn
(z)2 = 0.
In Foh, take
Proof.
zl = z2 = 0, z3 = z, z4 = -z
a1 = ns, a2 = -ns,
a3 = a4 = 0.
Then
I
o=
jEB
now for any
eU(j) 4 4 [ns+nj] (0) ' [-n$+njl (0)'9'[n.] (z) 't9'[nj) (-z)
A E
2Z
2g
/&[a+Xl (z) = exp(27Tita' A')'S'[al (z),
so
45'[-ns+n
(0) = 4T[ns+n2nsl (0) = exp(-4Trit(r1;+n
)
ns) z9'[6 S+n
and
'&(nj l (-z) = e* (nj) 49'[nj l (z) .
But
(wo{'}o{=
-1')
so putting all this together, we have the formula (7.4).
QED
(0),
3.113
Corollary 7.5:(
*[in l (z) 2
$[n3 J (O)
0
jEU
O
O
z
)
1
In (7.4), set T = U U {-}, hence S =
Proof:
.
We now apply Frobenius' identity to refine (5.3) above:
Theorem 7.6:
As in (5.3) consider the map
space of monic polyn.
Jac C - 0
U(t) of degree g
W
D
W
,
t
UD (t)
Then for all finite branch points ak, 1 < k < 2g+l, and for all
V c
such that
D
U (ak) _ -
'TT
iEV
#V = g+1, k E V
[nu,V+nk1 (0) 19 [+fk] (z)
(ak`ai)
i9 [nUeVl (O)
i'k
where z = J
w
2
'c9[Sl
and the sign is given by
D
(-1)
Proof:
(-'1)
We make a partial fraction expansion:
(t)-
UD(t)
TT(t-ak)
kEV
D
=
C
k,
kEV t-ak
Then UD(t), is monic but otherwise arbitrary, so
= 1 but
k,V
In particular, if IdkXk V = 1,
I
AD
k
otherwise the
ak,V are arbitrary.
P
3.114
for all D, then
dk = 1, all k.
',dow by (5.3),
UD (ak)
ck
2
D
k,V
iUV(ak-ai)
(19-[6+nkl (z)
1
(z)
)
(ak-ai
II
it v
ilk
i34k
On the other hand, by (7.4), with T = V U
(-), S = UoV,
tnU
,9'[nkl (z)
(*)
exp (4 ri
1 =
2
9'[nUoV+nkl (0)
)
15)_
kEV
[nUoV[ (O) - &101(z)
Using the definition of
19'-functions with characteristic, it
follows:
2
[d+nk) (z)
9[d] (z)
t
exp Wri d
2
'
k)
nk)
'
(z+Std'+d")
Since z is arbitrary in (*), we can replace it by z+Sld'+d"
and find:
2
(0)
1=
kEV
kEV
J
Thus the coefficients of
ck = exp (41ri
) f (ak-ai)
iEV,i#k
.
TT
iEV,i k
(a -a )
D
k,V
ck
lkV are all 1, hence
t9[nuov+nk1 (0)
( & [nUoVl (0)
l
[dl (z) (z)/
[nUovl (0)
p,
U°Vkl O1 2
exp (47ri.... ) . ((fl01(0)
for all D.
(+flkd
)
2
QED
3.115
A second application is to the explicit solutions (xk(t),yk(t))
of Neumann's system of differential equations,
xk
Yk
Yk = -akxk + xk(jaix
where
- iyi)
a1< --- < ag+1 are fixed real numbers and
lxkYk = 0.
jxk = 1
and
We saw that
(xkyk-xkyk)2
2
a
k k
k
k
are integrals of this motion and set up the following maps:
TT (Sg+1
subvariety
F1 (x,Y)=cl
(Fg;;(,;og+i)
n
3g+l
space of polyn.
U(t) , V (t) , W (t)
s.t.
F
(Jac C--O)
g
Z
f--V2 = U-W
Here TT(Sg+l) is the affine variety of x,y such that
1xiyi = 0,
rr(x,Y) _ (U,V, W) where
2
U (t) = fl (t)
tXkk
V (t) = fl (t)
2
tkakk
W(t) = fl (t) - ( Yk + l
t-ak
2
= 1,
-
zeroes of
(z,St)
3.116
fl (t) =
g+l
TT
(t-a)
f2 (t)
ck
fl (t) ' It-a
,
f (t) = f1 (t) f2 (t). We shall
k
k=1
assume for simplicity that the constants ck are all chosen to be
positive.
means that
The other cases may be treated quite similarly.
This
sign f2(ak) _ (--1)g+l+k, hence the zeroes
b
a2
< b2 < ..... < age bg< ag+1
Graph of f:
We assume that the cycles Ai,Bi on the curve C given by s2 = f(t)
are chosen as in §5, with respect to the linear ordering of the
branch points on the real axis.
Neumann's equations are the equations given by the Hamiltonian
vector field XH on TT(Sg+l), which is tangent to the above subvariety.
We have seen that
Tr*XH = -2/ -D.
and that the vector field D.
-
on (Cg, where if
and
on Jac C is given by
ei(a/azi)
wi are the normalized 1-forms on C,
fi(t) = eitg-1
wi = 4i(t)dt/s
Therefore the solutions (xk(t),yk(t)), t E 7R,
3.117
of Neumann's equations project to curves on Jac C-0, which lift to
the straight lines:
z0 + 2/T t e,
t E ]R,
e = (e1,
.
eg) .
Moreover:
in (Cg.
xk =
TI (ak-aQ) -l U (ak)
Qk
+/y
-&101 (0)
f\
2
(0)
(Z)
t4'[ 6 } (Z)
by (7.6), where in (7.6) we choose V = U
=
(i.e., corresponding to the branch points al,-,a a+1 ), and
D
z =
J
W, D = (divisor defined by U(t) = 0, s = V(t)').
The n2k-1
goo
appears because ak is the (2k-1) st branch point in the linear
ordering.
The sign becomes +1 if we put the characteristic S
back into a translation by
2
xk = +
I:
$[n2k-l] (O) . [n2k-l] (z-1) 2
X9'[0] (0) . 19'[0]
(see proof of (7.6)). Now note that whereas (t' [n2k-1] (z) /YO] (z) ) 2
is periodic with respect to L0, 'i92k1 ] (z)/9'[O] (z) is not.
In fact,
19[n2k-1] (z+On+m)
'r[O] (z+Stn+m)
(Ch. II,
1 and Def.
(5.7)).
mk+nl+ +nk-l
19'[n2k_lI (z)
't9' [0] (z)
Thus let L be the sublattice in L.
3.118
of index 2g+1 defined by
L=
SZn+m
n,m E ag and ml,m,+n1,....
,m +n +...+n
1
g
These ratios are La-periodic.
g--1
,n
g
1
even
So if we consider the torus CCg/L
which covers CCg/L,, it follows that we can complete the previous
diagram as follows:
subvariety
F1(x'y) cq
T (Sg+1)
(zeroes of
CCg/L
Fg+1(x , y) = Cg+1
3g+1
space of polyn.
U(t),V(t),W(t)
s.t.
4
-
F14
f-V2 = Uw
if we define the upper arrow by
t' ['12k_11 (O) .
xk
& [ol (0)
.
h2k-il (z
'i9' [d] (-_J)
Tg/L St
_
(zeroes of
[Sl (z,)
3.119
Note that the action of the group Ln/L = (?L/22Z)g+l on
Cg/L -(zeroes of 1l9[S1) corresponds to the action of the elementary
2-group (xk,yk
, (Ekxk,Ekyk), El,...,Eg+1E{+1} on
Now the liftings of straight lines in (rg/Ln
(rg/L
(Sg+1).
T
are straight lines in
so the solution to Neumann's equations are:
&1n2k-11 (O) " &[n2k-l3 (zo-4+2 t e)
19'[0] (z0--1+2/T t e)
'l4'[O] (0)
xk (t) _
Finally, if we want
,
t E JR
xk(t) to be real, we have seen that this means
that the divisor b given by U(t) = 0, s = V(t) should consist in
g points (P1,...,Pg)
with
bi < t(Pi) < ai+l' s(Pi) pure imaginary.
In this case
(bi,0)
P.
z O-L
_
I
j'
00
-
j
Co
Pri
w=
i- i
w
J
(bi0)
3.120
Thomae's formula and moduli of hyperelli tic curves
§8.
As a consequence of the formula expressing the polynomial UU(t)
in terms of theta functions, we can directly relate the cross-ratios
of the branch points ai to the "theta-constants" '&[n1(0,Q).
Beitrag zur bestimmun
result goes back to Thomae:
This
von 19'(0,---,0)
durch die Klassenmoduln algebraischer Funktionen, Crelle, 71 (1870).
We claim;
For all sets of branch points B w {a11...ra2g+l,'},
Theorem 8.1.
there is a constant c such that for all S c B--, #S even,
17'[nSl (C)
4
#SoU 4 g+1
if
0
=
(ai-aj)-1
c-(-1)#$SU
.
if #SoU = g+1
'iT
iESoU
jEB-SoU-o
The result looks more natural if we don't distinguish one branch
point by putting it at infinity.
Let B =
{al,.-.,a2g+2}
be all
finite, put the ai's on a simple closed curve in this order and
choose Ai,Bi as before.
Then for all S c B, #S even:
0
(8.2)
1g'[ns1 (o)
if
#SoU
g+l
4
(- 1) #Sf1U
"ff
iESoU
j%SoU
(a1-a]) -1 if #SoU = g+l
3.121
In fact Thomae evaluated
(8.3)
The answer is:
c too.
a)-2 .
q' [ns] (C) 4 = +(det
TI
(ai-a
i<j
2iV'-_1
(I
j=1
wi =
i<j
i,jgSoU
i,jES0U
where
(ai-a
iI
)
j
)
vJ
For a proof of this, see Fay, op. cit., p. 46.
We can deduce (8.2) from (8.1) by making a substitution
ai =
=
Aa'+B
---'
Ca +D
Aa2g+2+B
or
CO
Ca2g+2+D'
The numbers
< i < 2 g+l
1
- - D
C
.
&[-n s](0) are not affected but the RHS changes to
a
c- {_11 #Sf1U
I
-1
r
- a.
) (Ca+D j
iES U 1 (Ca
jEB-SoU-=
(C.
a 2g+2
^
2g+2
tT (Ca! +D) 9+1) (-l) #Sf10
!
i=1
iESoU
1
jEB-SoU-0O
.'TT(Ca"+D) g+11 (-1) #Sf1U
lJ
(a'-a')
i 3
TT
iESoU
j0oU
_l
iT (Ca'+D)
iESoU
(a' -a') -1.
a
i
-1
3.122
To prove (8.1), we substitute
a. - g into 7.6.
D =
1<i<2g+1
Then
uD(t) = TI (t-a.)
Y
iEV'
where
V' =
and
UD(ak) _
UT
iEV'
But
D
B eV if g is odd, D B eVU{-}
D
I (D) = I (e rrov )
So
if g is even.
eU
g odd
eUo{oo}
g even
; eUOV +
I (eu)
+
f
I (e Uo {-}
g odd
)
g even
(211 UOV+nU0V) + (S261+611).
Therefore replacing the argument by a theta characteristic:
I" +nkl(I(D)
l (1 (D) )
hence (7.6) reads:
2
)
exp
t (s'*nUoV) -rill)
[n oV+nk1 (o)
(
..S
[nl
UOV(0)
2
3.123
(8.4)
iT, (ak-at)
IT (a-a.)
k Y
(-,
4 (tn U
oVnip
k-
tnto
n' ) [nUoV+nk] (4) 4
UoVk
qKnU,V) (0) 4
iEV-{k}
By (6.3), the sign is
(_1) # (UoV) fl{k,ao}
Since k E V, this is +1 if
k E U, -1 if k 0 U.
Now in (8.1), we
may choose c to make the formula correct for S = 4,, and then prove
it for any S by decreasing induction on the number of elements. in
(Sou)r, U:
if (SoU) = U, then S = 4
and we are done.
Formula (8..4)
is just the ratio of the 2 cases of (8.1):
S = UoV
and
S = C(Uo(V-k+-)).
This is straightforward although somewhat painstaking to verify.
Therefore applying .(8.4) twice, we obtain the ratio of (8.1) for
S = UoV,
and
S = Uo (V_k+k) ,
if
k E V, k % V.
For these SoU is respectively V and V-k+k, hence, step by step,
we can move from the formula in the case SoU = U to S0U = (any V
with #V = q+l).
QED
We now introduce a moduli space in order to formulate our
results more geometrically:
3.124
(8.5)
set of pairs
C a hyperellintic
{1,2,---,2g+l,°}=a B a
curve,
mod
bijection where B c C are the branch
points of
n : C ..-> 3P 1
°Q, 52 )
We are merely defining
isomorphism.
dVg2) as a set here, the set of isomorphism
classes of hyperelliptic curves with "marked" branch points.
However because the image of the branch points in ]PI determine
d1o
the curve,
d'K.g2) can be described equivalently as:
set of sequences
(2)
(8.6)
g
Since we can normalize P,,
dt g
to be
co
{
mod projective
equivalence
PGL(2,IC)
, we can also say:
set of sequences a1, a2, --,a2g+1
(2) _
(8.7)
Pw
{of distinct points of IP1
of distinct complex numbers
}
mod affine
equivalence
ai i
, Aal+u
hence further normalizing, e.g., P1 to be 0, P2 to be 1:
(8.8)
(2)
JrV
(open subset of C2g-1 of points (a3'a4'...,a2g+1))
such that ai
g
This makes °Qg2d),Dinto an affine variety.
aj, ai # 0,1
In terms of the 2nd
description of di(g), if t is the coordinate on ]Pthe affine
ring of
d'[.(2) is generated by the cross-ratios
9
3.125
t (Pi) -t (Pk)
(P
t(P-t(Pk)
t(Pi)-t(PQ)
In terms of the 3rd description (8.7) of
{i(92), with one point
normalized, the affine ring is generated by the functions
a.
i
--
a.
--
J
ak
Now consider the universal covering space
Letting Aij c CCn be the diagonal zi = zj, and let
the base point
B =
g
of Q 2)
b Edg(2) be
we can describe it concretely
as follows via (8.8):
space of maps
(8.9)
g
[((-(O,1))2g_l_iuj"ij]
c: [0,11-
such that
QU(O) = b = (2,3,'-',2g)
modulo homotopy: o - Oj if
3(p:
[0,112 -
[((r-(O,1))2g-l- UM..]
4)(0,t) = 0 (t) , (D (l,t) _ 1 (t)
0(s,0) = b,
c(s,l) indep. of s.
The projection
(2)
is the map
qs
cb (1) ,
and the covering group r =
Sri (°g
is the group of loops in (C-(0,l))2g-1- UAij mod homotopy.
(
92)
This is
essentially what E. Artin called braids with 2g+l-strands except
3.126
that 2 strands are normalized at 0,1 and each strand comes back to
its starting place ("pure braids").
Here is an example:
is easily shown to be 7r1 (G2g+1_L'oij) , the
group of all pure braids.) We call r the group of normalized
(In fact 2Z
pure braids.
let G =
We can describe
a bit differently as follows:
r
group of all orientation preserving
homeomorphisms
0:
fl O) = 0, c (1) = 1,
]P1 > ]Pl
$ (00)
such that
= -, topologized in
the compact-open topology.
let g =
subgroup of
cp
such that
c(i) = i,
'i = 0,1,...,2g,-
Then we have a map
inducing a bijection:
Tr :
G
(2) , Tr $ _ { cp (0)
3.127
(8.10)
(2)
G/Kg
The following lemma is easy:
Lemma 8.11:
For all
E ((r-(0,1))2g-l- UDij, there
are disjoint discs Di about Pi and a map
: D2x...xD2g
such that
of
xi.
G
Thus
iP
is a local section
TF
Cr
hence
IffW l (IIDi ) N =
HDi x Kg
by the group structure.
The lemma can be proven by use of suitable families of
homeomorphisms of ]Pl which are different from the identity only
near one point P and move P a little bit in any desired direction.
The lemma implies that
rr
has the homotopy lifting property and
hence the following map p is bijective:
3.128
Equiv. relation O evof if
30: (0,112 - G,
space of gaps
[0,1)----a G,)
0(0,t)0 (t) r(D
0)=id en
(D(s,0)=id., Tr(D(s,l)indep. of s
tity
(t)
p
f
space of maps
[011)--a(a!_(0,1))28-1--k/Aij
00) = b
homotopy of paths
A with fixed initial
and end points
oy.,
g
(In fact, the surjectivity of p is just the lifting of a path:
G
i
V(2)
g
and the injectivity of p is a lifting of the type
a10(2)
di9
3.129
Starting with
4:
[0,1] --- > (a-(0,1))2g-1- UAij, lifting
a:
where
p(1), we get a map
G and taking
[0,1]
Kg
°
g ----> G/K0
is the path-connected component of the identity in Kg
i.e., { E Kg j
(0,11 -- Kg such that (0) = e, i)(1) =
From the homotopy lifting property of
that a
to
7
,
}.
it follows immediately
c relates to our other constructions by
is continuous.
a commutative diagram
G/Kg
g
(2)
a G/Kg
g
equivariant for the homomorphism:
a*:
Kg/Kg
r
(Kg/Kgo, acts by right multiplication on G/K°g) given as follows:
Vy:
y
M°0,1)) 2g-1_UAi. , with
via p to cp: [0,11 -- G. Then
[0,1]
Tr
Ltd T,jyl = d()
y (l) = b, lift
(1) = b, i.e., 0(l) E Kg
y (O) =
All this follows formally from the definition of
a
.
in fact,
it can further be shown, but this is not merely formal,that
is homeomorphic to G/Kg, hence
we don't need it.
r
Kg/K;;
g
but we omit this because
The reason we have defined
a so carefully is
3.130
that we wish to use a to define the global period map:
2- dg
4i
In fact, given a point of °9g, we have 2g+2 branch points
: A'> P
B = {0,1,a2,....a2g'Col C 1Pl and a homeomorphism
such that p (O) = 0, $ (1) = 1, 4 (i) = ai, 2 < i < 2g and
given up to replacing
to the identity.
points B
.
by
4o
,
i
(°°)
_
00,
fixing the i's and isotopic
Let C be the hyperelliptic curve with branch
induces a homeomorphism of the standard
Then
with C.
hyperelliptic curve C0 with branch points
Taking the standard homology basis A.,Bi on CO, we obtain a
on C, hence normalized1-forms
homology basis
W7 =
such that
w3.
S2aj =
This
J
f
f(Bi
(Ai)
defines the map
Sid, hence finally
wi
0
It should be mentioned that all the topology on the last
3 pages was traditionally compressed in the following few sentences:
to each choice of branch points P, we associate a period matrix S2(B).
As B varies, we move the paths Ai,Bi continuously.
S2ij(B) is locally
in this way a single-valued holomorphic function on the space of B's.
Globally, if we replace the space of B's by its universal covering
space
SZi.(B) is still a holomorphic function by analytic continuation.
I'm not sure whether this "sloppy" way of talking isn't clearer!
3.131
Note that the map
0
is equivariant with respect to a
homomorphism of discrete groups:
r
S2* :
) / (+I)
Sp (2g,
To define S2*, let
--- M- (011)) 2g-1 " UAi7
[0,1)
(0) =
be a braid.
Lift
0 to
D:
Then
(0,11
Lift
carrying
c(l) to a homeomorphism
acts on H (C,2L)
symplectic matrix
G.
>
c(1) is a homeomorphism of 1P1
to itself.
`Y
0 (1) = b
`'
of C0 itself.
Then
tke
,
in its basis {Ai,Bi}, by4a 2gx2g integral
The equivariance of
n
is clear (see Ch. 21, §4).
An interesting side-remark in this connection is:
Lemma 8.12.
of
y
The image of
S2* is the level two subgroup
r
2
E Sp(2g,2Z)s'.t. Y = I2g(mod 2).
Proof:
Note that if
Xi E Hl(C0-B,7L/2ZZ) is the loop around
the ith branch point, then the image of
ai in 1P1-B goes twice
around the ith branch point, hence is zero in H1(1P1-B,2L/22G).
Therefore we have a diagram:
3.132
H1 (C0-B,2Z/22Z)/r -
,Xi, - > =
HI (CO,2L/22Z )
K!
(]P1-B, 2Z/22Z)
and it's easy to check that
is injective*.
K
point of B to itself, it maps a loop
branch point to a homologous loop.
on
Hl (]Pl-B, 2z/22Z) .
Thus In
Thus
T
'(1) carries each
p i in ]P1 going around the ith
Thus
(D(1) acts by the identity
acts by the identity on
I~1 (CO, 2Z/2ZZ) .
SZ* c r2.
To prove the converse, recall from the Appendix to §4, Ch. II,
that
r2, or rather its image in the group of automorphisms of
is generated by the maps
H1(C0,2Z),
x i> x + 2(x,e)e
where
e
is one of the elements
Ai,Bi, Ai+Aj, Ai+Bj or Bi+Bj.
To lift these generators in the braid group
r
,
consider the
following simple closed curves in 7P1-B:
This is the purely topological version of the description of
2-torsion on Jac C0 by even subsets of B. H1(]Pl -B,TL/22Z)is the
free group on loops ui around the branch points mod E}.ti- 0,21ji- 0.
One checks that K(Ai)
This
P2i-l+u2i' K(Bi) _ "2i+...+u 2g+l'
proves K injective and identifies H1(C0,2Z/22Z) with even subsets
S of B (mod S - B-S):
let a,S correspond when
K (a> _
I pi,
iES
3.133
P
A; -Z
lo
2;
j MI
xy rl
rz
Each of these lifts in C0 to 2 disjoint simple closed curves and a
little reflection will convince the reader that they lift as follows:
i)
Cij lifts to C j U C a, Cij
Ai+A3, Cij - -(Ai+Aj)
ii)
Dij lifts to D. U Dij, D.
