Modular Units

360 Pages·1981·0.45 MB·English

by  Daniel S. Kubert

Grundlehren der
mathematischen Wissenschaften 244
A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S. S. Chern 1. L. Doob A. Grothendieck
E. Heinz F. Hirzebruch L. Hormander S. Mac Lane
W. Magnus C. C. Moore 1. K. Moser M. Nagata
W. Schmidt D. S. Scott 1. Tits
B. L. van der Waerden
Managing Editors
B. Eckmann S. R. S. Varadhan
Daniel S. Kubert
Serge Lang
Modular Units
Springer Science+Business Media, LLC
AMS Subject Classifications: IOD99, 12A45
Library of Congress Cataloging in Publication Data
Kubert. Daniel S.
Modular units.
(Grundlehren der mathematischen Wissenschaften; 244)
Bibliography: p.
Includes index.
I. Algebraic number theory. 2. Class field
theory. 3. Modules (Algebra) I. Lang, Serge,
1927- II. Title.
QA247.K83 512'.74 81-824 AACR2
© 1981 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1981
Softcover reprint of the hardcover 1st edition 1981
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer Science+Business Media, LLC.
9 8 7 6 5 4 3 2 I
ISBN 978-1-4419-2813-9 ISBN 978-1-4757-1741-9 (eBook)
DOI 10.1007/978-1-4757-1741-9
Contents
Introduction IX
Chapter I
Distributions on Toroidal Groups
§1. The Cartan Group 2
§2. Distributions 4
§3. Stickel berger Distributions 8
§4. Lifting Distributions from Q/Z 11
§5. Bernoulli-Cartan Numbers 12
§6. Universal Distributions 17
Chapter 2
Modular Units 24
§1. The Klein Forms and Siegel Functions 25
§2. Units in the Modular Function Field 34
§3. The Siegel Units as Universal Distribution 37
§4. Thc Precise Distribution Relations 42
§5. The Units over Z 48
§6. The Weierstrass Units 50
Chapter 3
Quadratic Relations 58
§1. Formal Quadratic Relations 58
§2. The Even Primitive Elements 62
§3. Weierstrass Forms 66
§4. The Klein Forms 68
§5. The Siegel Group 75
Chapter 4
The Siegel Units Are Generators 81
§1. Statement of Results RI
§2. Cyclotomic Integers li4
v
Contents
*3. Remarks on q-Expansions 87
*4. The Prime Power Case 90
*5. The Composite Case 94
*6. Dependence of ~ 103
§7. Projective Limits 104
Chapter 5
The Cuspidal Divisor Class Group on X(N) 110
§1. The Stickelberger Ideal 111
*2. The Prime Power Case, p ~ 5 115
§3. Computation of the Order 118
§4. Eigencomponents at Level p 122
§5. p-Adic Orders of Character Sums 126
§6. Proof of the Theorems 131
§7. The Special Group 133
*8. The Special Group Disappears on Xdp) 140
§9. Projective Limits 141
Chapter 6
The Cuspidal Divisor Class Group on Xl (N) 146
§ 1. Index of the Stickelberger Ideal 147
§2. The p-Primary Part at Level p 151
§3. Part of the Cuspidal Divisor Class Group on XdN) 152
§4. Computation of a Class Number 159
§5. Projective Limits 165
§6. Projective Limit of the Trivial Group 168
Chapter 7
Modular Units on Tate Curves 172
§1. Specializations of Divisors and Functions at Infinity 173
§2. Non-Degeneracy of the Units 181
§3. The Value of a Gauss Sum 186
Chapter 8
Diophantine Applications 190
§1. Integral Points 190
§2. Correspondence with the Fermat Curve 193
§3. Torsion Points 197
VI
Contents
Chapter 9
Unramified Units 211
§1. The Invariants ()(c,c') 211
§2. The Index of the Siegel Group 213
§3. The Robert Group 214
§4, Lemmas on Roots of Unity 216
§5. A Refined Index 218
Chapter 10
More Units in the Modular Function Field 224
§1. Transformation of the Klein Forms 224
§2. Klein Forms and Weierstrass Functions 227
§3. More Expressions for Modular Units 229
Chapter II
Siegel-Robert Units in Arbitrary Class Fields 233
§1. Siegel-Ramachandra Invariants as Distributions 233
§2. Stickel berger Elements 241
§3. Ideal Factorization of the Siegel Numbers 246
§4. The Robert Group in the Ray Class Field 252
§5. Taking Roots 260
§6. The Robert Group under the Norm Map 266
Chapter 12
Klein Units in Arbitrary Class Fields 269
§1. The Klein Invariants 269
§2. Behavior under the Artin Automorphism 277
§3. Modular Units in K(I) as Klein Units 285
§4. Modular Units in K(f) as Klein Units 298
§5. A Description of Emo,,(K( i ))12hwN(fl 303
Chapter 13
Computation of a Unit Index 311
