Introduction to Elliptic Curves and Modular Forms

By: Neal I. Koblitz

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Hardcover | 3 May 1993 | Edition Number 2

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Hardcover
270 Pages

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Revised

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24.77 x 16.51 x 1.91

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The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.

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Preface to the First Edition
Preface to the Second Edition
From Congruent Numbers to Elliptic Curves	p. 1
Congruent numbers	p. 3
A certain cubic equation	p. 6
Elliptic curves	p. 9
Doubly periodic functions	p. 14
The field of elliptic functions	p. 18
Elliptic curves in Weierstrass form	p. 22
The addition law	p. 29
Points of finite order	p. 36
Points over finite fields, and the congruent number problem	p. 43
The Hasse-Weil L-Function of an Elliptic Curve	p. 51
The congruence zeta-function	p. 51
The zeta-function of E[subscript n]	p. 56
Varying the prime p	p. 64
The prototype: the Riemann zeta-function	p. 70
The Hasse-Weil L-function and its functional equation	p. 79
The critical value	p. 90
Modular forms	p. 98
SL[subscript 2](Z) and its congruence subgroups	p. 98
Modular forms for SL[subscript 2](Z)	p. 108
Modular forms for congruence subgroups	p. 124
Transformation formula for the theta-function	p. 147
The modular interpretation and Hecke operators	p. 153
Modular Forms of Half Integer Weight	p. 176
Definitions and examples	p. 177
Eisenstein series of half integer weight for [actual symbol not reproducible](4)	p. 185
Hecke operators on forms of half integer weight	p. 202
The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem	p. 212
Answers, Hints, and References for Selected Exercises	p. 223
Bibliography	p. 240
Index	p. 245
Table of Contents provided by Blackwell. All Rights Reserved.

Introduction to Elliptic Curves and Modular Forms

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

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