Table Of ContentLecture Notes in
Mathematics
Edited by A Dold and B. Eckmann
476
Modular Functions
of One Variable IV
Proceedings of the International Summer School,
University of Antwerp, RUCA,
July 17 - August 3, 1972
Edited by B. J. Birch and W. Kuyk
Spri nger-Verlag
Berlin· Heidelberg· New York 1975
Editors
Dr. Bryan J. Birch
Mathematical Institute
University of Oxford
24-29 8t. Giles
Oxford OX1 3 LB/England
Prof. Willem Kuyk
Leerstoel Algebra
Rijksuniversitair Centrum Antwerpen
2020 Antwerpen
Middelheimlaan 1
Holland
(Revised)
Library of Congress Cataloging in Publication Data
Main entry under title:
Modular functions of one variable.
(Lecture notes in mathematics 320, 349-350,
476)
A NATO Advanced Study Institute; held University
of Antwerp, RUCA, July 17-August 3. 1972.
Includes bibliographies.
1. Functions, Modular--Congresses. I. Kuyk,
Willem, ed. -II. NATO Advanced Study Institute.
III. Series: Lecture notes in mathematics (Berlin)
320. [etc.]
QA3.L28 [QA343] no. 320, etc. 510'.8s [515'.9]
73-78427
AMS Subject Classifications (1970): lOC15, 10 D05, lOD25, 14K22,
14K25
ISBN 3-540·07392-2 Springer·Verlag Berlin' Heidelberg· New York
ISBN 0-387-07392-2 Springer-Verlag New York' Heidelberg· Berlin
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© by Springer-Verlag Berlin' Heidelberg 1975
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Preface
This is Volume 4 of the Proceedings of the International
Summer School on
"Modular functions of one variable and
arithmetical applications"
which took place at RUCA, Antwerp University, from July 17
till August 3, 1972. The preceding three volumes have
been published as issues 320, 349 and 350 of the Lecture
Notes Series.
This final volume contains papers by B. Birch, P. Deligne
and H.P.F. Swinnerton-Dyer, a letter from Tate to Cassels,
and several numerical tables. Its overall theme is the
arithmetic of elliptic curves.
For the acknowledgements regarding financial and material
support, the reader is referred to the preface of volume
320. Once more, the editors wish to thank their major
financial sponsors, NATO, IBM Belgium and Rank Xerox Belgium
for their support.
B.J. BIRCH
W. KUYK
CONTENTS
1. Foreword 1
2. H.P.F. SWINNERTON-DYER~B.J. BIRCH, Elliptic Curves and 2
Modular Functions
3. J. TATE, Algorithm for determining the Type of a Singu 33
lar Fiber in an Elliptic Pencil
4. P. DELIGNE, Courbes Elliptiques: Formulaire (d'apres J. Tate) 53
5. Sources and reliability of the tables 75
Remarks on isogenies 78
6. Table 1 81
7. Table 2 114
8. Table 3 116
9. Table 4 123
10. Table 5 135
11- Table 6 142
12. Corrigenda to Volumes I-III 145
13. List of Adresses of Authors 151
FOREWORD
It was felt that the Antwerp conference would serve as an excellent
pretext for the publication of various documents and tables that have
proved useful for the study of elliptic curves and modular functions, but
which for one reason or another have never been properly published.
The volume begins with four lectures about the arithmetic theory of
elliptic curves parametrised by modular functions, which were shared at
Antwerp between Swinnerton-Dyer and Birch. They are followed by Tate's
letter to Cassels, Deligne's Formulaire, and Swinnerton-Dyer's tables of
elliptic curves of low conductor. Tate's letter contains the standard ac
count of his algorithm for computing the conductor and Neron models of
elliptic curves; by courtesy of the author and the Xerox Corporation it
has been rather widely circulated, though some of the earlier copies are
beginning to fade. For reasons of bulk, Swinnerton-Dyer's tables have
had rather less circulation, though they are potentially very useful. We
have produced, as Table 1, a presumably complete version for conductors up
to 200. (The 1966 version missed a fair number of curves, and several iso
genies; we believe that all the gaps have been filled in the present ver
sion). Table 1 is preceded by a short section in which the reliability
and sources of the various tables, particulary of Table 1, are discussed.
There follows a table of the generators of the Mordell-Weil group for those
curves in Table 1 for which this group has positive rank; a table prepared
by Velu of the Frobenius eigenvalues for primes up to 100 for the curves in
Table 1; and a complete table extracted from F.B. Coghlan's Manchester the
sis of elliptic curves whose conductor has the form 2a3b. Table 5 descri
bes how the homology of H/ro(N) is split into rational eigenspaces of the
Hecke algebra, for N ~ 300; this is an improvement by D.J. Tingley of an
earlier table due to A.O.L. Atkin. The final table, also provided by
Atkin, lists the supersingular equation for primes up to 307.
