À la recherche de petites sommes d'exponentielles
Annales de l'Institut Fourier, Tome 52 (2002) no. 1, pp. 47-80.

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 x=1 n exp2 π i f ( x ) / n soit en O(n 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 x=1 n exp2 π i f ( x ) / n is O(n 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.

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
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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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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