On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of ℚ(−7)

Abstract

For a modular elliptic curve [Formula: see text] and its quadratic twists [Formula: see text], we give equivalent conditions such that the [Formula: see text]-Selmer group [Formula: see text] is minimal, namely, it is of order [Formula: see text]. One of these conditions is described by the [Formula: see text]-value [Formula: see text]. The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of [Formula: see text]). Also we prove the full Birch–Swinnerton-Dyer conjecture when the equivalent conditions are satisfied. This generalizes a result by J. Coates, Y. Li, Y. Tian and S. Zhai.