One-level density of zeros of Dirichlet $L$-functions over function fields

TL;DR

This paper calculates the one-level density of zeros of certain Dirichlet L-functions over function fields, confirming predictions from Random Matrix Theory and identifying the symmetry type as unitary.

Contribution

It provides explicit calculations of zero distributions for specific families of Dirichlet L-functions over function fields, including lower order terms beyond RMT predictions.

Findings

Main terms match RMT predictions

Lower order terms are explicitly computed

Symmetry type confirmed as unitary

Abstract

We compute the one-level density of zeros of order $\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting ($q\not\equiv1\pmod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and lower order terms not predicted by RMT. We also confirm the symmetry type of the families is unitary, supporting Katz and Sarnak's philosophy.