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

Ricotta, Guillaume

Annales mathématiques Blaise Pascal, Tome 26 (2019) no. 1, pp. 101-117.

Dans [13], l’auteur démontre, sous des hypothèses très générales, que les sommes courtes des fonctions traces $\ell$-adiques sur des corps finis de centre variable convergent en loi vers une variable aléatoire gaussienne ou un vecteur aléatoire gaussien. Les ingrédients principaux sont le théorème d’équirépartition de P. Deligne, les travaux de N. Katz et les résultats présentés dans [3]. Ceci s’applique en particulier au sommes de Kloosterman ${\mathsf{Kl}}_{2,{𝔽}_q}$ de dimension $2$ étudiées par N. Katz dans [6] et [7] lorsque le corps ${𝔽}_q$ grandit.

Dans cet article, on considère le cas des sommes courtes des sommes de Kloosterman normalisées de module une puissance d’un nombre premier ${\mathsf{Kl}}_{p^n}$, lorsque $p$ tend vers l’infini parmi les nombres premiers et $n\ge 2$ est un entier fixé. Sous des hypothèses très naturelles, on démontre la convergence en loi vers une variable aléatoire gaussienne réelle standard.

In [13], the author 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 [3]. In particular, this applies to $2$-dimensional Kloosterman sums ${\mathsf{Kl}}_{2,{𝔽}_q}$ studied by N. Katz in [6] and in [7] when the field ${𝔽}_q$ gets large.

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.

Classification : 11T23, 11L05
Mots clés : Kloosterman sums, moments

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