On the Growth of Selmer Groups of an Elliptic Curve with Supersingular Reduction in the Z₂-extension of Q

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

On pairs of p-adic L-functions for weight-two modular forms

“…the elliptic curve case) quite well in the supersingular case when p is odd 3 . The formulas also match the algebraic ones of Kurihara and Otsuki when p = 2 [KO06]. For the unknown cases (in which p = 2), the formulas thus serve as a prediction of how…”