On the growth of Selmer groups of an elliptic curve with supersingular reduction in the \(\mathbb Z_2\)-extension of \(\mathbb Q\) (Q851479)

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). If \(E\) has good ordinary reduction at a prime \(p\), the growth of Tate-Shafarevich groups (and Selmer groups) of \(E\) in a \(\mathbb{Z}_p\)-extension can be understood by usual Iwasawa theory. If \(E\) has supersingular reduction at an odd prime \(p\), the most basic case was dealt with in [\textit{M. Kurihara}, Invent. Math. 149, No. 1, 195--224 (2002; Zbl 1033.11028)], under the assumption that \(p\) does not divide the \(L\)-value \(L(E,1)/\Omega_E\) (where \(\Omega_E\) is the Néron period). In this paper the authors determine the precise structure of the 2-adic Selmer groups of \(E\) over the cyclotomic fields \(\mathbb{Q}_n:= \mathbb{Q}(\mu_{2^{n+1}})\), in the case \(a_2(E)= \pm 2\) under the assumption that the normalized \(L\)-value \(L(E,1)/\Omega_E\) and Tamagawa factor \(\text{Tam}(E)\) are prime to 2. Note that the main assumption in this case is just \(2\nmid L(E,1)/\Omega_E\), since Birch and Swinnerton-Dyer conjecture would imply that \(2\nmid\text{Tam}(E)\). A notable distinction from the case of odd \(p\), is that the 2-adic Selmer group of \(E\) over \(\mathbb{Q}_n\) has positive corank (namely, 1). The Selmer group is computed in terms of the modular elements of Mazur and Tate.