Modular Units
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
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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, 1927II. 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 76 5 4 3 2 I ISBN 978-1-4419-2813-9 DOI 10.1007/978-1-4757-1741-9
ISBN 978-1-4757-1741-9 (eBook)
Contents
Introduction
IX
Chapter I
Distributions on Toroidal Groups §1. §2. §3. §4. §5. §6.
The Cartan Group Distributions Stickel berger Distributions Lifting Distributions from Q/Z Bernoulli-Cartan Numbers Universal Distributions
2 4 8 11 12
17
Chapter 2
Modular Units §1. §2. §3. §4. §5. §6.
The Klein Forms and Siegel Functions Units in the Modular Function Field The Siegel Units as Universal Distribution Thc Precise Distribution Relations The Units over Z The Weierstrass Units
24
25 34 37
42 48 50
Chapter 3
Quadratic Relations §1. Formal Quadratic Relations §2. The Even Primitive Elements §3. Weierstrass Forms §4. The Klein Forms §5. The Siegel Group
58 58 62 66 68
75
Chapter 4
The Siegel Units Are Generators
81
§1. Statement of Results
RI
§2. Cyclotomic Integers
li4
v
Contents
*3. *4. *5. *6. §7.
Remarks on q-Expansions The Prime Power Case The Composite Case Dependence of ~ Projective Limits
87 90 94 103 104
Chapter 5
The Cuspidal Divisor Class Group on X(N) §1. *2. §3. §4. §5. §6. §7. *8. §9.
The Stickelberger Ideal The Prime Power Case, p ~ 5 Computation of the Order Eigencomponents at Level p p-Adic Orders of Character Sums Proof of the Theorems The Special Group The Special Group Disappears on Xdp) Projective Limits
110
111 115 118 122 126 131 133 140 141
Chapter 6
The Cuspidal Divisor Class Group on Xl (N) §1. Index of the Stickelberger Ideal §2. The p-Primary Part at Level p
§3. §4. §5. §6.
Part of the Cuspidal Divisor Class Group on XdN) Computation of a Class Number Projective Limits Projective Limit of the Trivial Group
146 147 151 152 159 165 168
Chapter 7
Modular Units on Tate Curves
172
§1. Specializations of Divisors and Functions at Infinity §2. Non-Degeneracy of the Units §3. The Value of a Gauss Sum
173 181 186
Chapter 8
Diophantine Applications §1. Integral Points
§2. Correspondence with the Fermat Curve §3. Torsion Points VI
190 190 193
197
Contents Chapter
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