Local Statistics for Zeros of Artin-Schreier L-functions

TL;DR

This paper investigates the local zero statistics of Artin-Schreier L-functions over finite fields, comparing empirical results with predictions from random matrix theory for three different families.

Contribution

It provides explicit zero-density computations for three families of Artin-Schreier L-functions and confirms their alignment with random matrix models.

Findings

1-level zero-density matches unitary/symplectic models

2-level density agrees with random matrix predictions

Results apply to ordinary, polynomial, and odd-polynomial families

Abstract

We study the local statistics of zeros of $L$-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier $L$-functions: the ordinary, polynomial (the $p$-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.