# Roots of $L$-functions of characters over function fields, generic linear independence and biases

## TL;DR

This paper investigates the distribution and biases of special sums and angles in function fields, establishing uniformity, linear independence of eigenvalues, and extending sieve methods to handle complex ramification scenarios.

## Contribution

It introduces new results on the distribution of Kloosterman and Birch sums, analyzes biases in Gaussian prime angles, and extends large sieve techniques to wild ramification cases.

## Key findings

- Joint uniform distribution of Kloosterman and Birch sums over finite fields.
- Identification of biases in the distribution of Gaussian prime angles.
- Extension of large sieve methods to wild ramification scenarios.

## Abstract

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick--Waxman and Keating--Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenvalues of $\ell$-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties.