Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture

Abstract

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.

33 pages

Cited by in corpus (6)

• Low-lying zeros of quadratic Dirichlet $L$-functions: Lower order terms for extended support
• Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture
• Low-lying zeros in families of holomorphic cusp forms: the weight aspect
• A conditional determination of the average rank of elliptic curves
• One-level density of quadratic twists of $L$-functions
• Lower order terms of the one level density of a family of quadratic Hecke $L$-functions