{Proceedings of the International Congress of Mathematicians
August 16-24, 1983, Warszawa
D. W. MASSEB
Zero Estimates on Group Varieties
1 . Introduction
'The aim of this article is to give an account of a class of estimates that
have in recent years been found useful in the theory of transcendental
numbers. As our space is limited, and the estimates are quite technical
in nature, wç can provide no hints of proofs, and we prefer to illustrate
the results with examples rather than supply full statements. We also
mention some of the applications.
One basic method in transcendence theory can be described in general
terms as follows. Suppose it is desired to establish a certain proposition.
Usually it will be possible to associate with the problem a collection of
meromorphic functions ft, ...,fnm
such a way that assuming the falsity
of the proposition imposes strong arithmetical conditions on the functions.
These enable us to construct a polynomial P =P(x1, ..., xn) such that the
function (f> =P(f1, ...,fn) is not identically zero but has a large number
of zeroes. By means of analytic interpolation arguments it can then be
«deduced that ^ takes many small values, and by using once more the
.arithmetical conditions we conclude that (/> has even more zeroes than
initially arranged. But often merely knowing upper bounds for the degree
of P leads to upper bounds for the number of zeroes of ^, and if we are
lucky these yield the contradiction that proves the original proposition.
In this article we shall limit attention to the last stage of this argument, as it is here that the most interesting recent progress has taken
place. A simple example will suffice to illustrate the sort of estimates
involved. To show that w and ew cannot both be algebraic for w ^ 0,
suppose they are. Then we end up with a non-zero polynomial P = P(x,y),
of degree at most L > 1 in x and of degree at most Jf ^ 1 in y, such that
the function
</>(z) = P(s,<f)
(1)
[493]
494
Section 3: D. W. Masser
has a zero of order at least T at the points 0, w,..., 8w for some 8. I t
is not hard to prove (see for example [7]) that if
TS>àLM,
T>2M,
(2)
then in fact P = 0. Thus if we can make T and 8 so large as to satisfy (2),.
we obtain our desired contradiction.
We shall refer to the above type of statement as ^ zero estimateIn the last few years such results have been systematized and put into
a general context, whereas before this, many ingenious ad hoc arguments,.
often involving manipulation of determinants or Kummer theory, were
found necessary to finish the proofs.
The essential tool in the recent developments has been commutative
algebra. This was introduced in the fundamental paper of ISTesterenko
[25]. His results were designed to apply to Siegel E-îunctions, and so
they treat the case when/ 1? ...,jfw are functions of a single complex variable satisfying linear differential equations over C(z). Later on Brownawell and the author [8], following the broad outline of [25] but changing:
the details somewhat, were able to treat non-linear differential equationsover C (see also Brownawell [6]). All this work is directed essentially
at bounding the order of a single zero of <f>.
2. Bounds without multiplicities
If instead we want to estimate the number of zeroes of ^ on a given setcounted without multiplicity, then provided the set has suitable translation properties, it is appropriate to consider translation formulae rather
than differential equations. A general context (though not the most general) where such formulae are available is that of group varieties. In
this case f 1 9 . . . , fn are more or less the coordinates of the associated exponential map, so it is possible to work directly on the group variety without
mentioning meromorphic functions. In the paper [19] of Wüstholz and
the author the following situation is studied.
Let G he sb commutative group variety of dimension n > 1, assumed
embedded in projective space PN of N ^ 1 dimensions, and let P be a finitely generated subgroup of G. For each integer r with 1 < r < n define
pr =*pr(r9Q) =minrank(P/rnfl"),
where the minimum is taken over all algebraic subgroups S of G of dimension n—r (with pr = rankP if no such PL exists). Let
p. =}i(r,G) = min (prlr).
35ero Estimates on Group Varieties
495
Let yx,..., ym be generators of P and for 8^ 0 write r(8) for the set of
linear combinations s ^ H - ... +sOTym as s19..., sw run over all integers
with 0 < 5X, ..., s m < #. The main theorem of [19] states that there is
•a constant e, depending only on G and the embedding, with the following
property. If P = P(XQ, ..., XN) is a homogeneous polynomial of degree
P ) > 1 vanishing on P(#) with
(8jnY>cD
(3)
then P vanishes identically on <?. The exponent [i in (3) can be shown to
be best possible for any given G and P.
