Proceedings of the International Congress of Mathematicians
Helsinki, 1978
Langlands9 Conjecture for GL (2)
over Functional Fields
V. G. Drinfeld
Introduction
0.1. Let k be a global field of characteristic p>09 91 its adele ring, W the Weil
group of k (W consists of elements of Gal (Jc/k) whose image in Z belongs to Z,
the topology on W being induced by the embedding W cz+Gal (k/k)XZ). If v is
a place of k9 kv will denote the completion of k at v and Wv will denote the Weil
group of kv.
For each number field E denote by S^E) the set of isomorphism classes of compatible systems of absolutely irreducible 2-dimensional A-adic representations of
W (A belongs to the set of places of E which do not divide °° and p). Compatibility
of a system {QX} means, by definition, that for every place v of k and every A£ Wv
Tr Qx(h) belongs to E and does not depend on A. Put
Zx=z lini^fjg).
EcQ
Denote by S2 the set of isomorphism classes of irreducible representations of
GL (2, 91) over ß which occur in the space of cusp forms.
Let EaQ9 [E: ß]<°°, Q={QX}£Z1(E)9 nÇ.Z2- n is said to be compatible with Q
if for some A and almost all places v of k L(s~\9 7c„)=[det (1—JF„ q~s9 öA)]"1
where L denotes Jacquet-Langlands' L-function, Fv£ W is a geometric Frobenius
•element of v9 qv is the order of the residue field of v (of course, the words "for some
A", can be replaced by "for all A"). Now let Q£Z19 n£Z2- 7c is said to be compatible
with g if there exists a subfield EczQ of finite degree over Q and an element
QE^Z^E) such that Q is the image of QE and n is compatible with QB. Denote by r
the set of pairs (Q9 TE) €2^X2^ such that n is compatible with Q.
0.2, In this report I shall sketch a proof of the following theorem,
566
THEOREM
V. G. Drinfeld
A
(LANGLANDS' CONJECTURE),
r is the graph of a bijection
Z^Z^.
The following theorems are easy consequences of the proof of Theorem A.
THEOREM B (PETERSSON'S CONJECTURE). Let % be an irreducible unitary representation of GL (2, 91) which occurs in the space of cusp forms. Then for everyplace v ofk
nv does not belong to the complementary series.
Recall that the complementary series for GL (2, kv) consists of unitary representations of GL (2, kv) having the form %{n9 v), where /* and v are quasicharacters of
Zr* and not characters.
THEOREM C. Let % be an irreducible unitary representation of GL (2, 91) which
occurs in the space of cusp forms. Suppose that either one of the local components
ofn belongs to the discrete series, or every field finitely generated and of transcendence
degree 3 over Fp has a smooth projective model. Then all the zeroes ofL(s9 n) are on
the line Re,y=-|.
Recall that the discrete series for GL (2, kv) consists of supercuspidal and special
representations.
0.3. Let us show that theorem A is implied by the following proposition.
THEOREM A'. For each n£Z2 there exist a number field E containing the field of
definition of n and 2-dimensional X-adic representations QX of W (for all places A of
E except those dividing °° and p) such that for almost all places v of k and all A
M*--!). ^=t d e t 0~ F v<lv\QÙY
-i
First of all, the representations Qk in Theorem A' are absolutely irreducible (because for every quasicharacter co of 9l*/fc* L(s9 n <8> co) is holomorphic). In order
to prove that {QA}£ZI(E) it remains to show that for everyplace v of k the restriction of the character of QX to Wv belongs to E and does not depend on A. In fact
the results of [1] imply that it is enough to show this for almost all v9 which is trivial.
We have shown (using Theorem A') that the projection r~+Z2 is surjective.
The surjectivity of the projection T^Z1 was proved by P. Deligne [1] using the
results of [5] (he also proved that if QÇZ^E) is compatible with n£Z2 then for all
places of k the local L- and e-functions of Q and it coincide).
The projection T-+Z1 is injective because an irreducible representation of
GL (2, 91) which occurs in the space of cusp forms is uniquely determined by almost
all of its local components. The injectivity of the projection r~+Z2 is an easy consequence of Chebotarev's density theorem.
0.4. The proof of Theorem A' can be considered as a noncommutative generalization of Lang's theory [10]. On the other hand it is closely connected to EichlerShimura's theory.
Langlands' Conjecture for GL (2) over Functional Fields
567
1. F-sheaves and their moduli schemes
1.1. Let Fq be the field of constants of k9 denote by X the smooth projective curve
over Fq corresponding to k. We shall write Y®Z9 YXZ instead of Y®FqZ9
YXFZ.
If S is a scheme over Fq, Fr s will denote the Frobenius endomorphism of S (over Fq).
Instead of FrSpecB we shall write FrB.
DEFINITION. Let S be a scheme over Fq. A left (right) F-sheaf over S of rank d
is, by definition, a diagram
,(id,XFr s )*^
(respectively
(idxXFrs)*JSPv
\ J5")
x
se
se'
where 3? and «F are locally free sheaves of ^X(S-modules of rank d9 Coker / is the
direct image of an invertible sheaf on the graph of a morphism a: S-+X, Coker j
is the direct image of an invertible sheaf on the graph of a morphism ß : S-+X. a
is called the zero of the F-sheaf, ß is called the pole of the F-sheaf. An (a9 ß)-F-sheaf
is an F-sheaf whose zero equals a and whose pole equals ß.
The role of rank 2 F-sheaves in the proof of Theorem A' is similar to that of
elliptic curves in Eichler-Shimura's theory.
REMARK. Suppose that a and ß are disjoint (this means, by definition, that the
graph of a doesn't intersect the graph of ß). Then a left (a, /?)-F-sheaf is, in fact,
the same as a right (a, /?)-F-sheaf. In this situation the word "left" or "right" will
be omitted. When a and ß are disjoint an (a, /?)-F-sheaf can be imagined as an
"isomorphism" from (id^XFr^)*^ to S£ which has "singularities" (a "simple
zero" at a and a "simple pole" at ß).
REMARK. Instead of "F-sheaf (id^XFr s )*Jöf->^^J^" we shall often say "FsheafJSf".
Let JSf be an F-sheaf (left or right) over S of rank d. Suppose that the zero and
pole of S£ are disjoint of D (this means, by definition, that the zero and pole are
morphisms from S to X—D). Denote by ££D the restriction of J5f to DXS. The
F-structure on ££ induces an isomorphism / : (id^XFr^)*^^«^. In this situation the notion of level D structure on JS? is introduced.
DEFINITION. A level D structure on j£f is an isomorphism h: &D~®éxs such
that the diagram
(id D xFr s )*A
(id D XFr s )*^ — ^
. o*xS
is commutative.
Let us agree that there are no level D structures if the zero or the pole is not disjoint of D. It is easy to show that a level D structure always exists if S is replaced
by an étale covering of S (provided the zero and pole are disjoint of D).
568
V. G. Drinfeld
It is quite natural that 2-dimensional vector bundles appear in the proof of Theorem A': the set of isomorphism classes of 2-dimensional vector bundles over X can
be identified with U\GL (2, 9Ï)/GL (2, k) (where U is the standard maximal compact
subgroup of GL (2,91)), while GL (2, 9l)/GL (2, k) can be identified with the set
of isomorphism classes of 2-dimensional vector bundles over X whose restrictions
to all finite subschemes of X are compatibly trivialized.
1.2. THEOREM 1.1. For every finite subscheme DaX and positive integer d, the
coarse moduli scheme of left (right) F-sheaves with level D structure exists.
These schemes will be denoted by »Jt (for left F-sheaves) and Mé (for right
F-sheaves). &M and Mß are schemes over (X—D)2=(X—D)X(X—D); the structure
morphism M$>-+(X-D)2 or &M-+{X-D)2 maps an (a,/?)-F-sheaf to (oc9ß).
THEOREM 1.2. There exist open subschemes UiCiMé (/—l, 2, 3,...) such that
UidUî+l9\JiUi=Mé
and for every i U{ is a quotient of a quasiprojective smooth
scheme over (X—D)2 of relative dimension 2d—2 by a finite group. This is also
true for &M.
1.3. We shall define some natural morphisms between the moduli schemes constructed above.
(1) Let 7u: &M -+X29 n: M&-+X2 be the structure morphisms, denote by A the
diagonal in X2. Then n^iX2—A) can be identified with n^ÇX2 —A); this follows
from the remark in 1.2 (after the definition of F-sheaf). Moreover ^M can be identified
with^jj.
