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The relationship between arithmetic objects (such as global elds, or varieties over global elds) and the analytic properties of their L- functions poses many deep and dicult questions. The theme of this paper is the Birch and Swinnerton Dyer conjecture, and certain renemen ts that were proposed by Mazur and Tate. We will formulate analogues of these conjectures over imaginary quadratic elds involving Heegner points, and explain how the fundamental work of V.A. Kolyvagin sheds light on these new conjectures. x1 Preliminaries. The relationship between arithmetic objects (such as global elds, or varieties over global elds) and the analytic properties of their L-functions poses many deep and subtle questions. The theme of this paper is the Birch Swinnerton-Dyer conjecture, which concerns the case where the arithmetic object in question is an elliptic curve dened over a global eld. Let E be an elliptic curve dened over the rational numbers. The conjecture of Shimura-Taniyama-Weil asserts that E is modular, i.e., is equipped with a rational map ' : X0(N) ! E; where X0(N) is the modular curve of level N, dened over Q, which parameter- izes elliptic curves with a distinguished cyclic N-isogeny. We assume that E has this property. (For a specic E this can be checked by a nite computation.)

Euler systems and refined conjectures of Birch Swinnerton-Dyer type