A refined conjecture of Mazur-Tate type for Heegner points

In [MT1], B. Mazur and J. Tate present a “refined conjecture of Birch and Swinnerton-Dyer type” for a modular elliptic curve E. This conjecture relates congruences for certain integral homology cycles on E(C) (the modular symbols) to the arithmetic of E over Q. In this paper we formulate an analogous conjecture for E over suitable imaginary quadratic fields, in which the role of the modular symbols is played by Heegner points. A large part of this conjecture can be proved, thanks to the ideas of Kolyvagin on the Euler system of Heegner points. In effect the main result of this paper can be viewed as a generalization of Kolyvagin’s result relating the structure of the Selmer group of E over K to the Heegner points defined in the Mordell-Weil groups of E over ring class fields of K. An explicit application of our method to the Galois module structure of Heegner points is given in section 2.2. Acknowledgements: I wish to thank Massimo Bertolini with whom I have had many fruitful discussions on the topics of this paper. I am also grateful to my advisor Benedict Gross for guiding me towards this topic. This research was funded in part by a Natural Sciences and Engineering Research Council of Canada (NSERC) ’67 award, and by a Sloan doctoral dissertation fellowship.