Bi-B., Dij
Bj-Bi
Ai+Bi, E
(Ai+Bi).
iii)
E.
lifts to E'
U E',
Ei
For every simple closed curve F in IP 1-B,
"Dehn twist" 8(F) E G:
Then
there is a so-called
take a small collar Fx [-e,+e] around F.
8(F) is the homeomorphism which is the identity outside the
collar and rotates the circles Fx{s), - c< s < e, through an angle
rr(s+e
varying from 0 to 21T
Dehn twist
as s varies from -e
to +e.
S(Cij) lifts to a homeomorphism of C0 which is
Now the
3.134
6(C,j)o8(Cj).
S(F), for a path F in CO, acts
And the Dehn twist
on homology by
xi
Thus
S (C i7 )
acts on H1 (CO, 2Z)
xi
>
x + (x.F) F.
>
by
x + 2 (x,Ai+Aj) (Ai+Aj) .
Likewise:
ij
6(Ai
S (D
)
)
act s b y
x
--- > x
act s b y
x
i
+ 2 ( x. B i -B ) (B i -B i )
' x + 2 ( x.
S (B
i
)
ac ts
by
x
i
>
S (E
i
)
ac t s b y
x
:
;
all of which generate
r2.
i
A
x + 2 ( x. B
i ) .A
i) B i
x + 2 ( x. Ai +B i ) (A i +B i )
Finally, all Dehn twists
induced by braids,t.e.S(S) = cs*(y) mod Kg:
S(S) are
making S the boundary
of a disc, one shrinks S to a point obtaining an isotopy of
homeomorphism
S(S) with the identity.
This may move 0, 1, and
but by a unique projectivity, one can keep putting them back.
we find
(D:
[O,l) -k G with
is a braid in
Toy
r
'
inducing
(O) = e,
c (l) =
S (S) .
g(2)
g
n E 2
2Z 2g,
Then
and the map
we can reformulate Thomae's theorem more geometrically.
for all
Thus
QED
S(S).
Finally, having set up the spaces
°,
we have the holomorphic map:
In fact,
Q
3.135
T
->
W
g
W 4 [n] (O,sZ(P))
Either by the functional equation for 19'(z,S2), or by Thomae's
formula, we see that the functions
j&[n] (O,s2)4
3.[O] (0,52)
on
dig are
4
r-invariant, hence are holomorphic functions on
depending only on
S E
Corollary 8.13:
27Z2g/2Z23.
Thomae's formula implies:
The affine ring of °Q g2) is generated by the
nowhere zero functions:
+4
i[ns] (O,SZ)1
S
(7PI-0-1
Proof.
such that #UQS = g+l.
B
Normalizing one branch point to
oo
,
and letting
al,a2,...,a2g+l be the others, we must check that each
ratio
ak-aR/ak--am is a polynomial in these 4th powers.
We use
the identity:
ak-aQ
\ak-am
If we write
2
(
a,-a
2
Cak-a-)
ml
+1
- \ak-am)
=
2
ak-am
{l,2,...,2g+l} = vy1.V a{k}, #V1 = #V2 = g,
t
2
then by
3.136
Thomae's formula
4
(ak-al)
M)
Tr
iEV
iT
(ak-ai)
[n
2
Write instead {1,2,
,2g+1} = V3 .u. V4 u{k,i,m}, #V3 = #V4 = g-1
and apply (8.14) to the pairs
V1 = V3+m, V2 = V4+Q.
V1 = V3+i, V2 = V4+m
and then to
Dividing we find
ak-ai 2
(ak_am)
for suitable ni.
oUl (O, S2) 4
(V1+k)
&In114 n9-[n214
19'[n3l4
14'[n414.
(8.14)
QED.
The relations among these generators presumably may all be
derived from
various specializations of Frobenius' identity.
3.137'
Characterization of hyperelliptic period matrices
§9.
The goal of this section is to prove that the fundamental
Vanishing property of
§6 characterizes hyperelliptic Jacobians.
The method will be to show that any abelian variety
XQ
has the Vanishing property must have a covering of degree
which
2g+1
which occurs as an orbit of the g commuting flows of the Neumann
dynamical system.
To state the result precisely, we fix, as above, the
following notation:
B = fixed set with 2g+2 elements in it
U c B, a subset of g+l elements in it.
group of subsets
n:
T c B, #T even
1 292 /2L2g
mod T - Cr
where
is any isomorphism satisfying
n
t(2n' (T1)) . (2n' (T2))]mod 2
(2n' (T1)) . (2n" (T2)) -
# (T1nT2)
# (TOU) - (g+l)
2
where
by
n (T)
t (2n' (T)) . (2n" (T)) mod 2
Cn'(T), n" (T)) .
We shall subsequently abbreviate -n(T)
nT.
Theorem 9.1:
Assume
S2
E
ITIT] (0, 2) = 0
Then
S2
g satisfies
E--
# (ToU) 54 g+l
s the period matrix of a smooth hyperelliptic curve
of genus g.
3.138
First of all, to write our formulae with
Proof:
unambiguous signs, we are forced to make a choice of lifting
of nT from 1 2Z2g/2Z2g to
1 2Z2g
2
2
To do this, we choose a fixed
element - E B-U, and choose
ni
for all i E B--.
lifting
nT E 2zz
2g
E
ni ` nco}mod
We also set n = 0.
2g by
_
nT
2Z2`
Then forAT define a
ny-
iIT
The "standard" choice, if B = {l,--.,2g+l,co}, is
0...0 2 0...0
n 2i-l
...
2
2
0 0---0
_
ice` place
V.
0...0 2 0...0
n2i
l
2
1 1 0...0
2 2
but there is no need to get that specific.)
The first part of the proof is to investigate the differential
of the theta function -9 [nT] (z,2) at z = 0. The tool at our
disposal is Frobenius' formula, and we propose to differentiate
it, and substitute so that very few terms remain.
In formula
3.139
(Fch) in theorem 7.1, replace
xl by
x1+y, x2 by x2-y, take
the differential with respect to y and set y
assuming
Jai =
0.
We get,
Jzi = 0:
(F h) EsU(j) (d49[al+n] (zl). Q[a2+njl (z2) jEB
d9[a2+nj] (z2) a[al+rljl (z1), ,*(a3+7111 (z3) 4[a4+f j1 (z4)
0.
Note first that ,9[n (z,S2) is an even function if
# (SoU)
E g+l mod 4
# (SoU)
E g-l mod 4.
and is odd if
Therefore
d-9(nsl (0,52) = 0
if #(SoU)
case
= g+l mod 4 and we may restrict our attention to the
#(SoU)
# (SoU)
=
g-l mod 4, and, replacing S by CS if necessary,
< g-l.
Lemma 9.2:
d.-
hS] (O,c2) = 0
if
(g-1) mod 4 and
# (SoU)
# (SoU) < (g-1).
Proof:
all i.
Let
T c B satisfy #T = g-5.
Moreover, take A,B,C c B-T
each, let
In
(Fch), let zi = 0,
3 disjoint sets of 3 elements
a E A be one of its elements and set
3.140
al =
+ na
nTCU
a2
n(T+A+B)oU + na
a3
n(T+A+C)OU + na
a4 = -a1-a2-a3
=
n(T+B+C)oU
+ na mod ZZ
2g
Then all terms with a factor 9[a 1 +ni](O) are zero, so the d,%9
goes with the
1st
For the last 3 factors to be non-zero,
factor.
we need
#(T+A+C-a)0{i} = g+l,
#(T+A+C-a)o{i} _ g+l,
#(T+B+C+a)o{i} = g+l.
and
This only happens if i = a.
d$[nToU] (0)'9[n(T+A+B)oTP (0)
So the formula reduces to
Since the last three are non-zero, the 1st is zero.
argument shows that
d
[nTQU](0) = 0 if T c B
A similar
satisfies
QED
#T = g-9, g-13, etc.
For all R c B, #R = g-2, and elements
Lemma 9.3.
0.
[n(T+A+C)°T. _4[n(T+B4C)°U}-(0)
a,b,c E B-R,
there is a relation
a d [ n (R+a) oU} (0) +
where
aiiv
34
0.
(R+b) aU]
(0) + v -d°[n (R+C)
OU
(0) = 0
3.141
Proof:
f E R.
Let d,e be 2 elements of B-R-{a,b,c}
In (Fch), let
and let
zi = 0, all i, and let
a1 = n(R-f)oU + of
n (R--f+d+e) oU + of
a2
a3 = n (R--f+a+b+c+d) bU + of
a4 = -a1-a2-a3
n(R-f+a+b+c+e)oU + of
Then all terms with a factor
term the d9
.
9[a1+ni) (0) are zero, so in each
goes with the first factor.
For the last 3 factors
to be non-zero, we need
# (R+d+e) a{i}
= g+l
#(R+a+b+c+d)o{i} = g+l
#(R+a+b+c+e)o{i} = g+l.
This happens if i = a, b, or c, giving 3 remaining terms and a
formula just as required.
Lemma 9.4:
dR [fi TOU]
Let
(0) E
QED
S,T c :B, #S = g, #T = g-1.
Then in
span of differentials 4[r1 (S-i) OU] (0)
for all i E S-SIT
T
'0,
3.142
Prove this by (n duction on # (SST) .
Proof:
If, to start with,
S = R+a+b, T = R+C, (a,b,c E B-R distinct, and #R = g-2), then
In general, choose
the result is precisely lemma 9.3.
a,b E S-S S T
and
c E T-SST.
=
1:1
By lemma 9.3,
X d,9 fn (T-c+a) oU] (0) + V .d a & (T-c+b) oU1 (0) .
Now (T-c+a) and (T-c+b) both have one more element in common
with S than T did, so by induction
d'9[n(T-c+b)oU](0)
so is
are both in the required span.
d9 [nTQU I(0) .
Let
Lemma 9.5:
and
d,9[n(T-c+a)oU1(0)
Therefore,
QED
S c B, #S = g.
Wa = d-5 fn (S-a) oU 1(0) ,
a E S, span
Then the g differentials
TXS2' 0.
We use the fact that the abelian variety
Proof:
X0
is
embedded in projective space by the functions -9 [n] (2z,S2) , when n
runs over coset representatives of
2ZZ2g/2Z2g
It follows that the whole set of differentials
span
TX
,
,0
(see Ch. 2,
d9[nS] (0,c2) must
as S runs over all even subsets of B.
we may as well assume #(SOU) = g-1.
space is still spanned by
).
§
By lemma 9.2
By lemma 9.4, the whole
d9[n5](0,Q)'s where S = (S0-{a})*U,
S0 is any one set of g elements of B and a runs over the elements
of S0.
Lemma 9.6.
QED
For all a E B, there is a unique vector Da E TX
S2'
,up to a scalar, such that for all
T c B, #T = g-l
Da9 fnToU] (0) = 0 <=> a E T.
0'
3.143
Proof:
Fix a subset
S c B with #S = g, a E S.
all
Da19[n(S-b)oU] (0) = 0,
determines
Da
up to a scalar by
Then the requirement
bES-{a}
lemma 9.5.
To see
that Da9[fTuU](0) = 0 if a E T, #T = g-l, use lemma 9.4.
On the other hand, if Da9 [nTQU ] (0) = 0 when a T, #T = g-l,
we would have that the differentials dd9 [ nT+a--b] (0) , all b E T+a,
were linearly dependent, contradicting lemma 9.5.
We now concentrate on the vectors Dk, k E U, and D,,.
9.5, no g of the vectors
{Dklk E U)
By
lie in a hyperplane, so
we may normalize the whole set up to multiplication of the
whole set by a single scalar by requiring
D
kEU k
Then define scalars
= 0.
ak, k E U, by:
Da, =
GakDk.
Note that for fixed D,, {Dk}, the ak are determined up to a
substitution ak f-- ak+u; and if D., {Dk} are changed by
scalars, the ak change by an affine substitution
So far the proof is quite natural.
ak I--a ak+'µ.
We must, however, normalize
D.O,{Dk} a bit more and for this a rather ad hoc Corollary of
Frobenius's formula is needed.
Lemma 9.7:
e
47rin
n.
R
For all j,k,Q.E U distinct,
.D,,,,41
n
[njl(0)
(0) -D
47ritn ,
e
nk
19(nk](0)";D-Sfn{Q,7}1(o)DQg[n{R,j}](o)
3.144
Start with the formula
'Proof:
e WO [ni1(0)2.4[ni](z)2 = 0
i ETJ+co
19
(Corollary 7.4).
Replace z by z +
O'n {j,k,Y,,oo} + n{j,k,k,°°}
and it becomes:
- 4 v r ] tt
E,,(i)e
n{
j r kr Z I
-9 [nil(0)
iE U + -
Differentiate this first by
47ri n{7rk,Z,°°}* ni,9
L.a
iEU+=
2
D,,, second by DQ
2
and set z = 0:
[nil (0)2-[D, [ni+n{j,k,i }] (0) D [fi+fl{akg,°°}](0)
+9[ni+n{j,k,t,co}](0).D-D11'9[ni+n{jrk,i 4(0)1 = 0
Since the sets
Uo{j,k,i}a{i}
i[ni+ n{j,kirk ,°°} ](0) = 0
for all i E B. Moreover, if i E
U+oo,
To get a non-zero
term in the above formula, we also need
4
Uo {j,k,i}e{i}
which narrows down the possibilities to i = j or k.
4Tri
e
t
We get
1
nk9[nk ](0)2.Da°° p[n
e
J
.
all have at most g-l elements,
#(Uo{j,k,9}0{i}) = g-1 only if i = j,k,i or
CO,Q
0.
{j
Q}
3(0)-DQ P [n{irQ}](0)
n{k,k}]
Q
,
0](O) = 0
3.145
Using the fact that
4Tri
-1 = (-])
and
j,00}0U)--1
= e4Tritn
+1 = (-1)
3
nrr
# ({k,oo}oU)-g-1
+1 =
(--1)
2
e
4TrinQ-nk
the two signs may be replaced by e
4Trit
e
and
j, respectively.
QED
Corollary 9.8.
9fol(o)
for all
by
X E (C*, we can assume
4Tritnk-nQ
2
e
[nQl(0)2 S[nkl(0)2
Q,k E U.
Proof:
pair
Replacing D00
If we choose D., suitably, this will be true for one
QO,ko.
Now vary k.
QO and all k.
By lemma 9.7, the formula is true for
Now interchange QO and k.
Since
I D. = 0, we get
jEU
o=
I Dj8[,j{Q,k} ] (0) =
(o) + DX& [n{Q,k} ] (d) .
jEU
4Tritri
e4Tri
e
'k
n"
r
nrr
= (-l) #
the formula is also true for k and Y.0.
always true.
_l '
k
QED
Now varying RU, it is
3.146
The next step in the proof is an elegant and quite important
consequence of Frobenius's formula:
Proposition 9.9.
Let
T c B satis
3 distinct elements in B-T.
-9 [nTOU+ n{a,c}] (0)
#T = g-l and let a,b,c
Then
hToU+ n{b,c}] (0) [Dc na] (z) i [nb) (z) -D[ nb] (z) $[na] (z)]
= a. D c [nTQU](0) 9[fToU+n{a,b}] (o) 9 [nc) (z) -9[n{a,b,c,-} ] (z)
where 6= e
41Titna .n
b. e
4Trin'OU.nf
T
c= +1.
In formula (Fch) , set zl = z4
Moreover, set
a1 =
nc
Proof:
0, z2
--=
z, z3 =
Z.
a2 = 0
a3 = n{a,b}
a4 = -al-a2-a3 2 fToU+n {a,b,c, ,,}mod 2L2g
Finally, evaluate the differential on the vector Dc.
The coefficients in the jth term of
2h)
are, up to constants
Dc&[nToU+fc+nj ](0) v4[nToU+n{a,b,c,oo}+nj ](0)
and
9 [nTQU+nC+nj l(o) .9[nToU+n{a,b,c,-}+nj ](o) .
.
3.147
For the 1st to be non-zero we need
# (T+c-) o{j }
by lemma 9.6.
= g-l,
c t (T+c)o{j}
This means j = c.
For the 2na to be non--zero,
we need
# (T+c) a{j }
= g+l
= g+l
#(T+a+b+c)o{j}
which means j = b or c.
Writing out the three non-zero terms
and evaluating the sign with some pain, we get the result.
We are now ready for the key point of the proof.
a 2g+l-sheeted covering X of
X,,
QED
We define
and a morphism
yS : X - V (9 f 0 ]) ------ > T2g+2
as follows:
X = (rgn
S2
S2
r
n
L =t,SLe+q, Ip,q E ZLg and tniq tnip E 72,
H xi, i E U; y,, i E U} are coordinates on
i defined by
xi
all
i E U}
c2g+2
-9[ni ](0) Ofni] (z)
,6f01
(0)
i E U
.
9
Note that
2L
L' C L
and
2g+1, and that L
is precisely the lattice with respect to which all the functions
3.148
i E U, are periodic. Moreover, by Corollary 7.5,
the image of 0 lies in the affine variety
-9 r r1 i l (z) J$ [ 0 l (z) ,
1.2
hence by differentiating, the image lies in
Ixiyi = 0
(1x)-l =
- called TT(Sg)
in §4, the complexified tangent bundle to Sg.
What we shall prove now is that the vector fields
Dk, k E U, on
X are mapped to half the Hamiltonian vector fields
.T2g+2
defined in §4.
XF on
k
It will follow that 0 is an isogeny of
the torus X onto one of the tori obtained by simultaneously
integrating the
which by the theory of
XF k,
94 are precisely
2g+1-fold covers of the hyperelliptic jacobians.
follow easily that, in fact,
XQ
It will then
is isomorphic to the
corresponding jacobian.
Recall that
F
k
Fk(X,Y), k E U, are the functions:
(x,y) = x k2
+
Z
£Ok
kEU
(xkyk_xkyk)2
ak-ak
(the ak here are the same ak defined earlier in this proof).
The corresponding vector fields X
are given by:
,
k
3.149
2(xjyp xQyk)
XF (xP, ) _
^ 2(
x
L,
=
pkk
if .
."k,
ak _aX
k
k
(_x
a.K _a p
k
if 2 = k
p
PEU
and
2 (c aZYk)
2
yk +
XF (yk)
k
2
(_ak)
_Y,)
+ 2xk (xk-1)
(
P
if f,
k
if Q=k
P
PEU
(See 94, Proof of Theorem 4.7.)
Note that
= X£F
IXF
k
= Xl = 0.
Now let capital Xi be the
k
function on
X
1
(z)
and let capital Yi by D,,Xi, again a function on
claim is that if we substitute Dk for Xf
then if
k 74 k, bk(X9), Dk(Y9)
formulae on
X:
k,
What we
Xk forxk, Yk for yk,
are given by almost the same
3.150
Lemma 9.10.
k, then on
If Q
XI:
(XkYz-XP. Yk)
ak-as
Dk (XQ)
(X Y Q-X Y k)
Dk(YQ) =
IDk = 0, so
But
formulae as
21XPk
Dk
k-a
k
1XFk (y
),
2
to the vector field
Yk + XQX
tk
are also given by the same
Hence the lemma implies:
k ).
The differential of
Corollary 9.11:
field
ka
Dk(Xk),D k(Yk)
(x
Xk
carries the vector
0
1X, ,
k
Before proving lemma 9.10, we shall evaluate XkYi-XtYk
in simpler terms:
Lemma 9.11.
If i
k,
(0)
XkYQ-XiYk = e
Proof:
,9 [0](0)
[n{k,Z} ](z)
4[0](z)
XkYP,XiYk = Xk -(XQ/Xk)
-i[nk] (0)'9[nX] (0)
nk] (z) .ID J[n2] (z)--[nR] (z)
4[o](0),
Using Proposition 9.9, with
(z)
$[ (0)](z)
T = U-{k,&}, c
the second term on the right equals
a = k, b = k,
3.151
e4rrit ni
n
D,1-4[f{k,Q(0) 4[2n{k,P.}] (0)
k
nk+2fQ
nR+ nk
0
t9[11{k,R}] (z)
0
0
z
Simplifying the characteristics and working out the sign, this
gives Lemma 9.11.
QED
Proof of 9.10:
If k -/ k,
'[nk1(0)
9[0] (z) D [nt] (z)-W [fP] (z) D [0] (z)
,s_` ]moo "
Dk (X&)
2
[ 07 (z)
Using Proposition 9.9 with T = U-{k,P}, c = k, a = Q, b = -,
the second term on the right equals
e47ritnk,Rnk
DkLn{k,R}(0) -9[k2n1 (0)
9[2 'plc+2n ](0)
[2nk +n ](0
-
'9[C (z)9[n{k,Q}] (z)
,q [0] (z)
hence
k (X)
Q
D
4rritnI
if
Dk$[n{k,Q}] (0)
e
[rtkl (0)
[nkl (z)
$[01(2)2
-9g(0](0)
On the other hand,
D=
p
apDp -
EU
PEU
(ap--a.) Dp
hence
(0)
=
EU(ap--a.)Dp9[n{k,k}l (0) .
p
Q,}
] (z)
3.152
if
DP-G [n{k,Q}] (O) = 0
But
p 76 k, k
(because p E Um{k, It}) , so
D Cn{k,0](0) =
Therefore,
D
Q
k (X)
-
ak
Q
19 [0] (0)
XkYk-Xi Yk
by
Xk
ak-aZ
.
.
N Il (0) L[nk] (z)
(z)
Dj[11{k,Q}] (0)
e4Tritfl .fl
(
\
[0] (0)
-9 [0] (z)
LO] (z)
Lemma 9.11.
Finally,
Dk(YQ) = DCO (Dk(XQ) )
XkYXQYk
D
9k- z
(Xk
XkYQ.TX£Yk
ak-ai
Xk
+
D00
(XkYQ-XQYk);
ak-a!C
hence it remains to prove
D,(XkYi-XQYk) = (ak-aL)XQXk.
