Inventiones mathematicae

, Volume 163, Issue 3, pp 581–655 | Cite as

The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II

  • Gergely Harcos
  • Philippe Michel


We prove a general subconvex bound in the level aspect for Rankin–Selberg L-functions associated with two primitive holomorphic or Maass cusp forms over Q. We use this bound to establish the equidistribution of incomplete Galois orbits of Heegner points on Shimura curves associated with indefinite quaternion algebras over Q.


Quaternion Algebra Heegner Point Level Aspect Shimura Curve Galois Orbit 
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  1. 1.
    Bertolini, M., Darmon, H.: Heegner points of Mumford–Tate curves. Invent. Math. 126, 413–456 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bernstein, J., Reznikov, A.: Analytic continuation of representations and estimates of automorphic forms. Ann. Math. 150, 329–352 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bernstein, J., Reznikov, A.: Periods, subconvexity and representation theory. To appear in J. Differ. Geom. 69 (2005)Google Scholar
  4. 4.
    Blomer, V.: Shifted convolution sums and subconvexity bounds for automorphic L-functions. Int. Math. Res. Not. 2004, no. 73, 3905–3926Google Scholar
  5. 5.
    Blomer, V., Harcos, G., Michel, P.: A Burgess-like subconvex bound for twisted L-functions (with Appendix 2 by Z. Mao). To appear in Forum Math.Google Scholar
  6. 6.
    Blomer, V., Harcos, G., Michel, P.: On the subconvexity problem for L-functions of modular forms with nebentypus. In preparationGoogle Scholar
  7. 7.
    Burgess, D.A.: On character sums and L-series, Proc. Lond. Math. Soc. 12, 193–206 (1962); II, ibid. 13, 524–536 (1963)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cogdell, J.: L-functions and Converse Theorems for GLn. To appear in Park City Lecture Notes (2004), available at∼cogdell/Google Scholar
  9. 9.
    Cogdell, J., Piatetskii-Shapiro, I.I.: The arithmetic and spectral analysis of Poincaré series. Perspect. Math., vol. 13. Boston, MA: Academic Press 1990Google Scholar
  10. 10.
    Darmon, H.: Rational points on modular elliptic curves. CBMS Regional Conference Series in Mathematics, vol. 101. Providence, RI: American Mathematical Society 2004Google Scholar
  11. 11.
    Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982/83)Google Scholar
  12. 12.
    Duke, W.: Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73–90 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Duke, W., Friedlander, J., Iwaniec, H.: Bounds for automorphic L-functions. Invent. Math. 112, 1–8 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Duke, W., Friedlander, J., Iwaniec, H.: A quadratic divisor problem. Invent. Math. 115, 209–217 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Duke, W., Friedlander, J., Iwaniec, H.: Bounds for automorphic L-functions. II. Invent. Math. 115, 219–239 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Duke, W., Friedlander, J., Iwaniec, H.: Bilinear forms with Kloosterman fractions. Invent. Math. 128, 23–43 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Duke, W., Friedlander, J., Iwaniec, H.: Representations by the determinant and mean values of L-functions. Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), pp. 109–115. Lond. Math. Soc. Lect. Note Ser., vol. 237. Cambridge: Cambridge Univ. Press 1997.Google Scholar
  18. 18.
    Duke, W., Friedlander, J., Iwaniec, H.: Bounds for automorphic L-functions. III. Invent. Math. 143, 221–248 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Duke, W., Friedlander, J., Iwaniec, H.: The subconvexity problem for Artin L-functions. Invent. Math. 149, 489–577 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Duke, W., Iwaniec, H.: Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes. Math. Ann. 286, 783–802 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Duke, W., Schulze-Pillot, R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99, 49–57 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Erdélyi, A., et al.: Tables of integral transforms, Vol. I, [based on notes left by H. Bateman]. New York: McGraw-Hill 1954Google Scholar
  23. 23.
    Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Harcos, G.: An additive problem in the Fourier coefficients of Maass forms. Math. Ann. 326, 347–365 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Harcos, G.: New bounds for automorphic L-functions, Ph.D. thesis. Princeton University 2003Google Scholar
  26. 26.
    Harcos, G., Michel, P.: On some shifted convolution problems. In preparationGoogle Scholar
  27. 27.
    Hoffstein, J., Lockhart, P.: Coefficients of Maass forms and the Siegel zero (with an appendix by D. Goldfeld, J. Hoffstein and D. Lieman). Ann. Math. 140, 161–181 (1994)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Huxley, M.: Introduction to Kloostermania. In: Elementary and Analytic Theory of Numbers, pp. 217–306. Banach Center Publ., vol. 17. Warsaw 1985Google Scholar
  29. 29.
