Let p be an odd prime number, E an elliptic curve defined over a number field. Suppose that E has good reduction at any prime lying above p, and has supersingular reduction at some prime lying above p. In this paper, we construct the plus and the minus Selmer groups of E over the cyclotomic Zp-extension in a more general setting than that of B.D. Kim, and give a generalization of a result of B.D. Kim on the triviality of finite Λ-submodules of the Pontryagin duals of the plus and the minus Selmer groups, where Λ is the Iwasawa algebra of the Galois group of the Zp-extension.

No abstract

No abstract

Let E be an elliptic curve defined over Q. If E has good ordinary reduction at a prime p, the growth of Tate-Shafarevich groups (and Selmer groups) of E in a Z p -extension can be understood by usual Iwasawa theory. But if E has supersingular reduction at p, the growth of Selmer and Tate-Shafarevich groups is more complicated. For an odd prime p, the most basic case was dealt with in [6] where the main assumption was that p does not divide the L-value L(E, 1)/Ω E (where Ω E is the Néron period). The aim of this paper is to study the case p = 2 under the same assumption on theFor a prime number p, we consider the cyclotomic Z p -extension Q ∞ /Q whose n-th layer we denote by Q n , namely Q n is the intermediate field with [Q n : Q] = p n . For an odd p, the condition p | L(E, 1)/Ω E implies rank E(Q ∞ ) = 0 (see [6]), but for p = 2 this does not hold. We will see that for p = 2 the condition p = 2 | L(E, 1)/Ω E would imply that the Selmer groups over Q n always have positive corank for n ≥ 1, hence would imply rank E(Q n ) > 0 if we assume the Birch and Swinnerton-Dyer conjecture (see Corollary 1.2). So the situation is different.As usual, put a p = p + 1 − #E(F p ). In the following, we suppose p = 2 and E has good supersingular reduction at 2. When a 2 = 0, we have two nice Iwasawa functions which describe the p-adic L-function of E by Pollack [11], and we can define ± Selmer groups as in Kobayashi [5], and can study them by the same method as for p > 2. In this paper, we consider the case a 2 = 0 (so a 2 = ±2). Let Sel(E/Q n ) be the Selmer group of E over Q n of E[2 ∞ ] (cf. 2.1). We will determine the Galois module structure (and the structure as an abelian group) of Sel(E/Q n ) completely in the case a 2 = ±2 under the assumption

For an odd prime number p, we study the number of generators of the unramified Iwasawa modules of the maximal multiple Zp-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.