But
47rit.n ..nZ D -9[71 {k,i}] (0)
D00 (,Vk XRYk) - e
-9[0](0)
9 (0l (z)D (n{k,i}] (z)
[01 (Z)
[n{k,9}]
00
[0](z)
3.153
NOW in the proof of 9.11, we deduced as a special case of 9.9
that
.9 [rk] (z) DJ4[nk] (z) - .9[n.] (z)
(z)
4Tritn it * [ n{k, k) 1(0) -.&(o 1(o)
T1
e
[nk](0) (Tit 1(o)
z+Qnk+nk
Substituting
for z and rewriting the theta functions,
this gives
4[o}(z)-D [n{k,Q}1(z) -
00
D,-9[n{k,i}1(0)
[o](0)
Lnnk
R] (z) S[nk] (z)
(o)
o
(The minus sign comes from
41rit
e
4Rltr.01 to
r .nn
Q
fn{k,i}+nkl(z) = e
This gives
47H
t
nQ D
XQYk)
X1 1 ( 0 )
[ nk] (a)9
4Tr3.
(ak-ad e
)
k- nQ
11
z
(,)z 3 (o)
[nk] (z) -t9[nz] (z)
t9 [01 (Z)
2
D° [n {k. R.} ] (o) D ki9[ n{k,L}] (0 ),9 [0] (0) 2
'& [ nk] (0)2,9[n Z] (0) Z
which by corollary 9.8 is
(ak-a.)
XR,.
cxx
QED
3.154
The rest of the proof is now simple.
it follows that the
image of 0 is contained in one of the orbits of the g flows
XF
,
i.e., in one of the complex varieties
k
Fk = ck,
k E U.
But these are affine pieces of 2k1-sheeted coverings Yc of
jacobians Jc of hyperelliptic curves (or of generalized jacobians
of singular limits of hyperelliptic curves).
Since the
differential of 0 carries the invariant vector fields on X'
to the invariant vector fields of the algebraic groups Yc,
0 must extend to an everywhere-defined homomorphisms
0: X
with finite kernel.
> Yc'
In particular, Yc is also compact, hence
is a covering of a jacobian of a smooth hyperelliptic curve.
Next, the finite group
Ker (X
>X
Ker (Yc
> Jc )
and the finite group
both act on the coordinates xi,yi by sign changes
(xi,yi)
1
),
(sixi'siyi)' hence 0 descends to
00: XS
But by construction 00
c.
`XS2-V (-9 [ O ])) c Jc-0 , hence 95-1 (9) = V (1 10 ])
,
if 00 had a kernel, the divisor V09[01) would be invariant under
a non-trivial translation, which it is not.
Therefore 00 is an
isomorphism of XR with the jacobian Jc and V(.9[0]) is isomorphic to 8.
QED
3.155
510.
The hyperelliptic p-function
On any hyperelliptic jacobian Jac C, there is one meromorphic
function which is most important, playing a central role in the
When g = 1, this function is
function theory on Jac C.
Weierstrass'
p-function, so, at the risk of precipitating
some confusion in notation, we want to call this function p(a)
too.
We fix a hyperelliptic curve C, and let:
B = branch points of C
= E B
a set of g+l points
U c t-(*
t = tangent vector to C at w
This defines for us
on Jac C.
a) an invariant vector field D.
{wi}
and
is a basis of
swi(=),t>
b)
F(c ), zi =
= ei, then
D,
'Wi
_
Namely, if
are coordinates on Jac c,
lei a/azi.
a definite theta divisor 0 a Jac C.
Namely, 0
is
the locus of divisor classes
g-1
ID .
1
c)
the
1
p-function.
(IQ-2°°)
QEU
Namely, let 0 be given by .9(z) = 0,
then
P (z) = D log .9 (z) .
Note that
v
is La -periodic, hence is a rational function on
the variety Jac C.
More intrinsically, p(z) is characterized
3.156_
up to an additive constant as the unique rational function f'
on Jac C such that
For all U c Jac C open,
for all hoj.omorphic g on U, g vanishing to order 1
on e n U and nowhere else,
f = D2log g + holo.'fcn. on U.
(In fact, we can even construct
p(z) in characteristic p!
Start with a Zariski-open covering {U } of Jac C and local
equations fa
Then
of 0 n Ua in Ua.
fa/f
is a unit in
Ua n US, hence
Dcofa
D,, fs
fa
f$
is a 1-cochain in 0Jac
But
C.
DO: H1 (d J) ---a H1 ((Di)
is zero,
hence
D
D
f
CO
ga
f
ga
a
How much does
additive constant in
depend on the given data?
p
p
depends on the choice of .9
i.e., the choice of homology basis {Ai},{Bi}.
p
1st, the
will be replaced by c-p.
If U is changed,
itself,
if t is changed,
p(z) is replaced
3.157
p(z+a), a E Jac C2.
by
on C and
,
must be made.
Thus
p really depends essentially only
though to get a definite
Note that
p(-z) =
p
p(z).
many further choices
We can easily identify
2g+l
in our affine model.
p
Let
be given by
C
s2 = f(t) =
fl
(t-ai).
1
Proposition 10.1.
Jac C - 0 =
In the affine model of Jac C:
{(U,V,W) f=
= U-W, degrees as before }
let
U (t) = tg+d1 tg^1+- - .
W (t) = tg+1+WOtg+- - -
Note that
2g+1
U,+W0 = -
ai. Assume t chosen so that D,,"U = V.
i=I
Then
for some constant d.
Proof:
Recall that
f
1
(t) =
II
(t-a.) and
iEU
2
2
U (t) _
xk
f1
kEU t-ak
V (t)
f1 (t)
If we expand
U(t) _ I TT(t-aQ) -xk2
k Rk
xkyk
kEU t-ak
W (t)
1
yk
kEU t-ak
3.158
we see that
jakx]c - jak
U1 =
Similarly,
W(t) =
(t-a
II
)
+
2
=
tg+l +
I II (t-a 2 ) y 2
k
k 2#k
tlyk - jaJ
t'+
hence
WQ = lYk - jak
Therefore
Ul-W0
2
2
= lakx k
- iy k
which proves the lst equality.
Now
D0U = V, so we find
2
D.(xk
_ /T xkyk
)
or
Yk
i.e.,
D
=
2
x (the derivative of Neumann's dynamical system).
Now in Neumann's system
:k = -akxk + xk Q a2x2
2)
y2
2
hence
+a2x22 Now in terms of theta functions
2
y2,
xk
3.159
xk = c
-9 111 k7
k
Consider the difference
a
[ 03) - 1 (
DOG (log.
A- U
4 - jU
ye
2)
The first term equals
D2'3[0]
D.,9[0] 2
(
0
and the second equals
+1
t
oxk
2
1
8 ak
xk
Working this out,
2
(i
y
2 2)
2_1Y
SLo]) -8
DAkI - DoD.
%,' E0
a, lkl
1 V11dO '7
2
1 D IX01
= z 09[1k7
which has at most simples poles at V('9101) U V (3 [rik 1). Since
this is true for every k, the difference has in fact only poles
at V(4(01).
But the only functions with only simple poles at
V(-9[0]) are constants, and this proves the second equality in
the Proposition.
QED
More generally, we can relate all the functions on Jac C
defined b the coefficients of U,V on the one hand, and by
derivatives-of log.9[0](z) on the other:
3.160
Proposition 10.2.
The two vector spaces of rational functions
(I)
on Jac C spanned by
a) D DO* log -9 [0] (z) ,
all invariant vector fields D,
and 1
b) the coefficients Ui of U(t) including UO = 1
are equal.
This space has dimension g+1, and consists of even
functions with at most double poles at 0.
(II) Likewise, the two vector spaces of rational functions
on Jac C spanned by
are equal.
D22log-9[0](z), all invariant vector fields D
a)
D
b)
the coefficients Vi of V(t)
This space has dimension g, and consists of odd
functions with poles of order exactly three on 0.
In fact, for suitable constants c,c' and d
D
a 0 log,-,9 [0 ] ( z ) = CX kD + d k
k
D
Proof:
kr
ak D
CO
2
log-9 [0 ] (z) = c
We calculate
Da D
k
Dak D2 log -9 [0 ] =
2
PD
,
log-9 [01 as follows:
Dak (U1-WO )
8
Dak(U1) - BDak(UI+WO
2g+l
But
ai
UI+W0 = 1
for all k E U.
is constant on Jac C, and
by Prop. 10.1
3.161
D ak U
=
V (ak) U (t) -U (ak) V (t)
t _ ak
hence
Da U1 = V(ak) = cluk
k
for some constant c1 depending only on C.
This proves
b ak D Z log -& [0 l= c 1ak
and hence proves (II).
fk =
Moreover, as Dakak = Uk' it proves that
is a function on Jac C killed by b..
But fk has poles only on 0 and either fk is a constant or
Dc,
must be everywhere tangent to
0
.
As this latter is not
the case, fk is a constant, which proves (I).
QED
We now come to the main point of this section:
we ask
whether we can coordinatize Jac C by using the function P(z)
and its derivatives along D,,:
p (k) (z)
only.
Dk p (z)
The fact that this is possible was discovered by
McKean-Val. Moerbeke in their beautiful paper*.
Not only
is this possible, but this leads to an affine embedding of
Jac C-0
governed by a quite intricate algebra.
To be precise, we fix n and consider the morphism
0
:
Jac C - 0
z
---- Tn
> (P (z) , p (1) (z)
r ...,; (n-1) (z) )
*The spectrum of Hill's equation, Inv. Math., 30, 1975
3.162
Theorem 10.3.
If n = 2g, 0n is an embedding, hence
(i) (z), 0 < i < 2g-1, generate the affine ring of Jac C - 0.
In fact, we may
and for
solve for Ui,Vi,Wi
in terms of
ir(k) and
ak,
ir(k) in terms of Ui,Vi,Wi and ak by means of "universal
polynomials".
Proof:
We shall not find the formulae relating the
{Ui,Vi,WiI and
{p(k)I
directly, but rather via a third set
of variables {U,V*,W I .
Our first job is to introduce these.
We convert the identity
f = UW + V2
a)
between polynomials in t to an identity between polynomials
in t-1.
f (t)
b)
t2g+l
(t)
_ (U tg
)(W
(t)) + 1(V (t) 12
t tg )
tg+l
A polynomial in t-1, with constant term 1, has-a unique square
root in the ring of power series, with constant term 1, so we
write
2g+1
f (t)
a.=1
t2g+1
alt-1+a2t-2....... ) 2
(1-ait-I) - (1 +
for suitable constants
and write
0(t-l) = 1 +
alt-1
+ alt-2 +.......
Thus (b) can be written:
c)
l =
f9 l
O(t 1)
2
.
O(t)
t-
l(V(t)-t-
`
56 ct
g
3.163
U(t)-t-g
U*(t-1)def
-
Let
(t1)
=
V*(t-1)
1+Uit-1+U2t-2+......
V(t) - t-g
de
$6 (t-1)
vt-1+v;t-2........
=
W*(t-1)
do
0(t-1)
1 + w*t-1+W1t`2........
so that c) becomes
U*(t-1).W*(t-1)+t-1-V*(t-1)2
1 =
d)
Note that the (U*,V!,W*) and the (Ui,Vi,Wi) determine each other
given the
a i, by the universal polynomials obtained by equating
coefficients of t-n in:
1+a2t 2+-..)
(1+U1t-1+U2t2+---) =
V1t-]+V2t 2+,..)
e)
(
V1t+1+v2t-1+
(l+Wpt-
e.5..
U, Ul+al
V1 = V1
WO
w0+a1
3.164
(d) written out gives a recursive procedure
On the other hand,
from (U*,V*), viz.
W
for finding the
U*+W* = 0
U2+U*W0*+w* = 0
f)
U3+u*w*+u*w*+w*+Vi2 = 0
Un+Wn-1+ (univ. polyn. in
0
Note that
-W0).=(U1-WD) =
The flow D.0
1.
can be easily written in terms of U*,V*,W*.
It
comes out as
= it(-W*
W* =
-(1-8p.t-l)V*
or
g)
Vi =
W
=
f(-Wi+U* i+1-2U1UI*.)
-Vi+1+2U*.Vi
1
3.165
These give us, by induction, the formulae:
*
= U1
4p
4p(1)
_ U* = V*
1
1
(W1+UZ-2U' )
4}x 2) = V1 =
2
= U2
h)
3
12
(using W1 =
4x(3) = U* - 3U*U'*
2
11
= V2 - 3UiVi
p(2k)= U*+1 + (Polyp. in
p(2k+1)=
U1,-..,Ukfl,Vll...,
in
Vk+1 +
We may solve these backwards:
U1 = 4p
V*
= 4.p (1)
U2 = 4p(2)+24p2
i)
V* = 4p (3)+48
*
_
Uk+1-
(2k)
"'lyn'
(1)
(1)
r ... ,z (2k-1) r call this
Fk+l(p"P(1),.*.,p(2k)
(polyn. in p,p (1) , - ..'p (2k))
Vk+l ,p (2k+1)+/polyn.
i
, call this
).
3.166
It is easy to set up a recursive procedure which determines the
sequences {Fk},{Gk}.
First of all, as
it follows
(10.4)
Gk(p,...,p(2k-1))
The dot here means this;
= Fk(,p,...rV(2k-2))
-
F(p,p(1),---,P (n)) is any polynomial,
if
then:
*
(1)
... J, (n+l))=
(k+l)
aF
k =o
ap (k)
= V*k
(-W*k + Uk+i - 8p.U ).
2((Vk+1_gVk)
+ Vk+1
V*
-
4,p-Vk - 4 (VU*)*
hence
(10.5)
Fk + 4,i-Fk + 4 (p-Fk)
Gk+l
Fk+l
Then (10.4) and (10.5) determine the polynomials {Fk},{Gk},
given the extra facts that Fk,Gk have no constant terms and that
the map
(1).p(2)
..I
has no kernel except for constants.
IV,P(1) ,p(2)
(E
r...}
We also note for future use
3.167
that W* is given in terms of the p(k) by the 2n,d equation in (g):
Wi = Ui+1 - 2V
1
- 8p-Uz
= Fi+l - 2Gi -- 8p -Fi
Algebraically, we have shown that the 2 polynomial rings
RUrVrW = T [Ul,U2,
R
.;V
I
, V2 r
...
1r
.1 /identities (f)
[p,p (1) ,V (2) , ...]
p
are isomorphic, by an isomorphism that carries the derivation
of RU*VrW
(k)
defined by (g) to the derivation
of R1P
given by
_ P(k+l), and carrying the subring
*,g
* , .. .rU*g rV1
* ' .. .rVgrW0'
*
* ..
CC[Ul
RU
g-
1/(First g identities1
in (f)
)
to the subring
Rg = T[P,V (l) , ...,P (2g-1) I.
To finish the proof of the Theorem, note that by (e), the
functions
of Jac C-e
,...,vg
and hence the whole affine ring
are polynomials in
what we have just said, polynomials
Thus
hence by
p,
p(1)r...rP(2g-1)
02g is an embedding.
QED
r
3.168
Still imitating the algebra of McKean and Van Moerbeke,'
we can go further and explicitly describe the equations in
p.P(l)I.--,V(2g) that define
o2g+1(Jac C-0) C T2g+1.
The
result is this:
I) There are unique 12olynomials without constant
Theorem 10.6:
term
E[P,p (1) IV (2)
Hkk
r ... I
such that
Hkk = Gk.Fi.
In fact,
(n)
Hk,kE
II)
1, n = max(2k-2,2k-3),
and
then
¢(t-1) = 1 + alt-1 + alt-2
if
Hk,O = Fk.
02g+1(Jac C-0) is defned by
g+l
I
k=1
Proof:
ag+l-kHkk =
k+l+g
0
k < g.
The method we follow is the most direct one, but it
unfortunately requires a rather nasty computation at the end.
If
is any power series, write
F(t-1) =
ag+2t-1 +......
8F = ag+l +
t_g-l-terms,
for the "tail" of F starting at the
a0+a1t+...+agt-g
F =
-1
and let
3.169
be the "head" of F.
Thus
F = F + t-g-1.6F.
Now
U(t) = O(t-1)U*(t-l)
t
= ('+t-g-1. 6O) is a polynomial of degree g in tr1.
the * and write
V, 6V,fnl, 6W
U,6U
for
For simplicity, we drop
U*,5U*, and similarly for
It follows:
below.
6( -[T) +
60-10
mod t-9-1,
hence
a)
6U
-Vl (6 (i-U) + 60 )mod t-g-l.
This formula enables one to solve for the terms in U* between
t-g-1
and t^2g-1 using the terms between 1 and t-g, given that
U* comes from a polynomial U in t of degree g.
Similarly, we
get formulae
b)
6V
+
c)
6W
E -r (6 ('W) +
mod
t-g-1
-W 9) mod t-g-l.
(In (c), the fact that W has degree g+l makes the formula
have an extra term.)
Now start with values of
they give a point of
we define the numbers
and ask whether
C--G).
From these values of
3.170
Uir...FUg'V1'...'Vg,W0,....W9
by the universal polynomials of Theorem 10.3, hence the
polynomials 6,f and W as well as the one extra number W.
These in turn define unique polynomials
U(t),V(t),W(t)
such
that
CU (t) t gl ` U mod t.-g-l
O(t1)
V
.
(t) t
O
V mod
(t-1)
(W (t) t
g
1
+
t-g-1
Wgt-g-l
mod t-g-2.
56 (t-1)
The condition that we have a point of Im 56
2g+1
is that
f = UW+V2
d1)
But we can rewrite this condition as
d2)
(Uwt-gl(W(t)t -g-11 + t-1 (V(t)t -g
56 (t)
0(t-1 )J` 0(t ) )
1
mod t-
Now, mod t-2g-2, we have seen that
U (t)
t-g
t -g-lr' CS ( U)+aO-[)mod
-
(t-1)
t-2g-2
56
V(t)t-9
(t-1)
F-
t-g-1 -1
nl -
t-g-1 -1
S (gSV) +Sq . ) mod
t-2g-2
S6
t-g-1
W (t)
0(t-1 )
=_
tS in1 +S .
9 )mod
t-2g-2
2g-2
3.171
Therefore equation (d2) is equivalent to
(a3) 1 = v-W+t`2-t_`1_1[ . S O-V) +2w"v.s9va CO.%
U- g+2t-'v-a ( i)+2t_'
cOjmod t-2g-2
As the terms in t0,t-l,"--,t-g cancel automatically by definition
of the universal polynomials for the W*, this reduces to
(d4)
a (-Wt-'
,
)
2604.6 (97.-U) 4:U-(6 0-R) -W
+2t'v-6 ( v)mcd t9-1
First look at the constant terms in this equation.
To calculate this, note that 8(U*.W*+t-lV*2) = 0, hence
constant term in
d (f 1.FS+t-l72)
-constant term in
6 (U*W*-f3'-V+t-l (V*2 _V 2)
(U9+1+Wg )
Altogether the constant terms give us:
-(U*9+1 +Wg*) = 2a g+l +(alU*g+---+agU)+(a1Wg-1+...+ag W*)-Wg'
Since Wg = Wg+cx1W i_1+ . +agWO+rxg+l
(e1)
ag+1
this reduces to
..+ agUi
Ug+1 + alu*,+
which is to be the equation in (II) for k = 0, i.e., set
Hk,0 ` Fk and then (e1) is:
g+1
(e2)
k1l ag+1-kHk,o
g+l
3.172
To get the remaining equations, substitute into (d4)
Wg = W g * + a1Wg_k + - - - + agW0 - Ug+1 _ a1Ug
.- _ agU*
and write (d4) as
Z as bus + !U (S (- w _ (W*+- - .+ag 0 U-* --... -a Tj*) )
2
+ t-1-V6 ( V5 - rS (UW+t Y) mod t 2g-2
Expand this into
(f 2)
- I0 ak+1.+gt Q
g+lag+1--kg7ck
10
k--
1
(",p' .....V (2g))-t-Z
so that the coefficients of each t-P' give us the remaining
equations in the form required.
It remains to check that
This should have a conceptual proof but it is
not too hard to check by directly differentiating.
Hk Q
= Gk-Fi .
We use D for
-
in this calculation.
Thus to start, note that
D (U) = V
2D(V) = -t(W + t-gr1W**) + t(U + tg-lU**
9
D(W) = -V + Gp-t-1(V - V*t-5)
from which one deduces
D(U-W)+t1V) =
Likewise, one has
V-Wgt-g-l Wg+1t-9-1
9+1
)--BV U
3.173
(a+8pt-1(q Vg -9)) a (U) + W a
D (U S ( W)) = 'V-6W) + U- S[
( V+8pt(V
D (2ta (sue)) = (- w+t-g-'W9*) + (u+t-'g`'v9*+1) - 8 t-rv) 5 (OV)
+t
Adding these up, we get a lot of cancellation, leading to
d(UW+t-
)
g
= V- (Ug+l
g)
-t-16 (S6-V)]
-8C7[ a
+ v La (QSti) -t xa
I v[ 6 (W)
-t-l(tu)
6
V'(U9+1+alUg+...+ag+l) - U
grl
ag+l-k(VQU
0 k=0
Vk-U*Z) -t-
12.=
Thus
D (-2NkRWk* R.) = QUk - VkUQ
which proves
D (H
k Q) _ U QVk = FR - Gk .
QED
The cases g = 1,2 are given explicitly in the table below.
Note that the equation= 0 gives
v(2g)
as polynomial in the
lower derivatives, so that substituting this, we have exactly
g equations for 02g(Jac C-0) in C2g
usual cubic equation in p,4'.
