Vanishing and Non-Vanishing Dirichlet Twists of L-Functions of Elliptic Curves

TL;DR

This paper investigates the conditions under which the central values of twisted L-functions of elliptic curves vanish or not, focusing on Dirichlet characters of specific orders, to understand their behavior and implications.

Contribution

It provides new insights into the vanishing patterns of L(E,1,χ) for elliptic curves when twisted by Dirichlet characters of fixed order.

Findings

Characterizes when L(E,1,χ) vanishes or not

Identifies patterns based on the order of Dirichlet characters

Contributes to understanding the Birch and Swinnerton-Dyer conjecture

Abstract

Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,\chi) of the twisted L-function as \chi ranges over Dirichlet characters of given order.