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)

@article{AIF_2002__52_1_47_0,
     author = {Fouvry, \'Etienne and Michel, Philippe},
     title = {\`A la recherche de petites sommes d'exponentielles},
     journal = {Annales de l'Institut Fourier},
     pages = {47--80},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {1},
     year = {2002},
     doi = {10.5802/aif.1876},
     mrnumber = {1881570},
     zbl = {1014.11048},
     language = {fr},
     url = {https://www.numdam.org/articles/10.5802/aif.1876/}
}

TY  - JOUR
AU  - Fouvry, Étienne
AU  - Michel, Philippe
TI  - À la recherche de petites sommes d'exponentielles
JO  - Annales de l'Institut Fourier
PY  - 2002
SP  - 47
EP  - 80
VL  - 52
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://www.numdam.org/articles/10.5802/aif.1876/
DO  - 10.5802/aif.1876
LA  - fr
ID  - AIF_2002__52_1_47_0
ER  -

%0 Journal Article
%A Fouvry, Étienne
%A Michel, Philippe
%T À la recherche de petites sommes d'exponentielles
%J Annales de l'Institut Fourier
%D 2002
%P 47-80
%V 52
%N 1
%I Association des Annales de l’institut Fourier
%U https://www.numdam.org/articles/10.5802/aif.1876/
%R 10.5802/aif.1876
%G fr
%F AIF_2002__52_1_47_0

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. https://www.numdam.org/articles/10.5802/aif.1876/

Bibliographie
Cité par

[BH] R. C. Baker; G. Harman The Brun-Titchmarsh Theorem on average, Analytic Number Theory, Proceedings of a Conference in honor of H. Halberstam (Progress in Mathematics), Volume 138 ; vol. 1 (1996), pp. 39-103 | Zbl

[BFI] E. Bombieri; J. Friedlander; H. Iwaniec Primes in Arithmetic Progressions to Large Moduli. II, Math. Annalen, Volume 277 (1987), pp. 361-393 | DOI | MR | Zbl

[Da] H. Davenport Multiplicative Number Theory, Graduate Texts in Mathematics, 74, Springer-Verlag, 1980 | MR | Zbl

[Del] P. Deligne Application de la formule des traces aux sommes trigonométriques, SGA4 1/2 (Lecture Notes in Math.), Volume 569 (1977) | Zbl

[Deu] M. Deuring Die Typen der Multiplikatorenringe elliptischer Funktionnenkörper, Abh. Math. Sem. Univ. Hamburg, Volume 14 (1941), pp. 197-272 | DOI | JFM | MR | Zbl

[DFI] W. Duke; J. B. Friedlander; H. Iwaniec Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math (2), Volume 141 (1995), pp. 423-441 | DOI | MR | Zbl

[E] N. Elkies The existence of infinitely many supersingular primes for every elliptic curve over $ℚ$ , Inv. Math., Volume 89 (1987), pp. 207-220 | MR | Zbl

[EH] P. D. T. A. Elliott; H. Halberstam A conjecture in prime number theory, Symp. Math. (INDAM Rome, 1968--69), Volume 4, pp. 59-72 | Zbl

[EMOT] A. Erdelyi; W. Magnus; F. Oberhettinger; F. G. Tricomi Higher Transcendental Functions, vol II, Mc Graw-Hill Company, Inc., 1953 | MR | Zbl

[F] E. Fouvry Théorème de Brun-Titchmarsh ; Application au théorème de Fermat, Invent. Math., Volume 79 (1985), pp. 383-407 | DOI | MR | Zbl

[FI] E. Fouvry; H. Iwaniec On a theorem of Bombieri-Vinogradov type, Mathematika, Volume 27 (1980), pp. 135-152 | DOI | MR | Zbl

[FM] E. Fouvry; M. R. Murty On the Distribution of Supersingular Primes, Canadian Journal of Mathematics, Volume 48 (1996), pp. 81-104 | DOI | MR | Zbl

[H-B] D. R. Heath-Brown Artin's conjecture for primitive roots, Q. J. Math. Oxford II, Volume 37 (1986), pp. 27-38 | DOI | MR | Zbl

[H-BP] D. R. Heath-Brown; S. J. Patterson The distribution of Kummer sums at prime arguments, J. reine u. angewandte Math., Volume 310 (1979), pp. 111-130 | MR | Zbl

[I1] H. Iwaniec Rosser's sieve, Acta. Arith., Volume 36 (1980), pp. 171-202 | MR | Zbl

[I2] H. Iwaniec Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17, American Mathematical Society, 1997 | MR | Zbl

[K1] N. M. Katz Monodromy groups attached to families of exponential sums, Duke Math. J. (1987), pp. 41-56 | DOI | MR | Zbl

[K2] N. M. Katz Gauss Sums, Kloosterman Sums and Monodromy Groups, Annals of Maths. Studies, 116, Princeton University Press | MR | Zbl

[K3] N. M. Katz Exponential sums and differential equations, Annals of Math. Studies, 124, Princeton University Press | MR | Zbl

[K4] N. M. Katz Exponential sums over finite fields and differential equations over the complex numbers, some interactions, Bull. Am. Math. Soc., Volume 23 (1990), pp. 269-309 | DOI | MR | Zbl

[KS] N. M. Katz; P. Sarnak Random Matrices, Frobenius Eigenvalues and Monodromy, Colloquium Pub., Volume 45 (1999) | Zbl

[LT] S. Lang; H. Trotter Frobenius in $G L_{2}$ -extensions, Lecture Notes in Math., 504, Springer-Verlag, 1976 | MR | Zbl

[Mi1] P. Michel Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman I, Invent. Math., Volume 121 (1995), pp. 61-78 | DOI | MR | Zbl

[Mi2] P. Michel Minoration de sommes d'exponentielles, Duke Math. J., Volume 95 (1998), pp. 227-240 | DOI | MR | Zbl

[Mo] L. J. Mordell On some exponential sums related to Kloosterman sums, Acta Arith., Volume 21 (1972), pp. 65-69 | MR | Zbl

[Sa] H. Salié Über die Kloostermannschen Summen $S (u, v; q)$ , Math. Zeitschr., Volume 34 (1931), pp. 91-109 | DOI | Zbl

[Sp] F. Spitzer Principles of Random Walk, The University Series in Higher Mathematics, Van Nostrand Company, 1964 | MR | Zbl

[Su] M. Sugiura Unitary Representations and Harmonic Analysis -- An Introduction, North-Holland Mathematical Library, 1990 | MR | Zbl

[V] R. C. Vaughan Mean Value Theorems in Prime Number Theory, J. London Math.Soc (2), Volume 10 (1975), pp. 153-162 | DOI | MR | Zbl

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math., Volume 115 (1982) | MR | Zbl

Kowalski, E.; Soundararajan, K. Exponential Sums, Twisted Multiplicativity, and Moments, Analysis at Large (2022), p. 299 | DOI:10.1007/978-3-031-05331-3_13
PERRET–GENTIL, CORENTIN Gaussian distribution of short sums of trace functions over finite fields, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 163 (2017) no. 3, p. 385 | DOI:10.1017/s0305004117000020
Patterson, S. J. The Asymptotic Distribution of Exponential Sums, I, Experimental Mathematics, Volume 12 (2003) no. 2, p. 135 | DOI:10.1080/10586458.2003.10504489
Fouvry, Etienne; Michel, Philippe Sommes de modules de sommes d’exponentielles, Pacific Journal of Mathematics, Volume 209 (2003) no. 2, p. 261 | DOI:10.2140/pjm.2003.209.261

Cité par 4 documents. Sources : Crossref