    Iwaniec, H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87, 385–401 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Iwaniec, H.: Introduction to the spectral theory of automorphic forms. Biblioteca de la Revista Matemática Iberoamericana, Revista Matemática Iberoamericana. Madrid 1995Google Scholar
  31. 31.
    Iwaniec, H., Sarnak, P.: Perspectives in the analytic theory of L-functions. Geom. Funct. Anal. 2000, Special Volume, Part II, pp. 705–741Google Scholar
  32. 32.
    Jacquet, H.: Automorphic forms on GL(2). Part II. Lect. Notes Math., vol. 278. Berlin, New York: Springer 1972Google Scholar
  33. 33.
    Jutila, M.: Transformations of exponential sums. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori 1989), pp. 263–270, Univ. Salerno. Salerno 1992Google Scholar
  34. 34.
    Jutila, M.: A variant of the circle method. Sieve methods, exponential sums and their applications in number theory, pp. 245–254. Cambridge University Press 1996Google Scholar
  35. 35.
    Jutila, M.: Convolutions of Fourier coefficients of cusp forms. Publ. Inst. Math., Nouv. Sér. 65, 31–51 (1999)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Kim, H.: Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 (with Appendix 1 by D. Ramakrishnan and Appendix 2 by H. Kim and P. Sarnak). J. Am. Math. Soc. 16, 139–183 (2003)Google Scholar
  37. 37.
    Kim, H., Sarnak, P.: Appendix: Refined estimates towards the Ramanujan and Selberg Conjectures. J. Am. Math. Soc. 16, 175–181 (2003)Google Scholar
  38. 38.
    Kim, H., Shahidi, F.: Cuspidality of symmetric powers with applications. Duke Math. J. 112, 177–197 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Kowalski, E., Michel, P., VanderKam, J.: Rankin–Selberg L-functions in the level aspect. Duke Math. J. 114, 123–191 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Krötz, B., Stanton, R.J.: Holomorphic extension of representations: (I) automorphic functions. Ann. Math. 159, 641–724 (2004)zbMATHGoogle Scholar
  41. 41.
    Linnik, Y.V.: Ergodic properties of algebraic fields [translated from the Russian by M.S. Keane]. Ergeb. Math. Grenzgeb., Band 45. New York: Springer 1968Google Scholar
  42. 42.
    Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on PSL2(Z)∖H. Publ. Math., Inst. Hautes Étud. Sci. 81, 207–237 (1995)zbMATHGoogle Scholar
  43. 43.
    Meurman, T.: On exponential sums involving the Fourier coefficients of Maass wave forms. J. Reine Angew. Math. 384, 192–207 (1988)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Michel, P.: The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. Ann. Math. 160, 185–236 (2004)zbMATHCrossRefGoogle Scholar
  45. 45.
    Michel, P.: Analytic number theory and families of L-functions. To appear in Park City Lecture Notes (2004), available at∼michel/Google Scholar
  46. 46.
    Motohashi, Y.: A note on the mean value of the zeta and L-functions. XIV. Proc. Japan Acad., Ser. A 80, 28–33 (2004)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Olver, F.W.J.: Asymptotics and special functions. New York: Academic Press 1974Google Scholar
  48. 48.
    Popa, A.: Central values of Rankin L-series over real quadratic fields. Ph.D. thesis. Harvard University 2003Google Scholar
  49. 49.
    Ramakrishnan, D.: Modularity of the Rankin–Selberg L-series, and multiplicity one for SL(2). Ann. Math. 152, 45–111 (2000)MathSciNetGoogle Scholar
  50. 50.
    Sarnak, P.: Integrals of products of eigenfunctions. Int. Math. Res. Not. 1994, no. 6, 251 ff., approx. 10 pp. (electronic)Google Scholar
  51. 51.
    Sarnak, P.: Estimates for Rankin–Selberg L-functions and quantum unique ergodicity. J. Funct. Anal. 184, 419–453 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Selberg, A.: On the estimation of Fourier coefficients of modular forms. Proc. Sympos. Pure Math., vol. VIII, pp. 1–15. Providence, RI: Am. Math. Soc. 1965Google Scholar
  53. 53.
    Venkatesh, A.: Sparse equidistribution problems, period bounds, and subconvexity. Preprint 2005. arxiv:math.NT/0506224Google Scholar
  54. 54.
    Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. 60, 374–484 (1981)MathSciNetGoogle Scholar
  55. 55.
    Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge: Cambridge University Press 1944Google Scholar
  56. 56.
    Zhang, S.: Heights of Heegner points on Shimura curves. Ann. Math. 153, 27–147 (2001)zbMATHGoogle Scholar
  57. 57.
    Zhang, S.: Gross–Zagier formula for GL2. Asian J. Math. 5, 183–290 (2001)zbMATHMathSciNetGoogle Scholar
  58. 58.
    Zhang, S.: Elliptic curves, L-functions, and CM points. Current developments in mathematics, 2001, pp. 179–219. Somerville, MA: Int. Press 2002Google Scholar
  59. 59.
    Zhang, S.: Gross–Zagier formula for GL(2). II. Heegner points and Rankin L-series. Math. Sci. Res. Inst. Publ., vol. 49, pp. 191–214. Cambridge: Cambridge University Press 2004Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentThe University of Texas at AustinAustinUSA
  2. 2.I3M, UMR CNRS 5149Université Montpellier II CC 051Montpellier Cedex 05France

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