\'Equir\'epartition de sommes exponentielles (travaux de Katz)

TL;DR

This paper investigates the distribution of exponential sums over finite fields, such as Gauss and Kloosterman sums, as the underlying characters vary over larger field extensions, extending previous work by Katz and Deligne.

Contribution

It extends Katz's results to new cases, analyzing the equidistribution of exponential sums via monodromy and Tannakian methods for fixed sheaves over larger fields.

Findings

Established equidistribution results for exponential sums over finite fields.

Connected monodromy groups to the distribution of sums.

Extended Katz's multiplicative case to broader settings.

Abstract

Many exponential sums over finite fields, including Gauss sums and Kloosterman sums, arise as the Fourier transform with respect to a character of the trace function of an $\ell$-adic sheaf on a commutative algebraic group. We study the equidistribution of these sums when the sheaf is fixed but the character varies over larger and larger extensions of the finite field. For the additive group, monodromy governs equidistribution by a theorem of Deligne. A few years ago, Katz solved the multiplicative variant of the question in a work where Tannakian ideas play an essential role.