(2) If «5? is an F-sheaf with level D structure then det Se is an F-sheaf with level
D structure. Therefore we have morphisms det: M&-+M}>9 det: »M^M—M^.
(3) If Se is a left (right) (a, j8)-F-sheaf then SP* is a right (left) (ß, a)-F-sheaf;
level D structure on Se induces level D structure on if*. Therefore we have an
isomorphism * : M^DM
such that the diagram
<===
M$
^- •- &M
I
!
(X-Df
• (X-Df
is commutative.
(4) If jSf-i-^-MidxXFrsfjS? is a left (a,ß)-F-s\\es£ then
(id x XFr s )*^
tid xFrs)
*
*'' , (idxXFrs)*J§? J- SF
is a right (Fr^oa./O-^-sheaf. If (id x XFr s )*i?-^ y **- & is a right (a, ß)-F-sheaf
then
(id x XFr s )*^ • t , d * * w (id xX Fr s )*if - L y
Langlands' Conjecture for GL (2) over Functional Fields
569
is a left (a, Fr^o/?) F-sheaf. In both cases level D structure on Se induces level D
structure on &. Therefore we obtain morphisms F x : &M-+M& and F 2 : M&-+&M
such that the diagrams
are commutative. Note that FXF2=Fr(^)?
F 2 F 1 =Frß 1/Ä ).
Moral. M^ looks like a product of two schemes over X(in fact it is not a product).
(5) If D'ziD there are obvious morphisms $M-+i>M9 M^^M».
(6) GL (^,91) acts on limP jM and ]imDMp. We shall not give the precise
definition. The definition is based on the following construction. Let j£? be an F-sheaf
over S of rank d with level D structure, let P be a quotient module of @£, Put
JP=i\n*P where % is the projection DXS-+D and i is the embedding DXS(+XXS.
The level X) structure on JS? induces an epimorphism S£-+P whose kernel has a
natural F-structure.
REMARK. The action of GL (d9 91) on the set of isomorphism classes of rf-dimensional vector bundles over X compatibly trivialized over finite subschemes (recall
that this set can be identified with GL (d, 9I)/GL (d9 k)) is described by a similar
construction.
2. Lang's theory
2.1. Let us consider the case d=l. Denote by Pic n X the moduli scheme of
invertible sheaves on X whose restriction to D is trivialized. Recall that (1) Picn X
is a group scheme over Fq; (2) there is an exact sequence 0->Picï> X-*Picn X^ deß Z->0,
where PiCp X is the connected component of PicB X; (3) Pic?) X is an algebraic variety (i.e. of finite type); (4) PicJ, X is called the generalized Jacobian variety of X
of conductor D. There is a natural mapping X—Z>->PicnX (the image of
ue(X-D)(Fq) in PicDZ(Fg) will be denoted by w).
There is a Cartesian square
Ml=lM
-+ Pic n X
Let yeMi(Fq);y = (a9ß9l), where a9ße(X-D)(Fq)9 l£(PieDX)(Fq)9 Fv(l)-J=ß-Öt.
Then *;/ = (/?, a, - / ) , Fx(^) = (Fr (a),/?, / - a ) , j F , ^ = (a,Fr(]8), 1+ß). PutKc,, JT=
= (PicD JT)(Ffl). Picj3 X acts on .4$: if (a, ß9 l)£Mb(Fq)9 mel?icD X9 then/77 • (a, /?, /) =
= def (a,/?, / + w). Therefore 9l*//c*=limZ)PicJ?X acts on }imPM}i. We have described the action of 91* on lim^ Mp mentioned in 1.3.
2.2. Theorem A' follows from a more precise statement (Theorem 4.1) describing
the cohomology of the compactified moduli space of F-sheaves of rank 2. We are
570
V. G. Drinfeld
going to discuss a similar theorem for d= 1 (Theorem 2.1), which is a rather awkward
reformulation of a part of Lang's theory.
Let DaX be a finite subscheme. Recall that Picjo Xacts on M& (see 2,1). Choose
a splitting of the exact sequence O-^Pic^Z^Pic^ J*"-> deß Z-0: P i c I ) Z = P i c ï ) i r x / ,
/c*Z. Denote by 7c the natural morphism M^/J-^(X—D)2. Consider rc^öj (in the
étale sense). The action of Pic^ X/J on M^/J induces an action of Picj, X/J on n^Qt.
