Heegner Modules and Elliptic Curves

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.

List of contents

Preface.- Introduction.- Preliminaries.- Bruhat-Tits trees with complex multiplication.- Heegner sheaves.- The Heegner module.- Cohomology of the Heegner module.- Finiteness of the Tate-Shafarevich groups.- Appendix A.: Rigid analytic modular forms.- Appendix B.: Automorphic forms and elliptic curves over function fields.- References.- Index.

Summary

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.

Product details

• Authors: M. L. Brown, Martin L Brown, Martin L. Brown
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2007
• EAN: 9783540222903
• ISBN: 978-3-540-22290-3
• No. of pages: 517
• Weight: 780 g
• Series: Lecture Notes in Mathematics; Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; B, Algebraische Geometrie, Mathematics and Statistics, Algebraic Geometry, Number Theory, finite field, Tate Conjecture, Tate-Shafarevich Groups