We should note here that Moreau [23] has given a much shorter version of the proof of this result, using geometric arguments in place of the
-commutative algebra (see also [24]). But it is not clear if his work extends
to multiplicities.
When G = A is an abelian variety defined over Q, this result has
been used in [18] to prove certain lower bounds for the Néron-Tate height
on the set of points of A defined over Q. When G = 6?™ is the n-ìold product of the multiplicative group Gm, it has been used by Waldschmidt
[37] to prove some generalizations of certain transcendence results on
numbers of the form euv (see also [39]). In particular these answer questions of Serre and Weil on characters (see [38]), and they have interesting
•consequences for ^p-adic regulators (see [40] and [41]).
We illustrate the above zero estimate when G is the product of Gm
by the additive group Ga. First of all embed G in P2 as the set of points
(x0, xt, a?2) with x0x2 ^ 0. If (y0, yx, y2) is another such point, their group
sum is (xQy0, x0y1 + x1y0, x2y2). The constant c of [19] turns out to be 1.
If P is generated by the point y = (1, w, ew), it is not difficult to see that
ju,(r,G) =1/2 provided wj2ni is irrational. So in this case let P(x,y)
be a polynomial of total degree at most B > 1 such that the function (1)
vanishes at the points 0, w,..., 8w for
#>2D2.
Then the homogeneous polynomial
P(X0, Zl9 X2) = X^P(X1IX0, X2/X0)
(4)
vanishes on r(8), and we conclude that P = 0.
Similarly by considering G = 6?^ in Pn we can obtain zero estimates
for (f>(z) =P(e w i 0 ,..., eunz). Sometimes these can be proved by means
of Tijdeman's powerful analytic methods [33], [35], which by virtue of
their simplicity and elegance have very often been found indispensable
496
Seetion 3: D. W. Masser
for problems involving exponential polynomials. But the strength of thealgebraic method lies in its generality; for example, if p(z) is a Weierstrass elliptic function we can obtain zero estimates for
P[p(%xz),...
..., p(Una)) simply by considering G = Mn for an appropriate elliptic
curve B. Such estimates were used in [20] to establish elliptic analogues
of some algebraic independence results for numbers of the form euv. Thus.
if p(z) has algebraic invariants and no complex multiplication, and u19..+
...,un as well as vx,...,vm
are complex numbers linearly independent
over Q, then provided
mn^2m
+ én
(5)
at least two of the numbers
p(^iVj)
(l<i<w,l<j<m)
(6)
are algebraically independent. The exponential analogue of this had.
been obtained independently by Brownawell [5], Smelev [32], WaldSchmidt [36] and Wallisser (see also Tijdeman [34]).
By suitably refining these zero estimates similar results can be given
for larger transcendence degrees. Thus Wüstholz and the author show
in [21] that, with mild linear independence measures on ux, ...,un and
^u-'-j^mj *^ e condition
mn^2k+1(Jc
+ 7)(m + 2n)
is enough to ensure that at least Jc of the numbers (6) are algebraically
independent. This is an analogue of a result of Ohudnovsky [9] (see also
Warkentin [42], Eeyssat [31], Philippon [27], Endell [11] and JSTesterenko
[26]).
3 . Multiplicities in a single direction
So far we have considered two types of zero estimate: those for a single
high order zero, and those for many zeroes without multiplicity. In [22]
these are combined (see also the announcements in [19]). For functions
of a single complex variable the natural concept is that of a one-parameter
subgroup of a group variety G. This is a non-zero analytic group homomorphism A from C to G. If as before G is embedded in PN and
P =P(X0,...,
XN) is a homogeneous polynomial it is easy to define
for any g in G the order of vanishing of P at g along A. With the notation
above, one of the results of [22] can be stated as follows. There is a constant c, depending only on G and the embedding, with the following prop-
Zero Estimates on Group Varieties
497
erty. If P has degree D > 1 and vanishes at each point of P(#) to order
at least T along A with
>cDr
(1 < r < n),
(7)
(8jnfr > cDr~l
(1 < r < n),
(8)
T(8/nfr
then P vanishes on g+ A(C) for some g in (?. The conditions (8) are not
natural and they reflect a technical difficulty in the proof. Note also that
the conditions (7) for T — 1 reduce essentially to (3).