§ 1. The Regulator Map and the Inertia Group 311
§2. An Index Computation 317
§3. Freeness Results 321
VIl
Contents
§4. The Index (EI/: Emod(H)) 323
95. More Roots of Unity Lemmas 327
§6. Proof of Theorem 4.2 329
Appendix
The Logarithm of the Siegel Functions 339
Bibliography 351
Index 357
Vlll
Introduction
In the present book, we have put together the basic theory of the units and
cuspidal divisor class group in the modular function fields, developed over
the past few years.
Let i) be the upper half plane, and N a positive integer. Let r(N) be the
subgroup of SL (Z) consisting of those matrices == 1 mod N. Then r(N)\i)
2
is complex analytic isomorphic to an affine curve YeN), whose compactifi
cation is called the modular curve X(N). The affine ring of regular functions
on yeN) over C is the integral closure of C[j] in the function field of X(N)
over C. Here j is the classical modular function. However, for arithmetic
applications, one considers the curve as defined over the cyclotomic field
Q(JlN) of N-th roots of unity, and one takes the integral closure either of
Q[j] or Z[j], depending on how much arithmetic one wants to throw in.
The units in these rings consist of those modular functions which have
no zeros or poles in the upper half plane. The points of X(N) which lie at
infinity, that is which do not correspond to points on the above affine set,
are called the cusps, because of the way they look in a fundamental domain
in the upper half plane. They generate a subgroup of the divisor class group,
which turns out to be finite, and is called the cuspidal divisor class group.
The trivial elements are represented precisely by the divisors of units as
above, called modular units.
Investigation of units and divisor, or ideal, class groups is a classical
activity. In the cyclotomic case, some basic problems remain open (Iwasawa
Leopoldt conjecture and Kummer-Vandiver conjecture-cf. Cyclotomic
Fields). Both in this case, and the case of the cuspidal divisor class group,
the class group is a module over a group ring Z[ G], where G ~ (Z/ NZ)* in
the cyclotomic case, and over a suitable Cartan group in the modular case.
Contrary to the cyclotomic ideal class group, it is now possible to exhibit
IX
Introduction
the cuspidal divisor class group as a cyclic module, since one can immediately
identify the cusps with elements of the Cartan group. The kernel is then the
analogue of the Stickel berger ideal, and corresponds to the divisors of units,
mentioned above, which can be completely described. These units are the
analogues of cyclotomic units.
The classical cyclotomic numbers e2ni/N - 1 satisfy certain relations,
arising from the identity
n
«(T - 1) = TN - 1.
,N=
1
These relations can be axiomatized, and are called distribution relations.
In Chapter 1, we have summarized this algebraic theory, independent of all
applications. We emphasize the universal properties and the general algebraic
structure, as distinguished from the p-adic properties which were emphasized
by Iwasawa and Mazur. The general theory has applications going beyond
the present book, and to other fields.
It is worthwhile here to review some facts from the theory of cyclotomic
units in light of our present approach. For x E Q/Z, x i= 0, we define
g(x) = e21tix - 1.
We view the map
x ~ g(x)
as a map Q/Z - [O} --> C*, that is into the multiplicative group of complex
numbers. We call the numbers g(x) cyclotomic numbers. They have the
following properties.
(1) If the denominator of x is composite, then g(x) is a unit, in the sense of
Dirichlet. If the denominator of x is a prime power pn, then g(x) is a p-unit.
(2) The map satisfies the distribution relation, that is for any positive integer
N, we have
n
g(y) = g(x).
Ny=x
Furthermore, the distribution 9 is essentially even, that is
g( -x) = g(x)(
for some root of unity (. If we view its values as lying in C* III (where
J1 is the group of all roots of unity) then it is an even distribution.
(3) Up to 2-torsion, 9 is the universal even distribution. (Cf. [Bass J.)
x