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International Mathematics Research Papers (2006) Vol. 2006 : article ID 35090, 78 pages, doi:10.1155/IMRP/2006/35090
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

A new bound K2/3+{varepsilon} for Rankin-Selberg L-functions for Hecke congruence subgroups

Yuk-Kam Lau, Jianya Liu and Yangbo Ye

Department of Mathematics, The University of Hong Kong Pokfulam Road, Hong Kong E-mail address: yklau{at}maths.hku.hk
Department of Mathematics, Shandong University 227 Shanda Nanlu, Jinan, Shandong 250100, China E-mail address: jyliu{at}sdu.edu.cn
Department of Mathematics, The University of Iowa 14 MacLean Hall, Iowa City, IA 52242-1419, USA E-mail address: yey{at}math.uiowa.edu

Let f be a holomorphic Hecke eigenform for {Gamma}0(N) of weight k, or a Maass eigenform for {Gamma}0(N) with Laplace eigenvalue 1/4 + k2. Let g be a fixed holomorphic or Maass cusp form for {Gamma}0(N). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f {otimes} g) is proved in the k-aspect: L(1/2 + it, f {otimes} g) <<N,g,t,{varepsilon} k2/3+{varepsilon}, while a convexity bound is only << k1+{varepsilon}. The dependence of the implied constant on t and the level N is polynomial. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to Re s > –1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.



References

  1. Bernstein J., Reznikov A. Analytic continuation of representations and estimates of automorphic forms. Annals of Mathematics. Second Series (1999) 150(1):329–352. MR1715328, 0934.11023.
  2. Blomer V. Shifted convolution sums and subconvexity bounds for automorphic L-functions. International Mathematics Research Notices (2004) 2004(73):3905–3926.[Abstract/Free Full Text]
  3. Blomer V. Rankin-Selberg L-functions on the critical line. Manuscripta Mathematica (2005) 117(2):111–133.[CrossRef][Web of Science]
  4. Blomer V., Harcos G., Michel P. A Burgess-like subconvexity bound for twisted L-functions. to appear in Forum Mathematicum.
  5. Conrey J. B., Iwaniec H. The cubic moment of central values of automorphic L-functions. Annals of Mathematics. Second Series (2000) 151(3):1175–1216.
  6. Deshouillers J.-M., Iwaniec H. Kloosterman sums and Fourier coefficients of cusp forms. Inventiones Mathematicae (1982) 70(2):219–288.[CrossRef][Web of Science]
  7. Good A. Beiträge zur theorie der Dirichletreihen, die Spitzenformen zugeordnet sind. Journal of Number Theory (1981) 13(1):18–65.[CrossRef][Web of Science]
  8. Good A. Cusp forms and eigenfunctions of the Laplacian. Mathematische Annalen (1981) 255(4):523–548.[CrossRef][Web of Science]
  9. Good A. The square mean of Dirichlet series associated with cusp forms. Mathematika (1982) 29(2):278–295. (1983).[Web of Science]
  10. Good A. The convolution method for Dirichlet series. In: Contemporary Mathematics—Hejhal D., Sarnak P., Terras A., eds. (1986) 53. The Selberg Trace Formula and Related Topics, 1984: Brunswick, Maine. Rhode Island: American Mathematical Society. 207–214.
  11. Harcos G., Michel P. The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points II. Inventiones Mathematicae (2006) 163(3):581–655.[CrossRef][Web of Science]
  12. Hoffstein J., Lockhart P. Coefficients of Maass forms and the Siegel zero. Annals of Mathematics. Second Series (1994) 140(1):161–181.
  13. Huxley M. N. On stationary phase integrals. Glasgow Mathematical Journal (1994) 36(3):355–362.[Web of Science]
  14. Huxley M. N. Area, Lattice Points, and Exponential Sums. In: London Mathematical Society Monographs. New Series (1996) 13. New York: The Clarendon Press, Oxford University Press. xii+494.
  15. Ivic A. The Riemann Zeta-Function. The Theory of the Riemann Zeta-Function with Applications. In: A Wiley-Interscience Publication (1985) New York: John Wiley & Sons. xvi+517.
  16. Ivic A. On sums of Hecke series in short intervals. Journal de Théorie des Nombres de Bordeaux (2001) 13(2):453–468.
  17. Iwaniec H. Spectral Methods of Automorphic Forms. In: Graduate Studies in Mathematics (2002) 53, 2nd. Madrid: American Mathematical Society, Rhode Island; Revista Matemática Iberoamericana. xii+220.
  18. Jacquet H. Automorphic Forms on GL(2). Part II. In: Lecture Notes in Mathematics (1972) 278. Berlin: Springer. xiii+142.
  19. Jutila M., Motohashi Y. Uniform bounds for Hecke L-functions. Acta Mathematica (2005) 195:61–115.[CrossRef][Web of Science]
  20. Kim H., Sarnak P. Appendix 2: refined estimates towards the Ramanujan and Selberg conjectures. Journal of the American Mathematical Society (2003) 16:175–181.
  21. Kowalski E., Michel P., VanderKam J. Rankin-Selberg L-functions in the level aspect. Duke Mathematical Journal (2002) 114(1):123–191.[CrossRef][Web of Science]
  22. Krötz B., Stanton R. J. Holomorphic extensions of representations I: automorphic functions. Annals of Mathematics. Second Series (2004) 159(2):641–724.
  23. Lau Y.-K., Liu J., Ye Y. Subconvexity bounds for Rankin-Selberg L-functions for congruence subgroups. Journal of Number Theory (2006) availabe online at http://www.sciencedirect.com.
  24. Liu J., Ye Y. Subconvexity for Rankin-Selberg L-functions of Maass forms. Geometric and Functional Analysis (2002) 12(6):1296–1323.[CrossRef][Web of Science]
  25. Liu J., Ye Y. Petersson and Kuznetsov trace formulas. In: Lie Groups and Automorphic Forms (2006) Rhode Island: International Press. 147–168. American Mathematical Society.
  26. Meurman T. On the order of the Maass L-function on the critical line. In: Colloquium of Mathematical Society János Bolyai—Györy K., Halász G., eds. (1990) 51. Number Theory, Vol. I, 1987: Budapest. Amsterdam: North-Holland. 325–354.
  27. Michel P. The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points. Annals of Mathematics. Second Series (2004) 160(1):185–236.
  28. Peng Z. Zeros and central values of automorphic L-functions (2001) New Jersey: Princeton University.
  29. Sarnak P. Integrals of products of eigenfunctions. International Mathematics Research Notices (1994) 1994(6):251–260.[Free Full Text]
  30. Sarnak P. Estimates for Rankin-Selberg L-functions and quantum unique ergodicity. Journal of Functional Analysis (2001) 184(2):419–453.[CrossRef][Web of Science]
  31. Sarnak P. Lecture at the Ohio State University. March 2003.
  32. Watson G. N. A Treatise on the Theory of Bessel Functions (1966) 2nd. Cambridge: Cambridge University Press.
  33. Ye Y. The fourth power moment of automorphic L-functions for GL(2) over a short interval. Transactions of the American Mathematical Society (2006) 358(5):2259–2268.[CrossRef][Web of Science]

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