À la recherche de petites sommes d'exponentielles

Fouvry, Étienne; Michel, Philippe

Annales de l'Institut Fourier, Tome 52 (2002), p. 47-80 / Harvested from Numdam

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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.

Publié le : 2002-01-01

MR 1881570 | Zbl 1014.11048

DOI : https://doi.org/10.5802/aif.1876
Classification: 11L05, 11L07, 11L20, 11T23, 14D05
Mots clés: sommes d'exponentielles sur un corps fini, sommes de Kloosterman et de Salié, monodromie, loi de Sato-Tate, grand crible

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     author = {Fouvry, \'Etienne and Michel, Philippe},
     title = {\`A la recherche de petites sommes d'exponentielles},
     journal = {Annales de l'Institut Fourier},
     volume = {52},
     year = {2002},
     pages = {47-80},
     doi = {10.5802/aif.1876},
     mrnumber = {1881570},
     zbl = {1014.11048},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_2002__52_1_47_0}
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Fouvry, Étienne; Michel, Philippe. À la recherche de petites sommes d'exponentielles. Annales de l'Institut Fourier, Tome 52 (2002) pp. 47-80. doi : 10.5802/aif.1876. http://gdmltest.u-ga.fr/item/AIF_2002__52_1_47_0/

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