On short sums of trace functions

Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe; Raju, Chandra Sekhar; Rivat, Joël; Soundararajan, Kannan

Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 423-449.

Nous considérons des sommes de fonctions oscillantes sur des intervalles contenus dans un groupe fini cyclique, de taille proche de la racine carrée du cardinal du groupe. Nous démontrons tout d’abord des bornes non-triviales pour tout intervalle de longueur à peine plus grande que cette racine carrée (améliorant l’inégalité de Polyá-Vinogradov) pour les fonctions bornées dont la transformée de Fourier est bornée. Nous démontrons ensuite que l’existence d’une borne non-triviale pour un intervalle de taille un peu plus petite que la racine carrée est une propriété stable par transformation de Fourier. Nous donnons des applications liées aux fonctions trace sur les corps finis.

We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the “Polyá-Vinogradov gap” in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform. We then give applications related to trace functions over finite fields.

Classification : 11L07, 11L05, 11T23
Keywords: Short exponential sums, trace functions, van der Corput lemma, completion method, Riemann Hypothesis over finite fields
Mot clés : Somme exponentielle courte, fonction trace, lemme de van der Corput, méthode de complétion, hypothèse de Riemann sur les corps finis

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