Non-vanishing Theorems for Quadratic Twists of Elliptic Curves

TL;DR

This paper demonstrates that for certain elliptic curves over the rationals, many quadratic twists have non-vanishing L-series at s=1, supporting aspects of the Birch-Swinnerton-Dyer conjecture through modular symbols.

Contribution

It establishes non-vanishing results for quadratic twists of elliptic curves using modular symbols, providing explicit examples where the BSD conjecture's 2-part holds.

Findings

Large class of quadratic twists with non-zero L-series at s=1

Verification of the 2-part of BSD conjecture for these twists

Application of modular symbols to non-vanishing theorems

Abstract

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.