As an example, take G = Ga x Gm in P 2 as above, and put A(z)
= (l,z,e*).
For a homogeneous polynomial P = P ( X 0 , X 1 ? X a ) and
a point # = (x0, x19 x2) on (? the order of vanishing of P at g along /I turns
out to be simply the order of zero at z = 0 of the function P(x0, w0z +
+ xlf xzez). Thus if the function (1) has a zero of order t at z = sw for some
integer s, then the polynomial P defined by (4) vanishes at sy to order t
along /l, where sy = (1, sw, esw). Hence we obtain multiplicity estimates
for such functions.
The elliptic analogues lead to corresponding generalizations of the
above algebraic independence results. For example, the condition (5)
may be relaxed to mn ^ 2m + 2n as long as we adjoin the numbers u19 ...
..., un to (6). If p(z) has complex multiplication over a quadratic field Jc,
the same result holds provided now ux,..., un are linearly independent
over Jc. A corollary is that if ß is cubic over Jc and u ^ 0 is such that p (u)
is algebraic, then p(ßu) and p(ß2u) are algebraically independent. This
is the elliptic analogue of Gelfond's result on a? and aß%.
The work of [22] includes some other refinements. Firstly the results
are stated for multihomogeneous polynomials; this allows us to have
different degrees in different variables, as in the original example (1).
Secondly, a more delicate measure of the distribution of P with respect
to algebraic subgroups is introduced; for example, if P consists only of
torsion points then px = ... = pn = [i = 0 and so the conditions (3),
(7), (8) are too restrictive. This last refinement is especially valuable
because Wüstholz has shown how in certain circumstances it can be used
to eliminate the technical conditions (8). Furthermore in [43] he deduces
the following remarkable consequence for an elliptic function p(z) with
algebraic invariants and complex multiplication over Jc; if
a19...,an
are algebraic numbers linearly independent over Jc, then p(ax),...,
p(an)
are algebraically independent. This is of course the analogue of the celebrated Lindemann-Weierstrass theorem of 1885. I t had previously been
proved by Ohudnovsky [10] for n = 1 , 2 , 3 .
498
Section 3: D. W. Masser
We should also mention that at about the same time Philippon [29]
found a different though related approach to this result, still using zero
estimates. Furthermore in [28] (see also [30]) he obtains slightly weaker
results for the numbers p(u), p(ßu), ..., p(ßd~1u); nevertheless these
are still far in advance of the known exponential analogues.
4. Arbitrary multiplicities
We end by discussing briefly the most exciting recent developments;
namely, the extension by Wüstholz [44] of the multiplicity estimates
to several variables. If G is a group variety of dimension n, and d is a fixed
integer with 1 < d < n, a ^-parameter subgroup of G is an analytic group
homomorphism A from Cd to G whose Jacobian is not identically zero.
The main result of [44] then directly generalizes, the earlier multiplicity
estimate to such A.
Now when this result is interpreted in terms of meromorphic functions,
it involves partial differentiation with respect to d variables. The case
d = n —1 is especially important, since it corresponds to Baker's method
in transcendence theory. For example, with G = G^ embedded in Pn
as the set of points (x0,..., xn)*vnth. xQ...xn ^ 0 , let ßx9*.*9ßn-i be
algebraic numbers with 1, ßx,..., ßn__1 linearly independent over Q, and
take A(zx,..., V-i) = (h e% •••* a*1"1» eß&+"+ßK-i*n-i). Let ax,..., an_x
be non-zero algebraic numbers with logarithms lx,..., ln„x not all zero.
If a£i... ofc[i is algebraic, Baker's method [1] yields a non-zero polynomial P =P(xx,..., xn), of total degree at most JD>1, such that the
function
<f>(zx, ..., V-l) = P ( A ..., <?n~\ eßl*l+~+ßn-lzn>-l)
has zeroes of order at least T at the points
(%,..., zn_x) =s(l1,...,ln_1)
(0<s<#)
with
Tn-18>cDn,
T>cD
for any large constant c. The main result of [44] then implies that P = 0.
