On the coefficients of the Taylor expansion of $L$-functions of elliptic curves

TL;DR

This paper studies the behavior of Taylor coefficients of elliptic curve $L$-functions, proving nonvanishing under GRH for large quadratic twists and providing unconditional lower bounds.

Contribution

It establishes nonvanishing results for Taylor coefficients of elliptic curve $L$-functions in quadratic twist families, both conditionally and unconditionally.

Findings

Coefficients are nonvanishing under GRH for large discriminants.

Unconditional lower bounds are derived for the number of nonvanishing coefficients.

Results rely on moments of derivatives of quadratic twist $L$-functions.

Abstract

In this paper, we investigate the coefficients of the Taylor expansion of the complex $L$-series of any elliptic curve over $\mathbb{Q}$. We prove that, in the family of quadratic twists by all the discriminants $d$, these coefficients are nonvanishing under GRH when $d$ is sufficiently large. Unconditionally, we obtain a general lower bound for the number of nonvanishing coefficients in the family of quadratic twists, through a series of results from the moments of the central values of the derivatives of quadratic twists of modular $L$-function.