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Geometric and Functional Analysis

, Volume 25, Issue 2, pp 580–657 | Cite as

Algebraic twists of modular forms and Hecke orbits

  • Étienne Fouvry
  • Emmanuel Kowalski
  • Philippe MichelEmail author
Open Access
Article

Abstract

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the -adic Fourier transform introduced by Deligne and studied by Katz and Laumon.

Keywords and phrases

Modular forms Fourier coefficients Hecke eigenvalues Hecke orbits horocycles -adic Fourier transform Riemann Hypothesis over finite fields 

Mathematics Subject Classification

11F11 11F32 11F37 11T23 11L05 

References

  1. Bea10.
    A. Beauville: Finite subgroups of \({{\rm PGL}_2 (K)}\) . Contemporary Math. A.M.S 522 (2010), 23–29.CrossRefMathSciNetGoogle Scholar
  2. BHM07.
    V. Blomer, G. Harcos and Ph. Michel. Bounds for modular L-functions in the level aspect. Ann. Sci. École Norm. Sup. (4) (5)40 (2007), 697–740.zbMATHMathSciNetGoogle Scholar
  3. BH08.
    V. Blomer and G. Harcos. Hybrid bounds for twisted L- functions. J. reine und angew. Mathematik 621 (2008), 53–79.zbMATHMathSciNetGoogle Scholar
  4. Byk98.
    V.A. Bykovski. A trace formula for the scalar product of Hecke series and its applications. J. Math. Sciences 89 (1998), 915–932.CrossRefGoogle Scholar
  5. CI00.
    J.B. Conrey and H. Iwaniec. The cubic moment of central values of automorphic L-functions. Ann. of Math. (2) (3)151 (2000), 1175–1216.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Del77.
    P. Deligne. Cohomologie étale, S.G.A 4\({\frac{1}{2}}\) , L.N.M 569, Springer, Verlag (1977).Google Scholar
  7. Del80.
    P. Deligne. La conjecture de Weil, II. Publ. Math. IHÉS 52 (1980), 137–252.CrossRefzbMATHMathSciNetGoogle Scholar
  8. DF13.
    P. Deligne and Y.Z. Flicker. Counting local systems with principal unipotent local monodromy. Ann. of Math. (2) (3)178 (2013), 921–982.CrossRefzbMATHMathSciNetGoogle Scholar
  9. DI82.
    J-M. Deshouillers and H. Iwaniec. Kloosterman sums and Fourier coefficients of cusp forms. Invent. math. (2)70 (1982/83), 219–288.Google Scholar
  10. DFI93.
    W.D. Duke, J. Friedlander and H. Iwaniec. Bounds for automorphic L-functions. Invent. math. 112 (1993), 1–8.CrossRefzbMATHMathSciNetGoogle Scholar
  11. DFI94.
    W.D. Duke, J. Friedlander and H. Iwaniec. Bounds for automorphic L-functions II. Invent. math. 115 (1994), 219–239.CrossRefzbMATHMathSciNetGoogle Scholar
  12. DFI02.
    W.D. Duke, J. Friedlander and H. Iwaniec. The subconvexity problem for Artin L-functions. Invent. math. (3)149 (2002), 489–577.CrossRefzbMATHMathSciNetGoogle Scholar
  13. EMS84.
    P.D.T.A Elliott, C.J. Moreno and F. Shahidi. On the absolute value of Ramanujan’s \({\tau}\) -function. Math. Ann. 266 (1984), 507–511.CrossRefzbMATHMathSciNetGoogle Scholar
  14. EMOT55.
    A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi. Higher transcendental functions, Vol. II, McGraw Hill (1955).Google Scholar
  15. FKM13.
    É. Fouvry, E. Kowalski and Ph. Michel. An inverse theorem for Gowers norms of trace functions over \({\mathbf{F}_{p}}\) . Math. Proc. Cambridge Philos. Soc. (2)155 (2013), 277–295.Google Scholar
  16. FKM14.
    É. Fouvry, E. Kowalski and Ph. Michel. Algebraic trace functions over the primes. Duke Math. J. (9)163 (2014), 1683–1736.Google Scholar
  17. FKM.
    É. Fouvry, E. Kowalski, Ph. Michel. On the exponent of distribution of the ternary divisor function, Mathematika (to appear). arXiv:1304.3199 .
  18. FG14.
    É Fouvry and S. Ganguly. Orthogonality between the Möbius function, additive characters, and Fourier coefficients of cusp forms. Compos. Math. (5)150 (2014), 763–797.Google Scholar
  19. FI85.
    J. B. Friedlander and H. Iwaniec. Incomplete Kloosterman sums and a divisor problem, with an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) (2)121 (1985), 319–350.CrossRefzbMATHMathSciNetGoogle Scholar
  20. GR94.
    I.S. Gradshteyn and I.M. Ryzhkik. Tables of integrals, series and products. 5th ed. (edited by A. Jeffrey). Academic Press (1994).Google Scholar
