Lower bound for the 2-adic valuations of central $L$-values of elliptic curves with complex multiplication

Abstract

Let $E_{-D}$ be the elliptic curve $y^2=x^3+Dx$ defined over $K=\mathbb{Q}(i)$ for $D\in K$ which is coprime to 2. In this paper, we give a lower bound for the 2-adic valuation of the algebraic part of the central value of Hecke $L$-function associated to $E_{-D}$.