In search of small exponential sums

Étienne Fouvry[1]; Philippe Michel[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 (2002)

  • Volume: 52, Issue: 1, page 47-80
  • ISSN: 0373-0956

Abstract

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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 exp 2 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.

How to cite

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Fouvry, Étienne, and Michel, Philippe. "À la recherche de petites sommes d'exponentielles." Annales de l’institut Fourier 52.1 (2002): 47-80. <http://eudml.org/doc/115980>.

@article{Fouvry2002,
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 $\sum _\{x=1\}^n \exp \big ( 2\pi i f(x)/n\big )$ soit en $O(n^\{\{1\over 2 \}-\beta _f\})$, où $\beta _f &gt;0$ 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.},
affiliation = {Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex (France); Université Montpellier II, Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex (France)},
author = {Fouvry, Étienne, Michel, Philippe},
journal = {Annales de l’institut Fourier},
keywords = {exponential sums over a finite field; Kloosterman and Salié sums; monodromy; Sato-Tate law; large sieve},
language = {fre},
number = {1},
pages = {47-80},
publisher = {Association des Annales de l'Institut Fourier},
title = {À la recherche de petites sommes d'exponentielles},
url = {http://eudml.org/doc/115980},
volume = {52},
year = {2002},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 1
SP - 47
EP - 80
AB - 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 $\sum _{x=1}^n \exp \big ( 2\pi i f(x)/n\big )$ soit en $O(n^{{1\over 2 }-\beta _f})$, où $\beta _f &gt;0$ 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.
LA - fre
KW - exponential sums over a finite field; Kloosterman and Salié sums; monodromy; Sato-Tate law; large sieve
UR - http://eudml.org/doc/115980
ER -

References

top
  1. C. Camacho, P. Sad, Invariant varieties through singularities of holomorphic vector fields, Annals of Math. 115 (1982) Zbl0503.32007MR657239
  2. R. C. Baker, G. Harman, The Brun-Titchmarsh Theorem on average, Analytic Number Theory, Proceedings of a Conference in honor of H. Halberstam 138 ; vol. 1 (1996), 39-103, Birhäuser Zbl0853.11078
  3. E. Bombieri, J. Friedlander, H. Iwaniec, Primes in Arithmetic Progressions to Large Moduli. II, Math. Annalen 277 (1987), 361-393 Zbl0625.10036MR891581
  4. H. Davenport, Multiplicative Number Theory, 74 (1980), Springer-Verlag Zbl0453.10002MR606931
  5. P. Deligne, Application de la formule des traces aux sommes trigonométriques, SGA4 1/2 569 (1977), Springer-Verlag Zbl0349.10031
  6. M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionnenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272 Zbl0025.02003MR5125
  7. W. Duke, J. B. Friedlander, H. Iwaniec, Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math (2) 141 (1995), 423-441 Zbl0840.11003MR1324141
  8. N. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over ℚ , Inv. Math. 89 (1987), 207-220 Zbl0631.14024MR903384
  9. P. D. T. A. Elliott, H. Halberstam, A conjecture in prime number theory, Symp. Math. (INDAM Rome, 1968--69) 4, 59-72 Zbl0238.10030
  10. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, vol II (1953), Mc Graw-Hill Company, Inc. Zbl0052.29502MR58756
  11. E. Fouvry, Théorème de Brun-Titchmarsh ; Application au théorème de Fermat, Invent. Math. 79 (1985), 383-407 Zbl0557.10035MR778134
  12. E. Fouvry, H. Iwaniec, On a theorem of Bombieri-Vinogradov type, Mathematika 27 (1980), 135-152 Zbl0469.10027MR610700
  13. E. Fouvry, M. R. Murty, On the Distribution of Supersingular Primes, Canadian Journal of Mathematics 48 (1996), 81-104 Zbl0864.11030MR1382477
  14. D. R. Heath-Brown, Artin's conjecture for primitive roots, Q. J. Math. Oxford II 37 (1986), 27-38 Zbl0586.10025MR830627
  15. D. R. Heath-Brown, S. J. Patterson, The distribution of Kummer sums at prime arguments, J. reine u. angewandte Math. 310 (1979), 111-130 Zbl0412.10028MR546667
  16. H. Iwaniec, Rosser's sieve, Acta. Arith. 36 (1980), 171-202 Zbl0435.10029MR581917
  17. H. Iwaniec, Topics in Classical Automorphic Forms, 17 (1997), American Mathematical Society Zbl0905.11023MR1474964
  18. N. M. Katz, Monodromy groups attached to families of exponential sums, Duke Math. J. (1987), 41-56 Zbl0643.12004MR885774
  19. N. M. Katz, Gauss Sums, Kloosterman Sums and Monodromy Groups, 116, Princeton University Press Zbl0675.14004MR955052
  20. N. M. Katz, Exponential sums and differential equations, 124, Princeton University Press Zbl0731.14008MR1081536
  21. N. M. Katz, Exponential sums over finite fields and differential equations over the complex numbers, some interactions, Bull. Am. Math. Soc. 23 (1990), 269-309 Zbl0727.11057MR1032857
  22. N. M. Katz, P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, Colloquium Pub. 45 (1999), Amer. Math. Soc., Providence, Rhode Island Zbl0958.11004
  23. S. Lang, H. Trotter, Frobenius in G L 2 -extensions, 504 (1976), Springer-Verlag Zbl0329.12015MR568299
  24. P. Michel, Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman I, Invent. Math. 121 (1995), 61-78 Zbl0844.11055MR1345284
  25. P. Michel, Minoration de sommes d'exponentielles, Duke Math. J. 95 (1998), 227-240 Zbl0958.11056MR1652005
  26. L. J. Mordell, On some exponential sums related to Kloosterman sums, Acta Arith. 21 (1972), 65-69 Zbl0249.10027MR306133
  27. H. Salié, Über die Kloostermannschen Summen S ( u , v ; q ) , Math. Zeitschr. 34 (1931), 91-109 Zbl0002.12801
  28. F. Spitzer, Principles of Random Walk, (1964), Van Nostrand Company Zbl0119.34304MR171290
  29. M. Sugiura, Unitary Representations and Harmonic Analysis -- An Introduction, (1990) Zbl0697.22001MR1049151
  30. R. C. Vaughan, Mean Value Theorems in Prime Number Theory, J. London Math.Soc (2) 10 (1975), 153-162 Zbl0314.10028MR376567

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