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Perret-Gentil, Corentin

Roots of \(L\)-functions of characters over function fields, generic linear independence and biases. (English) Zbl 1459.11228

Algebra Number Theory 14, No. 5, 1291-1329 (2020).
Summary: 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 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.

Cited in 6 Documents

MSC:

11T23 Exponential sums
11J72 Irrationality; linear independence over a field
11N36 Applications of sieve methods
11R58 Arithmetic theory of algebraic function fields

Keywords:

exponential sums; linear independence; \(L\)-functions; large sieve; characters; function fields; Kloosterman sums
PDFBibTeX XMLCite
\textit{C. Perret-Gentil}, Algebra Number Theory 14, No. 5, 1291--1329 (2020; Zbl 1459.11228)
Full Text: DOI arXiv

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