Independence of the zeros of elliptic curve L-functions over function fields

TL;DR

This paper demonstrates that a function field analogue of the Linear Independence hypothesis for zeros of elliptic curve L-functions holds generically within certain families, with strong quantitative bounds on independence.

Contribution

It establishes a generic linear independence result for zeros of elliptic curve L-functions over function fields, extending the understanding of zero distributions in this setting.

Findings

Proves a function field analogue of LI holds generically.

Provides quantitative bounds on the number of elliptic curves with independent zeros.

Extends results to symmetric powers of elliptic curve L-functions.

Abstract

The Linear Independence hypothesis (LI), which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as was pointed out by Rubinstein and Sarnak. In this paper, we prove that a function field analogue of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite field, we give strong quantitative bounds for the number of elements in the family for which the relevant L-functions have their zeros as linearly independent over the rationals as possible.