À la recherche de petites sommes d'exponentielles  [ In search of small exponential sums ]
Annales de l'Institut Fourier, Volume 52 (2002) no. 1, p. 47-80

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.

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.

DOI : https://doi.org/10.5802/aif.1876
Classification:  11L05,  11L07,  11L20,  11T23,  14D05
Keywords: exponential sums over a finite field, Kloosterman and Salié sums, monodromy, Sato-Tate law, large sieve
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {1},
     year = {2002},
     pages = {47-80},
     doi = {10.5802/aif.1876},
     zbl = {1014.11048},
     mrnumber = {1881570},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2002__52_1_47_0}
}
Fouvry, Étienne; Michel, Philippe. À la recherche de petites sommes d'exponentielles. Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 47-80. doi : 10.5802/aif.1876. http://www.numdam.org/item/AIF_2002__52_1_47_0/

[BH] R. C. Baker; G. Harman The Brun-Titchmarsh Theorem on average, Analytic Number Theory, Proceedings of a Conference in honor of H. Halberstam, Birhäuser (Progress in Mathematics) Tome 138 ; vol. 1 (1996), pp. 39-103 | Zbl 0853.11078

[BFI] E. Bombieri; J. Friedlander; H. Iwaniec Primes in Arithmetic Progressions to Large Moduli. II, Math. Annalen, Tome 277 (1987), pp. 361-393 | Article | MR 891581 | Zbl 0625.10036

[Da] H. Davenport Multiplicative Number Theory, Springer-Verlag, Graduate Texts in Mathematics, Tome 74 (1980) | MR 606931 | Zbl 0453.10002

[Del] P. Deligne Application de la formule des traces aux sommes trigonométriques, SGA4 1/2, Springer-Verlag (Lecture Notes in Math.) Tome 569 (1977) | Zbl 0349.10031

[Deu] M. Deuring Die Typen der Multiplikatorenringe elliptischer Funktionnenkörper, Abh. Math. Sem. Univ. Hamburg, Tome 14 (1941), pp. 197-272 | Article | JFM 67.0107.01 | MR 5125 | Zbl 0025.02003

[DFI] W. Duke; J. B. Friedlander; H. Iwaniec Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math (2), Tome 141 (1995), pp. 423-441 | Article | MR 1324141 | Zbl 0840.11003

[E] N. Elkies The existence of infinitely many supersingular primes for every elliptic curve over , Inv. Math., Tome 89 (1987), pp. 207-220 | MR 903384 | Zbl 0631.14024

[EH] P. D. T. A. Elliott; H. Halberstam A conjecture in prime number theory, Symp. Math. (INDAM Rome, 1968--69), Tome 4, pp. 59-72 | Zbl 0238.10030

[EMOT] A. Erdelyi; W. Magnus; F. Oberhettinger; F. G. Tricomi Higher Transcendental Functions, Mc Graw-Hill Company, Inc. Tome vol II (1953) | MR 58756 | Zbl 0052.29502

[F] E. Fouvry Théorème de Brun-Titchmarsh ; Application au théorème de Fermat, Invent. Math., Tome 79 (1985), pp. 383-407 | Article | MR 778134 | Zbl 0557.10035

[FI] E. Fouvry; H. Iwaniec On a theorem of Bombieri-Vinogradov type, Mathematika, Tome 27 (1980), pp. 135-152 | Article | MR 610700 | Zbl 0469.10027

[FM] E. Fouvry; M. R. Murty On the Distribution of Supersingular Primes, Canadian Journal of Mathematics, Tome 48 (1996), pp. 81-104 | Article | MR 1382477 | Zbl 0864.11030

[H-B] D. R. Heath-Brown Artin's conjecture for primitive roots, Q. J. Math. Oxford II, Tome 37 (1986), pp. 27-38 | Article | MR 830627 | Zbl 0586.10025

[H-BP] D. R. Heath-Brown; S. J. Patterson The distribution of Kummer sums at prime arguments, J. reine u. angewandte Math., Tome 310 (1979), pp. 111-130 | MR 546667 | Zbl 0412.10028

[I1] H. Iwaniec Rosser's sieve, Acta. Arith., Tome 36 (1980), pp. 171-202 | MR 581917 | Zbl 0435.10029

[I2] H. Iwaniec Topics in Classical Automorphic Forms, American Mathematical Society, Graduate Studies in Mathematics, Tome 17 (1997) | MR 1474964 | Zbl 0905.11023

[K1] N. M. Katz Monodromy groups attached to families of exponential sums, Duke Math. J. (1987), pp. 41-56 | Article | MR 885774 | Zbl 0643.12004

[K2] N. M. Katz Gauss Sums, Kloosterman Sums and Monodromy Groups, Princeton University Press, Annals of Maths. Studies, Tome 116 | MR 955052 | Zbl 0675.14004

[K3] N. M. Katz Exponential sums and differential equations, Princeton University Press, Annals of Math. Studies, Tome 124 | MR 1081536 | Zbl 0731.14008

[K4] N. M. Katz Exponential sums over finite fields and differential equations over the complex numbers, some interactions, Bull. Am. Math. Soc., Tome 23 (1990), pp. 269-309 | Article | MR 1032857 | Zbl 0727.11057

[KS] N. M. Katz; P. Sarnak Random Matrices, Frobenius Eigenvalues and Monodromy, Colloquium Pub., Amer. Math. Soc., Providence, Rhode Island, Tome 45 (1999) | Zbl 0958.11004

[LT] S. Lang; H. Trotter Frobenius in GL 2 -extensions, Springer-Verlag, Lecture Notes in Math., Tome 504 (1976) | MR 568299 | Zbl 0329.12015

[Mi1] P. Michel Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman I, Invent. Math., Tome 121 (1995), pp. 61-78 | Article | MR 1345284 | Zbl 0844.11055

[Mi2] P. Michel Minoration de sommes d'exponentielles, Duke Math. J., Tome 95 (1998), pp. 227-240 | Article | MR 1652005 | Zbl 0958.11056

[Mo] L. J. Mordell On some exponential sums related to Kloosterman sums, Acta Arith., Tome 21 (1972), pp. 65-69 | MR 306133 | Zbl 0249.10027

[Sa] H. Salié Über die Kloostermannschen Summen S(u,v;q), Math. Zeitschr., Tome 34 (1931), pp. 91-109 | Article | Zbl 0002.12801

[Sp] F. Spitzer Principles of Random Walk, Van Nostrand Company, The University Series in Higher Mathematics (1964) | MR 171290 | Zbl 0119.34304

[Su] M. Sugiura Unitary Representations and Harmonic Analysis -- An Introduction, North-Holland Mathematical Library (1990) | MR 1049151 | Zbl 0697.22001

[V] R. C. Vaughan Mean Value Theorems in Prime Number Theory, J. London Math.Soc (2), Tome 10 (1975), pp. 153-162 | Article | MR 376567 | Zbl 0314.10028

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math., Tome 115 (1982) | MR 657239 | Zbl 0503.32007