Heegner Modules and Elliptic Curves

Martin L. Brown (auth.)9783540222903, 3-540-22290-1

Heegner points on both modular curves and elliptic curves over global fields of any characteristic form the topic of this research monograph. The Heegner module of an elliptic curve is an original concept introduced in this text. The computation of the cohomology of the Heegner module is the main technical result and is applied to prove the Tate conjecture for a class of elliptic surfaces over finite fields; this conjecture is equivalent to the Birch and Swinnerton-Dyer conjecture for the corresponding elliptic curves over global fields.

Table of contents :
1. Introduction….Pages 1-11
2. Preliminaries….Pages 13-30
3. Bruhat-Tits trees with complex multiplication….Pages 31-74
4. Heegner sheaves….Pages 75-103
5. The Heegner module….Pages 105-222
6. Cohomology of the Heegner module….Pages 223-327
7. Finiteness of Tate-Shafarevich groups….Pages 329-434
Appendix….Pages 435-505
References….Pages 507-510
Index….Pages 511-517