Arithmetic on Elliptic Curves with Complex Multiplication
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776 Benedict H. Gross
Arithmetic on Elliptic Curves with Complex Multiplication With an Appendix by B. Mazur
Springer-Verlag Berlin Heidelberg New York 1980
Author Benedict H. Gross Mathematics Department Princeton University Princeton, NJ 0 8 5 4 4 USA
A M S Subject Classifications (1980): 10B10, 10D25, 12A25, 1 4 K 2 2 ISBN 3-540-09743-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-09743-0 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data Gross, Benedict, H 1950Arithmetic on elliptic curves with complex multiplication. (Lecture notes in mathematics; 7?6) Bibliography: p. Includes index. 1. Curves, Elliptic. 2. Multiplication, Complex. I. Title. I1.Series: Lecture notes in mathematics (Berlin); 776. QA3.L28 no. 776 [QA567] 510s [516.3'5] 80-334 ISBN 0-387-09743-0 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
To my father -on his
75 t h b i r t h d a y
T a b l e of Contents
0. I. 2. Chapter i: 3. 4. 5. 6. 7. 8. Chapter 2: 9. I0. ll. Chapter 3: 12. 13. 14. Chapter 4: 15. 16. 17. 18. 19. 20. Chapter 5: 21. 22. 23. 24. Appendix
Introduction . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . Notation and Conventions . . . . . . . . . . . .
i 2 3
The t h e o r y of complex m u l t i p l i c a t i o n Elliptic curves . . . . . . . . . . . . . . . Elliptic curves over ~ a n d ~ . . . . . . . . . . The a n a l y t i c t h e o r y of complex m u l t i p l i c a t i o n . . . . . Elliptic curves over p-adie fields . o • , g-adic G a l o i s r e p r e s e n t a t i o n s • . • • T h e arithmetic t h e o r y of c o m p l e x m u l t i p l i c a t i o n . . . . .
4 8 12 14 17 20
A classification Curves over H . . . . . . . . . . . . . . . . D e s c e n d e d curves . . . . . . . . . . . . . . . ~-curves . . . . . . . . . . . . . . . . . .
23
29 32
Local arithmetic A c l a s s i f i c a t i o n over F . . . . . . . . . . . . A rational p-isogeny . . . . . . . . . . . . . . L o c a l invariants and global t o r s i o n . . . . . . . . .
34 38 42
Global a~ithmetic R e s t r i c t i o n of Scalars . . . . . . . . . . . . . The ~ - r a n k . . . . . . . . . . . . . . . . . The first descent . . . . . . . . . . . . . . . A f a c t o r i z a t i o n of the L-series . . . . . . . . . . The sign in the functional e q u a t i o n . . . . . . . . . ~-curves and m o d u l a r forms . . . . . . . . . . . . The ~-curve
45 49 53 57 60 64
A(p)
Periods . . . . . . . . . . . . . . .
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