The subconvexity problem for GL2
Publications Mathématiques de l'IHÉS, Tome 111 (2010), pp. 171-271.

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.

@article{PMIHES_2010__111__171_0,
     author = {Michel, Philippe and Venkatesh, Akshay},
     title = {The subconvexity problem for {GL2}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {171--271},
     publisher = {Springer-Verlag},
     volume = {111},
     year = {2010},
     doi = {10.1007/s10240-010-0025-8},
     mrnumber = {2653249},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-010-0025-8/}
}
TY  - JOUR
AU  - Michel, Philippe
AU  - Venkatesh, Akshay
TI  - The subconvexity problem for GL2
JO  - Publications Mathématiques de l'IHÉS
PY  - 2010
SP  - 171
EP  - 271
VL  - 111
PB  - Springer-Verlag
UR  - http://www.numdam.org/articles/10.1007/s10240-010-0025-8/
DO  - 10.1007/s10240-010-0025-8
LA  - en
ID  - PMIHES_2010__111__171_0
ER  - 
%0 Journal Article
%A Michel, Philippe
%A Venkatesh, Akshay
%T The subconvexity problem for GL2
%J Publications Mathématiques de l'IHÉS
%D 2010
%P 171-271
%V 111
%I Springer-Verlag
%U http://www.numdam.org/articles/10.1007/s10240-010-0025-8/
%R 10.1007/s10240-010-0025-8
%G en
%F PMIHES_2010__111__171_0
Michel, Philippe; Venkatesh, Akshay. The subconvexity problem for GL2. Publications Mathématiques de l'IHÉS, Tome 111 (2010), pp. 171-271. doi : 10.1007/s10240-010-0025-8. http://www.numdam.org/articles/10.1007/s10240-010-0025-8/

1. J. Arthur, Eisenstein series and the trace formula, in: Automorphic Forms, Representations and L-functions, Part 1, (1979), Am. Math. Soc., Providence | MR | Zbl

2. J. Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compos. Math. 40 (1980), p. 87-121 | Numdam | MR | Zbl

3. I. N. Bernšteĭn, All reductive 𝔭-adic groups are of type I, Funkc. Anal. Prilozh. 8 (1974), p. 3-6 | MR | Zbl

4. J. Bernstein, A. Reznikov, Sobolev norms of automorphic functionals, Int. Math. Res. Not. 40 (2002), p. 2155-2174 | MR | Zbl

5. J. Bernstein and A. Reznikov , Subconvexity bounds for triple L-functions and representation theory, arXiv: math/0608555v1 , 2006. | MR | Zbl

6. V. Blomer, Rankin-Selberg L-functions on the critical line, Manusc. Math. 117 (2005), p. 111-133 | MR | Zbl

7. V. Blomer, G. Harcos, The spectral decomposition of shifted convolution sums, Duke Math. J. 144 (2008), p. 321-339 | MR | Zbl

8. V. Blomer, G. Harcos, Hybrid bounds for twisted L-functions, J. Reine Angew. Math. 621 (2008), p. 53-79 | MR | Zbl

9. V. Blomer, G. Harcos, Ph. Michel, Bounds for modular L-functions in the level aspect, Ann. Sci. École Norm. Supér. (4) 40 (2007), p. 697-740 | MR | Zbl

10. C. J. Bushnell, G. Henniart, An upper bound on conductors for pairs, J. Number Theory 65 (1997), p. 183-196 | MR | Zbl

11. N. Burq, P. Gérard, N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), p. 445-486 | MR | Zbl

12. D. A. Burgess, On character sums and L-series. II, Proc. Lond. Math. Soc. (3) 13 (1963), p. 524-536 | MR | Zbl

13. M. Cowling, U. Haagerup, R. Howe, Almost L 2 matrix coefficients, J. Reine Angew. Math. 387 (1988), p. 97-110 | MR | Zbl