For g = 1, this is the
Moreover, higher derivatives
3.174
Tables
F0=1
G00
F1 = 4p
G1 = 4p
F2 = 4p"+24p2
G2 =
F3 = 4 piv+40 (.,')
G3 = 4pv+l60,p' ,p"+80
H10 = 4p
H1,1=8p2
H
20
= 4p"+24p2
+480p2.V
'
,1= 16p p"-8 (p') 2+64;3
.prr?+8(p")
H30 =
73,1 =
H12 2 = 8(p') 2+32;x3
H2 2 = 8 (p")
2
2+96x2-p"+288;4
= 16p "piv-8 (p I I I) 2 +96;2 pxv-192pp I p "'+256; (p") 2+192 (p') 2p If
Curve of genus in 1 in C3 embedded by
-a2 = al- (4p)+(4p"+24p2)
a3 = al-
(8p2)+(16p.p"-8(p')2+64;3)
Abelian surface in C5 embedded by p,p',pof p fit ,piv
-a3 = a2-'(4,p)+a1- (4p"+24p2) + (4piv+40(;')2+80pp"+160;3)
-a4 =
+8(p")2+320p2p"+480;4)
-a5 = a2(8(p')2+32;3)+a1(8p")2+96p2p"+288;4)
+ 16p"p iv-8WIT)2
+ 192(p'),p"+1920p
2
3 p"+2304; 5
"'+256;. (p") 2
3.175
V(n),
n >2g, are expressed recursively in
, ... , (2g-1) by
repeated differentiation of the equation k = 0, and substitution
of previous expressions for p(m), 2g < m <n.
Likewise, the other
vector fields on Jac C can be given by elementary explicit
formulae.
We sketch this.
We use the basis Dk' 1 < k < g, introduced in §3, i.e.,
D
g
=
P
I
k
k=l
for all P E C-(oo), a = t(P).
working with.
Here D1 is the D
we have been
We showed in §3 that
(viuVjui)
DkUQ =
i+j =k-t Q-1
max (k,2 )
j<min(k,i)-1
Thus in the sum for DkUl we have only the one term j = 0,
and
D U = VkUO-VOUk
k 1
= Vk
.
Therefore
(10.7)
4(DkJ) = DkU1
= Vk
= Vk+a1Vk_l+...+ak-1V1
k
QI
Gi(p,p(1),...).ak-i
3.176
Thus, in yet another basis Ek, 1 < k < g, the invariant vector
fields on Jac C are just given by
1
(10.7) '
(Here
EkP(n)
EkP
(n)
k(p,p(1),...).
is given by the rule
= EkDnp = DnEk
=
G(n)(p,t(1),...).)
At this point, we have found the link to the famous Kortewegde Vries equation.
E
2
Namely, we have
P = P (3) + 12p.p (1)
This means that if
.
V is restricted to a 2-plane in Jac C
tangent to the vector fields E2 and E1 = D = ', it gives a
solution of the KdV equation
3
atf (x, t) = a 3f (x, t) + l2f (x, t) - axf (x, t)
ax
We want to explore this link further in the last section.
3.177
The Korteweg-deVries dynamical system.
§11.
As with the Neumann dynamical system, our purpose now is
to introduce a dynamical system interesting in its own right,
and then to show that it can, in some cases, be integrated
explicitly by the theory of hyperelliptic Jacobians.
More
precisely, we can, following the ideas in the previous section,
define an embedding of Jac C in an infinite dimensional space:
vector space R1 of analytic functions
(Jac-O) --- k ( f (x) defaped in scene neigh. of 0 E cr
00
z0
----> p (z0+xe)
p tn) (z0)
n0
n
. L,
On Rj, we consider a simple class of vector fields X:
which assign to f a tangent vector in Xf E TR
f = R.
those
given by
1'
Xf = P(f,f,.,.,f(n)), P a polynomial.
Integrating this vector field means finding an analytic function
f(x,y) s.t.
ay
P tf, ax,
aff
axn
By the Cauchy-Kowalevski Theorem, for all f(x,O) analytic in
c, there exists
jxj <
f(x,y) analytic in jxj,jyj < n solving
What we want to do is to set up a sequence XI,X2'...
this.
of such vector fields called the Kortweg-de Vries hierarchy
which
and
1
a) commute[Xi,X.] = 0 - we must define this carefully
b) are Hamiltonian in a certain formal sense, such that
c) for all g, and for all hyperelliptic curves C of genus g:
3.178
Im(Jac-0) =
orbit of all flows Xn,
i.e., all n are tangent to image
and a codimension g subspace of
Icn n are even 0 on Image
In fact, d) in some sense "fixing the value of these Hamiltonians"
we will merely state some results
gives the orbits of the Xn's:
of this type without proof.
Thus {Xn}
may be considered an
infinite-dimensional completely integrable Hamiltonian system.
We first investigate what it means for 2 such vector fields
Let
to commute.
Xf =
Yf = Q(f,f,---,f(m)).
Then, starting at a function f, the path through f obtained by
integrating Xf is
n
f(x,t) = f+tP(f,
22
I a(k)
k=0 of
(f,...).(d)kP(f,...) +......
(because the t-derivative of the RHS is
tP (f, f,
+t
)
of (k)
(d) P (f, f,
-
)---
P (f+ tp, f+t
P, ..... ) mod t) .
-
To go in 2 directions at once, one must be.able to define the
ts.t)-term unambiguously, i.e., the coefficient of s.t
t P (f+sQ, f+sQ, - .... )
and
in
3.179
s Q (f+tP, f+tP,
must be equal.
This means
ap
of (k)
Theorem 11.2:
)
dx
k
d
aQ
£a f
Q
(k)
dx
)k
P
(11.1) holds if and only if for all f E R1,
(for IxI,IsI,ItI<s) such
there is an analytic function f(x,s,t)
that
at = p(f,f,...)
of
of
as
Proof:
=
Q(f,f,...).
The existence of f implies (11.1) by working out
the meaning of the equality of mixed derivatives
a2f
ap
as
s=t=O
asst
_
s=t=0
aQ
at
s=t=0
Given (11.1), we define f as follows:
a)
let
b)
let
R3 = 0![XO,X1,X2,...]
R3
be a polynomial ring
be the map
X1 = x f
Thus R3 is a differential ring if we let Xi = Xi+l, and - is
the homomorphism of differential rings carrying X0 to f.
3.180
c)
Let
D:
R3
be the derivation such that
-> R3
D(X0) = P(XO,XI,...,gn)
= D (a) -
D (a)
be the derivation such that
E: R3
d) Likewise, let
E(X0) =
Q(XO1XI,...,Xm)
E(a)*.
E(a)
e) Let
t (s,t) _
DIE3 (X
I
i,ja0
)
--_-, ; ti si ER 3 ([S, t1l.
Note that (11.1) means precisely that (D,EI = 0 as derivations
of R3, and we check:
at
as=Es.
Moreover,
a
(...,(k),...)) _
axe
at(
ap
aXk
=
at
D t(k)
k
= D(P(...,4) (k),---)).
Now both P(---,cD (k)---) and at
T = DT,
asT
= Es
hence they are equal and
at
and
satisfy the equations
Tlc
= P(...,Xk,...),
-0
3.181
Likewise,
as
Therefore
DIES (X
f (x, s, t)
iii!
)
0
t1 s3
satisfies
at -
p(..-.(x)
k
k
as
Q(...,(d--) p,...).
QED
Thus the differential ring R1 is very convenient for
integrating flows.
However, the Hamiltonians that define these
flows do not exist on R1.
instead, we need
R2 = f(differential) ring of C°° functions
f(x) with compact support
R2
has the advantage that there is a large class of functions
(usually called "functionals")
Op:
R2
T
,
namely
+co
O (f) =
J
P(x,f,f,-.-,f(n))dx
where P is a polynomial in f1...,f(n), whose coefficients are Ca'
functions of x, and whose constant term has compact support.
These functionals may be called C"O-functionals because by the
calculus of variations they have excellent derivatives: i.e.,
3.182
lim
e-> 0
5p (f+cg) -O (f)
n
ap
i
=
e
k-0 of (k)
_oo
+00
_
(
g
(k) dx
k
1(ko(-1)k(dx)(afp(k)))'g(x) dx
(integration by parts).
Define
(-1)k (d d )k (of ap(k) )
Sf
k=o
to be the variational derivative of p.
We want to set up a
co-symplectic structure on R2, and define vector fields V56
for
p
these Hamiltonians.
These
will turn out to be examples of the
same type of vector fields that we considered on R1: but on R1,
they can be integrated locally, on.R2 they come from Hamiltonians.
At the very end of this §, we will mention briefly yet another
approach:
that of McKean, Van Moerbecke and Trubowitz, who used
R4 = ring of periodic
C00
-functions f(x), and could do both at once.
However, the clearest and most elegant way to bring in the
co-symplectic structure is in a much larger vector space:
space of differential operators.
a
This approach goes back to
Lax and Gel'fand--Dikii and has been highly developed by M. Adler
and LebedeV-Manin*.
Up to a point, we can develop the theory for
any of our differential rings R, but later we will restrict to R.
P4ark . ;,Adler, Ih a Trace Ptmctional for Formal Pseudo-Differential (aerators
and the Symplectic Structure of the Korteweg-DeVries Tyne Equations, Inv. Math.,
50, (1979), p. 219;
J.I. Manin,. Algebraic Aspects of non-linear differential equations, Modern problems
in Mathematics, (VINITI, 1978)
3.183
The central idea is to associate to f E R the differential
operator
(dd{) 2
+ f (x)
and to consider R as part of an even bigger space, viz.
RCD ] =
vector space of all differential operators
d
an(x)Dn
I
n=
D = d
dx
'
`an (x) E R
In fact, we put this in a yet bigger space:
R{D} =
vector space of "pseudo-differential
operators symbols"
d
I
an(x)D',
an(x)E R
IL--00
In R{D}, we can introduce a ring structure as follows:
Note that
D(fg) = fDg + gDf.
Thus as operators on R,
(11.3)
DOf = f*D + f
.
Taking this as our golden rule, we get a ring structure
on R[D] such that:
n
(k )
n+m-k
k=O
n
=
ak (fDn) *ak (gDm)
ky0 k T
3.184
or more generally
Co
(11.4 )
I
XOY =
k
k=0
* = multiply by
aky* aky
as though £,D
11
commute.
R{D} too, if we extend (11.1) via
In fact, this extends to
fop-1 = D-1of + D°foD
foD-1
D-1of =
hence
feD-l
=
-
D-lofoD-1
foD- 2
--
DofoD2
+
feD-1 - f°D-2 +
D-1afOD-3
-
f6D-3
i.e.,
(11.3)'
D-lOf =[foD-l
foD-2
-
+
faD-s+...+(-l)kf(k)aD-k-1+.....a
Note that again
D-1 f =
k0 k! ak(D-1)*ax(f)
so the general rule for mult. is still (11.2):
Co
XoY =
I
k' ;nX * axY
k=0
Associativity is very easy to check:
(X°Y)°Z =
aD(I
Q
akX * akY) * akZ
k!
k!p! (i-p) !
Xa(YoZ) _
1
!
aD +p
k * k (I
aDX
ax
1
Z ! p ! (k-p) !
1
(Q-p)!p!k!
*x
X * axaD- p
1-! aDY * ax Z)
aDX * aD ax-pY *
ak+p
D
Z-pak
*
x} Z
*
axZ
3.185
Proposition 11.5:
For all d E ZZ,
every element
X = Dd + a1Dd-1 +... E R{D}
has an inverse
X-1 - D-d + b1D--d-l.......
E R{D}.
Corollary 11.6.
The set of elements
1 + a1D-1 + a2D-1 +........
in
Lie
R{D}
is a group , celled the Volterra group by Lebedev-Manin.
{a1D-1+a2D-2+ Proof of Prop.:
have
b1,...,b
-}
is a Lie algebra under
Construct D-1 by induction.
[ ,
3
Suppose we
such that
1+cD-n-1......
Then it follows that
(D-d+b1D-d-1+...+bnD-d-n-CD-d-n-1)
m (Dd+a1Dd-l+ , - ) = 1 + (terms in D-n-2 or lower).
QED
For instance, one checks that
(D2+q)-1
= D-2
-
qD-4 +
24D-5
+.....
The following Proposition is due*, in fact, to I. Schur in 1904,
as P.M. Cohn pointed out to me:
I. Schur, uber vertauschbare lineare Differentialausdrficke,
Berliner Math. Ges. Sitzber. 3 (Archiv der Math. Beilage (3)
(1904), pp. 2-8.
8)
3.186
Proposition 11.7:
For all
a1Dd-1 +.....
X = Dd +
X has a unique d-
d > 1 and all
E R{(D)}
root
x1/d = D + b1 + b2D-1 +.... E R{D}
and the commutator Z(X) of X in R{D} is the ring of Laurent series
n
cix1/d,
i
c E T.
The main point is the calculation:
Proof:
Lx, cDm I = d61)
d+m-1
+ lower terms, c E R.
From this it follows by easy induction that Z(X) has, mod scalars
and lower order terms, a unique element of each degree m E ZZ and
that it has the form (cDm+lower terms) c E (C. If
Y E Z (X)
has
degree 1, Y'E z(x) has degree -1, it follows that YOY' = c+W,
c E T, c
0
and deg W< 0.
Therefore
CO
Y-1 = !Y
c
-
I
i=0
(-1) 1W1'/c1 E Z (X) ,
hence
n
Z(X) D {ring of Laurent series
I
ciYi}
i=-OD
hence "=" holds here because each side has one new element in
each degree.
in z (X) so
Thus Z(X) is commutative.
Finally, X itself is
3.187
X=
d
I
i=-00
ci E o, cd
ciY1,
0
and, in a ring of Laurent series, such an element has a unique
dth
root (up to root of unity):
X
1/d
1/d
= cd YO (1 +
cd-1Y -l + cd-2 -2
Y
1/d
cd
C
d
where the last term can be expanded by the binomial theorem.
QED
Returning to the 2nd order operator X = D2 +q, we can calculate
in terms of the universal polynomials introduced in §10.
fact, expand:
17_
9n(q,4,...)l
CO
D
2+q) -n
n=0
where
fngn are universal polynomials without constant term,
except for f0 = 1.
(Also go = 0).
Now
9 O (D2+ q) n+1 =
O (fnD"2n)
00
o(D2+q)
=
(D2+q)oV5-1+-q
_
(D2+q)
.-g
,
o(D2+q)-n
(fnD_g2) 0(D2+q) n+1
0
+
r
l
(fnDI-2f x02-fnq 2 - g D)° (D2 ) n
hence
0 =
gn
2
-n
f
((f -g )D + 2f (D+
n n
n 2 q) - 2f nqT n4 - -7-) O(D +q)
In
3.188
hence
gn = fn
(11.8)
l
__
gn
ng + 2fnq + 4
fn+i
Thus, relating this to §10, if q = 2pr, then
4nfn(q.q,...)
= Fn(2, S.,....)
Gn(2, ..... )
.
One may while away an hour or more calculating this out a ways:
n = D + (q
-
) e (D2+q)
-1 +
-
13q
- 96 ja (D2 t q) -2 +-
2
- D-2 + (q$ ) D`3 + (6q-) D
16
4
D+
m
to be the ring R2 of c -functions
We now choose our R
on ]R with compact support.
This enables us to integrate elements
of R as well as differentiate them.
Many of our conclusions will
however be quite formal and for these we may afterwards go back
to the original R.
in calculations in R2{D}, we find that the coefficient of
D-1 has a very important special property, viz.:
Theorem 11.9 (Adler):
of D-1 in
X,Y
For all X,Y E R2{D}, the coefficient a-1
is the derivative of a polynomial in the
coefficients of X and Y,
hence
3.189
Proof:
By linearity, it suffices to consider the case
X = aDk,
Y = bDQ.
Clearly, if k+Q < -1 or if k > O,Q > 0 there is nothing to check.
We may as well suppose k > 1,Q < -1 and use:
XoY =
n=0
(n) Dk+Q-n
(k
n ab
with term
Likewise,
YGX.
is the coefficient of D-1 in
ab(k+Q+1)+(-l)k+Qa(k+Q+1)b
The difference is
the derivative of
lab (k+Q) - a b (k+ Q-1)
+ab
which is
(k+Q-2)-
,.
1)k+Qa(k+Q) bl.
QED
We define
tr:
R2{D}
p- (C
by
+CO
tr X =
J a1(x)dx,
if
X =
akDk
-ao
Now put the vector spaces
R2[ D 1,
Lied
[ X,Y7
= tr(XoY)
in duality by
.
In particular, if
X
d
I a Dn,
n=0
n
CO
Y =
I
n=0
D-n-1b
n
3.190
then
d
f
<X,Y>
=
so that
functions
(anbn)dx
I
J
n=0
R2[D] is isomorphic to a subspace of (Lie )*, the linear
k
on Lie g
.
Thus the lie algebra Lie 3 acts on Lie.
by the adjoint representation adX(Y) = [X,YI and on R2[Dl
by
Explicitly, for all Y E Lie
the co-adjoint representation.
define
ad*:
R2[D ] > RJDI
by
<ad* (X),Y2> = -<X,adY (Y2)>
1
1
-<X,[Y1,Y21 >.
Let
+:{R 2 ID
>R 2
d
be the projection
(
n
a D)+
n
n=..-oo
I
E a
n= O
n
D.
n
ad*(X) = [Y,XI+
Corollas 11.10:
Proof:
=
D
<ad* (X),Y2>
_
-<X,[Y1,Y2 ]>
1
-tr(XOYImY2-XOY2DY1)
-tr((XoY1-Y1OX)oY2)
_ -tr ([X,Y1I +0Y2)
_ < [Y1,X I+, Y2>.
since deg 2 < -1
QED
3.191
We now recall a very general construction due to Kostant
and Kirillov which has many important applications.
an ordinary Lie group and
Then j
dimensional).
Let G be
its Lie algebra (which is finite
has a "co-symplectic structure" on it.
We explain this quite carefully to facilitate the infinitedimensional version to be used below:
a)
Yx E
*, identify Ti*,X =*
hence
T*
Then for all a,S E T**,x
define
92*(a,$) =<x,[a,9]a
Thus
S2
is a skew-symmetric bilinear form on T**
I ,X.
b) Now for all functions f,g on n*, we get
namely
dfx,dgX E T**,x
-f (x)
< y,df X > = lim f (x+ey)
E
y,dgx> = lim
g(x-FE6)(W)
Hence we define the Poisson bracket:
{f,g}X =
S2* (dfX,dgX)
= < x, [dfx,dgx ]
3.192
c)
For all functions f ont1*, this gives a vector field
Vf on U. Namely, if x ET, then (Vf)x E T%*,x =
is given by
<(Vf)x,s>
(dfx,
=
S2
=
<x,[dfx,8]>
=
<x, addf W>
x
= -<ad*
(x),S>
x
or
(Vf)x
-ad* f (x) .
x
Note that for all functions
g
on
*,
V* (g) def <Vf,dg>
=
S2* (df,dg)
{f,g}
,
hence
{f,g}x
d
+C(Vf)x
0
d)
E=0
Moreover, given any 2 vector fields V1,V2, we get their
bracket (V1,V2) = V31 which may be defined equivalently as
V3(f) = V1(V.2(f))-V2 (V1(f)), all functions f on
or directly by:
V 3,x0 = lim
where xl = Vl,xo,
V2,xo+sxl V2,xo
E
x2 = V2,xo.
- lim Vl,xo+ex2Vl,xo
£
3.193
The basic result -that this is a "good" co-symplectic
structure- is that
{f,{g,h}}+ {g,{h,f}}+ {h,{f,g}} = 0
or equivalently
[V'ffv9] = V{f,g}
.
We prove this in 2 steps:
Step I:
For all
a E 41 ,
let Ra be the linear fcn. on
given by Qa (x) = <x,a>. Then (dia)x = a for all x E
and
the definition tells us immediately
Thus, Jacobi's identity on tl gives us (11.9) when
are
i
a
f,grh
's.
Step II:
We prove (11.9) at a point x0 under the assumption
that (df) x
= 0.
It merely states the equality of the mixed
0
2nd
derivatives of f:
i.e., let (d?f)x
be the 2na derivative:
2
(d2f)x(Y,z) = aeart f(x+Ey+ r1z)
I
(0.0)
Then
{g,{h,f}} =
ae({h,f}(x0+E(Xg)x
0
a2
asarj
,f (x0+e(Xg)x0 + n(Xh)x0+s(X9)x J)
s0, (pt. of -t)}.* depending
in C
way on E
a 2
=
))Ie=D
aEar,(xo+C(g)xx +"(Xh)x +
0
0
ignore this because
dfx = 0
(d2f)x
(
0
(Xg)x r (Xh)x ).
0
0
3.194
Thus
{g,{h,f}} = {h,{g,f}}
and
df
x
= 0 = {f,{g,h}} = 0.
0
QED
Rather surprisingly, all of this works without essential change
for the infinite-dimensional case.
R2[D]
= Lie y
r
provided we restrict ourselves to an appropriate class of functions
on R2[D].
We use the maps
R2 [D ] ---?P CC
+Go
f P (x, - - -,k)...)dx,
OP(X)
_CO
if x =
d
I akDk
0
P a CCO function
depending on a
finite number of
the aa)
k ,s
The main point is that, as above, ¢p is sufficiently differentiable:
OP(X+EY) =
is a coo function of
J P(x,...,(ak+Ebk) (Q),...)dx
e; and:
dg OP (X+EY)
=
a
J
I
k,iBak
b (PI) dx
a
J
_ <Y,
aPW bkdx
ak
D
k
((_l)L()i aP
rr))>'
Q
3.195
hence
(dop)X
=
((l).L(d)Q 'PM)
I D-k
k
aak
By (11.10), the corresponding vector field V
(Q)
a
J(x m {
k, k
+
-k (-l) QaQ
D'
(
)
The Jacobi identity for {
is just:
2(&)!C
DI
IX,
(y6P )x
P
ELie 3.
,
}
(V0p,V5Q I =
-x
ap
r
a
`plc
x
QD
1dkaQ
(-1) (E) aq(Q>
1
and the formula
V{56prs6Q}
are proven exactly as before.
We now specialize all this to the submanifold M of R[DI:
M = {D2+q q ER} .
