Vol. 14, No. 5, 2020

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

Corentin Perret-Gentil

Vol. 14 (2020), No. 5, 1291–1329
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 ${\mathbb{𝔽}}_{q}^{×}$, 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 and Waxman, and Keating and 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.

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Keywords
exponential sums, linear independence, L-functions, large sieve, characters, function fields, Kloosterman sums
Mathematical Subject Classification 2010
Primary: 14G10
Secondary: 11J72, 11N36, 11R58, 11T23
Milestones
Received: 9 May 2019
Revised: 14 September 2019
Accepted: 16 December 2019
Published: 13 July 2020
Authors
 Corentin Perret-Gentil Zürich Switzerland