The action of Ft on M^/J induces an isomorphism (Frx_DXidx^.Dy
n^Qi^n^Q^
2.3.
(X-D)2
A PF-sheaf of sets on (X— Z>)2 is an étale sheaf of sets & on
with an isomorphism
ÇFrx-.DXidx-D)*&r~&
DEFINITION.
REMARK.
Note that (Fr x _ 1) XFr x _ I) )*J 5 '=«^' for any étale sheaf & on
(X-D)2.
THEOREM 2.1. The category of constructible locally constant PF-sheaves of sets
on (X—D)2 is equivalent to the category of finite n1(X—D)Xn1(X—D)-SQts.
REMARK. The category of constructible locally constant étale sheaves of sets on
(X—D)2 is equivalent to the category of finite ^{(X— Z>)2)-sets. But the natural
morphism->n1((X—D)2)-+7i1(X—D)Xn1(X—D)
is not an isomorphism: if D = 0 it
is injective but not surjective (for instance, Tü1(P1XP1)=Z^ZXZ);
if ZM0 it
is even not injective.
2.4. Let us return to the situation of 2.2. 7c*ß/=(7i;*Zj)<g>z,ßj> n*Zi =
lininTi*(Z//nZ) and n*(Z/lnZ) is a constructive PF-sheaf of sets. Therefore
n*Qi corresponds to an /-adic representation of n1(X—D)X7i1(X~D).
Y\cDXjJ acts
on 71*6;, therefore we have, in fact, a representation of
n1(X-D)Xn1(X-D)X(PicD
X/J);
denote it by Vx.
THEOREM 2.2.
Vj (g) Ql =
Qj
©
QzQQ^QX
z € Horn (Pic D X/J, Q*)
where QxÇ.Hom(n1(X—D)9Qi)
is compatible with % (this means that for every
closed point v£X—D Qx(Fv)=%(v)9 where F^n^X—D)
is the geometric Frobenius element of v and v is the canonical image of v in Y\cD X/J).
3. Compactification
3.1. Denote by Kthe fraction field of k®k9 put
ArD=M^®xy>xK=ßM®x^xK.
We are going to describe the nature of nonquasicompactness of JfD.
For every n£Z denote by JTrf the open subscheme of JfD corresponding to
sheaves of degree /?; then ^ = I L I ^ D " - For every m9n£Z denote by J/S1*11 the
open subscheme of Jfj}9 corresponding to F-sheaves if which do not contain invertible subsheaves of degree more than m\ J^^czJ^1*7'
and ^ i " = U»j^/Dm,n-
Langlands' Conjecture for GL (2) over Functional Fields
571
3.1. Jf$un is a quasiprojective surface. If 2m—n^2 then (*#S®KR) —
" ® K K ) is isomorphic to the countable disjoint sum of affine lines.
THEOREM
(^D
W,
3.2. If 2m—n^29Jf^iiX
has a canonical compactification dff>n. We are going
to describe its properties.
(1) J/n'n is a proper algebraic space over K (I don't know whether it is a scheme),
Jfm*n is an open dense subspace of J/$'n.
(2) J%n'n is normal. If Y is a fibre of a singularity resolution of J^'^^KK
then
H°(Y9Qi) = Qi9 H*t(Y9Qi)=0 (in other words, YTed is either a point or a curve
whose irreducible components are rational and whose intersection graph is a tree).
(3) If m'^m there is a commutative diagram
(4) Put Fg'w = J g , l ' w - > 3 H ' " (Fg'" is considered as a reduced subspace). The
morphism r s + 1 ' " - r g ' " induced by jfr+1>n-^Jg1*» is radical; this means that it
is finite and its geometric fibres contain exactly one point (but they may not be
reduced).
(5) Let m'^m. The morphism o¥gx'tn®KK-^J/gìtìl^KK
maps each irreducible
component of J/g'*n C^xR-Ag1*" ®>KK onto a point of rngn®KK (different components correspond to different points).
(6) If D'r>D the morphism J/g>"->Jfgx>n is the restriction of a finite morphism
(7) The action of GL (2, 91), F l 5 F 2 and * on lim P J/^ is the restriction of an
action of GL (2, 91), Fl9F2 and * on jj„lim p ^.yfff1'" (the projective limit is understood formally).
3.3. DEFINITION. The .R-category is the category of schemes localized by radical
morphisms.
rngn considered as an object of the J?-category will be denoted by rnD (because it
doesn't depend on m). We are going to describe Fg.