Hence afi... a&Li1 is transcendental.
But now the generality of the algebraic approach means that the
analogues for any commutative group variety can be proved in the same
Zero Estimates on Group Varieties
499
way. Some of these are worked out in [45] (see also [46]); for example,
if p(z) has algebraic invariants and ax,..., a n _ n ß19...,
ßn„x are as above,
it is shown that a?*... aßntr^eVlUl... ev™v™ is transcendental for any algebraic yx,...,
ym and any complex numbers ux, ...,um such that p(ux), ...
..., p(um) are algebraic. In terms of linear forms, this result mixes ordinary logarithms with elliptic logarithms.
By refining his zero estimates to deal with torsion points, Wüstholz
settles two other linear forms problems that seemed hopeless a few years
ago. The first of these was raised by Baker [1]. Let px(z), ..., pn(z) have
algebraic invariants, and let £x(z),..., £n(z) be the corresponding Weierstrass zeta functions. Pick periods co19..., con of px(z), ..., pn(z) respectively, and put
Vi = ft(* + û><)-W»)
(1 < * < « ) .
In [47] it is shown that any linear form in cox, . . . , con, r\x, ..., r\n and
2ni with algebraic coefficients is either zero or transcendental. This was
previously known only for n = 1 , 2 . Furthermore Wüstholz determines
when such a linear form can vanish; this had nofc even been settled for
n == 2 (see [16] for a more detailed history). The proofs involve the group
G =G'xGm,
where G' is an extension of a product of elliptic curves
by Ga.
The second problem was raised by Bertrand [2], Take now a single
pair of functions p(z), £(#) and numbers co, rj as above. Write A (a) == coÇ(z) —
— riz, and let ux,..., un be such that p(ux),..., p(un) are algebraic. Then
in [48] it is shown that any linear form in co, ?y, X(ux), ..., h(un) with
algebraic coefficients is either zero or transcendental, and again the two
possibilities can be distinguished. For n = 1, 2 these results had been
proved by Laurent [14]; see also the very interesting related work of
Bertrand [3] and their joint article [4]. This time the group G is an extension of an elliptic curve by 6?™ x Ga. As a consequence, one can now decide
whether a period of an arbitrary differential on an elliptic curve is transcendental or not, provided both are defined over Q.
Finally the techniques of [44] also lead to quantitative results good
enough for applications to diophantine problems. For example, when
combiued with a method of Lang [13], they provide a new proof of Siegel's theorem for arbitrary curves. Furthermore, the only ineffective step
in the proof consists of the determination of a basis of the corresponding
Mordell-Weil group. Previously this approach could only be made to
work in the case of complex multiplication [15].
36 — Proceedings...
500
Section 3: D. W. Masser
References
[1] Baker A., Transcendental Number Theory, Cambridge, 1975.
[2] Bertrand D., Abelian Functions and Transcendence, Journées
Arithmétiques
1980 (ed. J. V. Armitage), London/Math. Soc. Lecture Notes 56, Cambridge, 1982,
p p . 1-10.
[3] Bertrand D., Valeurs de fonctions thêta et hauteurs #-adiques, Sem. Theorie
Nombres Paris [Sém. Delange-Pisot-Poitou) 1980/81, Progress in Math. 22, Birkhäuser, Boston-Basel-Stuttgart, 1982, p p . 1-11.
[4] Bertrand D. and Laurent M., Propriétés de transcendance de nombres liés aux
fonctions thêta, O. E. Acad. Sei. Paris 292 (1981), p p . 747-749.
[5] Brownawell W. D., The Algebraic Independence of Certain Values of the Exponential Function, K. Norshe VidensJc. Selsk. Skr. 23 (1972), 5p.
[6] Brownawell, W. D., Zero Estimates for Solutions of Differential Equations,
Approximations diophantiennes et nombres transcendants, Luminy 1982 (eds.
D. Bertrand and M. Waldschmidt), Progress in Math. 31, Brikhäuser, BostonBasel-Stuttgart, 1983, pp. 67-94.
[7] Brownawell W. D. and Masser D. W., Multiplicity Estimates for Analytic F u n c tions I, J. reine angew. Math. 314 (1980), pp. 200-216.
