Elliptic Functions and Transcendence

157 Pages·1975·1.779 MB·English

by  Dr. D. W. Masser (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .31 Eckmann
734
David Masser
Elliptic Functions
dna Transcendence
Springer-Verlag
Berlin.Heidelberg. New York 91 5 7
.rD .D W. Masser
Dept. of Mathematics
ytisrevinU of mahgnittoN
ytisrevinU Park
mahgnittoN NG7 2RD/England
Library of Congress Cataloging in Publication Data
Masser, David William, 19L8-
Elliptic functions and transcendence.
(Lecture notes in mathematics ; 437)
Bibliography: p.
Includes index.
I. Punctions, En_liptic. 2. N~o~?oers~ Transcendental.
I. Title. II. Series: Lecture notes in mathematics
(Berlin) ; 437.
QA3.L28 no. 437 cQA343~ 510'.8s~15~.3537
74-32365
AMS Subject Classifications 1( 970): 10D25, 10F35, 33A25
ISBN 3-540-07136-9 Springer-Verlag Berlin • Heidelberg • New York
tSBN 0-387-07136-9 Springer-Verlag New York • Heidelberg • Berlin
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CONTENTS
Introduction
Chapter I. A transcendence measure . . . . . . . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . I
1.2. Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . I
1.3. Proof of Theorem I . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter II. Vanishing of linear forms without comp,!ex multiplication 16
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2,2. Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . 16
2.3. The main lemma . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4. The auxiliary function . . . . . . . . . . . . . . . . . . . . . 26
2.5, Proof of Theorem II . . . . . . . . . . . . . . . . . . . . . . 32
Chapter III. Vanishing of linear forms with complex multiplication . 36
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2. The upper bound . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3. Proof of Theorem III . . . . . . . . . . . . . . . . . . . . . . 41
Chapter IV. An effective proof 0f a theorem of Coates . . . . . . . . 44
Chapter V. A lower bound for nonyvanishing linear forms . . . . . . . 49
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2. The auxiliary function . . . . . . . . . . . . . . . . . . . . . 50
5.3. Proof of Theorem IV . . . . . . . . . . . . . . . . . . . . . . 61
Chapter VI. Lemmas on elliptic functions with complex multiplication 63
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2. Multiplication formulae . . . . . . . . . . . . . . . . . . . . 63
6.3. Estimates for algebraic points . . . . . . . . . . . . . . . . . 68
Chapter VII. Linear forms in algebraic points . . . . . . . . . . . . 77
7,1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Four lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.3 A simplification . . . . . . . . . . . . . . . . . . . . . . . . 86
7.4 The auxiliary function . . . . . . . . . . . . . . . . . . . . . 88
7.5 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.6 The case n = 2; a postscript . . . . . . . . . . . . . . . . . . 111
IV
Appendix I
A non-analytic modular function . . . . . . . . . . . . . . . . . . . 113
Appendix II
Zeros of polynomials in several variables . . . . . . . . . . . . . . 123
Appendix III
A transcendence theorem for algebraic points . . . . . . . . . . . . . 132
Appendix IV
Rational points on curves of genus one with complex multiplication. .137
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
INTRODUCTION
These Notes are concerned with some new transcendence
properties, or more exactly linear independence properties,
of certain numbers associated with elliptic functions.
The purpose of this general introduction is to describe
the results and provide a historical context for them.
Essentially we treat special periodic functions f(z)
of a single complex variable z. The methods of transcen-
dence theory are most likely to succeed when these functions
have an algebraic addition theorem. Such functions are
rather strictly delineated by an old theorem of Weierstrass,
which says that up to algebraic dependence f(z) must be one
of the following.
First, f(z) can be the exponential function , ~z e and
we take ~ as an algebraic number so that the differential
equation of f(z) is defined over the field ~ of algebraic
numbers. Then it is a classical result of Lindemann that
the fundamental period 2~i/e of f(z) is a transcendental
number.
