International Mathematics Research Notices

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm162, 35 pages, doi:10.1093/imrn/rnm162 published on February 11, 2008

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Lau, Y.-K. Articles by Wu, J. Search for Related Content Social Bookmarking          What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

A Large Sieve Inequality of Elliott–Montgomery–Vaughan Type for Automorphic Forms and Two Applications

Y.-K. Lau¹ and J. Wu^2,3,

¹ Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
² Institut Elie Cartan Nancy (IECN), Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes, B.P. 239, 54506, Vandœuvre-lès-Nancy, France
³ School of Mathematical Sciences, Shandong, Normal University, Jinan, Shandong 250014, China

Correspondence: Correspondence to be sent to: yklau{at}maths.hku.hk

In this paper, we establish a large sieve inequality of Elliott–Montgomery–Vaughan type for Fourier coefficients of newforms. As applications, we give a significant improvement on the principal result of Duke and Kowalski on Linnik's problem for modular forms and prove the upper part of the first conjecture of Montgomery–Vaughan in the context of automorphic L-functions.

References

1. Cogdell J., Michel P. On the complex moments of symmetric power L-functions at s = 1. International Mathematics Research Notices (2004) 31:1561–1618.
2. Deligne P. Publications Mathematiques Institut de Hautes Etudes Scientifiques. (1974) 48:273–308.
3. Deligne P. La conjecture de Weil 2. Publications Mathematiques Institut de Hautes Etudes Scientifiques (1981) 52:313–428.
4. Duke W., Kowalski E. A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. Inventiones Mathematicae (2000) 139(no. 1):1–39.[Web of Science]
5. Elliott P. D. T. A. Probabilistic Number Theory 1: Mean-Value Theorems (1979) 239. Berlin: Springer. xxii–359. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science].
6. Elliott P. D. T. A. Probabilistic Number Theory 2: Central Limit Theorems (1980) 240. New York: Springer. xviii–341. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
7. Goldfeld D., Hoffstein J., Lieman D. An effective zero-free region. Annals of Mathematics (2) (1994) 140(no. 1):177–81.[CrossRef][Web of Science]
8. Graham S. W., Ringrose. Lower bounds for least quadratic nonresidues. In: Analytic Number Theory (1990) Boston: Birkhäuser. 269–309. Progress in Mathematics 85.
9. Granville A., Soundararajan K. The distribution of values of . Geometric and Functional Analysis (2003) 13(no. 5):992–1028.[CrossRef][Web of Science]
10. Hoffstein J., Lockhart P. Coefficients of Maass forms and the Siegel zero. Annals of Mathematics (2) (1994) 140(no. 1):161–81.[CrossRef][Web of Science]
11. Iwaniec H., Luo W., Sarnak P. Low lying zeros of families of L-functions. Publications Mathematiques. Institut de Hautes Etudes Scientifiques (2000) 91:55–131.[CrossRef]
12. Kohnen W. On Hecke eigenvalues of newforms. Mathematische Annalen (2004) 329:623–28.[Web of Science]
13. Kowalski E. Variants of recognition problems for modular forms. Archiv der Mathematik (Basel) (2005) 84(no. 1):57–70.[CrossRef]
14. Kowalski E. The principle of the large sieve. (2006) preprint http://front.math.ucdavis. edu/math.NT/0610021.
15. Kowalski E., Michel P., Vanderkam J. Rankin–Selberg L-functions in the level aspect. Duke Mathematical Journal (2002) 114:123–91.[CrossRef][Web of Science]
16. Lau Y.-K., Wu J. A density theorem on automorphic L-functions and some applications. Transactions of the American Mathematical Society (2006) 358:441–72.[CrossRef][Web of Science]
17. Lau Y.-K., Wu J. Extreme values of symmetric power L-functions at 1. Acta Arithmetica (2007) 126(no. 1):57–76.[Web of Science]
18. Lau Y.-K., Wu J. On the least quadratic non-residue. International Journal of Number Theory. (forthcoming).
19. Liu J.-Y., Royer E., Wu J. On a conjecture of Montgomery–Vaughan on extreme values of automorphic L-functions at 1. Proceedings, Anatomy of Integers (2006) preprint arXiv.org/math.NT/0604334.
20. Montgomery H. L., Vaughan R. C. Extreme values of Dirichlet L-functions at 1. In: Number Theory in Progress—Györy K., Iwaniec H., Urbanowicz J., eds. (1999) Vol. 2. Berlin: de Gruyter. 1039–52.
21. Murty Ram M. Congruences between modular forms. In: Analytic Number Theory—Motohashi Y., ed. (1997) Cambridge: Cambridge University Press. 309–20. London Mathematical Society Lecture Note Series 247.
22. Royer E., Wu J. Taille des valeurs de fonctions L de carrés symétriques au bord de la bande critique. Revista Matematica Iberoamericana (2005) 21:263–312.[Web of Science]
23. Sarnak P. Letter to Zeev Rudnick. http://www.math.princeton.edu/sarnak/RudnickLtrSept2002.pdf (accessed April 2007).
24. Sengupta J. Distingushing Hecke eigenvalues of primitive cusp forms. Acta Arithmetica (2004) 114:23–34.[Web of Science]
25. Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory (1995) Cambridge: Cambridge University Press. xvi–448.

CiteULike    Connotea    Del.icio.us    What's this?

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Lau, Y.-K. Articles by Wu, J. Search for Related Content Social Bookmarking          What's this?