Denote by Y*'3 the fibred product of XXXXX
by Pic* XXPtâ X over
0
0
Piç°ZXPiç°X, the morphisms XXXxX^Piç
XX&ç X and Pic* XXPiç' X-+
Pic°XxPiç°A' being given respectively by (oL9£>9ß)y-^(Z—ÖL9ß—l) and F r - 1 .
The morphism yW-yM-LJ-i given by (a, t,ß9 a9 &)-(a, Fr(£), ß, a+l9 b-Ç)
(where a, £, ߣX(Fq)9 ae^X(Fq)9
Z>£PiçJ* X(Fqj) is radical, therefore in the
JÊ-category y'-"-' doesn't depend on /. The result is: rn0 — Yiin'~i®x^xK9 the morphism YUH-t-+XXX being given by (a9l;9ß)>-+{<x9ß).
The description of F£ for 2)^0 is more complicated; the main difference from
the case Z)^=0 is that TnD is not ^-isomorphic to a smooth curve.
572
V. G. Drinfeld
3.4. The construction of J/g*n is based on the notion of degenerate F-sheaf
Let F be a field, a£X(F), ߀X(E)9 oc^ß.
DEFINITION. Let S£ be a 2-dimensional locally free sheaf on X®E. An (a, /?)transform of Se is a subsheaf Se'aSffß)
such that Supp JSf (jß)/i?' = {a,0} and
aimEH°(Sf(ß)/Sf')=2
(we identify a and ß with the corresponding points of
X®E).
DEFINITION. A degenerate (a, ß)-F-sheaf over E of rank 2 is a triple (S£9 S£'9 <p)
consisting of a 2-dimensional locally free sheaf S£ on X ® F , its (a,/?)-transform
S£' and a morphism 9 : (id x XFr £ )*JS?-^i?' such that (1) cp has rank 1 at each
point of Jf<g>F, (2) I m p e t i , (3) p| ( i dx xFr £ mnim</0^0.
In this situation put ^=j£?nlm<p. <p induces an injection <p': (id A -XFr x )^-^
Im <p — A(ß). Supp Coker p ' consists of a single point £.
DEFINITION. £ is called the degenerator of Se.
REMARK.
The third projection r'gn-+X
is given by the degenerator.
4. Proof of Theorems A, B and C
4.1. Choose a subgroup Jc:9I* such that the composition J-+$t*/k*-+dcgZ is
an isomorphism. Let /^/? be a prime number. The action of GL (2, 91) on
Ilncz limo,». ^S" , B induces an action of GL(2, 91) on II„ l i m ^ i ^ * (J£"' M <g>* X, Qt);
put f?=[n f I HmD> in Hìt ÇAg"' "<g)KK9 Qi)]J. Note that the representation of GL (2, 91)
on V2 is not admissible,
Put F Gal(X/^) = {y€AutK|3m,w6Zy| k 0 f c = Fr2'(g)FrX}/{F4|«€Z}. The action of F1 and F 2 on UHhmDf„l.yf'g"'" and the action of Gal (K/K) on K induce an
action of F Gal (K/K) on Pf1'. There is a canonical homomorphism / : F Gal (JK/iQ-»Gal(£//c)XGal(/c//c): an element of F Gal (K/K) represented by y £ Aut K is
mapped to (y F r ^ e i , y F r f l i a x ) , where y|*®fc=Fr?®Frj[. The image of /
contains WX W. One can deduce from theorem 2.1. that the action of FGal(K/K)
on Vi is trivial on Ker/. Therefore we have a representation of WX J^XGL (2, 91)
on Vf.
Denote by Vjcusp the greatest GL(2, 9ï)-invariant subspace of V* contained in
0n M liigD^5(^ ( l I + 8 ) / B L n iQi)] J r - *ï0UBp i s »^X R^-invariant. The representation of
GL (2, 91) on Vf™* is already admissible.
THEOREM4.1. PîCU8p®GIGj=©w€2£e*®£*®rc> where Z( = {7ieZ2\ the restriction
of 71 to J is trivial}, it denotes the representation contragradient to n, Qn is a 2-dimensional representation of W over Qx compatible with n.
Theorem A' follows from Theorem 4.1.