ELLIPTIC CURVES AND MODULAR FUNCTIONS
,by H. P. F. SWINNERTON-DYER and B.J. BIRCH
CONTENTS
O. Introduction 3
1. The zeta-function of an elliptic curve 4
2. Mellin transforms and Weil's theorem 7
3. The theorems of Eichler and Shimura 11
4. Cusp forms of weight 2 15
5. The structure of XO(N) 21
6. Rational points on E 28
References 31
International Summer School on Modular Functions
Antwerp 1972
3
ELLIPTIC CURVES AND MODULAR FUNCTIONS
o. Introduction.
The purpose of this article is to describe the relations, partly proved
and partly conjectural, between the set of elliptic curves defined over
Q and the set of fields of modular functions for the groups ro(N). The
existence of such relations depends on four things :
(i) The exact shape of the conjectured functional equation for
the global zeta-function of an elliptic curve over Q, and
the associated L-series. This is a special case of a much
more general conjecture, on the functional equation for the
global zeta-function of a non-singular variety, which ~ay be
found in Deligne [2]; but the connection with modular forms
provides evidence for it of a kind which is not available in
the more general case.
(ii) The theorem of Weil [12] which gives necessary and sufficient
conditions for a Dirichlet series to be the Mellin transform
of a cusp form for rO(N). According to the conjecture in (i),
these conditions are satisfied by the zeta-function of an el
liptic curve defined over Q, for some N which depends on the
curve, and the resulting form has weight 2.
(iii) The theorem of Eichler-Shimura; see [9] and [10], Ch. VII.
This implies that if an elliptic curve E defined over Q is
parametrized by modular fUnctions whose q-series expansions
have rational coefficients, so that the canonical differen
tial on E corresponds to a modular form of weight 2, then
the Mellin transform of this modular form is essentially the
zeta-function of E.
4
(iv) The work of Atkin and Lehner [1] which classifies cusp forms
for fO(N).
Taken together, these imply that if an elliptic curve defined over Q has
geometric conductor N (in the sense defined in Tate's article in this
volume) then it can be parametrized by modular functions for rO(N); in
particular it is, up to isogeny, a factor of the Jacobian of the modular
curve XO(N) which corresponds to the group rO(N). Since there is an ef
fective way of decomposing this Jacobian into its simple factors in any
particular case, this enables one to find all the elliptic curves of gi-
ven conductor -assuming certain conjectures.
This parametrization of an elliptic curve E by modular functions also
enables one to reformulate some of the Birch-Swinnerton-Dyer conjectures.
These conjectures relate the group of rational points on E to the beha-
viour of the zeta-function of E near s = 1; in particular they state
that the group of rational points is infinite if and only if ~E(l) = O.
But by the Mellin transform formula ~E(l) is, up to unimportant factors,
the integral over (O,i-) of the modular form of weight 2 associated with
E; so the conj ectures imply that the group of rational points of E is in
finite if and only if the image of (O,i-) in E is homologically trivial.
1. The zeta-function of an elliptic curve.
Let E be an elliptic curve defined over Q, by which we mean a curve of
genus 1 with a distinguished point, both the curve and the point being
defined over Q. We can write E in the affine form
(1)
with the point at infinity as the distinguished point; and we may assume
that this is a minimal model in the sense of Neron -that is, the a. are
~
5
integers and the absolute value of the discriminant ~ is as small as
possible. For any prime p we denote by E = Ep the curve
defined over GFCp), the finite field of p elements, where ai denotes
the image of ai in GF(p). If P does not divide ~ then E is an elliptic
curve and its L-series is defined as
where al ,a2 are the characteristic roots of the Frobenius map
(x,y) ~ CxP,yp) regarded as an element of the ring of endomorphisms of E.
This implies ala2 = p and lall = la21 = pl/2. Moreover if Nv denotes
E
the number of points on defined over GFCpv), including the point at in-
finity, then
which provides an alternative definition of al and a2• In particular we
have
from which they can easily be computed.
If P divides ~ then E is a curve with one singular point; this is neces-
sarily defined over GF(p) and may be
Ci) a double point at which each of the tangent directions is defi-
ned over GFCp), or
Cii) a double point at which the tangent directions are conjugate
over GFCp), or
Ciii) a cusp.
6
The L-series of E is then defined to be (l_u)-l in case (i), (l+u)-l in
case (ii) and 1 in case (iii). As a mnemonic it may be noted that in
the non-singular case and the three singular cases we have
L(E,u)
The global zeta-function of E is now defined to be
where the product is taken over all primes p. This is certainly defined
in Re(s) > 3/2; it is conjectured that it can be analytically continued
to the entire s-plane except for simple poles at the negative integers,
and that it satisfies a functional equation similar to that of the Rie-
mann zeta-function.
To write down the functional equation we must define the geometric conduc
tor of E; this is
where m(E,p) is the number of irreducible components of the fibre of the
Neron model of E over GF(p). For more details see Neron [6] and Ogg
[7]. It is known that m(E,p) = 1 if ~ is non-singular, so that the pro
duct need only be taken over the primes dividing 6. Moreover p exactly
divides N if ~ has a double point; while if ~ has a cusp p2 always divi
des 6 and p2 exactly divides 6 if p > 3. Now write
then it is conjectured that Z(s) can be analytically continued to the en-
tire s-plane as a holomorphic function, and that it satisfies the func-
tional equation