  21. Hea86.
    D. R. Heath-Brown. The divisor function \({d_3(n)}\) in arithmetic progressions. Acta Arith. (1)47 (1986), 29–56.zbMATHMathSciNetGoogle Scholar
  22. Hea97.
    D. R. Heath-Brown. The density of rational points on cubic surfaces. Acta Arith. 79 (1997), 17–30.MathSciNetGoogle Scholar
  23. Iwa87.
    H. Iwaniec. Fourier coefficients of modular forms of half-integral weight. Invent. math. (2)87 (1987), 385–401.CrossRefzbMATHMathSciNetGoogle Scholar
  24. Iwa90.
    H. Iwaniec. Small eigenvalues of Laplacian for \({\Gamma_0(N)}\) . Acta Arith. (1)56 (1990), 65–82.zbMATHMathSciNetGoogle Scholar
  25. Iwa95.
    H. Iwaniec. Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matematica Iberoamericana, Revista Matematica Iberoamericana, Madrid (1995).Google Scholar
  26. Iwa97.
    H. Iwaniec.Topics in classical automorphic forms. Grad. Studies in Math. A.M.S 17, (1997).Google Scholar
  27. IK04.
    H. Iwaniec and E. Kowalski. Analytic number theory. A.M.S. Coll. Publ. 53 (2004).Google Scholar
  28. Kat80.
    N.M. Katz. Sommes exponentielles, Astérisque 79, Soc. Math. France (1980).Google Scholar
  29. Kat88.
    N.M. Katz. Gauss sums, Kloosterman sums and monodromy groups, Annals of Math. Studies 116, Princeton Univ. Press (1988).Google Scholar
  30. Kat90.
    N.M. Katz. Exponential sums and differential equations, Annals of Math. Studies 124, Princeton Univ. Press (1990).Google Scholar
  31. KS03.
    H. Kim and P. Sarnak. Refined estimates towards the Ramanujan and Selberg conjectures. J. American Math. Soc. 16 (2003), 175–181.CrossRefMathSciNetGoogle Scholar
  32. Kow14.
    E. Kowalski. An introduction to the representation theory of groups. Grad. Studies in Math. A.M.S 155 (2014).Google Scholar
  33. KRW07.
    E. Kowalski, O. Robert and J. Wu. Small gaps in coefficients of L-functions and \({\mathfrak{B}}\) -free numbers in small intervals. Rev. Mat. Iberoamericana 23 (2007), 281–326.CrossRefzbMATHMathSciNetGoogle Scholar
  34. Lau87.
    G. Laumon. Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil. Publ. Math. IHÉS 65 (1987), 131–210.CrossRefzbMATHMathSciNetGoogle Scholar
  35. MV10.
    Ph. Michel and A. Venkatesh. The subconvexity problem for \({{\rm GL}_2}\) . Publ. Math. I.H.É.S 111 (2010), 171–271.CrossRefMathSciNetGoogle Scholar
  36. Mot.
    Y. Motohashi. On sums of Hecke-Maass eigenvalues squared over primes in short intervals, preprint arXiv:1209.4140v1.
  37. Mun13.
    R. Munshi. Shifted convolution sums for \({GL(3) \times GL(2)}\) . Duke Math. J. (13)162 (2013), 2345–2362.CrossRefzbMATHMathSciNetGoogle Scholar
  38. Pit95.
    N. Pitt. On shifted convolutions of \({\zeta(s)^3}\) with automorphic L-functions. Duke Math. J. (2)77 (1995), 383–406.CrossRefzbMATHMathSciNetGoogle Scholar
  39. RR05.
    D. Ramakrishnan and J. Rogawski. Average values of modular L-series via the relative trace formula, Special Issue: In memory of Armand Borel. Part 3. Pure Appl. Math. Q. (4)1 (2005), 701–735.CrossRefzbMATHMathSciNetGoogle Scholar
  40. Sar91.
    P. Sarnak. Diophantine problems and linear groups. In: Proceedings of the I.C.M. 1990, Kyoto, Springer (1991), 459–471.Google Scholar
  41. Ser71.
    J-P. Serre. Représentations linéaires des groupes finis, 2ème Édition, Hermann, (1971).Google Scholar
  42. Str04.
    A. Strömbergsson. On the uniform equidistribution of long closed horocycles. Duke Math. J. (3)123 (2004), 507–547.CrossRefzbMATHMathSciNetGoogle Scholar
  43. SU.
    P. Sarnak and A. Ubis. The horocycle flow at prime times. Journal Math. Pures Appl., to appear.Google Scholar
  44. Ven10.
    A. Venkatesh. Sparse equidistribution problems, period bounds and subconvexity. Ann. of Math. (2) 172 (2010), 989–1094.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Étienne Fouvry
    • 1
  • Emmanuel Kowalski
    • 2
  • Philippe Michel
    • 3
    Email author
  1. 1.Laboratoire de MathématiqueUniversité Paris SudOrsay CedexFrance
  2. 2.ETH Zürich – D-MATHZürichSwitzerland
  3. 3.EPFL Chaire TANLausanneSwitzerland

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