In general, one cannot restrict a co-symplectic structure from
a space. N to a submanifold M unless for all x E M, the 2-form
Sx factors through
T*
x
Sly TNT
N
- -- T
r xX
\PX
71
f
TM,xxTM,x
But if we compute
we find the following.
a,s ELie*
2+q(a,S) _ <
D2+q,
lot,a l >
dx.
-
For all
3.196
Let
..
a=D_
D-ls0+D-2$1+...
then
D-1a0D-1 S0
[a,al =
-
D-1$0D-1ao +....
=
D-3
.
higher terms,
so
,*2+q (a, ) =
J(&060-0a0)dx
= 2 f &00 dx.
This depends only on a0,$0, which give the restriction to
to linear functions on TM,D2+q.
structure on M.
a,
Thus we have a co-symplectic
In fact, it is non-degenerate now.
This
non-degenerate 2-form was discovered by Gardner and Greene.
Now for all functionals on M:
OP(D2+q)
f P(x.q,q,...,q(n))dx
=
we see that using the variational derivative 6P/6q
D-1.
(d96p)D +q =
{OP.0Q}D2+q
(V56
P)D+q
2f
[o2+q,
= D6P
Sq
dq
D -1 -6P
6ql+
6 - 1 6 P D2
Sq
defined above:
3.197
We will also have occasion to compute the bracket of 2 vector
fields on M directly.
Suppose
Pi(x,q,q,-,q(n))
and
define 2 vector fields V1,V2 by the rule
(Vl) D2+q =
P1(x,q,q,...,q(n.))
(V2)D2+q =
P2(x,g,Q,--.,q(n))
then we can compute [V1,V2] directly as in (d) above:
P
[VlV2 )D2+q = lim
2
(x,q+eP (x,q)) -P2 (x,q)
P1(x,q+eP2(x,q))-P1(x,q))
- lim
E
e
(11.12)'
ap2
0o
(k)
k0 aq Pl
aPl
_
aq(k)
(k)
P2
which is a vector field of the same form.
Formulae (11.12) have the following consequence:
Proposition 11.13.
Given
if there exists a polynomial
Coq)
then
and Q(x,grq,---,g(m)
H(x,q,4,-..,q(k)) such that
Caq)
{OP,giQ} = 0.
don't involve x, the converse is true.
as follows.
This can be proven
We use a purely formal result of the variational
calculus in the differential ring
3.198
R3 = (r[XO'X
Theorem 11.14.
The se uence
R3
3
is exact, i.e., for all
olynomials
Sf
some g.
d X= 0 '"---> f=
Working over Rz, we see that
Sketch 9_f _proof:
Of M 0
f = g
f(XO'X1'--.,Xn)'
derivative Sq of 0f is zero.
Since this is purely formal, it holds in R3 too.
converse,
To prove the
use induction on the order n of the highest derivatives
in f, and argue like this:
8x =
0 f = Xn.f1+f3
3g s.t. f-g E T[XO,...,Xn-1]
f1, f2 E C [X0, .. , Xn-l ]
Corollary 11.15.
If
QED
P(q,4,--.,q(n)), Q(q,q.---,q(n))
are
polynomials, then
{opIOQ} = 0
s polynomial Fi(q,q,-..,q(k)) such that
Lp
CB
gI-(Sq)
Proof:
{op,oQ} = 0 implies
d{op,oQ} = 0, i.e.,
((5q'
aq
hence H exists by the Theorem.
\8q) ) - O
QED
3.199
This completes our general discussion of the co-symplectic
We now introduce the Korteweg-de Vries
structure on M.
Hamiltonians:
tr ((D2+q) n+1/2)
Expanding / +q
as above, we see that
Hn = n+
tr ((D2+q) n (D2+q) n)
I tr (fkD-2k)°(D2+q) n`k.
n+ k=0
Note that n-k > 0, the kt term is a differential operator, hence
has'no trace and if n-k < 2, the kth term involves D-3 and lower,
so has no trace.
H
Therefore
tr ( (fn+1D_ gT+1)a(n2+q) -1>
n
n
1
m+
n+3z
ffn+l (q,4,---)dx
= 56 (fn+1/n+h)
which is a function on M of the type we are considering.
want to calculate the derivative of H
Lemma 11.17.
(D2+q)n+'J_
dq
n
:
(n+/)[(D2+q
l
1
1
6fn+l
8q
or
- (n+h)f n
+q = D
--1
ofn(q,q,..).
'
We
3.200
Proof:
This amounts to saying that for all a(x)E R,
n+35 ,.
de
(n+/)tr((D2+q)n-koa)
_
Write
(D2+q+sa)/ = E+e El (mod E2) .
Then
m
de tr(En a (E+eE1)m)' _
tr(EnoEO...oEmE1OEo---OE)
m tr
=
(En+m-1p El)
'
especially for m = 2, this says
tr(Enoa) = ds tr(Eno(D2+q+Ea)) = 2 tr(En+1oE1)
so
dE
tr(Eno(E+EE1)m) = z tr(En+m-2oa).
In particular, if n = 0, we get
m-
tr((D2+q+sa)m/2/
dE
Theorem 11.18.
b)
a)
m tr(( D2+q)) 22
0a).
n
(VH)D
+q = 2gn (q,q, - - - )
QED
_ - f (D2+q) ,
[(D2+q)
{Hn,Hm} = 0, all n,m.
Proof:
and for the
In fact, the 1s- part of (a) is just the lemma,
nd
[D2+q, (D2+q)
n-/
-ID 2 +q,(D2+q) C/
-[D +q, (D2+q) nl
[D
2+q, D_1. fn
(q,q,
-(Do.fn-D-1fnD2)+
_ -(fn+fn)
= -2g
n.
]+
)] +
n-/
l
+]
3.201
As for (b)
{Hn,Hm}D2+q =
/IVH
,
dHm>D2+q
n
- (n+2) [D2+q, (D2+q)n`" ], (m+2) (D2+q)dn
1>
_ -(n+z) (m+Z)tr ( [D2+q, (D2+q)+-k ]o (D2+q)mbut
tr ([A,B ]oC) = tr (ABC-BAC) = tr (B (CA-AC))
These flows VH
0 if [A,C ] = 0.
are the KdV dynamical system.
Note that
n
they are defined by universal polynomials 2gn so in fact they
make sense for any differential ring R:
(VH
)D2+q =
1
(VH2' D2+q =
(VH
3
u2+q
(
q+g*
6q._
4
(which integrates to q(x+t)
i.e., it is just transl.)
)
+q (5)
=
etc.
We want to elaborate on the conclusions that we have drawn.
First of all, notice that combining the last Theorem with
Corollary 11.15, we have reproven the conclusion of §10:
for all k,Q, there is a polynomial
such that
Hk2 = FkGi
Alternatively, we could have used this to prove {Hi,H.} = 0.
Secondly, notice that the conclusion
3.202
]=0
[ VH , VH
i
a
makes sense over any differential ring R even when the Hi don't.
Namely the vector fields VH
may be defined by part (a) of
Theorem 11.18, and their commutativity may be expressed by
(11.12)' by the polynomial identity
agi
(gi )
aq (k)
k10
00
(k)
CO
=
ag3 ,
k0 aq (k)
(gi) (k)
In particular, over the ring R1 of analytic functions, it follows
that starting with any analytic function f(x) defined by
Ixj
< e, we can integrate any finite set of the flows VH
getting an analytic
f(x,tIt...,tn)
defined for 1xI,1tl1,." ,1tn1
such that
of =
atl
6f f-f(3)
of
at
of
2
=
4
1 < i < n.
The seond form of these equations given in Theorem 11.18 is called
a Lax equation.'
and
t - X (t)
In general, if S is a vector space of operators X,
is, a 1-parameter of operators and
(D:
S
is a way of transforming one operator into another, then a Lax
equation for the family x(t) is an equation:
atx(t) = [x(t) ,(X(t) )).
S
n
3.203
The importance of such equations is that they say that the
operators X(t) are infinitesimally conjugate to each other, i.e.,
X(t0+St) =
(X(t0)) ) mod 6t2
In good cases, this implies that any sort
of spectrum of
X(t) is independent of t.
It is evident that this whole collection of flows on M
mirrors the flows on Jac C, as defined in §10.
The precise
link is this:
Theorem 11.19:
For a
genus g, let C be a smooth hyperelliptic
curve of genus g, and let B,TJ,-,fix
the vector field D
be defined as usual.
on Jac C be written
lei a/azi.
Let
Define
an embedding
Jac C -- e
zo
I-
M
>
(the operator
(ax) 2 +2p (z0+xe)
Then all the flows VEn
on M
)
are tangent to the image and VEn
restricts to the flow 4En on Jac C.
Proof:
We simply combine the results of §10 and §11.
Note that
Ek (2p) =
Ck (3p,p, - -)
2
= 2. k-1 gk (2, 2#, ... )
= 4 k-l (V)D2+2
.
3.204
E M, the vectors
At the point
Corollary 11.20.
span a finite-dimensional space of dimension g.
VH
In terms of
n
the moduli
ai of C defined in §10, for all
(VHk) D2+2px +
Proof:
4 (VHk-1)
k > g
D2+2p +.....+ 4k-l (VH1) D2+V = 0
Combine Theorem 11.19 and (10.7).
For all
Corollary 11.21.
C,z0, there is a differential
operator of degree 2g+l which commutes with D2+2p, namely
g+l
I
£=l
Proof:
a
g
+l-Q.4Q-g-1(D2+2z)+-
Combine Cor. 11.20 and Theorem 11.18a.
One case where the KdV dynamical system has been explored
much more deeply, first by McKean-Van Moerbeke, then by
McKean-Trubowitz, is over the ring
R4 = C°° periodic real functions on IR.
We sketch their theory very briefly.
The operators
xq = (a) 2 + q (x)
with q periodic can be analyzed by the Floquet theory (cf. Magnus,
Hill's equation).
In particular, for all
so-called h-spectrum:
h E C*, they have a
3.205
h-spectrum =
set of
(
s.t. there is an eigenfunction
l
f (x) with
f(x)+q(x)f(x) = Xf(x)
f (x+1) = hf(x)
The fact that the K-dV flows can be written in the Lax form
q-
)2+gt(x))+_3
(dx)2+gt(x),((d
Ed
or
at
Xq
t
[Kq ry gt
t
shows by standard results that the h-spectrum is constant as
a.function of t.
We may now consider
I(q) = {(h,A)
A
h-spectrum} c (r 2
which is readily seen to be a 1-dimensional complex analytic
subset such that the projection
In fact, for each
a
J(q)
D(A-plane) is 2-1.
,
f0 (x, a)
let
f1(x,A)
Then
be the 2'solutions of
f+gf = of
f0(0'A)
f0(0,A)
f1(0,A) = 0.,
0
f1(0,A) == 1.
with
f (x+l,A)
f0(x+l,A)} are again.2 solutions, so we can write them
3.206
f0(x+1,X) = a(X)f0(x,X)+b()L)f1(x,X)
f1(x+l,X) = c(X)f0(x,X)+d(X)f1(x,X)
a,b,c,d entire analytic functions of X such that
ad-bc
_
1.
J(q) is defined by
Then
h2_ (X)+d (X))h + 1 = 0.
Let
I(q), i.e., the 2 sheets
J*(q) be the normalization of
separated at double zeroes of the discriminant 0(X) =
if any.
Thus a "hyperelliptic curve"
(a+d)2-4
J*(q) usually of
infinite genus is associated to this situation.
The basic result
of the theory of McKean and collaborators is that for all
cc of finite or infinite genus the following sets
J*(q)
are equal:
1)
1*(q1) _(:, I*(q)
{qlI the branch points of
are the same, hence
1*(ql)
--c'
2)
{qll j(q1) =
3)
the orbit of the KdV flows through q
4)
the set of all q
1
}
J*(q)
J(q) as subsets of T2}
1
such that the IrdV Hamiltonians
1
r
JF(q1i1i.)dx
Fn(q,q,...)
=
are equal.
J
0
0
Moreover, this set is canonically isomorphic to a distinguished
component of the subgroup of real points on the Jacobian of C.
3.207
fl.
The Prime Form E(x,y).
Given an arbitrary compact Riemann surface X, of genus g,
wouldn't it be handy if we had a holomorphic function
such that E(x,y) = 0 if and only if x = y?
E: X x X --> T.
Although such a function doesn't exist, it turns out that it
To'understand part of the problem and how to
"'almost" does!
fix it, let's look at the simplest case:
Example.
Let
The function x-y works on IP1--{-I but
X = IP1.
not on all of IP1.
So consider instead the "differential":
x-y
E (x,y) =
where
dx, dy
/dx /dy
are defined as follows:
Choose a line bundle square root L of
isomorphism L02
such that
-
1.
5211
Choose a section
Then
dx'
-'-dx
x
so define v by
Then if
/dx C r (Ipl- {=},L)
( dx)2 = dx (under this isomorphism). To check that this
is finite on all of IP1xIP1, let x' = 1/x
IP1- {0 }.
and an
x' = l/x, y' = 1/y,
be a coordinate on
3.208
1
Y
1
dx/d-y -dy'
R(x,y) _ -x
y-
x
=
x'
X
=
y'
For a general compact Riemann surface X, we will have to
modify this approach in several ways.
L02
an isomorphism
-- Std
First, choose an L and
such that
h0(L)
In terms
= 1.
of divisors, this means we want to find a divisor class
d E Picg-1(X)
such that
We must show that such a
Lemma 1.
2
28
b)
181 = a single divisor 8
8
exists:
Picg-l(X) as above, i.e., there exists a
E
S
and
a)
nonsingular, odd, theta characteristic.
Proof.
theorem.
We use Riemann's theorem, and Lefschetz' embedding
We want to translate the conditions on
into theta
d
functions:
exists
S
and
[S" J E
3
2Z
2g/Z2g
s.t.
61
[8 )(0,S2) = 0
a)
-
b)
dz$'[a , J(o,O)
0
Now, Lefschetz' theorem states that:
2g-1,
g
z
F-----
( .. 4 [ a I (Z' 2) ....
8',6" E
' fig/ Zg
3.209
is an embedding.
In particular, the differentials
r
dz-9 [a ] (o,SZ)
must span the cotangent space. Note that
0 implies
61
61
[a ] is even, hence S fa if ] (z,c ) is invariant under z 4---> --z,
hence
](o,S2) = 0. Thus if [VII satisfies (b), it also
,
satisfies (a).
QED
a3[a;]
By Riemann's theorem, let
=
al
(o) u.
az
be the
r
unique 1-form which is zero on
above to
In fact,
6.
corresponds as
2&, since (S)-6 = (an effective
divisor in K--S, i.e., 6).
of L.
6, where
So
t = (/)2, where
/
is a section
We may think of x as a differential form of weight 2.
This
will take the place of v'-d-x.
Next we modify the numerator x-y in E, using a theta function
r
-'9 [a:,, ]) (f Xw)
for higher genus.
The prime form
Definition.
l
E (x,Y) _
where:
E(x,y) is given by
r
[arr ](fx w
(x) /57YF
a)
6
is a fixed, nonsingular odd theta characteristic.
b)
6
corresponds to
6r
c) TET,
Y7
are as above.
This is a holomorphic differential form of weight
X xX, where
X
is the universal cover of X.
on
3.210
A few remarks:
1.
E is not defined on
X xX
from x to y must be made.
since a choice of path of integration
To make this well-defined we simply pull
back to the universal cover.
2.
Note, however, that whether this is zero or not only depends
on the image of x,y in X: E(1) = 0 <=- E ( c2,y2) = 0 when
N N
x1,x2)' resp. ylry2
have the same projection to X.
Alternatively,
we can consider E(x,y) as a holomorphic section of a line bundle
on x x X.
The following properties make the prime form useful:
Pro2erties of E(x,y).
Let x,y E X, x,y
their images in X.
1.
E(x,y) = 0
2.
E has a first order zero along the diagonal A = X x X.
3.
E(x,y) _ - E(y,x).
4.
Choose a local coordinate t about x E X
x = y.
(This is its major property.)
such that
= dt.
Then
E (x,y) =
t (x) - t (y)
dt x
5.
(1
-
O (t (x) -t (y)) 2) .
If x or y is moved by an A-period, E(x,y) remains
invariant.
If x is moved by a B-period EmiBi to x',
y
E (x,y) = +E (x,y) exp (-iri tmc m + 27ri tm xf ri)
If y is similarly moved to y':
Y
E(x,y') = +E(x,y)exp(-iri tmcm - 2rri tm f -W).
x
3.211
The main lemma that we need to prove this is:
g-l
Lemma 2.
Given
6 as above,
Pi},
181
then
i=1
y
f w) = 0
>
a)
x= y
h)
x = some Pi or
c)
y = some Pi.
or
x
Proof.
By Riemann's theorem,
Sfa,,](
Now
Yfw)
x
=0 s
h°(EPi) = 1, so
Case 1.
y-x+6 y
# 0.
h°(y + EPi) = 1 or 2.
h°(y + EPi) = 1.
So
ly + EPii _ {y + EPi}
l Y- x+ EPi
34 0 4==a either
x = Y, or x = some Pi.
h°(y + EPi) = 2.
Case 2.
By Riemann-Roch,
h°(K - y - EPi) = 1, but
K - EPi - EPi
So
h° (EPi - y) = 1
sn
y = some Pi.
QED
The proofs of the properties above are now quite easy.
For instance, for (1):
W
V (-
From the lemma we know:
y
{ , ] ( f w) ) = V (x-y) U
x
(
i i
U P x X)
U
(
.u x x Pi) .
i
3.212
The fact that it vanishes to order one is left to the reader.
x), I:
But this is precisely why we divided by
()divisor =
I Pi
as divisors. For the others, (3), (5) are immediate,
QED
and (4) is just a local calculation.
so (E(x,y)) = A
As one. application of the prime form, we will construct all
meromorphic functions on X, as well as the basic differentials:
(a)
Pic X and suppose
f(x) =
n
E(x,ai)
II
E(x,b )
iOl
ai
such that
Eaj
for all i,j.
Then
E X
Given a1,...,an,b11...,b
bj
Dbi in
is a single-valued meromorphic function
i
on X with zeros =
Eai, poles =
Ebi.
etc., cancel out,
To prove this, note that all the
n
so you are left with
[s
' w)
IT
and now just check
i=1'9 T I(J'w)
invariance under A,B periods.
(b)
Construction of differentials of the 3rd kind.
We want
Wa-b (x)
=
the unique differential 1-form on x with
a) zero A-periods
b) single pole at.a with residue 1
single pole at b with residue -1.
In fact,
locally:
ma-b(x) = dx log E(x,b)'
To check this, look
3.213
wa-b (x) = dx log (t (x) -t (a) ) - dx log (t (x) -t (b) ) + holcmorphic
differential
=
(c)
t
dt (x)
-
(xx))- -t
t(a)
dt (x)
t(xx))- -t
t(b)
+ holomorphic differential.
Construction of differentials of the 2nd kind.
We want
na (x) = a 1-form on x with
a) zero A-periods
b) double pole at a E x.
Note that such an
na is unique up to a multiplicative
constant.
A
Consider:
w (x, Y) = dX# y log E (X, y)
This is a well-defined 2-form on
dX -dy log[ f (x,y) g (x) h (y) ]
x x x, since
= dx dY log f (x,y) .
For each fixed
y = a, by choosing a basis for the tangent space to x at a, it
restricts to a 1-form on x equal to the above
multiplicative constant.
n
a
(x) up to
a
In this manner, we can construct
differential 1-forms with any allowed divisor of zeros and poles.
3.214
92.
Fay's Trisecant Identity
We now come to a very fundamental identity between theta
functions that holds for the period matrices of curves, but not
for general period matrices.
Although the basic ideas behind this
identity go back to Riemann, it was not clearly isolated until
Fay made his beautiful and systematic analysis of the theory of
theta functions (J. Fay, Theta functions on Riemann surfaces,
Springer Lecture Notes 352, 1973).
Theorem
[Fay, op. cit., p. 34, formula 45].
Let X be a compact
Riemann surface, X its universal covering space,
,9(z)
associated theta function and E(x,y) its prime form.
its_,
Then for
all a,b,c,d E 5, z E Tg.
d
c
f w}E(c,b)E(a,d)
(z + f
+(z +
a
b
c
d
J w),E(c,a)E(d,b)
a
c+d
(z +
J
w). U
(z),E (c , d) E (a, b) .
a+b
This type of identity is very special.
The theta function
on general abelian varieties doesn't satisfy identities like
3
cig (z+ai). 9 (z+bi) = 0.
J.--L
The proof of the theorem falls into several steps, each of
which is straightforward but sometimes tedious.
3.215
Step I.
Check that all three terms satisfy the same functional
equations and are differentials of the same type.
This way all
three terms will be sections of the same line line bundle L on
the space X xX xX xX X Jac(X).
Step TI.
Next show that if both terms on the left are zero, then
the right hand side is also zero.
Step III.
Let Dl,D2
be the codimension one subsets where the
two terms on the left, respectively, are zero.
components D3 of
Step IV.
Step V.
intersection is generically transversal.
D1f,
H1 (X X X X X X X X Jac (X) ,
Assume:
Then for all
L-1) = 0.
X
smooth complete variety
L
line bundle on X such that H1(X,L-1) = 0.
t,sl,s2 E H0(X,L) global sections s.t.
a)
sl = 0, s2 = 0 are divisors D1,D2 without
multiplicity.
b)
For all components D, of D1
D2, the
intersection is generically transversal.
c)
Then:
2X1,X2
s1 (x) = s2 (x) = 0
t (x) = 0.
such that
t = X1s1 + X2s2.
Stet VI.
First, let a = b, secondly let b = c, to see that the
constants are one, finishing the proof.
3.216
We will not go through all the details but instead touch on
all the main points:
Ste
:
Everything is invariant under z 1j z+n, n E Mg.
