The equidistribution of L-functions of twists by Witt vector Dirichlet characters over function fields

TL;DR

This paper generalizes Katz's equidistribution results for L-functions of Dirichlet characters over function fields to include twists of arbitrary Galois representations, establishing broader distribution and independence properties as q increases.

Contribution

It extends Katz's results to L-functions of twists of any Galois representation, including independence of different representations, over function fields.

Findings

L-functions of twists are equidistributed as q approaches infinity.

Independence of L-functions of different Galois representations is established.

Generalization includes characters with squarefree conductor, extending previous results.

Abstract

Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary Galois representation by Dirichlet characters, including independence of the L-functions of twists of different representations by the same Dirichlet character. A similar generalization, without the independence statement, for characters with squarefree conductor, was proven by Hall, Keating, and Roditty-Gershon.