Duke Mathematical Journal

Algebraic trace functions over the primes

Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study sums over primes of trace functions of ℓ-adic sheaves. Using an extension of our earlier results on algebraic twists of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann hypothesis over finite fields, we prove general estimates with power saving for such sums. We then derive various concrete applications.

Article information

Source
Duke Math. J., Volume 163, Number 9 (2014), 1683-1736.

Dates
First available in Project Euclid: 12 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1402578556

Digital Object Identifier
doi:10.1215/00127094-2690587

Mathematical Reviews number (MathSciNet)
MR3217765

Zentralblatt MATH identifier
1318.11103

Subjects
Primary: 11N05: Distribution of primes
Secondary: 11N 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11N35: Sieves 11F11: Holomorphic modular forms of integral weight 11T23: Exponential sums 11L05: Gauss and Kloosterman sums; generalizations

Citation

Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe. Algebraic trace functions over the primes. Duke Math. J. 163 (2014), no. 9, 1683--1736. doi:10.1215/00127094-2690587. https://projecteuclid.org/euclid.dmj/1402578556


Export citation

References

  • [1] V. Blomer, Subconvexity for twisted L-functions on $\operatorname{GL} (3)$, Amer. J. Math. 134 (2012), 1385–1421.
    Mathematical Reviews (MathSciNet): MR2975240
    Digital Object Identifier: doi:10.1353/ajm.2012.0032
  • [2] A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
    Mathematical Reviews (MathSciNet): MR1102012
  • [3] J. Bourgain, More on the sum-product phenomenon in prime fields and its applications, Int. J. Number Theory 1 (2005), 1–32.
    Mathematical Reviews (MathSciNet): MR2172328
    Digital Object Identifier: doi:10.1142/S1793042105000108
  • [4] J. Bourgain, On the Fourier-Walsh spectrum of the Moebius function, Israel J. Math. 197 (2013), 215–235.
    Mathematical Reviews (MathSciNet): MR3096614
    Digital Object Identifier: doi:10.1007/s11856-013-0002-2
  • [5] J. Bourgain and M. Z. Garaev, Sumsets of reciprocals in prime fields and multilinear Kloosterman sums, preprint, arXiv:1211.4184v1 [math.NT].
    arXiv: 1211.4184v1
  • [6] J. Bourgain, P. Sarnak, and T. Ziegler, “Disjointness of Moebius from horocycle flows” in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math. 28, Springer, New York, 2013, 67–83.
    Mathematical Reviews (MathSciNet): MR2986954
    Digital Object Identifier: doi:10.1007/978-1-4614-4075-8_5
  • [7] P. Deligne, Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4 1/2), Lecture Notes in Math. 569, Springer, Berlin, 1977.
    Mathematical Reviews (MathSciNet): MR463174
  • [8] P. Deligne, Finitude de l’extension de $\mathbb{Q}$ engendrée par des traces de Frobenius, en caractéristique finie, Mosc. Math. J. 12 (2012), 497–514.
    Mathematical Reviews (MathSciNet): MR3024820
  • [9] H. Esnault and M. Kerz, A finiteness theorem for Galois representations of function fields over finite fields (after Deligne), Acta Math. Vietnam. 37 (2012), 531–562.
    Mathematical Reviews (MathSciNet): MR3058662
  • [10] E. Fouvry, Autour du théorème de Bombieri-Vinogradov, Acta Math. 152 (1984), 219–244.
    Mathematical Reviews (MathSciNet): MR741055
    Digital Object Identifier: doi:10.1007/BF02392198
  • [11] E. Fouvry, Sur le problème des diviseurs de Titchmarsh, J. Reine Angew. Math. 357 (1985), 51–76.
    Mathematical Reviews (MathSciNet): MR783533
  • [12] E. Fouvry, E. Kowalski, and Ph. Michel, Algebraic twists of modular forms and Hecke orbits, preprint, arXiv:1207.0617v4 [math.NT].
    arXiv: 1207.0617v4
  • [13] E. Fouvry, E. Kowalski, and Ph. Michel, Counting sheaves using spherical codes, Math. Res. Lett. 20 (2013), 305–323.
    Mathematical Reviews (MathSciNet): MR3151649
    Digital Object Identifier: doi:10.4310/MRL.2013.v20.n2.a8
  • [14] E. Fouvry and Ph. Michel, Sur certaines sommes d’exponentielles sur les nombres premiers, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 93–130.
    Mathematical Reviews (MathSciNet): MR1604298
    Digital Object Identifier: doi:10.1016/S0012-9593(98)80019-0
  • [15] E. Fouvry and Ph. Michel, Sommes de modules de sommes d’exponentielles, Pacific J. Math. 209 (2003), 261–288.
    Mathematical Reviews (MathSciNet): MR1978371
    Digital Object Identifier: doi:10.2140/pjm.2003.209.261
  • [16] E. Fouvry and Ph. Michel, Sur le changement de signe des sommes de Kloosterman, Ann. of Math. (2) 165 (2007), 675–715.
    Mathematical Reviews (MathSciNet): MR2335794
    Digital Object Identifier: doi:10.4007/annals.2007.165.675
  • [17] E. Fouvry and I. Shparlinski, On a ternary quadratic form over primes, Acta Arith. 150 (2011), 285–314.
    Mathematical Reviews (MathSciNet): MR2842959
    Digital Object Identifier: doi:10.4064/aa150-3-5
