Publications mathématiques de l'IHÉS

, Volume 111, Issue 1, pp 171–271 | Cite as

The subconvexity problem for GL2



Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL 1 and GL 2 automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.


Eisenstein Series Automorphic Form Sobolev Norm Automorphic Representation Cuspidal Representation 
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  1. 1.
    J. Arthur, Eisenstein series and the trace formula, in Automorphic Forms, Representations and L -functions, Part 1 (Proc. Sympos. Pure Math., XXXIII, Oregon State Univ., Corvallis, Ore., 1977), pp. 253–274, Am. Math. Soc., Providence, 1979. Google Scholar
  2. 2.
    J. Arthur, A trace formula for reductive groups. II. Applications of a truncation operator. Compos. Math., 40 (1980), 87–121. MATHMathSciNetGoogle Scholar
  3. 3.
    I. N. Bernšteĭn, All reductive \(\protect\mathfrak{p}\)-adic groups are of type I. Funkc. Anal. Prilozh., 8 (1974), 3–6. Google Scholar
  4. 4.
    J. Bernstein and A. Reznikov, Sobolev norms of automorphic functionals, Int. Math. Res. Not., 40 (2002), 2155–2174. CrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Bernstein and A. Reznikov, Subconvexity bounds for triple L-functions and representation theory, arXiv:math/0608555v1, 2006.
  6. 6.
    V. Blomer, Rankin-Selberg L-functions on the critical line, Manusc. Math., 117 (2005), 111–133. MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. Blomer and G. Harcos, The spectral decomposition of shifted convolution sums, Duke Math. J., 144 (2008), 321–339. MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    V. Blomer and G. Harcos, Hybrid bounds for twisted L-functions, J. Reine Angew. Math., 621 (2008), 53–79. MATHMathSciNetGoogle Scholar
  9. 9.
    V. Blomer, G. Harcos, and Ph. Michel, Bounds for modular L-functions in the level aspect, Ann. Sci. École Norm. Supér. (4), 40 (2007), 697–740. MATHMathSciNetGoogle Scholar
  10. 10.
    C. J. Bushnell and G. Henniart, An upper bound on conductors for pairs, J. Number Theory, 65 (1997), 183–196. MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    N. Burq, P. Gérard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J., 138 (2007), 445–486. MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. A. Burgess, On character sums and L-series. II, Proc. Lond. Math. Soc. (3), 13 (1963), 524–536. MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Cowling, U. Haagerup, and R. Howe, Almost L 2 matrix coefficients, J. Reine Angew. Math., 387 (1988), 97–110. MATHMathSciNetGoogle Scholar
  14. 14.
    L. Clozel, Démonstration de la conjecture τ, Invent. Math., 151 (2003), 297–328. MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    L. Clozel and E. Ullmo, Équidistribution de mesures algébriques, Compos. Math., 141 (2005), 1255–1309. MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Diaconu and P. Garrett, Subconvexity bounds for automorphic L-functions for GL(2) over number fields, preprint (2008). Google Scholar
  17. 17.
    W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic L-functions, Invent. Math., 112 (1993), 1–8. MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic L-functions. II, Invent. Math., 115 (1994), 219–239. MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    W. Duke, J. B. Friedlander, and H. Iwaniec, The subconvexity problem for Artin L-functions, Invent. Math., 149 (2002), 489–577. MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh, The distribution of periodic torus orbits on homogeneous spaces: Duke’s theorem for cubic fields, Ann. Math., to appear (2007), arXiv:0903.3591.
  21. 21.
    M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces I, Duke Math. J., 148 (2009), 119–174. MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    É. Fouvry and H. Iwaniec, A subconvexity bound for Hecke L-functions, Ann. Sci. École Norm. Supér. (4), 34 (2001), 669–683. MATHMathSciNetGoogle Scholar
  23. 23.
    J. Friedlander and H. Iwaniec, A mean-value theorem for character sums, Mich. Math. J., 39 (1992), 153–159. MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, with an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman, Ann. Math. (2), 140 (1994), 161–181. MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Supér. (4), 11 (1978), 471–542. MATHMathSciNetGoogle Scholar
  26. 26.
    S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, in Automorphic forms, representations and L -functions, Part 1 (Proc. Sympos. Pure Math., XXXIII, Oregon State Univ., Corvallis, Ore., 1977), pp. 213–251, Am. Math. Soc., Providence, 1979. Google Scholar
  27. 27.
    A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika, 29 (1982), 278–295. MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    A. Gorodnik, F. Maucourant, and H. Oh, Manin’s and Peyre’s conjectures on rational points and adelic mixing, Ann. Sci. École Norm. Supér. (4), 41 (2008), 383–435. MathSciNetGoogle Scholar
  29. 29.
    D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions, Invent. Math., 47 (1978), 149–170. MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    D. R. Heath-Brown, Convexity bounds for L-function, preprint (2008), arXiv:0809.1752.
  31. 31.
    G. Harcos and Ph. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points. II, Invent. Math., 163 (2006), 581–655. MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    A. Ichino, Trilinear forms and the central values of triple product L-functions, Duke Math. J., 145 (2008), 281–307. MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal., 19 (2010), 1378–1425. MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    A. Ivić, On sums of Hecke series in short intervals, J. Théor. Nr. Bordx., 13 (2001), 453–468. MATHGoogle Scholar
  35. 35.
