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2020 Roots of $L$-functions of characters over function fields, generic linear independence and biases
Corentin Perret-Gentil
Algebra Number Theory 14(5): 1291-1329 (2020). DOI: 10.2140/ant.2020.14.1291

Abstract

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field 𝔽 q , for almost all couples of arguments in 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 -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.

Citation

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Corentin Perret-Gentil. "Roots of $L$-functions of characters over function fields, generic linear independence and biases." Algebra Number Theory 14 (5) 1291 - 1329, 2020. https://doi.org/10.2140/ant.2020.14.1291

Information

Received: 9 May 2019; Revised: 14 September 2019; Accepted: 16 December 2019; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07244795
MathSciNet: MR4129387
Digital Object Identifier: 10.2140/ant.2020.14.1291

Subjects:
Primary: 14G10
Secondary: 11J72 , 11N36 , 11R58 , 11T23

Keywords: characters , exponential sums , function fields , Kloosterman sums , large sieve , L-functions , linear independence

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.14 • No. 5 • 2020
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