Introduction To Elliptic Curves And Modular Forms, Second Edition
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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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. The second edition of this text includes an updated bibliography indicating the latest, dramatic changes in the direction of proving the Birch and Swinnerton conjecture. It also discusses the current state of knowledge of elliptic curves. Booknews A text for a graduate mathematics course, covering the properties of elliptic curves and modular forms, emphasizing certain connections with number theory. Uses the congruent number problem as the basis, rather than any of the more advanced or more algebraically oriented approaches. First published in 1984, and updated to discuss current ideas and note recent publications. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Preface to the First EditionPreface to the Second EditionCh. IFrom Congruent Numbers to Elliptic Curves11Congruent numbers32A certain cubic equation63Elliptic curves94Doubly periodic functions145The field of elliptic functions186Elliptic curves in Weierstrass form227The addition law298Points of finite order369Points over finite fields, and the congruent number problem43Ch. IIThe Hasse-Weil L-Function of an Elliptic Curve511The congruence zeta-function512The zeta-function of E[subscript n]563Varying the prime p644The prototype: the Riemann zeta-function705The Hasse-Weil L-function and its functional equation796The critical value90Ch. IIIModular forms981SL[subscript 2](Z) and its congruence subgroups982Modular forms for SL[subscript 2](Z)1083Modular forms for congruence subgroups1244Transformation formula for the theta-function1475The modular interpretation and Hecke operators153Ch. IVModular Forms of Half Integer Weight1761Definitions and examples1772Eisenstein series of half integer weight for [actual symbol not reproducible](4)1853Hecke operators on forms of half integer weight2024The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem212Answers, Hints, and References for Selected Exercises223Bibliography240Index245
\ BooknewsA text for a graduate mathematics course, covering the properties of elliptic curves and modular forms, emphasizing certain connections with number theory. Uses the congruent number problem as the basis, rather than any of the more advanced or more algebraically oriented approaches. First published in 1984, and updated to discuss current ideas and note recent publications. Annotation c. Book News, Inc., Portland, OR (booknews.com)\ \
