Moments of Artin-Schreier L-functions

TL;DR

This paper computes moments of Artin-Schreier L-functions over finite fields, revealing their statistical behavior and confirming their unitary symmetry type as the genus grows.

Contribution

It provides explicit formulas for moments of Artin-Schreier L-functions and establishes their symmetry type, extending understanding of their distribution.

Findings

Exact formulas for moments of L-functions

Confirmation of unitary symmetry type

Analysis of moments as genus tends to infinity

Abstract

We compute moments of $L$-functions associated to the polynomial family of Artin--Schreier covers over $\mathbb{F}_q$, where $q$ is a power of a prime $p>2$, when the size of the finite field is fixed and the genus of the family goes to infinity. More specifically, we compute the $k^{\text{th}}$ moment for a large range of values of $k$, depending on the sizes of $p$ and $q$. We also compute the second moment in absolute value of the polynomial family, obtaining an exact formula with a lower order term, and confirming the unitary symmetry type of the family.