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

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.