Langlands' Conjecture for GL(2) over Functional Fields
573
REMARK. The proof of Theorem 4.1. gives in fact a complete description of Vt
for all i. It turns out that V?=V?=0, V? is the direct sum of K,cusp and its orthogonal
complement J^Eis, Vfis has an explicit description, the description of Vf and Vf
follows from the fact that the fibres of det: ^ ® x f - > ^ ( g ) X x ; f K are connected.
4.2. The proof of Theorem 4.1. is based on Lefschetz formula, Selberg trace
formula and an explicit description oîJ/j>(Fq) similar to Deuring's results on elliptic
curves over Fp. This method was used in [6] and [8]. It turns out that the structure
of the spaces Jfg^n is closely related to the structure of Selberg trace formula. For
instance, the fact that TnB is a scheme over XXXXX (not only XXX) corresponds
to the fact that Selberg trace formula involves integrals of the form ff(s)dIn L(s)9
where L(s) is an abelian Z-function (so that in order to compute the trace of the
Hecke operator corresponding to a place of k one has to use Z-functions containing
information on all places of k).
4.3. Let v and w be closed points of X9 v^w. The proof of Theorem 4.1 uses the
computation of the (-function of an open subscheme of J£ßXx*(vXw) (this
(-function is expressed in terms of cusp forms). Theorem B follows from the fact
that the poles of the (-function of a surface over Fq are on the lines Re,y=0, j 9
1, 2. This fact is a consequence of Weil's conjecture proved by Deligne and Abhyankar's theorem (resolution of singularities for surfaces).
If everyfieldfinitelygenerated and of transcendence degree 3 over Fp has a smooth
projective model, the zeroes and poles of the (-function of a 3-dimensional scheme
over Fq are on the lines Res=n/29 n£Z. Applying this to a suitable open subscheme of Jd%Xx*{XXv) for some closed point v^X—D9 we obtain the second
part of Theorem C.
In [2], [3] Theorem A' was proved assuming that n^ is in the discrete series. In
[3] the representation of W corresponding to n appears as a part of Hlt(F ® *£, QÌ)
for some scheme F over X of relative dimension 1. The zeroes and poles of ((F, s)
are on the lines Rts=n/29 n£Z9 because Fis a surface. This enables us to prove
the first part of theorem C.
4.4. The surface F mentioned in 4.3 is the moduli scheme of elliptic modules
(see [2]) with some additional structures. Elliptic modules are functional analogues
of elliptic curves, and the result of [3] mentioned in 4.3 is analogous to EichlerSliimura's Theorem.
On the other hand, elliptic modules are, in some sense, F-sheaves of special type
(this is not an obvious consequence of the definitions of elliptic module and F-sheaf).
The precise statement and proof can be found in [4], [9]. The method use
4]
is analogous to the method used by Burchnall, Chaundy, Baker and Kricever in
order to describe the commutative rings of differential operators (see [7], [9]).
574
V. G. Drinfeld: Langlands' Conjecture for GL (2) over Functional Fields
Bibliography
1. P. Deligne, Les constantes des équations fonctionnelles des fonctions L, Lecture Notes in Math.
vol. 349, Springer-Verlag, Berlin, 1973.
2. V. G. Drinfeld, Elliptic modules, Mat. Sb. 94 (1974).
3. V. G. Drinfeld, Elliptic modules II, Mat. Sb. 102 (1977).
4. V. G. Drinfeld, On commutative subrings of some non-commutative rings. Functional Anal.
Appi. 11 (1977).
5. H. Jacquet and R. P. Langlands, Automorphic forms on GL (2). Lecture Notes in Math., voli
114, Springer-Verlag, Berlin, 1970.
6. Y. Ihara, Hecke polynomials as congruence zeta functions in elliptic modular case, Ann. o,
Math. (2) 85 (1967), 267—295.
7. I. M. Kricever, The methods of algebraic geometry in the theory of non-linear equations, Uspehf
Mat. Nauk, 32 (1977).
8. R. P. Langlands, Modular forms and l-adic representations. Lecture Notes in Math., vol. 349,
Springer-Verlag, Berlin, 1973.
9. D. Mumford, An algebro-geometric construction of commuting operators and of solutions of the
Toda lattice equation, Korteweg De Vries equation and related nonlinear equations, Preprint, 1977.
10. J.-P. Serre, Groupes algébriques et corps de classes. Hermann, Paris (1959).