A refined conjecture of Mazur-Tate type for Heegner points
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- Darmon, H. Invent Math (1992) 110: 123. doi:10.1007/BF01231327
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In [MT1], Mazur and Tate present a “refined conjecture of Birch and Swinnerton-Dyer type” for a modular elliptic curveE. This conjecture relates congruences for certain integral homology cycles onE(C) (the modular symbols) to the arithmetic ofE overQ. In this paper we formulate an analogous conjecture forE over a suitable imaginary quadratic fieldK, 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 ofE overK to the Heegner points defined in the Mordell-Weil groups ofE over ring class fields ofK. An explicit application of our method to the Galois module structure of Heegner points is given in Sect. 2.2.