Euler Systems and Refined Conjectures of Birch Swinnerton-Dyer Type

The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their Lfunctions poses many deep and difficult questions. The theme of this paper is the Birch and Swinnerton Dyer conjecture, and certain refinements that were proposed by Mazur and Tate. We will formulate analogues of these conjectures over imaginary quadratic fields involving Heegner points, and explain how the fundamental work of V.A. Kolyvagin sheds light on these new conjectures. §1 Preliminaries. The relationship between arithmetic objects (such as global fields, or varieties over global fields) 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 defined over a global field. Let E be an elliptic curve defined over the rational numbers. The conjecture of Shimura-Taniyama-Weil asserts that E is modular, i.e., is equipped with a rational map φ : X0(N) −→ E, where X0(N) is the modular curve of level N , defined over Q, which parameterizes elliptic curves with a distinguished cyclic N -isogeny. We assume that E has this property. (For a specific E this can be checked by a finite computation.) 1991 Mathematics Subject Classification. Primary 11G40; Secondary 11G05. The ideas for this paper are part of the author’s Harvard PhD thesis; he gratefully acknowledges the support of Harvard University, and in particular of his advisor, B.H. Gross. Financial support was provided at various stages by the Natural Sciences and Enginnering Research Council and a Sloan Doctoral Dissertation Fellowship. Finally, this paper was written during a visit to the I.H.E.S. in the summer of 1991. 1 2 EULER SYSTEMS AND REFINED CONJECTURES The pullback of the Néron differential ω on E is a cusp form of weight 2 on X0(N), φω = cf(q)dq/q, where f(q) = ∑ n>0 anq n is normalized so that a1 = 1, and c denotes the Manin constant associated to the modular parametrization φ. Let K be a number field. (In the applications we discuss, K will be either Q, or a quadratic field.) Given a place v of K, let Kv denote the completion of K at v, and let kv denote the residue field if v is non-archimedean. Let S be a finite set of places of K, and let ES(K) denote the subgroup of finite index in E(K) which is defined by the exact sequence 0 −→ ES(K) −→ E(K) −→ ⊕v∈SEns(kv) ⊕ E/E0(K) −→ JS −→ 0, where Ens(kv) denotes the group of non-singular points in the special fiber of E at v, and where E/E0(K) is the group of connected components in the Néron model E/OK of E over SpecOK . §1.1 Arithmetic invariants. The triple (E, K, S) gives rise to the following arithmetic data: 1. The rank r of the finitely generated abelian groups E(K) and ES(K). 2. The order of the conjecturally finite Shafarevich-Tate group III(E/K). This is the group of elements in H(K, E) whose restrictions in H(Kv, E) are 0 for all places v of K. It arises naturally in descent arguments. 3. The Néron-Tate canonical height associated to the Poincaré divisor on E × E; it is a positive-definite bilinear pairing 〈 , 〉NT : E(K) × E(K) −→ R. It gives rise to a regulator term. We describe the general construction of the regulator suggested in [MT2]. While not strictly necessary for this section, the extra generality will be useful later. Let 〈 , 〉 denote a G-valued pairing on A×B, where G is an abelian group and A and B are subgroups of finite index in E(K). We embed G as the degree one elements in the graded algebra Sym(G) = ⊕r≥0Sym(G). If A and B are free, the regulator R(A, B) in Sym(G) is defined to be the determinant of the r × r matrix (〈Pi, Qj〉), where P1, . . . , Pr and Q1, . . . , Qr denote integral bases for A and B respectively which induce compatible orientations on E(K)⊗R. The element R(A, B) is homogeneous of degree r and can be viewed as belonging to Sym(G). If A and B are not free, one needs the hypothesis that there exist subgroups A ′ and B ′ of A and B which are free and of finite index, such that multiplication by [A : A ′ ][B : B ′ ] induces an isomorphism on G. This hypothesis is satisfied, for example, if G = R, or if G is finite and of order prime