14. L. Clozel, Démonstration de la conjecture τ , Invent. Math. 151 (2003), p. 297-328 | MR | Zbl

15. L. Clozel, E. Ullmo, Équidistribution de mesures algébriques, Compos. Math. 141 (2005), p. 1255-1309 | MR | Zbl

16. A. Diaconu and P. Garrett, Subconvexity bounds for automorphic L-functions for GL(2) over number fields, preprint (2008). | MR | Zbl

17. W. Duke, J. Friedlander, H. Iwaniec, Bounds for automorphic L-functions, Invent. Math. 112 (1993), p. 1-8 | MR | Zbl

18. W. Duke, J. Friedlander, H. Iwaniec, Bounds for automorphic L-functions. II, Invent. Math. 115 (1994), p. 219-239 | MR | Zbl

19. W. Duke, J. B. Friedlander, H. Iwaniec, The subconvexity problem for Artin L-functions, Invent. Math. 149 (2002), p. 489-577 | MR | Zbl

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 . | MR | Zbl

21. M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces I, Duke Math. J. 148 (2009), p. 119-174 | MR | Zbl

22. É. Fouvry, H. Iwaniec, A subconvexity bound for Hecke L-functions, Ann. Sci. École Norm. Supér. (4) 34 (2001), p. 669-683 | Numdam | MR | Zbl

23. J. Friedlander, H. Iwaniec, A mean-value theorem for character sums, Mich. Math. J. 39 (1992), p. 153-159 | MR | Zbl

24. J. Hoffstein, 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), p. 161-181 | MR | Zbl

25. S. Gelbart, H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Supér. (4) 11 (1978), p. 471-542 | Numdam | MR | Zbl

26. S. Gelbart, H. Jacquet, Forms of GL(2) from the analytic point of view, in: Automorphic forms, representations and L-functions, Part 1, (1979), Am. Math. Soc., Providence | MR | Zbl

27. A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29 (1982), p. 278-295 | MR | Zbl

28. A. Gorodnik, F. Maucourant, H. Oh, Manin’s and Peyre’s conjectures on rational points and adelic mixing, Ann. Sci. École Norm. Supér. (4) 41 (2008), p. 383-435 | Numdam | MR | Zbl

29. D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions, Invent. Math. 47 (1978), p. 149-170 | MR | Zbl

30. D. R. Heath-Brown, Convexity bounds for L-function, preprint (2008), arXiv: 0809.1752 . | MR | Zbl

31. G. Harcos, Ph. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006), p. 581-655 | MR | Zbl

32. A. Ichino, Trilinear forms and the central values of triple product L-functions, Duke Math. J. 145 (2008), p. 281-307 | MR | Zbl

33. A. Ichino, T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Funct. Anal. 19 (2010), p. 1378-1425 | MR | Zbl

34. A. Ivić, On sums of Hecke series in short intervals, J. Théor. Nr. Bordx. 13 (2001), p. 453-468 | Numdam | MR | Zbl

35. H. Iwaniec, The spectral growth of automorphic L-functions, J. Reine Angew. Math. 428 (1992), p. 139-159 | MR | Zbl

36. H. Iwaniec, Harmonic analysis in number theory, in: Prospects in Mathematics, (1999), Am. Math. Soc., Providence | MR | Zbl

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. | MR | Zbl

38. H. Jacquet, R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Mathematics 114 (1970), Springer, Berlin | MR | Zbl

39. H. Jacquet, I. I. Piatetski-Shapiro, J. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann. 256 (1981), p. 199-214 | MR | Zbl

40. M. Jutila, The twelfth moment of central values of Hecke series, J. Number Theory 108 (2004), p. 157-168 | MR | Zbl

41. M. Jutila, Y. Motohashi, Uniform bound for Hecke L-functions, Acta Math. 195 (2005), p. 61-115 | MR | Zbl

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), p. 139-183 | MR | Zbl

43. E. Kowalski, Ph. Michel, J. Vanderkam, Rankin-Selberg L-functions in the level aspect, Duke Math. J. 114 (2002), p. 123-191 | MR | Zbl