Under
z {-y z + Stm, M E Zg, the 3 terms are multiplied by
e-TrI tms2m- 2 Tr i tm (z+ ?w)
e-Tri tmS2m- 2 Tr i tm (Z+ bf W)
a
e rri txnm-27ri tm (z+lw)
e-7ri tmnm-2rri tm(z+bw)
and
c+d
e-7ri tmS2m-2Tri tm(z+a+bw)
e-Tri tmQm--27ri
respectively, which are equal because
c
fW +
a
on (X)4.
d
fW
b
c
=
d
+
fW
b
c+d
fW
a
a+b
Among the many substitutions in a,b,c,d, we consider only
c 1-> yc, y E Trl (x) .
Let
n+S2m
be the period defined by Y
Note that the half-order differentials
r;-FXT
are sections of a
line bundle on X, hence are invariant by all such substitutions.
Thus
E (Yc,b) = e- 7r1 tmS2m
r
c
e-2Tri tm ( w +6") -2Tri tn 6'
E (c,b) .
Collecting all the factors, you find that all 3 terms are multiplied
by
2c
e-2Tri tmRm-27ri tm6"-2Tri tn. S *_2Tri tm(z+A+bw)
3.217
Std:
One must look at all 16 combinations of one of the
4 factors of the 1st term with one of the 4 factors of the 2nd
term.
Most combinations are obvious, e.g.,
E(c,d) = 0
E(a,d) = 0, E(c,a) _.0
c+d
dr
A(z + Jw) = 0, E(c,a) = 0
f w) = 0
e(z +
b
a+b
A slightly less obvious case is when
c
c
e(z + Jw) = 0,
B(z + fW) = 0.
a
If
b
Dz is the divisor of degree g-l on X defined by z, this means
IDz+c
- al
34 0
1Dz+c-bj
7-4 0
Then either a = b, or IDz+cj is a pencil or IDz+c-a-bl
Therefore
either a = b, or JDzj
Therefore either
Step III:
c
E(a,b.) = 0 or
IDz+c+d-a-bl
0(z) = 0 or
0.
c+d
0(z + f w) = 0.
a+b
Let's look at .c
the generic transversality of
8(z + fW) = 0
and
8(z + fw) = 0.
a
codimension
0 or
0.
b
>2.
We can ignore loci of
Recall that the differential d
0
z
z = zo if and only if
single divisor
Fz
,
I D
z
is a pencil;
and if JD
this differential pulls back on
zero on
0
'
is a
zo
oI
0
unique 1-form wz
vanishes at
Fz
.
0
Thus the loci where
X
to the
3.218
c
d1e (z +
c
f w)
or
=0
dze (z + (w) = 0
a
b
are the loci where.
or
IDz+c-al
are pencils:
IDz+c-bi
have higher codimension and can be ignored.
that of the 3 alternatives:
(3)
IDz+c-a-b)
the
a = b,
(1)
we may also suppose
Let
1-forms on X zero on the divisor in
pencil and
IDz+cl
(2)
34 0, exactly one holds.
these
wa, resp. wb, be
ADZ+c-al, resp.
IDz+c-bI.
c
If
Wa 34 Wb, this means that the differentials of
e(z + fw)
a
c
and
6(z + fw)
in the z-direction are independent.
If a = b,
b
but
c
0, this means that in the a--direction
wa(a)
e(z + fw) has
a
c
non-zero differential while
6(z + fw)
b
has zero differential.
In
both cases, the intersection is transversal.
But now there are 3 possibilities
Case 1:
a = b,
IDz+cl is one divisor, IDz+c-a-bj = 0.
Case 2:
a 34 b,
ADZ+cl pencil, IDz+c-a-bl = 0.
Case 3:
a 76 b,
IDz+c) one divisor,
It is not hard to show that in case 1,
cases 2 and 3,
Step IV:
wa
IDz+c-a-bi # 0.
wa(a)
0
while in
wb.
Look at the projection
X XX XX XX xJ
p345
X xXxJ
where
a,b vary in the fibres.
L restricts to each fibre p315(z)
3.219
to a line bundle of the form
pll(M1)®p21(M2)
positive degree. By the Kfnneth formula,
where
Hi (L-1I
i = 0,1, hence by the Leray spectral sequence
Step V:
M1,M2 have
fib7re
) _ (0) ,
H1(L-1)
(0).
Use the exact sequence
0 - LT1
(sl,s
(D 0X
--->
0
$z)
s1
where
is the subscheme s1 = s2 = 0.
Step VI;
Obvious.
Next, we want to give a geometric interpretation of the
identity.
In fact:
a)
use
1201
to map
Jac(X) to projective space.
The image is called the Kummer variety.
b)
in this mapping, the trisecant identity will tell us
that the images of certain sets of three points
are collinear, i.e., the "Kummer
(C04 of them!)
variety" has
oo
4 trisecants.
First, we need to know what 1201 consists of:
Lemma.
1201
=
the set of divisors of the form
c'n
IE
Proof.
[n
)(zrz) = 0 )
r
for all (c) E C 2g
g/'Zg
First, it is easy to check that ([}z,2
)
= 0) E!201
3.220
[ ](z, 2)
We have 9 [n 01(z + n, z)
e-4Tri tnn. e-2Tri tmnm-4rri tmz-,[01{z,
and 9 [n ](z +nm,
n
z
Since the transition functions are the squares of those of l0I,
V(9[° ](z, a)) E
120
Next, we must check that these span 120l.
dimension
of H0(20):
it will be 29.
One way is to find the
Since the 9[o](z,2) are all
linearly independent, this shows they are a basis.
t
dimension, let
f(z) =
a(n)e2Tri
I
nz E H°(20).
To find the
Thus:
nEMg
f (z+nm) = e-2Tri tmnm-4Tri tmz f(z)
.
This gives us a formula for a(k):
a(k+2m) = a (k)
.
e-27r
tmom
.
k,m E Zg
.
This gives us an upper bound of 2g for the dimension, which is what
we wanted.
QED
Moreover,'recall from Chapter II, the fundamental:
Addition Theorem.
2-g
1
nE
g/Z
,&[07(x,
n
](y,2)
.
3.221
The geometric interpretation of the theorem is
Geometric Corollary.
Then,
Let
Jac (X) -->
define
1201
Va,b,c,d E x, the three points
c+d
a+c
z f w),
b+c
c(2 J W),
w)
a+d
b+d
a+b
are collinear.
Proof.
The map
0, by the lemma above, is given explicitly
as
We want to use the trisecant identity to give us one relation.
Let V = vector space spanned by-9103(Z' 2) ,
Let
Q = symmetric bilinear form on, V with these ,9 [n](z,
2)
as an
orthonormal basis.
Now note,
1)
For any
a E Qg, i9(z+a,c),i9(z-a,c) E V.
(For instance, use the addition formula to get this.)
2)
Let Ta .4(z,St) = -)g(z+a,S2). Then,
v f E V,
Q (Ta$ - T* \9,
f) = 2-g
-
f (a).
(Just check this on basis elements f (z) =.9 [n J (Z'
By the addition theorem, both sides are
Now we can apply the trisect theorem.
substitution
z f-->z - 2
.
J
a+b
We get:
2)
3(a,
.
2)).
In the theorem, make the
3.222
LHS = C 9(z + 2
+
2
b+c
b+c
a+d
a+d
J9(z_J)
a+c
a+c
b+d
b+d
J)(z_J)
c+d
c+d
RHS _ c3,9(z + 1 J
a+b
w)
S(z -- 2 j w)
a+b
91s are of the form of the
Note that these three products of
function in notes 1,2. Let f (z) =,g Cn ] (z,
for any
2)
Mg/jg
r1
2
Apply Q(_,f) to the equation to obtain
a+c
b+c
n
clg[o.](2
J
Wr 2) +
W.
J
b+d
a+d
c+d
c3 [TO) ](2
Jw
2
a+b
where cl,c2,c3 are independent of
ri.
QED
3.223
Corollaries of the identity
§3.
In this section we will study what happens to. Fay's identities
when the 4 points a,b,c,d come together in various stages.
The
result will be identities involving derivatives of theta functions.
For the following formulas, let
First, we need some notation.
a)
z E Cg
b)
a,b,c,d E X
c)
d)
with distinct projections to X
(z) the theta function of X
for every a E X,"and local coordinates t on X near a, we
expand the differentials of the 1st kind:
W
=
t
3
j ) dt
v
(
j=0
and let
vj = (vi,,...,vgj).
(Note that the mapping
>eg
X
-)x - J xw
a
is given near a by
CO
t 4--
j=0
+
V.
tj+l
( 7+l)
3.224
We let
Da = constant vector field
vo- az (i.e.,
D a = constant vector field
v1 3t
a
voi azi
v2
constant vector field
b
b
f
We abbreviate
e)
to
fw
J.
a
a
The identities we will prove are:
(1)
(Fay, Prop. 2.10, formula 38);
D
b
log
c
(Z+ 1)
= c + c2
-9 (z)
where
1
wa_c(b) = cldt
E (a, C)
E(b,c)E(a,b)
(2)
, (Z+ 1)
(z+ C) -
(Z)
(t a local coordinate near b)
= c 2 dt
(Fay, Cor. 2.12, formula 38):
DaDb log
where
(z) = c1 + c2
19 (z+ f) &(z+
w(a,b) = _cldta dtb
1
E (a,b)
2
=
c2 dta dtb
(z)2
(ta,tb local coord. near a,b resp.)
3.225
(3)
(Fay, Cor. 2.13, p. 27) :
D4
(z) )2 a log tiA (z) + 6[ D2log,'
a
+ 3 D' 2 log'- (z)
+
2DaD"
a log
(z)
cl D2
log- (z) + C2 = 0
a
where c1,c2 are constants depending on the Taylor
expansions of E(a,b) and c)(a,b).
As explained in Ch. II, there are 3 ways to get meromorphic
functions on
XQ from :
Tr9(z+ai)
--a as products
-!P.
if
7T-O(z+bi
I
ai -- 1 bi
(z+a)
(z)
as differences of logaritignic derivatives
D log
as 2nd logarit snic derivatives
D D' log ,9 (z) .
.
The above identities give basic identities between meromorphic
Identities (1) will appear as
functions formed in these 3 ways.
the limiting case of the trisecant identity when d >b, while
a,b,c are still distinct.
we let c --- * a while a,b
appear when finally we let
Identity (2) will appear when in (1)
are still distinct.
b
Identity (3) will
>a.
Before proving the formulas, we need the following lemma:
Lemma
:
a)
d x
E (x,b) I
E x,a)
=
x=b
1
E(b,a
3.226
E (x, a) d
a-b (x)
b)
W
c)
dx
E(x,b)
E (x,b)
x E(x,a)
= -w (a, b)
a_x(b)
x=a
(To prove (a), use the local expansion of E near x = b; (b) and
(c) are restatements of the definitions.)
Proof of formula one:
We want:
C
(lb)
C
- IDb, (z) IS (z + f
Dbo (z + f) -9 (z)
a
c
a
b
c
f)
(z) +c219(z+ f)e(z+ f )
b
a
a
Take the trisecant identity, and divide by E(c,b)E(a,d):
d
c
(z +
f )s(z
+
a
f
b
E(c,d)E(a,b)
E (c, b) E (a, d)
C
d
b
a
E (c, b) E (a, d) j lfz + J )+ f j
E (c,a) E (d b)
c+d
f
a+b
Now differentiate w.r.t d (as scalar functions on our
R.S..), and let d -->b:
3.227
CJ
E(x,b) I
+ E(c,a)
9 (z + ).D,(z)
b
E(c,b dX E a,x Ix=b -(z +
c
b
J)9(z + f)
b
a
E (a, b) d E (C, x)
E (c,b) x E(a,x)
a
c\
9 (z)9(z + f)
x=b
a
c
+-7(z)
f)
a
Now use the lemma (a),(b) to get:
c
c
z J)Db9(z)
+
a
E (c, a)
+ E(c,b)E(a,b)
19
z + J)t9(z +
b
c
a-c (b) J(z ),- (z +
1)
c
c
+ 9'(z)Dbi9(z + f)
a
a
This gives us our formula.
Proof of formula two:
We want:
(2b)
[
(z)] $(z) - Da& (z) .Db.9 (z)
cl-9(z)
z
b
a
+ c2'5 (z + f) (z + f)
a
Now, take formula (lb), differentiate w.r.t. c, let
b
c -.> a,
while noticing that cl,c2 in (lb) are not constants w.r.t.
and in fact both vanish when c = a:
c,
3.228
[DaDb.9
(z)'1,9(z)
-
Da-9 (z)
do W a-c (b)
I
c=a
S (z)
a
E (a, c)
1
+ do E (b,c)
E (a,b)
+
c=a
b
fb P(z + af)
But now use the lemma (a), (c), to get:
[D a D
O(z)),&(z) -- Db&(z) - Da,} (z)
a
= -w(a,b) ti9(z)2 +
1
E (a, b)
2
,4 `z
z
D
which is exactly what we wanted.
Proof of formula three:
From (2) we have:
(3b)
E (a,b) 2 [DaDb log 9(z)]9(z)2 = -W(a,b) E (a,b) 2
z
b
+-a
19
(z) 2
b
z
a
a
The idea now is:
Let a,b be in the same coordinate neighborhood,'t(a) = 0
t(b) = t
and expand(3b) in terms of t, and pick off the first non-trivial
term, which will be the t4 term!
-
3.229
(1) Locally we have:
wi =
(j=o vi3 t 1 dt
7
b
jwi = vi0t + 11
t2
2
t3
+ v.2 6 +...
a
b=
t3 11
i t(j-0 Vij
Da + t D'a
a
Jaz
x
+ 2 Da +---
1
E(a,b) = (t + clt3 + c2t5 +...)
-
dt F
OT
E (a,b)2 = (t2 + 2c1t4 + (2c2+c2) t6 +- .. ) dt
(o1
)
dt
Calculating from the definition, one easily checks:
w (a, b) _
(
- 2c1 + (6c1 -12c2) t2 +-
...
) dt (o) dt.
Hence
w(a,b)E(a,b)2 = 1 + (3c1 2 -- lOc2)t4 +......
(2)
Locally the term E(a,b)2 DaDb logi9(z) is
(t2 + 2ct4+- ..) (D2 log 14 (z) + t DaDalog 14 (z) +
2
t2
-
D2
log9(z) + t3
DaDa log &(Z)
+..
DaD. log i9 (z )
+ t4 [2c, Da log-9 (z) + 2 DaDa log 9 (z) l
+-.
)
3.230
(3)
Let
2
'9i
azaz.
1
az1'0 r -tlij
-& (z+6) --(z-S) -$(z)
16ii9i (z) + 3
+ 1
i 7 ij (z)
g
+ 61
i,jfk=1
g
g
6i 9 i (z) + 3
3=1
g
1r] rl
(z) 2 +
4
i,3=l
Then
66
LLLL
9(z) -
etc.
,
]
6i636i9
k ilk (z)+...I
ij
ij (z)
a i6.6kki.k(z) +---
6.6. (-8(z) -Si. (z) -
(z)cg. (z))
13
1
D
Sk6Qf12(z)'7ijki(z)
+
-
irj rkrlC=a
J
+
Now expand in t via 6i =
+.. .
Jw.(t)dt:
0
S(z)2+t J.
+t
ij
Y3
kr =1 v.o o kovko([2
J(vio31
v..+v.30Vil
2
-
ijkQ
(
irj=1
iov 2
6
3 i)3 jki + 4 aij
ilv'1
4
9
13
kQ)
vizv.o)
(,g
.
3
.
3.231
+t2(Vl
t4(112 D4&
a +
aD'log+
s
a - 3Da D3$
a
4
3
Now substitute what we have in (3b).
i <
(Da21og9 )_323
' 2+
(DaDalog t9)
+
Remarkably, the ti-terms for
cancel and the t4 terms give:
4
2.2c1 Dalog19 + 2DaDa log - (z)
2
9
= (10c2 Da!'
3'DagDad+
(Da2 log 5) _j2.
4
Use the following lemma:
Lemma.
D4
4
log f + 3[ D 2 log f l 2 -
2
2
Proof:
D
2 Df D3f + 3 (D2f) 2
f2
f2
2
f
Completely straightforward.
We have:
2c1 '
D2 log 'j +
DaD" log,4
6
(10c2-3c12) +
Da2
4
log-'
4
-
a )2
!(D2,
4
+ 12 Da log-
Da iog 9
+
2
QED
.
3.232
As in §2, these analytic identities have a geometric
interpretation in terms of the Kummer variety c(Jac(x)) c
Geometric
Corollaryy of (1) :
Then for all
a,b,c
Let 1201 define
Jac (X)
E X, then the ima es under c
IP(29`1)
a
]P(2g'"1)
of
c
1)
2)
the point
2
Jw
a
c
the infinitely near point
(2 f w) + s-Db
a
a+c
3)
the point
2
f w
2b
c
are collinear, i.e., there is a line in IP
c(1
O(Jac(x)) at
2 fw)
(2g-1)
a
,(2
1
0(Jac(X)) a
tangent.. to
..,.
_
_
along the direction
a+cf w)
Db
and meeting
also.
2b
Proof:
This is clearly the limiting form of the Geometric
Corollary in §2.
Dby) [3(+)
-
Alternatively, we can write (1) as
(z-y)
a
c
C
(zr+ 2 J).9(- z f) +
a
a+c
c2
(z+2
a
2b)
and
wi(b) = voidt
where
Db(y)
=
voi a/ayi
,
a+c
-2 f
2b
3.233
Applying the addition theorem and Q as in §2, this gives us
a+c
c
](y, 2)
=
cl9[°](z
)
cJw
a
a
+ c2.9 [° ](2
L,
J,
2)
where cl,c2 are independent of 1.
.
QED
c
When c approaches a,
approaches
J)
(O)
which is a
a
singular point of the Kummer Variety.
in
x(29-1)
at
(0) all pull back to even functions on Jac(X).
this situation, elements of
to
In fact, the local coordinates
In
define tangent vectors
Symm2(TJac(X),0)
4(Jac(X)) by the formula:
if (Aij) is a symmetric gxg matrix, let tA
vector at
cp(0) given by
=
to (f
be the
)
2
1
i,7
az9
ij
(0)
.
1
7
for all coordinate functions fa on IP (2g-1) near 4 (0) .
In particular, if
a,b
E X, then we get a tangent vector
t
a,b)
by
t(a,b) (f a) = D aD b (faoW0)
.
v.b) + v.a)
, where
(This is the case .A..
],0 v(b))
10
1,O ],O
1] = 12(v.a)
b) dtb).
a)dta, wi(b) = v(1,0
Wi(a) = v
, 1,0
Geometric Corollary of (2):
For all a,b E X, the point
a
the vector
the point
0( fw)
b
are collinear.
(0),
3.234
Proof:
To see that this is the correct limiting form of the
previous assertion, note that (2) can be rewritten:
Da(Y) DA(Y)
, (z+X) . 9 (z-y)
Y=O
b
=c15(z)2+c29(z+ J)i(z
b
- f).
a
a
Applying the addition formula and Q, we get
b
c2-&[on )(f ui. 201
](z,2)
z=0
where
a
cl,c2 are independent of
QED
n.
A different limiting case of (1) is when c approaches b rather
Analytically, the constants c1,c2 will approach -, but
than a.
geometrically the meaning is that (Jac(X)) will have a point of
b
inflection at 02 f w). This has been used very effectively by
a
Welters and Arbarello-de Conchini in their work on the Schottky
problem:
cf. Introduction.
Another interpretation of formula (1) shows how the Riemann
surface x is intertwined with the function theory of Jac(X).
a,c E X, let
For
Va,c be the vector space of second order theta functions
on Jac(X) spanned by the functions
c
c
19 (z + 2 J.i9z - 2 f )
a
c
a
c
c
+zf
- 2f
a
a
(z
1 < i < g.
az.
a
tg(z + 2 f ).
a
3.235
Lemma:
dim Va,c = g+l.
Proof:
D =
If not, then for some constant c and vector
there would be an identity:
I ai a/3zi
c
t
2 J)9(z - 2 f
(
a
c
c
c'
(z +
a
J).D(z - 2 f)
2 a
a
c
c
a
a
19(z-2f).D(z+zJY.
c
Let
Then
w = z
a
c
(w + J).D(w) = 0.
(w) = 0
a
c
Since
(w + f)
a
54
for almost all w such that -9 (w) = 0, this
0
means that -&(w) = 0
D 19(w) = 0
which we have seen never
QED
holds unless D = 0.
Using
Va,c and formula (1), we can recover X as follows:
locus of decomposable
Corollary:
V
a,c
f
(functions
(z+e)-9(z-e),eE1C
set of fungttions
l c+a
+2f )1z-2f
2b
b EX
cone over X.
2b
3.236
Proof:
This follows from identity (1).
n:
Suppose -9(z+e).-g(z-e) E Va,c. Note that if
c
+ 2
C
J)
_ S (z - z
J)
a
then all functions in Va,c vanish.
= 0
a
Therefore
c
c
(z + 2
(z - 2
_
J)
c
Substituting z + i
(z+e) = 0
= 0 ---
J)
or -9 (z-e)
= 0.
a
a
c
e -
for z and
J
for e, this says:
f
a
a
c
c
(z + J) _.(z) = 0
(z + e)
or 19 (z +
0
a
J
--
e) = 0.
a
b
c
We will show that if this holds, then e
.1.
b
or
e =
w
Jw
which,
a
substituting back, is what we want.
Our hypothesis can be written
0ftO(c1 =
(*)
0e U 0
a
where
Of
is
\
ifcaw-e)
(f w}
0 translated by f.
express this in terms of divisors.
Next, use Riemann's theorem to
To fix notation, let
define the divisor class D(z) by
z
J
W = z
D (z)
and let
6 be the divisor class of degree g--l such that
z EO
> ID(z)+61
0.
z E Mg
3.237
Let Dz = D (z) +6.
So, z E S n do
Dz
and I Dz + c - a l
0
a
Let W = set of divisors Dz such that
z E Ol ®c
.
