In number theory, the Birch and Swinnerton-Dyer (BSD) conjecture for a Selmer group relates the corank of a Selmer group of an elliptic curve over a number field to the order of zero of theâ€¦ (More)

Since the analytic functional equation holds for the Â±-p-adic L-functions constructed in [7], the algebraic functional equation for the Â±-Selmer groups is expected to hold as well. In this paper, weâ€¦ (More)

We show Kobayashiâ€™s plus/minus Selmer groups of elliptic curves for supersingular primes have no proper Î›-submodule of finite index. As an application we show the precise relation between theâ€¦ (More)

The branched crystal morphology of linear polyethylene formed at various temperatures from thin films has been studied by atomic-force microscopy (AFM), transmission electron microscopy (TEM),â€¦ (More)

By improving the techniques of [10] we prove some symmetric structure of the minus Selmer groups of elliptic curves for supersingular primes. This structure was already known for the Selmer groupsâ€¦ (More)