Title:

Arithmetic of certain padic Galois representations

In the first part of this work we prove a formula relating the special value of a certain Lfunction to the order of a certain Selmer group. More precisely, let E be an elliptic curve with complex multiplication by K, a quadratic imaginary extension of π of class number 1, and suppose that E is defined over K. Let I_{f} denote the group of fractional ideals of K prime to f, the conductor of E, and ψ the Grössencharacter attached to E. One may consider the Lfunctions attached to the homomorphisms ψ^{k} for positive integers k and ask whether analogues of the Birch and Swinnerton  Dyer conjecture hold. Our main result on Lfunctions and Selmer groups is the following. Theorem 0.0.1 Let k be a positive integer and p > k a prime where E has good supersingular reduction. Then with the understanding that both sides may be simultaneously infinite. Here H^{1}_{f }(K, A_{k}) is the Selmer group for the representation associated to ψ^{k} by Class Field Theory, and Ω is a certain period of E. We furthermore show that both sides of the above equality are finite if k ≥ 3, and extend this to the case k = 2 when E is assumed to be defined over π. In the following two sections we bound the Selmer group of the dual representation by using the functional equation, then deduce results over π when E is assumed to be defined over π. The method we use is modelled on Rubin's approach to the Birch and Swinnerton  Dyer conjecture and uses the two variable Main Conjecture proved by Rubin and Kato's explicit reciprocity law. After the work mentioned above was completed it came to the attention of the author that some similar results have been obtained independently by Han. We subsequently realised that his results together with results of Nekovár could be used to provide information about Chow groups of arbitrary codimension for self products of CM elliptic curves.