Clearly
W
Ja
g-2
contains the subset
Wa = {divisors Do + a:
Do =
Qi}.
our
i=l
hypothesis (*) tells us that:
Dz E W' - either
0
or
ID
+ c -- a - D(e) I
z
0.
is an irreducible set, it must lie entirely in one of
Wa
Since
I Dz+D (e) l
these two sets:
g-2
D = DO+a E W
either ID0+a + D ( e )
l
34 0
V D0 =
Qi
i=l
2
or ID0+c - D()I
0
V D0 =
1
The following lemma then finishes this proof.
If
Lemma:
D(e)
is a divisor of degree zero such that for all
g-2
Do
ID0+a+D(e)I 3 0
then
D(e) . b-a
Proof.
for some
b E X.
Left to the reader.
So, we have used formula (1) to construct the cone over X,
and hence X.
We can ask whether we can also use formula (2) to
construct X.
As a possible approach, start out as above, and let
0.
3.238
V
0
be the vector space spanned by the functions
-3(z) 2,
As above:
9 (z)
az.az.
1
J
a-a
1
a\9-
az
1 < i < 7 < g.
7
a)
V0 c vector space of second order $.functions
b)
dim VO < l + 9(9+1).
decomposition f unctions
Consider v0 (1
(z+a).
)
Formula 5.2 tells us that this contains the set:
b
br
{$(z +
1)
,3(z - f)
a
some a,b E X}
a
which is isomorphic to a cone over Symm2 X.
Question 1.
Are these two spaces equal?
This would follow, as above, from the following question:
Question 2.
If D is a divisor class of degree 0 on X such that
for all divisors E of degree g-l for which JEl is a pencil, then
either JD+EI # 0
or ID-El
D n a-b for some a,b E X?
0, then does it follow that
3.239
§ 4.
A
lications to solutions of differential equations
The corollaries of Fay's trisecant identity can be used to
construct special solutions to many equations occurring in
Mathematical Physics.
In this section we will consider the
following equations:
utt - uxx = sin u.
1)
Sine-Gordan:
2)
Korteweg-de Vries (K- dV) :
3)
Kadomtsev-Petviashvili (K-P) :
u,
+ uxxx + u ux = 0.
uYy
+ (ut+u
+u-ux) x
= 0.
Many other equations also have solutions constructed via
theta functions, such as
4)
Non-linear Schr6dinger: iut = uxx +
5)
Massive Thirring model,
i ux = v(l + uv)
i vy = u(1 + vu),
but we will not consider these here (for the non-linear Schrtdinger
equation, see the PhD thesis of E. Previato, Harvard, 1983).
We will give some solutions in terms of 9 -functions to the
first three equations.
In the last section, we will indicate
how one uses the generalized Jacobian to relate our solutions to
the famous "soliton" solutions to the K-dV equation.
The easiest solutions to obtain are some for the K-P equation.
For all a E X, 12 D2 log9 (z0 + xv0+ti/3yv1 -2tv2)+2c1
satisfies k-P, where:
Corollary.
c1
is the constant appearing in formula (3),
§3.
vi = (vij,...,vg7)
wi =
Ivi7 t3 dt,
(t a local coordinate near a)
3.240
Proof.
Take
Da
of formula (3) and set u(z) = D2 logS(z)
to get:
D4 u(z)+ 12D2u(z)-u(z) + 12(Dau(W))2 + 2c1D2a u(z)
- 2DaD"
a u(z) + 3 (Da) 2 u(z) = 0.
Let
v = 12u+2c1 = 12 D2log.9 (z) + 2c1; then:
3 D'2v(z) + Da(D3v(z) + v(z)-Dav(z) - 2 Da v(z)) = 0.
Finally, note that by definition,
Dau(z) = ax u(z +
D'au(z) = ay u(z + y -vl) ,
I-)
Dau(z) _
u(z + t-v2).
Thus
v(z0 + xv0 +
y v1 - 2tv2)
solves K-P, as wanted.
QED
In order to find solutions to KdV
to consider hyperelliptic curves.
T:
X
Let
>IP1 the double cover, and let
and Sine-Gordan, we need
X
be hyperelliptic,
is X
;,X
be the
involution.
Let
a E X
be a branch point of "?T and let t be a local
coordinate about a such that the hyperelliptic involution i is
just"
i*wj = -wj (see Ch. (IIIa, I2.) so if
wj = Vi (t) dt, vi (t) dt + vj (-t) d (-t) = 0. Thus vi is an even
function of t, hence vjl = 0 and D'a = 0.
t +--- e -t.
But
3.241
satisfies KdV
12 Da loge (0 + xv0 + tv2)) + 2c1
Corollary.
where:
are as in the previous corollary,
cl, v0,v2
X hyperelliptic, and the local coordinate t
a E X,
and
i*t = -t-
a satisfies
Proof.
Take Da of formula (3), and use the above fact that
DI = 0 to get the result.
Recall from Cb.
Next we would like to tackle Sine-Gordan.
a
b
a,b are branch points,
i.e., if
fw E 2 L
are branch points, then
a,b E X
that if
lila
=
fw
(n+cm)
for some n,m E Zg,
a
a,b E X branch points.
Let X be hyperelliptic
To solve Sine-Gordan:
b
Start with Formula (2).
and subtract the
Substitute z ---.;- z + f
a
original formula:
,
b
b
a
a
(z+ f) (z+ f -- 2 f )
g(z+2f }S(z)
a
a
-= c2
DaDblog
b
b
b
'9 (Z+P
b 2
--(z)2
g(z+ f)
a
It
b
fW
a
=
z
(n+.SZm)
and get, using the functional equation for
b
(z+ $)
DaDb log
(z)
(Z)
=
c2 e
-Tri*
n
-2Tri mz
i9 (z)b2
e
-0 (z+ f
2'-
a
19(Z+I 2
-e
Ti
2Tri mz .
e
a
'9 (z)
3.242
b
9 (z+ f
Let u(z)
2i log
b
)
(z) - 21r tm(z + 2 J); then
a
iu(z)-e_iu(z)
DaDb u(z) _ -
c2
tm Sam+ r i tmn
= -4c2 e- Tri -
So u(z) satisfies DaDbu(z) = C;-sin u(z).
Thus for any
zo, the
function
v(x,t) = u(z0 + x(a2
+ t(a2b))
satisfies
a2
where
a,b
v (x, t)
32
--
a
v (x, t) = cz
.
sin v (x, t) ,
are proportional to (w1(a),...,wg(a)) and (w1(b),...,wg(b))
respectively.
We pass over the interesting question of when v and
c; are real and what these solutions "look like".
3.243
§5.
The Generalized Jacobian of a Singular Curve and its
Theta Function
In this section we will define and describe the generalized
Jacobian of the simplest singular curves:
by identifying 2g points of IP
l
the curves obtained
in pairs.
We will then determine
their theta functions and theta divisors.
Finally, we will apply
this theory to understand analytically and geometrically the limits
of the solutions to the T(dV equation that were discussed in the
previous section, when the hyperelliptic curve becomes singular of
the above form.
Let C be a singular curve of genus g, and let S = Sing(C).
Suppose the singularities of C are only nodes pl,...,p9 and that C
has normalization
7: IPl
>C.
If
7-1 (pi) = {bi,ci},
i = 1...g,
this means that c is just IP1 with the g pairs of points {birci}
identified.
-
We assume
bi "i Vi.
Now, in general we define
group of divisors D = E nixi,
Pic C =
xi E C-S
if D = (f) for
mod:
D "S 0
scam
f E T (C) ,
f continuous and
finite, nonzero at each p.
In our case we can pull back to fl 1 and we get
group of divisors b = E nixi, xi E IPl -
Pic C
mod:
and
D
0
if
f(bi) = f(ci)
D = (f),
for all
f
7r-1(S)
E C(IP
i = 1...g
We define Jac(C) to be the piece Pic°(C) of Pic(C) corresponding
to divisors
n x
i i
of degree 0, i.e.,
Eni = 0.
The structure of
3.244
Jac(C) is easy to work out n
start with D of degree 0.
As a
divisor on IP1, it equals the divisor of zeroes and poles of some
rational function f.
The ratios
f(bi)/f(ci) represent the
obstruction to,D being zero in Pic(C).
It is easy to verify that
they set up an isomorphism of groups:
Jac(C)
>
(
*)g
f(b1) ,...,
> (f(c1)
DI
As in chapter IIla,
f(b )
f(cg /
we can add to any divisor D the divisor
x-x0 and get a family of divisors D+x-x0 depending on a point x near
x0.
Letting x approach x0 this gives a tangent vector to Jac C
near D, and as D varies, an invariant vector field Dx
on Jac C.
0
For later use we can work out this vector field in terms of
coordinates X1, ...
on (T*) q:
If D = (f (t)) , then D+x-x0 = (f (t) t_x ) ; hence the
0
coordinates of D+x-x0 in Jac C are
b.-x
f(bi).
f (ci) -
b i-x
0
c.-x
ci-x0
Then
f (bi)
ci-x0
T . brx0 -
?x0 -
b.-c.
=
x.
Y
=0
bi_i
(c
x)
x0 (c1- 0
(b-)
3.245
Thus the vector field Dxo is given by
Dxo
(bi_x0)ci_x0)
`i aai
c
it
Now, Jac C is not compact:
compactification of it.
however!
-
we want to construct a natural
This will no longer be a group
N.B.
It is clear what we need to do to compactify:
we need
to allow the support of our divisors to approach the singular
points.
But considering divisors
not work very well.
Enixi, arbitrary xi E C
does
We need to encode the "multiplicity" of the
singular point in a more subtle way.
This is done as follows.
In general let
set of coherent
Pic C
oC-module
=
up to isomorphism
Translating this to more down-to-earth language, this becomes
k
set of all divisors D =
n.x.
i=1I i
dC,xi modules Mx. c C(C),
if
xi
along with finitely generated
(xi E
C)
are arbitrary, where
is not singular, MX is simply t ni-
x
to
local coordinate near x
and if xi is singular, ni is determined via:
ni = dim
M
r ti---inoC'x.
- dim
(,C,x1
Mx MC,x.
By convention, b& = dC,x if x t {xl;...,xk}.
mod:
D N D" if 3 f
E
(C) such that x = f 1,
t1x E C
3.246
The module, Mx
can be thought of as a refined way of measuring the
Y
multiplicity ni at the singular points:
multiplicity modules.
Pic C
we will call them the
always has a natural structure of
projective variety but let's just think of it as a set.
In our case of g nodes, we know exactly what the Mx 's must
look like:
Lemma.
If
is an ordinary
Y
double point obtained by
p E C
glueing two points b,c in a smooth curve C, then for all
which are finitely generated
M c T(C)
OP ,C-modules, either:
a)
M= f -@p,C
for some
f E VC)
b)
M = f- r(j'p C
for some
f E VC), where
or
dP C = Ob,C.rnOc,(5,
normalization of (,P,CProof.
Mk'k = iciodule of functions f such that
Let
ordaf a k, ordbf >
M c Mk,Z
and
£,.
Take k,i
the largest integers so that
Then almost all functions
ordbf = Q.
So choose such an
f E M
f E M.
satisfy
ordaf = k
We have
f-(DP,C c M c Mk,R
But now Mk,i = f-M0,0 and M0,0 is just
is the subspace of
codimension 1 in
OP,C.
Moreover,
0P ,C
@rP,C defined as {gJg(b)=g(a)} so it has
0P,C.
Therefore
dim Mkrk/f (0P,C = dim f -UP,C/f qP
C
So either M = f-dP,C or M = Mk ,Z = f-P,C,
= dim P,C/OP,C = l.
as wanted.
QED
3.247
From this lemma, we get immediately:
T c {P1,...rP9}, let
For any subset
Corollary.
CT = [C with Pi separated into bi and i1 for i E T3 = CIP1
identified for
i
Pic C =
with b.,ci
Then, as a set:
T3.
fl Pic (CT)
(f = disjoint union)
T
Proof:
In fact, divide up all divisors D =
{Znixi,Mi}
according to whether their multiplicity modules are isomorphic to
(OPTIC
or
=
&Pi01C
For each subset
T
(Dbi,tP1
c {P1,. .
D whose multplicity module is
at each singular point.
ddiIIpI
fl
(T)
Pg,}, let
Pic(C)
bP C
exactly for
be the set of
Pi E T.
We
claim:
Pic (C) (T)
In fact, if
Pic (CT) .
D E Pic(C)(T), then when Pi
exists an f. such that PIP
a
T, Pi singular, there
= fide C It's not hard to see that
a.'
one can choose a single rational function
f
such that this holds for all such P
Let D' be defined by the multiplicity modules
a divisor on CT with "trivial" multiplicity
f-1. MP.
(np
i
singularities of CT.
Two such are equivalent in
only if they are equivalent in Pic(C)
,C
It defines
at all the
Pic((:) if and
because the condition
f (bi) = f(c) in the definition of equality in Pic (CT) is the same
as the condition
Q
equality in Pic(C).
C = (0P
C
included in the definition of
3.248
Actually, we can be much more explicit, and make the degree 0
component Jac C into a compact analytic space as follows:
(IPl) g/....
Theorem Jac(C)
(Wk
.Xl...., ...,C g g)
kth
with equivalence relation
.)
n
,.,
"0'... 1, g),
(
spot
for all k.
kth spot
(bib .) (ci-c . )
where
ij = (b-c 'i ) (ci-b.)
Sketch of proof:
.
Fix some n > g and let
S = nordered sets (xl, ... , n) : xi E IPl ; for each i, a at most one j s.t. xjE {bi,c.
Define two maps
by
n
7'1(xl,,..'xn)
nl
(IPA') g
Jac (C)
bl^xi
n
cl-xi
i=1
`
1T2(xlI...,xn)
_ (the divisor
b -x.
1j \ ,
cg
y
the multiplicity module is
and
where if xi = bj or c.,
(the maximal ideal of functions zero
i
at pi).
The following things are not hard to prove:
a)
if n is sufficiently large, e.g., 2g, then
7, . 72 are
surjective
b)
Tr2
is constant on the fibres of 7l so that there is a
unique map
gyp:
(IPl) g
>Jac (C)
satisfying
cp°'Tl = 7T2'
3.249
c)
(A is independent of n and defines an isomorphism of
(IP l) g/- with Jac (C) .
d)
(A
restricted to (C*)g is the isomorphism
(C*) g
-->Jac (C)
defined above.
e)
h = g-#T
and
T{1,---,g} is any subset,
if
More generally,
E:
T. --a{O,w}
any function; then p
restricted to
H {s(i)} x
iET
c (mil) g
H C*
iET
is the same isomorphism of (C*)h
_ ..Jac(CT)
up to multiplication by a constant in
(C*)h
The idea of the crucial step b is this:
Say
1(yl,...,yn), and
Tr1(xl,---,xn) = Tr
xi,yi E ]Pl - U {bk,ck}. Let
k
fi(t)
11
l<i<n
(t-y)
(t-x.)/
1
1 l<i<n
JH
yip
xi
then the hypothesis says that
f (bk) = f (ck) ,
all k
hence
(ixi - n-°°)
^'
(Eyi -
n-)
in
Pic (C) .
Jac(C)
3.250
In Step C, the w1 s come in because for .any x2 , - - - ,xn S IlP k{},
we have
_ IT 2(ck,x2' 2'..." x
v2(bk,x2,..-,xn)
b.
bk
n
and
n xibl
n xi-b
bk-b
.
\ bk-cl i=2
xi g
ratio
Wkl
The details of the proof are not central to the exposition and
are omitted.
Several points in this proof are useful below.
Firstly, note
that Jac C has one "most singular" point at infinity, namely the
point corresponding to (X1,...,X
We will call this P...
an analog of
dim Trl
Let
)
where all
Secondly, the map
0 for Jac C.
ki are either 0 or =.
Trl enables us to construct
To do this, let's calculate
Xg)
Tr1(xl,...,xg) = (al,...,ag).
constant, let
is omitted.
cp(t) = c
So deg ((p)
Up to an undetermined
TI(t-xi), where if
< g.
xi = =
Write (P (t) _ IcDit
that term
he
i=0
depend on xi,...,xg and determine
permutation.
Now,
uniquely up to
(pi
3.251
for
Ak
So
g
i =0
i
7rll(l1,-.-,1g)
k = l...g
b') = 0
for
.
k = 1...g.
is given by the set of solutions in
of these
equations so
dim Trll (al, - - . , ag) = g - rank (Akci - b )
i=0...g
k=1...g
In particular,
iT
is generically 1-1.
1
Next, let us determine the analog of the theta divisor 0
We want equations for the locus where the
using the above.
divisor
is effective.
iIl xi - W
From the discussion above,
this is exactly when deg 0 < g-l, i.e., g = 0.
there is such a
point
.. r
1-A 1
.
Over a given
if and only if:
1-A
. bg- gcg
b1-A1cl
= 0
det
bg-1 1
cg-1
11
.
.
g
cg-1
gg
This determinant is the analog of -9 and its zeroes, as a subset
of (]P1) g/- or via (P as a subset of Pic C) , are the analog of 0.
3.252
We shall call this function
TC has a useful expansion.
First recall the Vandermonde determinant
... 1
1
a1
a2
a2
a2
a
g
det
1
2
...
a2
g
II
(ai-a.)
i>j
g-l
a2g-l
a1
g-1)
a3 J
In'the above determinant, this enables us to work out the
coefficient of the
IT
i.>j
(c.-c.)
i,jES
where
1
3
H A. term:
iES 1
H (b-b.)
3
i>j
i,j S
(b.-c.)
. 7
II
icy
1
(-1)#S
-
a(S)
jES
a(S) = the sign of the permutation changing 1..g to
(S,{l..g)-S)
and preserving the order of each set (e.g., a({l,31)=-1).
TC, therefore, can be expanded:
(*)
TC =
IT
i<j
(b.-b.)
(-l)#S
1
c.-b
If
jES
(A.
fl
jai
-Z
bi -j
ij
]I
w
0.
i<j
i,jES
Note that the worst boundary point, p.0 =
correspondingly, det(0,...,0)
is not on
0, and
0.
I claim that this determinant is also a limit of theta
functions of our non-singular curves C.
link as follows:
Formally, we can see a
3.253
Let
0ij(t) be a family of period matrices in which
Tm Hii (t) -- > oG
as
t ----> 0,
1<
i < g,
and
Stij(t)
are continuous for It l< E, if
i
j.
Then consider .9(z, Q (t)) . The limit of this function as t -- >0
will be just 1.
A better thing to do is to translate the functions
by a vector depending on t first:
Let 80(t) = diagonal of H(t); then
(z - 6, (t) ,
eTri tmSkn+2iri m (z - 62 (t)
St (t) ) =
Y
II eTFl (mini)
mEXg i=l
03- ai(t)
IT
i<j
As t ->0, this function approaches:
II
m =(ml " ... ,mg)
e2Tri.mizijt2ij (0) e2Tri mz
i<j
m.=0
or 1
1
e2h1 ij (0)
I
IT
S={l..g} i<j
-
II
-
e27rizi
iES
i,jES
Now if
2Tr i S2
'J (0) =
(bi-b .) (ci-c . )
(bi--c (ci-bj
=
Wij
)
e27Ti m.mj"ij (t) .e2lra t
3.254
and
e2Trizi = _ 1
i
-
cl
IT
J#i
bi-b:
it equals TC up to a constant. In fact, if Ct is a family of smooth
curves of genus g "degenerating" to C, it can be shown that its
period matrix behaves exactly like this.
lattice
Correspondingly, in the
Lq (t) =229 + c2 (t)2Lg, the B-periods
infinity, but the A-periods
(Cg/7Lg+S2 (t)Zg
coordinates
e27'1zi,
S2(t) zg
2Zg remain finite.
go to
Thus
tends to Cg/Zg, which is just (C*) g with
We do not want to describe this in detail,
referring the reader to Fay, op. cit.,
Ch. 3.
In the limit, is there anything left of the quasi-periodicity
of -& with respect to its B-periods?
but there is, in fact, something.
At first it would seem not
In fact, the three methods by
which we formed from -9 meromorphic functions on Xn now give us
rational functions on the compactification Jac CC
which are
continuous maps
Jac C - (codim 2 set of indeterminacy) ---> ]Pl
The point is that the induced rational maps
(3P l)g
- (codim. 2 set) ----> IP l
are compatible with the equivalence relation - of the above theorem.
Let's check this for the second logarithmic derivative with
respect to the invariant vector fields
xi aa.
Xi a/aai, Xj a/aai, i.e.,
- j aa. (log TC(Xl,...,Xg)).
3
3.255
Note that this is the analog of
r
II
ap
of-b]
jai
1
lim
lien
l k -}W
k
.
aj
Let
a
J
1
irjr then
7
ai
Ak-roo
=
If
aa.1
a, -39X
J
a
a.
.
1
ax [log
]
a' + log S (-1)
-#S 1
iES
ilk
1rjES
=
ai aa. A. a (1og
1
#S
(-1)
i<j
kES
-a
a
[log
j
S with
#S+1
lim Xi aA
-
fl
i<j
i,jES
kE S
ak+0
iES-k
i,jES-k
(-l)
w
wi.
IT
S with
j
ij
w
11
iEs
1-l
ak
it
a'.]
iES-k
xi)
wgkAg)]
aj aa.[log
singular limit of hyperelliptic curves.
bk = -ck for all k.
coordinate on P1,
`
a.
Now let's apply this to give solutions of KdV.
be a
if kEs
wi. - IIa-j(1
i<j
We want C to
This occurs if
In fact, when this. is satisfied, if t is the
let
x = t2
y=t
]T
i=l
(t2-b. )
.
if ks
l
3.256
Then
x(bk) = x(ck), y(bk) = y(ck), and the 2 functions
the singular curve
c-{-}
x,y embed
The image is defined by
into a2.
which is a limit of equations y2 = f2g+l (x)
for smooth hyperelliptic curves of genus g.
y2 = x
-
II (x--b.) 2,
Recall from the beginning of this section that the invariant
vector field on Jac C associated to a point x E C
0
bi - ci
a
ci-x0
bi--x0
is
bk = -ck, then
If
In
D
x0
- 2-
i
. -.r
-2
-I-
0
i 1--b? x0-2 i
bia
= 2- (x-2
aai
biai
+...).
