Distribution of short sums of classical Kloosterman sums of prime powers moduli

Abstract

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums ${\mathsf{Kl}}_{2,{\mathbb{F}}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field ${\mathbb{F}}_q$ gets large. \par This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli ${\mathsf{Kl}}_{p^n}$, as $p$ tends to infinity among the prime numbers and $n\ge 2$ is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.