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

TL;DR

This paper establishes a lower bound for the 2-adic valuation of the algebraic part of the central value of Hecke L-functions associated with elliptic curves with complex multiplication over ield, specifically for curves of the form y^2=x^3+Dx.

Contribution

It provides a new lower bound for the 2-adic valuation of central L-values of CM elliptic curves over ield, advancing understanding of their arithmetic properties.

Findings

Lower bound for 2-adic valuation established

Results apply to elliptic curves y^2=x^3+Dx over ield

Enhances knowledge of L-value divisibility properties

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 $E_{-D}$.