[8] Brownawell W. D. and Masser D. W., Multiplicity Estimates for Analytic F u n c tions I I , Buhe Math. J. 47 (1980), pp. 273-295.
[9] Chudnovsky G. V., Some Analytic Methods in the Theory of Transcendental Numbers, Inst, of Math., Ukr. Acad. Soi., Preprint IM 74-8 (48 p.) and IM 74-9 (52 p.),
Kiev, 1974.
[10] Chudnovsky G. V., Algebraic Independence of the Values of Elliptic Functions
at Algebraic Points ; Elliptic Analogue of the Lindemann-Weierstrass Theorem,
Inventiones Math. 61 (1980), p p . 267-290.
[11] Endell K., Zur algebraischen Unabhängigkeit gewisser Werte der Exponentialfunktion (nach Ohudnovsky), Diplomarbeit, Düsseldorf, 1981.
[12] Greifend A. O., Transcendental and Algebraic Numbers, Dover, New York,
1960.
[13] Lang S., Diophantine Approximation on Toruses, Amer. J. Math. 86 (1964),
p p . 521-533.
[14] Laurent M., Transcendance de périodes d'intégrales elliptiques I I , J. reine angew.
Math. 333 (1982), pp. 144-161.
[15] Masser D. W., Linear Forms in Algebraic Points of Abelian Functions I I I , Proo*
London Math. Soc. 33 (1976), pp. 549-564.
[16] Masser D . W., Some Vector Spaces Associated with Two Elliptic Functions,
Transcendence Theory : Advances and Applications (eds. A.Baker and D.W. Masser),
Academic Press, London, 1977, p p . 101-119.
[17] Masser D . W . , On Polynomials and Exponential Polynomials in Several Complex
Variables, Inventiones Math. 63 (1981), pp. 81-95.
[18] Masser D. W., Small Values of the Quadratic P a r t of the Néron-Tate Height,
Sern. Theorie Nombres Paris (Sem. Delange-Pisot-Poitou) 1979/80, Progress in
Math. 12, Birkhäuser, Boston-Basel-Stuttgart, 1981, pp. 213-222; proofs t o
appear in Oompositio Math.
[19] Masser D. W. and Wüstholz G., Zero Estimates on Group Varieties I, Inventiones Math. 64 (1981), pp. 489-516.
Zero Estimates on Group Varieties
501
[20] Masser D. W. and Wüstholz G., Algebraic Independence Properties of Values
of Elliptic Functions, Journées Arithmétiques 1980 (ed. J. V. Armitage), London
Math. Soo. Lecture Notes 56, Cambridge, 1982, p p . 360-363.
[21] Masser D. W. and Wüstholz G., Fields of Large Transcendence Degree Generated
by Values of Elliptic Functions, Inventiones Math. 72 (1983), p p . 407-464.
[22] Masser D. W. and Wüstholz G., Zero Estimates on Group Varieties I I .
[23] Moreau J. C , Démonstrations géométriques de lemmes de zeros I, Sém. Theorie
Nombres Paris (Sém. Delange-Pisot-Poitou) 1981/82, Progress in Math., Birkhäuser, Boston-Basel-Stuttgart.
[24] Moreau J. C , Démonstrations géométriques do lem mes de zeros I I , Approxima*
tions diophantiennes et nombres transcendants, Luminy 1982 (eds. D. Bertrand
and M. Waldschmidt), Progress in Math. 31, Birkhäuser, Boston-Basel-Stuttgart, 1983, pp. 191-197.
[25] Nesterenko J . V., Estimates for the Orders of Zeroes of Functions of a Certain
Class and Their Applications to the Theory of Transcendental Numbers, lav.
Akad. Nauk. USSE, Ser. Mat. 41 (1977), p p . 253-284; Math. USSB Izv. 11 (1977),
p p . 239-270.
[26] Nesterenko J. V., On the Algebraic Independence of Algebraic Numbers to
Algebraic Powers, Approximations
diophantiennes
et nombres
transcendants,
Zuminy 1982 (eds. D. Bertrand and M. Waldschmidt), Progress in Math. 31,
Birkhäuser, Boston-Basel-Stuttgart, 1983, p p . 199-220.
