Drinfeld Modular Curves
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1231 Ernst-Ulrich Gekeler
Drinfeld Modular Curves
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Ernst-Ulrich Gekeler Max-Planck Institut fur Mathematik Gottfried-Claren-Str. 26, 5300 Bonn 3, Federal Republic of Germany
Mathematics Subject Classification (1980): 12A90, 10D 12, 10D07, 14H 25 ISBN 3-540-17201-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17201-7 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Table of Contents Introduction
o
Notations
I
Drinfeld Modules 1. Algebraic Theory
2
2. Analytic Theory
5
3. The Operation of
7
GL(r,Afl
4. The Modular Schemes for
r
and
2
9
II
III
1. Adelic Description of Lattices
10
2. Lattice Invariants
13
3. Morphisms of Lattices
15
Partial Zeta Functions 1. Relations with Lattice Sums
17
2. The Rational Function
20
3. Evaluation at IV
V
s
=
0
Z (Sl a,n and s
22
Drinfeld Modules of Rank 1 1. The Case of a Rational Function Field
25
2. Normalization
26
3. Some Lemmata
30
4. Computation of Lattice Invariants
33
5. Distinguished 1-D-Modules
38
Modular Curves over
C
1. The "Upper Half-Plane"
40
2. Group Actions
43
3. Modular Forms
47
4. Elliptic Points
50
5. Modular Forms and Differentials
51
Appendix: The First Betti Number of
r
54
IV
VI
VII
Expansions around Cusps 1. Preparations
58
2. Formulae
60
3. Computation of the Factors
61
4. The 6-Functions
65
5. Some Consequences
71
Modular Forms and Functions
1- The Field of Modular Functions 2. The Field of Definition of the Elliptic Points 3. Behavior of E (q-1) at Elliptic Points 4. The Graded Algebra of Modu La r' Forms
VIII
78 82 83 85
5. Higher Modular Curves
86
6. Modular Forms for Congruence Subgroups
92
Complements 1. Hecke Operators
94
2. Connections with the Classification of Elliptic Curves
96
3. Some Open Questions
99
Index
101
List of Symbols
102
Bibliography
104
Introduction The analogy of the arithmetic of number fields with that of "function fields"
(i.e. function fields in one variable over a finite field of
constants) has been known for a long time. This analogy starts with elementary things (structure of rings of integers, ramification theory, product formula ..• ), but reaches into such deep fields like for example -
(abelian and non-abelian) class field theory;
- Iwasawa theory; - special values of L-functions (conjectures of Birch and SwinnertonDyer and of Stark, relations with K-theory); - diophantine geometry (conjecture
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