Heegner modules and elliptic curves Heegner points on both modular curves and elliptic curves over global fields of any characteristic is 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 being equivalent to the Birch and Swinnerton Dyer conjecture for the corresponding elliptic curves over global fields. TOC:Preface. 1. Introduction. 2. Preliminaries. 3. Bruhat Tits trees with complex multiplication. 4. Heegner sheaves. 5. The Heegner module. 6. Cohomology of the Heegner module. 7. Finiteness of the Tate Shafarevich groups. Appendix A. Rigid analytic modular forms. Appendix B. Automorphic forms and elliptic curves over function fields. References. Index