[27] Philippou P., Indépendance algébrique de valeurs de fonctions exponentielles
^-adiques, J. reine angew. Math. 329 (1981), p p . 42-51.
[28] Philippon P . , Variétés abéliennes et indépendance algébrique I, Inventiones
Math. 70 (1983), pp. 289-318.
[29] Philippon P . , Variétés abéliennes et indépendance algébrique I I , Inventiones
Math. 72 (1983), pp. 389-405.
[30] Philippon P., Sous-groupes à n paramètres et indépendance algébrique, Approximations diophantiennes et nombres transcendants, Luminy 1982 (eds. D. Bertrand
and M. Waldschmidt), Progress in Math. 31, Birkhäuser, Boston-Basel-Stuttgart, 1983, pp. 221-234.
[31] Keyssat E., Un critère d'indépendance algébrique, J. reine angew. Math. 329
(1981), pp. 66-81.
[32] Smelev A. A., On the Method of A. O. Gelfond in the Theory of Transcendental
Numbers, Mat. Zametki 10 (1971), pp. 415-426; Math. Notes 10 (1971), p p . 672678.
[33] Tijdeman E., On the Number of Zeroes of General Exponential Polynomials,
Indagationes Math. 33 (1971), pp. 1-7.
[34] Tijdeman K., On the Algebraic Independence of Certain Numbers, Indagationes
Math. 33 (1971), pp. 146-162.
[35] Tijdeman K., An Auxiliary Kesult in the Theory of Transcendental Numbers,
J. Number Th. 5 (1973), pp. 80-94.
[36] Waldschmidt M., Indépendance algébrique des valeurs de la fonction exponentielle, Bull. Soc. Math. France 99 (1971), p p . 285-304.
[37] Waldschmidt M., Transcendance et exponentielles en plusieurs variables, Inventiones Math. 63 (1981), pp. 97-127. .
[38] Waldschmidt M., Sur certains caractères du groupe des classes d'idèles d'un
corps de nombres, Sém. Theorie Nombres Paris (Sém.
Delange-Pisot-Poitou)
502
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
£47]
[48]
Section 3 : D . W. Masser
1980/81, Progress in Math. 22, Birkhäuser, Boston-Basel-Stuttgart, 1982, p p .
323-335.
'
Waldschmidt M., Sous-groupes analytiques de groupes algébriques, Ann. of
Math. 117 (1983), p p . 627-657.
Waldschmidt M., A Lower Bound for the#-adio Eank of the Units of an Algebraic
Number Field, Coll. Number Theory, Janos Bolyai Math. Soc. Budapest, July
1981; Topics in Classical Number Theory 34.
Waldschmidt M., Dépendance de logarithmes dans les groupes algébriques,
Approximations
diophantiennes et nombres transcendants, Luminy 1982 (eds.
D. Bertrand and M. Waldschmidt), Progress in Math. 31, Birkhäuser, BostonBasel-Stuttgart, 1983, pp. 289-328.
Warkentin P . , Algebraische Unabhängigkeit gewisser p-adischer Zahlen, Diplomarbeit, Freiburg, 1978.
Wüstholz G., Über das abelsche Analogon des Lindemannschen Satzes, Inventiones Math. 72 (1983), p p . 363-388.
Wüstholz G., MulUplizitäts ab Schätzungen auf algebraischen Gruppen.
Wüstholz G., Some Remarks on a Conjecture of M. Waldschmidt, Approximations
diophantiennes et nombres transcendants, Lummy 1982 (eds. D . Bertrand and
M. Waldschmidt), Progress in Math. 31, Birkhäuser, Boston-Basel-Stuttgart,
1983, pp. 329-336.
Wüstholz G., Algebraic Values of Analytic Homomorphisms of Algebraic Groups,
Preprint.
Wüstholz G., Zum Periodenproblem, to appear in Inventiones
Math.
Wüstholz G., Transzendenzeigenschaften von Perioden elliptischer Integrale,
to appear in J. reine angew. Math.
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF NOTTINGHAM
UNIVERSITY PARK
NOTTINGHAM, U.K.