The second possibility is that f(z) is a Weierstrass
elliptic function ~(z) satisfying the differential equation
(~J(z)) 2 = 4(~(z)) 3 - g2~(z) - g3
for complex numbers g2, g3 with 32g ~ 27g32 . Once again
we take g2, g3 to be algebraic numbers (and this assump-
tion will be maintained throughout these Notes). This
is a doubly periodic function, and so we may choose a
fundamental pair of periods ~i, ~2 with the imaginary part
VI
of ~2/~1 positive. We also consider the associated Weier-
strass zeta function ~(z) defined by '~ )z( = -P(z) and
normalized additively to be an odd function of z. This
possesses an adequate addition theorem, and there exist
quasi-periods i~ and ~2 corresponding to ~i and ~2 such
that
~(z + ~i ) = ~(z) + Hi i( = 1,2).
Siegel was the first to investigate the arithmetical
properties of ~i and ~2; he proved in [26~ that they
cannot both be algebraic numbers. A few years later
Schneider in his fundamental researches improved on this
result by obtaining the transcendence not only of wl and
~2 but also of nl and ~2. An account of this can be found
in his book [25]. The exponential case above is naturally
included by considering the five numbers ~I, ~z, ~i, ~2 and
2~i together; thus we see that each is transcendental. The
most general theorem extending this statement would assert
the algebraic independence of these five numbers, but there
are two reasons why this cannot yet be proved. First, the
general techniques are not available, and secondly, the
Legendre relation
~2~i - ~i~2 = 2~i
shows that such a theorem would be false.
So we turn to the more fruitful question of linear
independence: are the above five numbers together with 1
linearly independent over the field A? The answer to
this involves a splitting of cases. We consider the set
of complex numbers I ~ 0 such that ~(z) and ~(~z) are
algebraically dependent functions. By the addition theorem,
this set together with I = O forms a field ~. If ~ ~
the field of rational numbers we say that ~(z) has complex
VII
multiplication; this is the exception rather than the rule
and then ~ must be a complex quadratic extension ~of Q.
Otherwise ~= ~ means that ~(z) is without complex multi-
plication and in this case an affirmative answer to the
above question had been conjectured.
Partial answers were first obtained by Schneider in
[25], who proved that 2~i/el, D1/Wl, and ~1+~1 are trans-
cendental for non-zero algebraic numbers ~ and .~ In
addition he showed that ~2/~I is transcendental under the
necessary condition that ~(z) has no complex multiplication;
for otherwise e2/~i lies in ~. These results were consid-
erably generalized by Baker with the aid of his many-
variable techniques. In two papers [21, 13[ he proved the
transcendence of any non-zero linear combination of el,
~2, ~i and ~z with algebraic coefficients, and shortly
afterwards Coates ~ obtained the extended result for
the five numbers ~i, ~2, ~l, n2 and 2zi. In Chapter II
of these Notes we prove that these five numbers are them-
selves linearly independent when ~(z) has no complex
multiplication; in conjunction with the result of Coates
this establishes the linear independence of i, ~i, ~2, ~I,
~2 and 2zi over ~. In other words, the vector space V
spanned over ~ by these six numbers is of maximal dimension
six.
The proof attempts to imitate the proof of the theorem
of Coates mentioned above. The main difficulty arises from
the periodicity of the auxiliary function ~; this means
that the zeros of ~ on the real diagonal do not carry as
much information as they normally would. Therefore we are
compelled to extrapolate further to deduce that ¢ is small
on a large part of the whole complex diagonal. After
VIII
using a device to replace ~ by a simpler function ~, we
may regard ~ as a polynomial, and as such it inherits the
smallness of ~ on a large subset ~ of ~3. To show that
is well-distributed in a rather weak sense is a matter of
diophantine approximation (this appears in a slightly dis-
guised form in the proof); we then require arguments
essentially in one complex variable to see that this im-
possibly restricts ~.