    H. Iwaniec, The spectral growth of automorphic L-functions, J. Reine Angew. Math., 428 (1992), 139–159. MATHMathSciNetGoogle Scholar
  36. 36.
    H. Iwaniec, Harmonic analysis in number theory, in Prospects in Mathematics (Princeton, NJ, 1996), pp. 51–68, Am. Math. Soc., Providence, 1999. Google Scholar
  37. 37.
    H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of L-functions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. (2000), Special Volume, pp. 705–741. Google Scholar
  38. 38.
    H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Mathematics, vol. 114, Springer, Berlin, 1970. MATHGoogle Scholar
  39. 39.
    H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann., 256 (1981), 199–214. MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    M. Jutila, The twelfth moment of central values of Hecke series, J. Number Theory, 108 (2004), 157–168. MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    M. Jutila and Y. Motohashi, Uniform bound for Hecke L-functions, Acta Math., 195 (2005), 61–115. MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, with appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak, J. Am. Math. Soc., 16 (2003), 139–183. MATHCrossRefGoogle Scholar
  43. 43.
    E. Kowalski, Ph. Michel, and J. VanderKam, Rankin-Selberg L-functions in the level aspect, Duke Math. J., 114 (2002), 123–191. MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    N. V. Kuznetsov, Sums of Kloosterman sums and the eighth power moment of the Riemann zeta-function, in Number Theory and Related Topics (Bombay, 1988), Tata Inst. Fund. Res. Stud. Math., vol. 12, pp. 57–117, Tata Inst. Fund. Res., Bombay, 1989. Google Scholar
  45. 45.
    J. Liu and Y. Ye, Subconvexity for Rankin-Selberg L-functions of Maass forms, Geom. Funct. Anal., 12 (2002), 1296–1323. MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    H. Y. Loke, Trilinear forms of \(\mathfrak{gl}_{2}\), Pac. J. Math., 197 (2001), 119–144. MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    W. Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for GL(n), in Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math., vol. 66, pp. 301–310, Am. Math. Soc., Providence, 1999. Google Scholar
  48. 48.
    T. Meurman, On the order of the Maass L-function on the critical line, in Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, vol. 51, pp. 325–354, North-Holland, Amsterdam, 1990. Google Scholar
  49. 49.
    Ph. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points, Ann. Math. (2), 160 (2004), 185–236. MATHCrossRefGoogle Scholar
  50. 50.
    Ph. Michel, Analytic number theory and families of automorphic L-functions, in Automorphic Forms and Applications (Park City, UT, 2002), IAS/Park City Math. Ser., vol. 12, pp. 179–296, Am. Math. Soc., Providence, 2007. Google Scholar
  51. 51.
    Ph. Michel and A. Venkatesh, Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik, in International Congress of Mathematicians, vol. II, pp. 421–457, Eur. Math. Soc., Zürich, 2006. Google Scholar
  52. 52.
    C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MATHGoogle Scholar
  53. 53.
    Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge Tracts in Mathematics, vol. 127, Cambridge University Press, Cambridge, 1997. MATHGoogle Scholar
  54. 54.
    Y. Motohashi, A functional equation for the spectral fourth moment of modular Hecke L-functions, in Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, vol. 360, p. 19, 2003. Google Scholar
  55. 55.
    W. Müller, The trace class conjecture in the theory of automorphic forms II, Geom. Funct. Anal., 8 (1998), 315–355. MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133–192. MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    A. I. Oksak, Trilinear Lorentz invariant forms, Commun. Math. Phys., 29 (1973), 189–217. CrossRefMathSciNetGoogle Scholar
  58. 58.
    D. Prasad, Trilinear forms for representations of GL(2) and local ε-factors, Compos. Math., 75 (1990), 1–46. MATHGoogle Scholar
  59. 59.
    A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, preprint, 2004, arXiv:math/0403437v2.
  60. 60.
    A. Reznikov, Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms, J. Am. Math. Soc., 21 (2008), 439–477. MATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Y. Sakellaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties, preprint, 2010. Google Scholar
  62. 62.
    P. Sarnak, L-functions, in Proceedings of the International Congress of Mathematicians, vol. I (Berlin, 1998), Documenta Mathematica (Extra volume), pp. 453–465. Google Scholar
  63. 63.
    P. Sarnak, Estimates for Rankin-Selberg L-functions and quantum unique ergodicity, J. Funct. Anal., 184 (2001), 419–453. MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    A. Venkatesh, Large sieve inequalities for GL(n)-forms in the conductor aspect, Adv. Math., 200 (2006), 336–356. MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    A. Venkatesh, Sparse equidistribution problems, period bounds, and subconvexity, Ann. Math., to appear, 2006. Google Scholar
  66. 66.
    A. Venkatesh, Notes on effective equidistribution, Pisa/CMI summer school 2007, unpublished, 2007. Google Scholar
  67. 67.
    J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math., 54 (1985), 173–242. MATHMathSciNetGoogle Scholar
  68. 68.
    H. Weyl, Zur abschätzung von ζ(1+it), Math. Z., 10 (1921), 88–101. CrossRefMathSciNetGoogle Scholar
  69. 69.
    D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1981), 415–437. MATHMathSciNetGoogle Scholar

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© IHES and Springer-Verlag 2010

Authors and Affiliations

  1. 1.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Université Montpellier IIMontpellierFrance
  3. 3.Stanford UniversityStanfordUSA

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