  • [18] J. B. Friedlander, K. Gong, and I. Shparlinski, Character sums over shifted primes (in Russian), Mat. Zametki 88, no. 4 (2010), 605–619; English translation in Math. Notes 88, no. 3-4 (2010), 585–598.
    Mathematical Reviews (MathSciNet): MR2882222
    Digital Object Identifier: doi:10.4213/mzm8692
  • [19] J. B. Friedlander and H. Iwaniec, Incomplete Kloosterman sums and a divisor problem, with an appendix by B. J. Birch and E. Bombieri, Ann. of Math. (2) 121 (1985), 319–350.
    Mathematical Reviews (MathSciNet): MR786351
    Digital Object Identifier: doi:10.2307/1971175
  • [20] B. Green, On (not) computing the Möbius function using bounded depth circuits, Combin. Probab. Comput. 21 (2012), 942–951.
    Mathematical Reviews (MathSciNet): MR2981162
    Digital Object Identifier: doi:10.1017/S0963548312000284
  • [21] G. Harman, Trigonometric sums over primes, I, Mathematika 28 (1981), 249–254.
    Mathematical Reviews (MathSciNet): MR645105
    Digital Object Identifier: doi:10.1112/S0025579300010305
  • [22] D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity, Canad. J. Math. 34 (1982), 1365–1377.
    Mathematical Reviews (MathSciNet): MR678676
    Digital Object Identifier: doi:10.4153/CJM-1982-095-9
  • [23] J. Heinloth, B.-C. Ngô, and Z. Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2) 177 (2013), 241–310.
    Mathematical Reviews (MathSciNet): MR2999041
    Digital Object Identifier: doi:10.4007/annals.2013.177.1.5
  • [24] L. K. Hua, Additive Theory of Prime Numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965.
    Mathematical Reviews (MathSciNet): MR194404
  • [25] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoam., Madrid, 1995.
    Mathematical Reviews (MathSciNet): MR1325466
  • [26] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004.
    Mathematical Reviews (MathSciNet): MR2061214
  • [27] H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of L-functions, Publ. Math. Inst. Hautes Études Sci. 91 (2000), 55–131.
    Mathematical Reviews (MathSciNet): MR1828743
    Digital Object Identifier: doi:10.1007/BF02698741
  • [28] A. A. Karatsuba, Sums of characters with prime numbers, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 299–321.
    Mathematical Reviews (MathSciNet): MR271040
  • [29] A. A. Karatsuba, Sums of Legendre symbols of quadratic polynomials with prime numbers (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 42, no. 2 (1978), 315–324, 470; English translation in Math. USSR-Izv. 12, no. 2 (1978), 299–308.
    Mathematical Reviews (MathSciNet): MR485724
  • [30] A. A. Karatsuba, Distribution of pairs of residues and nonresidues of special form (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 51, no. 5 (1987), 994–1009, 1117–1118; English translation in Math. USSR-Izv. 31, no. 2 (1988), 307–323.
    Mathematical Reviews (MathSciNet): MR925091
  • [31] N. M. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Ann. of Math. Stud. 116, Princeton Univ. Press, Princeton, 1988.
    Mathematical Reviews (MathSciNet): MR955052
  • [32] N. M. Katz, Rigid Local Systems, Ann. of Math. Stud. 139, Princeton Univ. Press, Princeton, 1996.
    Mathematical Reviews (MathSciNet): MR1366651
  • [33] K. Matomäki, A note on signs of Kloosterman sums, Bull. Soc. Math. France 139 (2011), 287–295.
    Mathematical Reviews (MathSciNet): MR2869308
  • [34] Ph. Michel, Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman, I, Invent. Math. 121 (1995), 61–78.
    Mathematical Reviews (MathSciNet): MR1345284
    Digital Object Identifier: doi:10.1007/BF01884290
  • [35] Ph. Michel, Minorations de sommes d’exponentielles, Duke Math. J. 95 (1998), 227–240.
    Mathematical Reviews (MathSciNet): MR1652005
    Digital Object Identifier: doi:10.1215/S0012-7094-98-09507-2
    Project Euclid: euclid.dmj/1077229697
  • [36] Ph. Michel, Autour des conjectures de Sato-Tate, Ph.D. dissertation, Université de Paris-Sud, Paris, France, 1995.
  • [37] H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, Berlin, 1971.
    Mathematical Reviews (MathSciNet): MR337847
  • [38] N. J. E. Pitt, On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms, J. Amer. Math. Soc. 26 (2013), 735–776.
    Mathematical Reviews (MathSciNet): MR3037786
    Digital Object Identifier: doi:10.1090/S0894-0347-2012-00750-4
  • [39] P. Sarnak, Mobius randomness and dynamics, Not. S. Afr. Math. Soc. 43 (2012), 89–97.
    Mathematical Reviews (MathSciNet): MR3014544
  • [40] J. Sivak-Fischler, Crible étrange et sommes de Kloosterman, Acta Arith. 128 (2007), 69–100.
    Mathematical Reviews (MathSciNet): MR2306565
    Digital Object Identifier: doi:10.4064/aa128-1-4
  • [41] J. Sivak-Fischler, Crible asymptotique et sommes de Kloosterman, Bull. Soc. Math. France 137 (2009), 1–62.
    Mathematical Reviews (MathSciNet): MR2496700
  • [42] Z. Yun, Examples of Kloosterman sheaves, in preparation.

Duke University Press

Duke Mathematical Journal

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more