44. N. V. Kuznetsov, Sums of Kloosterman sums and the eighth power moment of the Riemann zeta-function, in: Number Theory and Related Topics, Tata Inst. Fund. Res. Stud. Math. 12 (1989), Tata Inst. Fund. Res., Bombay | MR | Zbl

45. J. Liu, Y. Ye, Subconvexity for Rankin-Selberg L-functions of Maass forms, Geom. Funct. Anal. 12 (2002), p. 1296-1323 | MR | Zbl

46. H. Y. Loke, Trilinear forms of 𝔤𝔩 2 , Pac. J. Math. 197 (2001), p. 119-144 | MR | Zbl

47. W. Luo, Z. Rudnick, P. Sarnak, On the generalized Ramanujan conjecture for GL(n), in: Automorphic Forms, Automorphic Representations, and Arithmetic, Proc. Sympos. Pure Math. 66 (1999), Am. Math. Soc., Providence | MR | Zbl

48. T. Meurman, On the order of the Maass L-function on the critical line, in: Number Theory, Vol. I, Colloq. Math. Soc. János Bolyai 51 (1990), North-Holland, Amsterdam | MR | Zbl

49. Ph. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points, Ann. Math. (2) 160 (2004), p. 185-236 | MR | Zbl

50. Ph. Michel, Analytic number theory and families of automorphic L-functions, in: Automorphic Forms and Applications, IAS/Park City Math. Ser. 12 (2007), Am. Math. Soc., Providence | MR | Zbl

51. Ph. Michel, A. Venkatesh, Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik, in: International Congress of Mathematicians, vol. II, (2006), Eur. Math. Soc., Zürich | MR | Zbl

52. C. Moeglin, J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics 113 (1995), Cambridge University Press, Cambridge | MR | Zbl

53. Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge Tracts in Mathematics 127 (1997), Cambridge University Press, Cambridge | MR | Zbl

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. | MR | Zbl

55. W. Müller, The trace class conjecture in the theory of automorphic forms II, Geom. Funct. Anal. 8 (1998), p. 315-355 | MR | Zbl

56. H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J. 113 (2002), p. 133-192 | MR | Zbl

57. A. I. Oksak, Trilinear Lorentz invariant forms, Commun. Math. Phys. 29 (1973), p. 189-217 | MR

58. D. Prasad, Trilinear forms for representations of GL(2) and local ε-factors, Compos. Math. 75 (1990), p. 1-46 | Numdam | MR | Zbl

59. A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, preprint, 2004, arXiv: math/0403437v2 .

60. A. Reznikov, Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms, J. Am. Math. Soc. 21 (2008), p. 439-477 | MR | Zbl

61. Y. Sakellaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties, preprint, 2010.

62. P. Sarnak, L-functions, in Proceedings of the International Congress of Mathematicians, vol. I (Berlin, 1998), Documenta Mathematica (Extra volume), pp. 453–465. | MR | Zbl

63. P. Sarnak, Estimates for Rankin-Selberg L-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), p. 419-453 | MR | Zbl

64. A. Venkatesh, Large sieve inequalities for GL(n)-forms in the conductor aspect, Adv. Math. 200 (2006), p. 336-356 | MR | Zbl

65. A. Venkatesh, Sparse equidistribution problems, period bounds, and subconvexity, Ann. Math., to appear, 2006. | MR | Zbl

66. A. Venkatesh, Notes on effective equidistribution, Pisa/CMI summer school 2007, unpublished, 2007.

67. J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54 (1985), p. 173-242 | Numdam | MR | Zbl

68. H. Weyl, Zur abschätzung von ζ(1+it), Math. Z. 10 (1921), p. 88-101

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), p. 415-437 | MR | Zbl

Cité par Sources :