À la recherche de petites sommes d'exponentielles

Fouvry, Étienne; Michel, Philippe

doi:10.5802/aif.1876

Fouvry, Étienne ¹ ; Michel, Philippe ²

¹ Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex (France)
² Université Montpellier II, Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex (France)

Annales de l'Institut Fourier, Tome 52 (2002) no. 1, pp. 47-80.

Résumé
Abstract

Soit $f (x)$ une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers $n$ , ayant exactement deux facteurs premiers, tels que la somme d’exponentielles $\sum_{x = 1}^{n} exp (2 π i f (x) / n)$ soit en $O (n^{\frac{1}{2} - β_{f}})$ , où $β_{f} > 0$ est une constante ne dépendant que de la géométrie de $f$ . On donne aussi des résultats de répartition du type Sato-Tate, pour certaines sommes de Salié, modulo $n$ , avec $n$ entier comme ci- dessus.

Let $f (x)$ be a rational function, with integer coefficients, satisfying rather general assumptions. We prove the existence of infinitely many integers $n$ , with exactly two prime divisors, such that the exponential sum $\sum_{x = 1}^{n} exp (2 π i f (x) / n)$ is $O (n^{\frac{1}{2} - β_{f}})$ , where $β_{f} > 0$ is a constant only depending on the geometrical data of $f$ . We also give Sato-Tate type results for some Salié sums modulo $n$ , with $n$ an integer as above.

MR Zbl

DOI : 10.5802/aif.1876

Classification : 11L05, 11L07, 11L20, 11T23, 14D05
Mot clés : sommes d'exponentielles sur un corps fini, sommes de Kloosterman et de Salié, monodromie, loi de Sato-Tate, grand crible
Keywords: exponential sums over a finite field, Kloosterman and Salié sums, monodromy, Sato-Tate law, large sieve

Affiliations des auteurs :

Fouvry, Étienne ¹ ; Michel, Philippe ²

¹ Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex (France)
² Université Montpellier II, Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex (France)

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     title = {\`A la recherche de petites sommes d'exponentielles},
     journal = {Annales de l'Institut Fourier},
     pages = {47--80},
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     volume = {52},
     number = {1},
     year = {2002},
     doi = {10.5802/aif.1876},
     mrnumber = {1881570},
     zbl = {1014.11048},
     language = {fr},
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}

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Fouvry, Étienne; Michel, Philippe. À la recherche de petites sommes d'exponentielles. Annales de l'Institut Fourier, Tome 52 (2002) no. 1, pp. 47-80. doi : 10.5802/aif.1876. http://www.numdam.org/articles/10.5802/aif.1876/

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