The results of Schneider mentioned above on the
transcendence of 2~i/~i, ni/~i and w2/~i can be regarded
as first steps towards the theorem of Chapter II. The
next (and most recent) step was taken by Coates who showed
in ~3] that ~i, ~2 and 2~i are linearly independent over
~when ~(z) has no complex multiplication. The new idea
in the proof of this result involves an appeal to the
deep and extensive theory of Serre on division points of
~(z). However, for historical reasons the theory of
transcendence has always laid emphasis on the concept of
effectiveness; it requires, for example, that all constants
occurring in the proofs should in principle be capable of
explicit evaluation. Serre's theory was not constructed
for transcendence purposes and sometimes fails to satisfy
such criteria. Therefore it is of some interest to have
a more elementary proof of Coates' result which does have
an effective structure. We provide such a proof in Chapter
IV.
So far we have not discussed what happens when ~(z)
has complex multiplication. Since ~2/~i is an algebraic
number it is clearly untrue that the six numbers i, el, ~2,
~i, ~2 and 2~i are linearly independent over A. In
Chapter III we use a very simple argument to prove the
IX
slightly unexpected fact that there is another essentially
distinct linear relation between the six numbers. This
leaves four candidates for a basis of V, namely, i, el, Dl
and 2~i, and in the same chapter we show that these four
numbers are indeed linearly independent over ~. These
results imply that the dimension of V is four in the case
of complex multiplication.
After proving theorems of this type we consider a
secondary objective of transcendence theory. This is to
derive quantitative refinements in the form of measures;
these tend to be more important for applications to other
areas of number theory. In the situation outlined so far
these refinements are of the following kind. If the co-
efficients ~0, ~i, ~2, 81, B2 and y denote algebraic
numbers not all zero, our results show that the expression
A = ~0 + ~1~1 + ~2~2 + ~l~l + 82D2 + y.2zi
does not vanish (under additional conditions depending
on complex multiplication). The problem is then to find a
positive lower bound for the absolute value of A in terms
of the degrees and heights of the coefficients.
We shall give two examples of such estimates. In
Chapter I we treat the case ~0 = 81 = B2 = Y = O when there
is no complex multiplication; this is equivalent to a
transcendence measure for ~2/~i. Feldman [16] has obtained
a measure for a more general class of number~ but the lower
bound turns out to be too weak for the application we have
to make in Chapter II. Although our proof is just a
modification of Feldman's argument*, it is important to
Since wrote * I Notes, these Feldman has published paper a ni which he carries out
this modification himself. This appears ni Acta Arithmetica, ,42 477-489 and the re-
suit ti contains si slightly sharper mine. than
X
give the details: first because of the crucial improve-
ment in the lower bound, and secondly because all Feldman
explicitly obtains in his paper is a measure of irration-
ality, which would be trivial for our particular number
~2/~i (since it is not real).
In our other example we consider the general form
A when ~0 ~ O. Previously Baker ]4[ had given an estimate
in the case 18 = 28 = T = O of the form
IAI > C exp(-(log H)K),
where H is the maximum of the heights of co, el and e2,
C > 0 depends only on their degrees and the numbers ~I and
m2, and < is a large absolute constant. In Chapter V we
show that for the more general form < can be taken as any
number greater than 1 provided only e0 ~ O; more precisely
IAI > c exp(- log H(log log H)~+~),
where H is now the maximum of the heights of e0, el, e2,
81, 82 and y, and C > O depends only on their degrees, the
numbers ~I and ~2, and the arbitrarily chosen e > O. Here
the dependence on H is quite near best possible, for stan-
dard arguments show that the absolute value of A can
frequently be smaller than H K- for some positive absolute
constant ~.
This completes the outline of our study of the
periods associated with ~(z). The remaining chapters
investigate the arithmetical nature of a more general
class of numbers, the algebraic points of ~(z). These