+ x4
-
a
The vector field associated to the point at infinity is therefore:
D.
a.
and the singular
2
i
biai aa.
r
p-function is:
TC(al....,Ag) =
(
biai
,.) 21og
SCl..g
(-1)
#S
bi-b
i,7ES
To obtain a solution to KdV, we need merely substitute
(ei+bix-2b1 t)
Xi
e
2
b +bi
II (xi - II b b
II b b)3 - iES
jai
it j
3.257
b.+b.
for any el,...,eg; or absorbing the factor -
in the
n
j74i
2log
f(x,t)
_
(ax)
]
bi b. 2
(-L-1)
IT
i<7
)
ei,
1
(ei+bix-2b3 t)
e
iES
i,jES
These are precisely the g-soliton solutions of KdV.
The famous asymptotic properties of g-solitons (that for
t
<< 0, it splits up into g widely separated blobs, which interact
for moderate values of t, and which for
t >>O
split up again into
the same g blobs, with the same shape but with a phase shift) can
all be deduced very simply from the above formula and the fact that
D2 log
TC
extends to a continuous function on the compactification
Jac C described above of the generalized jacobian.
To get a real-
valued function f(x,t), assume that all bi are real, and define
c:
2
Jac C c Jac C
((C*) g
by
0(x,t)
(...
_
e +b x-2b 3t
i
b.-b.
]T
jai b3-+
e
Then
f (x,t) = (D2.log TC) (a (x,t)) .
As shown above, D. log TC extends to a continuous function on
Jac C.
In fact, it is zero at the "most singular" point P
given by letting all coordinates
To see this, write
ai on .((E*)g
tend to 0 or
3.258
aSXS
S-{i,",g}
Then if
I b.
b SS
iES
aS-(EaSbS?S)2
D2
logg TC (EaSaS)2
Note that all terms
X2S
in the numerator cancel out while for
every S, the denominator has a
a2S term since aS
0.
Thus
D2 log TC(P0) = 0.
Therefore, for all e > 0, there is a neighborhood UE
of
P.
in Jac C such that:
P E UE:
ID22 log TC(P)I < E.
Therefore, there is a constant c such that if
x-2bit I> C,
all 1< i < g
c (x, t) E UE
If(x,t)I <E.
Thus the effective support of f(x,t) is a set of g bands
x-2bi t
I
<C
representing "blobs" moving with distinct positive velocities 2bi.
3.259
Moreover, if
t ---a +oo
and we stay in the loth band, then
x--2bi t I
----a Co ,
i i
and lim a(x,t) will lie on the is h 1-dimensional strata
Ji
=
0
{(A1,...rX
g
E T*,
)Iai
but
ai C- {0,-}
if
i 34 i0}.
0
and let t - >+- . Then for
In fact, fix the value z.= x-2b?
10 t
(i # i0) ,
some choice of e iE {0,00},
3.260
(elf-.rei
lim a(x,t) =
t-r-OO
0
-l'
Ai
,
0
Ei +l1 ...rE9)
0
lim Cr (x, t) = (e 11 r ... rEil' 1' Ai r Ei1+1' ... feg
0
0
0
t->+co
where
By the theorem describing how
depends only on 2Z.
Ai
0
(]P'-)g
is "glued" together to produce Jac C, we see that for some
constant fl.
10
(E1r...,Ei 0_1, A, Ei
+11...re9)
0
n .
(Ellr...,Eil-l,
e i01,
0
ei1+1r..,re;1), all A
0
(- meaning equality in Jac C).
Therefore,
lim
f (x, t) =
t+-00
lim
t
f (x, t)
i°
x-2bt-,7-+
(r1 i °/i ° )
x-2b . ° t= x.
i.e., for t ->- or t -- -, f(x,t) has the same shape in each band except
for a phase shift.
The fact that this shape is a single "wave"
moreover is more or less a consequence of the simple fact that on
each 1-dimensional stratum Ji
,
the rational function
TC(A) tends
0
asymptotically to
(l+A)AS
i.e., up to the scale factor
(in a suitable coordinate
A E (C*),
AS, has a single negative zero.
When
you set A = ebx and take logarithmic derivatives, f will have a
single pair of complex conjugate poles closest to the real axis and
these give its wave'shape.
Jac(C)
give
More generally, the zeroes of
TC on
poles of f(x,t) but only for complex values of x,t
and f will have large values along the real points near these poles.
3.261
Resolution of algebraic equations
by theta constants
Hiroshi UMEMURA
The history of algebraic equations is very long.
The
necessity and the trial of solving algebraic equations existed
already in the ancient civilizations.
The Babylonians solved
equations of degree 2 around 2000 B.C. as well as the Indians
and the Chinese.
In the 16th century, the Italians discovered
the resolutions of the equations of degree 3 and 4 by radicals
known as Cardano's formula and Ferrari's formula.
1826, Abel [1]
However in
(independently about the same epoch Galois [71)
proved the impossibility of solving general equations of degree
5 by radicals.
This is one of the most remarkable event in
the history of algebraic equations.
Was there' nothing to do in
this branch of mathematics after the work of Abel and Galois?
Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can
solve the algebraic equation of degree 5'by using an elliptic
modular function.
Since
= exp((l/n)log a)
which is also
a
written as
exp((l/n)
(1/x)dx), to allow only the extractions
1
of radicals is to use only the exponential.
Hence under this
restriction, as we learn in the Galois theory, we can construct
only compositions of cyclic extensions, namely solvable extentions.
The idea of Hermite and Kronecker is as follows; if we
use another transcendental function than the exponential, we can
solve the algebraic equation of degree 5.
In fact their result
a
is,analogous to the formula
P-a = exp(l/n)
(1/x)dx).
1
In the
3.262
quintic equation they replace the exponential by an elliptic
modular function and the integral
integrals.
r (1/x)dx
by elliptic
Kronecker [15] thought the resolution of the equa-
tion of degree 5 by an elliptic modular function would be a
special case of a more general theorem which might exist.
Kronecker's idea was realized in few cases by Klein [11],
[13].
Jordan [10] showed that we can solve any algebraic equation of
higher degree by modular functions.
by Thomae's formula, §8 Chap.
III
Jordan's idea is clarified
(cf. Lindemann [16]).
In this
appendix, we show how we can deduce from Thomae's formula the
resolution of algebraic equations by a Siegel modular function
which is explicitely expressed by theta constants (Theorem 2).
Therefore Kronecker's idea is completely realized.
Our resolu-
tion of higher algebraic equations is also similar to the
a
formula
exp((1/n) 5
(1/x)dx). In our resolution the expo1
nential is replaced by the Siegel modular function and the
integral
` (1/x)dx
is replaced by hyperelliptic integrals.
The existance of such resolution shows that the theta function
is useful not only for non-linear differential equations but
also for algebraic equations.
Let us fix some notations.
covention of Chap. II.
2g+l
Let
F(X)
be a polynomial of odd degree
with coefficients in the complex number field
assume that the equation
that
Y2 = F(X)
Then
C
of
We follow in principle the
F(X) = 0
defines a hyperelliptic curve
and at
-.
Let
We
has only simple roots so
is a two sheeted covering of
F(X) = 0
C.
JP1
C
of genus
g.
ramified at the roots
be the roots
3.263
of
Let us set
F(X) = 0.
S, T
B' =
For two subsets
B', we put. SeT = S 0 T
of
n2i__1
S n T.
is defined
ithplace
2
and
as the 2xg matrix
Ti
2i is the 2xg matrix
ith place
(0...0
0...0
T c B, the sum
For all
by
Classically the period matrix
q T.
with respect to the normalized basis of
=.
Thus
of
S2
is calculated
C
in §5, Chap.
H1(C, 7Z)
is determined when we fix not only
Q
the order of its roots.
the subset of
but also
F(x)
U =
consisting of all the odd numbers of
B'
For row vectors
Finally we put
is denoted
nk
E
m1,m2 E
]Rg,
z
and a symmetric
c Cg
B'.
gxg matrix
with positive definite imaginary part, we define the theta
T
function 0[ml]
(Z' -U) =
m2
(+mi)t(Z+m2))
E
e2J g
m
where
e(x) = exp(27ix).
m
0[m](T)
be denoted by
The theta constant
0[m1](0,T)
will
2
for short.
2
Theorem 1.
The following equality holds;
x1-x3
x1-x2
(1
_ (e2
[
0
0
7
...
::i
0-p
(Q)4e lp 2 ....
0
0...0J
p 1
(Q)
40[
2
'2 0 ...
0, (a2)4
0J (0)4 + e
( Q ) 4) /
01
1 (Q) 4e 2 2
l0 ... 00 .
(20(2
0](0)4).
3.264
The theorem is deduced from Theorem 8.1, §8, Chap. III, by
carrying out precisely the calculation indicated in the proof
t
of Corollary 8.13 and form the formula 0[m1 + 1](z,T) = e(m1E2)
2
m
2
e[m1](z,T)
for
c
Z39
(see for example Igusa [9], Chap. I
2
§10,
with
(0,2) p-49).
In fact for a division
#V1 = #V2 = g, it follows from Theorem 8.1, §8, Chap. III,
(1.1) 6 [n (V +k) oU] (St) 4 = c (-1)
#(U-(V 2+k) )
IT
itN2+k, j ev1
2
because
by
(V2 + koU)oU = V2+k
V2+k.
(1.2)
B = V1 U V2 u {k}
Theorem 8.1,
e [n (V +k) oU]
(S2)
4
here the union
98, Chap. III for
= C (-l)
V2L{k}
is denoted
S = (V1+k)oU
#(U-(V1+k))
E
icv1+k,jcV2
1
(xi-x,)
gives
(xi - xj ) -l
Dividing (1.1) by (1.2), we get
0 In (v2+k) oU] (n)
e In (V1+k) oU] (0)
4
4
II
- (-1)
(xi-xm)
#(U-(V2+k))+#(U-V1+k)) iEVI+k,mcv2
JI
(xi-xj)
ieV2+k,jcV1
(xk-xi)
IT
iEV
2
TI
icV1
(x k- x i)
Let us consider a division
Putting
B' ={1,2,3}u{2nl2snsg}i.{2n+112sn5g}.
V3 = {2nj25nsg}, V4 = {2n+112sn5g}, we apply (1.3) for
k=l, V1=V3+2, V2= V4+3;
0 Ln
(V4+3+1)
oU] (S2)
e Ln (V3+2+1) oU] (S2)
4
IT
iEV4+3
II
icV3+2
(xx.)
1- 1
(x1 -x.1
3.265
Next (1. 3) for k =1, V1 =V 4 +2, V2 = V3+3 is
8 Cn
(1.5)
(V3+3+1) oUl (SZ)
e [n (V4+2+1) oUl (S2)
4
11
ieV3+3
4
=
iEV4+2
(x 1 -x i )
(x1 -x.)'
a.
Multiplying (1.4) with (1.5), we get
e [ n (V4+3+1) oUl (52) 46 [ n
+3+1)
1
(xl-x3) 2
(2)
(2)4
(x1-x2) 2'
] (Q)4
(V
For the above division
B' = {1,2,3},u. {2nj2!5n5g}u {2n+1j2snsg}
if we interchange 1 and 2, then (1.6) becomes
e [n (V4+3+2) oUl (SZ) 4e [n (V3+3+2) QUl (S2) 4
(1.7)
e n (V3+1+2) oU]
S
)
n (V4+1+2) oUl S)
4e
4
(x2-x3) 2
(x2--xl) 2
We notice the following identity,
x-x3
1
(1.8)
1{1 +
x1-x2 - 2
(
x 1 -x 3
x1-x2
) 2 -
x 2 -x 3)2}.
x2-x1
It follows from (1.6),(1.7) and (1.8)
(1.9) xl-x2 =
(6 In (V3+2+1) oUl (52)
+ e [n (V4+3+1) oU] (SZ)
46
4e
[n
(V4+2+1) oU] (S2)
4
[n (V3+3+1) oU] (S2) 4
- e [n (V4+3+2) oU] (SZ) 4e [n (V3+3+2) oU] (S2) 4)/
(V
The theta characteristics in (1.9) are half integral.
now follows from the following formula: for
in
Theorem
Mg,,
3.266
®[m 21 +11
2
(z,,r) = e(m1 2)e [m11 (z,T)
2
2
We notice that by the transformation formula, the right
hand side of the equality in Theorem 1 is a Siegel modular function of level 2
(see Igusa [9), Chap. 5 §1, Corollary).
A marvellous application of Theorem 1 is the resolution of
the algebraic equation by a Siegel modular function.
Theorem 2.
Let
(2.1)
a0
0, ai E T (Osisn)
be an algebraic equation irreducible over a certain subfield of
T, then a root of the algebraic equation (2.1) is given by
0:::i
.((1
(2.2)
(e [2
1 O...p
0)4010
0
.....
(Q)4)/(20{:
0
with F (X) = X (X - 1) (a0Xn +a 1 X n-l +...+ an)
F (X) = X (X - 1) (X - 2) (a0Xn +
More precisely let
S2
alxn-"l
+. -+ ate)
for
for
n
is the period matrix of the hyperelliptic curve
when the roots of
C
:
X2
odd
n
even.
be the roots of equation (2.1).
with respect to the classical normalized basis of
odd
1
is the period matrix of a hyperelliptic curve
= F (X)
Then
(2)4
(SZ)4eiT 2
0
0
2
and
0...0
0
+0 OJ(S2)46 2.... 0]()4
0
(0461
where
0.011.
01(S'Z)4
F(X) = 0
are ordered as"f"ollows
xl = 0, x2 = 1, xi+2 = ai (lsisn)
x2 = 1, xi+2 = ai (lsisn), xn+3 = 2.
(2.1) is given by (2..2).
and for
The root
n
a1
C
x1(C, 7Z)
:
for
even
n
xl
="01,
of equation
3.267
Proof.
It follows from the assumption that the equation
is irreducible over a subfield of
roots. Since
C, F(X) = 0
has only simple
(x1 -x3)/(x1 - x2) = x3 = al , Theorem 2 follows
from Theorem 1.
To determine the period
the algebraic equation.
S2
we have to number the roots of
Even if we don't know the precise roots
of the equation, the numbering can be done once we can separate
the roots of the algebraic equation.
The complex Sturm theorem
says that there exists an algorithm of separating the roots of
the algebraic equation (Weber [19], 1 §103, §104).
Therefore
Theorem 2 is a resolution of an algebraic. equation by a Siegel
modular function.
Compared with the formula
= exp((l/n)log a)
a
= exp((1/n):
(l/x)dx), in our theorem the exponential is re1.
placed by the Siegel modular function (2.2) and the integral
Sa1
(1/x)dx
05isg-1
is replaced by hyperelliptic integrals
which determine the period
Q.
Let u:s compare our Theorem with the result due Hermite [8],
Kronecker [15]. and Klein [12] on the resolution of the quintic
algebraic equation by an elliptic modular function.
Their
theory s,ti.cks to the modular variety of elliptic curves with
level five structure (cf. Chap. 1).
plane and
Fn
c
0, a
be the upper half
d = 1 mod n}.
in,usual way and the quotient variety
variety of elliptic curves with level
tion-field
H
be the principal congruence subgroup of level
{( a a).ESL2(2)Ib
H'
Let
(G (H/rn)
n
has, a model (Q (H/rn)
H/rn
rn
operates on
is the modular
structure.
over
n
W
The func-
and the
3.268
morphism
(see Deligne et Rapaport (41).
H/rn + H/F1
tion
descends giving an inclusion
Tr: H/ Fn -r H/ I'1
'-, Q(H/rn)
Therefore
H/I'5
The natural projec-
is a Galois covering with group
H/I'1
I'1/±In'
is a Galois covering with group
-r H/I'1
which is isomorphic to.the alternating group
Since
of degree 5.
ot5
The key point
is this family contains any Galois extension with group
To be more precise,c since
z5
in
has a subgroup of index 5,
°t5
there exists an extension (resolvent)
[F, Q(H/r1)1 = 5.
Q[H/r11
is a one para-
Q[j(w)], Q(H/I'5)/Q(H/I'1)
meter family of Galois.extensions with group 615.
with
r1/±r5
is a rational curve and its coordinate ring
is a polynomial ring
T.
Q (H/ T'1)
Q(H/I'5)
F
Q(H/I'1)
Moreover one can show among such re
solvents there is a particular one described explicitely by
using the Dedekind
degree 5 of
(2.3)
n
function
There exists a resolvent of
:
given by an equation
Q(H/I'5)/Q(H/F1)
w5 +b 1 w 4 +b 2 w 3 +b 3 w 3 +b 4 w + b5 = j (.w) ,
and the solutions wi(w)
of equation (2.3) are expli-
(15is5)
citely written by the Dedekind
function.
r1
quintic equation over a subfield
(2.4)
X5 + a
1
X
4
3
+ a2 X + a
3
X
2
bi c Q (lsis5)
k
of
Now given a general
T
+ a X + a5 = 0,
4
ai E k, (lsis5) .
Then it is easy to see that by a Tschirnhausen transformation
involving only the extractions of square and cube roots, the
resolution of the given equation (2.4) is reduced to that of
(2.5)
X5 + b
1
X
4
+b 2 X 3 + b 3 X 2 +b 4 X+ a5 = 0
where a is in a solvable extension of
k(ai)1<i,,5
by adjunction of square and cube roots (Weber [191,
obtained
1
§60, §80,
3.269
981) .
Next we look for a point
w0 c
I-i
such that a; = b5 - j (w0) .
This procedure depends on elliptic integrals.
elliptic curve
of
C
C : y2 = 4x3 - g2x - g3
is equal to
=
26.32.g3 /(g3
2
2
tractions of a square or cube root.
the elliptic curve
C
tic integrals
1/
4x
a,b.
3ax-b dx
Therefore
This is done by exw0
of
is calculated by ellip-
for suitable paths
(1s i s 5)
wi (w0)
j
T b5 - a;
Then the period
y2 = 4x3 - ax -b
Y
,7
j (w0) = b5 - a5.
:
the modular invariant
- 27g3We solve in
for unknowns
-27b 2)
Recall for an
y
and
are the solutions
of the equation (2.5) hence the given equation (2.4) is solved.
If we try to solve a quintic equation by Theorem 2, it is simpler
than the above mentioned classical method because in our theory
the Tschirnhausen transformation is not involved.
But we need
a modular function of genus 3.
Remark 3.
(3.1)
Let
f(X) = a X n +
a1Xn-l
an = 0
0
a0 34
0, ai c
CC
be a general algebraic equation of even degree n= 2g+2
over a subfield
"general".
multiplying
k
of
(X - 1 )
4
We do not want to clarify the word
C.
Then considering
X,
(0si5n)
or
f(X)
(X - 2) ,
itself as
F(X)
instead of
we can show that for f(X) =
F(X), the values of the modular function in Theorem 1 for all
the orders of the roots of
tension of (3.1) over
theorem is clear.
..,x2g+2)), C
k.
Let X
F(X) = 0, generate the Galois exin this form, the back ground of our
g(2)
be the moduli space of
a hyperelliptic curve of genus
g
and
(C,(xl,x2,
(xl,x2,
,x2g+2), the (ordered) set of the Weierstrass points as in §8
3.270
Chap. in.
operates on Age)
The symmetric group G2g+2
permutations of the Weierstrass points.
2.4 and §6, Proposition 6.1,
modular variety
TTC
§2, Lemma
is a subvariety of the
k(2)
of the principally polarized abelian varie-
M2
ties of dimension
By Chap.
as
with level 2 structure.
g
Let
be the
M1
modular variety of the principally polarized abelian varieties
of dimension
Then there is a canonical morphism
g.
Sp29 (2Z/22Z).
-} M1
This morphism is a Galois
of forgetting the level 2 structure.
covering with group
DI2
The Galois group of (3.1)
which is a subgroup of Cy2g+2, interchanges the Weierstrass
points of the hyperelliptic curve
tion of r2g+2
C
:
X2 = F(X).
This opera-
on the Weierstrass points induces a faithful
representation 8;2g+2 -> Sp (J (C) 2) = Sp2g (2Z/22Z)
by Chap. M §6,
Therefore the equation (3.1) is solved in a
Proposition 6.3.
specialization of the Galois covering
M2 -
M1.
The specializa-
tion involves the modular function in Theorem 1 and the hyperelliptic integrals.
Finally we notice that Theorem 2 is similar to
Remark 4.
Jacobi's formula
1/k
dx/
\J
(/,_x2)
:
(1-k
Setting
2x2
)
and
K = Sdx//l_x2) (l-k2x2), iK' =
w = iK'/K, we have
k= 610(0,w)/
1
e00(0,w).
= 0
Jacobi's formula solves a quadratic equation
by theta constants and elliptic integrals.
1-k2x2
3.271
Bibliography
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Coavriahted Material
David Mumford
Tata Lectures on Theta 11
The second in a series of three volumes surveying the theory
of theta functions, this volume gives emphasis to the special
properties of the theta functions associated with compact
Riemann surfaces and how they lead to solutions of the
Korteweg-de-Vries equations as well as other non-linear
differential equations of mathematical physics.
This book presents an explicit elementary construction of
hyperelliptic Jacobian varieties and is a self-contained intro-
duction to the theory of the
Jacobians.
It also ties together
nineteenth-century discoveries due to Jacobi, Neumann. and
Frobenius with recent discoveries of Gelfand. McKean, Moser,
John Fay, and others.
A definitive body of information and research on the subject
of theta functions, this %olume will be a useful addition to
indi,,idual and mathematics research libraries.
ISBN 0-6176-3110-0
Birkhauser
Boston Basel - Berlin