In [13], the author proved, under some very general conditions, that short sums of -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 -dimensional Kloosterman sums studied by N. Katz in [6] and in [7] when the field gets large.
This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli , as tends to infinity among the prime numbers and is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.
Dans [13], l’auteur démontre, sous des hypothèses très générales, que les sommes courtes des fonctions traces -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 de dimension étudiées par N. Katz dans [6] et [7] lorsque le corps grandit.
Dans cet article, on considère le cas des sommes courtes des sommes de Kloosterman normalisées de module une puissance d’un nombre premier , lorsque tend vers l’infini parmi les nombres premiers et est un entier fixé. Sous des hypothèses très naturelles, on démontre la convergence en loi vers une variable aléatoire gaussienne réelle standard.
Keywords: Kloosterman sums, moments
Guillaume Ricotta 1
@article{AMBP_2019__26_1_101_0, author = {Guillaume Ricotta}, title = {Distribution of short sums of classical {Kloosterman} sums of prime powers moduli}, journal = {Annales math\'ematiques Blaise Pascal}, pages = {101--117}, publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal}, volume = {26}, number = {1}, year = {2019}, doi = {10.5802/ambp.385}, language = {en}, url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.385/} }
TY - JOUR AU - Guillaume Ricotta TI - Distribution of short sums of classical Kloosterman sums of prime powers moduli JO - Annales mathématiques Blaise Pascal PY - 2019 SP - 101 EP - 117 VL - 26 IS - 1 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.385/ DO - 10.5802/ambp.385 LA - en ID - AMBP_2019__26_1_101_0 ER -
%0 Journal Article %A Guillaume Ricotta %T Distribution of short sums of classical Kloosterman sums of prime powers moduli %J Annales mathématiques Blaise Pascal %D 2019 %P 101-117 %V 26 %N 1 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.385/ %R 10.5802/ambp.385 %G en %F AMBP_2019__26_1_101_0
Guillaume Ricotta. Distribution of short sums of classical Kloosterman sums of prime powers moduli. Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 101-117. doi : 10.5802/ambp.385. https://ambp.centre-mersenne.org/articles/10.5802/ambp.385/
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