On a Generalization of Tate Dualities with Application to Iwasawa Theory

Let E be an abelian variety deened over a number eld K. Let p be a prime number. Let X(K;E) p 1 be the p-Tate-Shafarevich group of E and S class E p 1 (K) the p 1-Selmer group of E. Thirty years ago, Tate proved a local duality theorem for E and used it to establish a global duality for E, later called Cassels-Tate pairing 3, 14]. It states that there is a pairing between X(K;E) p 1 and the p-Tate-Shafarevich group X(K;E) p 1 for the dual abelian variety E of E and that this pairing is nondegenerate modulo the maximal divisible subgroups. In terms of the Selmer groups, it states that there is a pairing between S class E p 1 (K) and S class E p 1 (K) which is nondegenerate modulo the maximal divisible subgroups. Let fV l g be a compatible system of l-adic representations of Gal(K=K) which are ordinary at p. Let T p be a Gal(K=K)-invariant lattice of V p and deene A = V p =T p. R. Greenberg has recently deened the concept of a p 1-Selmer group for such an A. This concept is a generalization of the classical Selmer group for an abelian variety with good, ordinary reduction or multiplicative reduction at p(Theorem 5). We will prove a local duality theorem(Theorem 1) for such an A and use it to construct a Cassels-Tate type pairing for Greenberg's general Selmer groups (Theorem 2). Let K 1 be any Z p-extension of K. Greenberg 5, 6] also deenes a (strict) Selmer group S str A (K 1) for a compatible system as above. Greenberg uses S str A (K 1) and its nonstrict version to formulate his motivic Iwasawa theory. We will give an application of the general Cassels-Tate pairing to the study of S str A (K 1)(Theorem 3). After xing some notations and conventions in x1, we will state the main theorems in x2. Theorem 1 will be proved in x3. The proof of Theorem 2 will be sketched in x4. We will also discuss some examples there. In x5 we prove the result on Greenberg's strict Selmer groups. I wish to thank my advisor, R. Greenberg. He suggested this problem to me, and his encouragement and helpful conversations were essential to the completion of this work. I would like to thank M. Flach for sending his work …