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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Weighted averages of modular $ L$-values

Author(s): Andrew Knightly; Charles Li
Journal: Trans. Amer. Math. Soc. 362 (2010), 1423-1443.
MSC (2000): Primary 11F11, 11F30, 11F67, 11F70
Posted: September 25, 2009
MathSciNet review: 2563735
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Abstract | References | Similar articles | Additional information

Abstract: Using an explicit relative trace formula on $ \operatorname{GL}(2)$, we derive a formula for averages of modular $ L$-values in the critical strip, weighting by Fourier coefficients, Hecke eigenvalues, and Petersson norms. As an application we show that a GRH holds for these averages as the weight or the level goes to $ \infty$. We also use the formula to give explicit zero-free regions of the form $ \vert\operatorname{Im}(s)\vert\le \tau_0$ for some particular modular $ L$-functions.

References:

[Ak]
A. Akbary, Non-vanishing of weight $ k$ modular $ L$-functions with large level, J. Ramanujan Math. Soc. 14 (1999), no. 1, 37-54. MR 1700874 (2000e:11067)

[AS]
N. Abramowitz and I. Stegun, editors, Handbook of mathematical functions, Dover Publications, New York, 1965.

[Du]
W. Duke, The critical order of vanishing of automorphic $ L$-functions with large level, Invent. Math. 119 (1995), no. 1, 165-174. MR 1309975 (95k:11075)

[El]
J. Ellenberg, On the error term in Duke's estimate for the average special value of $ L$-functions, Canad. Math. Bull. 48 (2005), no. 4, 535-546. MR 2176151 (2007e:11057)

[FW]
B. Feigon and D. Whitehouse, Averages of central $ L$-values of Hilbert modular forms with an application to subconvexity, preprint (2007).

[Ka]
Y. Kamiya, Certain mean values and non-vanishing of automorphic $ L$-functions with large level, Acta Arith. 93 (2000), no. 2, 157-176. MR 1757188 (2001h:11066)

[KL1]
A. Knightly and C. Li, A relative trace formula proof of the Petersson trace formula, Acta Arith. 122 (2006), no. 3, 297-313. MR 2239919 (2007d:11042)

[KL2]
-, Traces of Hecke operators, Mathematical Surveys and Monographs, 133, Amer. Math. Soc., 2006. MR 2273356 (2008g:11090)

[Ko]
W. Kohnen, Nonvanishing of Hecke $ L$-functions associated to cusp forms inside the critical strip, J. Number Theory 67 (1997), no. 2, 182-189. MR 1486497 (98j:11037)

[Li]
C. Li, Kuznietsov trace formula and weighted distribution of Hecke eigenvalues, J. Number Theory 104 (2004), no. 1, 177-192. MR 2021634 (2004k:11059)

[Mi]
P. Michel, Analytic number theory and families of automorphic $ L$-functions, in: Automorphic forms and applications, 181-295, IAS/Park City Math. Ser., 12, Amer. Math. Soc., Providence, RI, 2007. MR 2331346 (2008m:11104)

[RaRo]
D. Ramakrishnan and J. Rogawski, Average values of modular $ L$-series via the relative trace formula, Pure Appl. Math. Q. 1 (2005), no. 4, 701-735. MR 2200997 (2007c:11096)

[Ro]
E. Royer, Facteurs $ \mathbf{Q}$-simples de $ J_0(N)$ de grande dimension et de grand rang, Bull. Soc. Math. France 128 (2000), no. 2, 219-248. MR 1772442 (2001j:11041)

[Ser]
J.-P. Serre, Répartition asymptotique des valeurs propres de l'opérateur de Hecke $ T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, pp. 75-102. MR 1396897 (97h:11048)

[Sh]
G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 1971. MR 0314766 (47:3318)

[Sl]
L. Slater, Confluent hypergeometric functions, Cambridge University Press, New York, 1960. MR 0107026 (21:5753)

[Sp]
J. Spouge, Computation of the gamma, digamma, and trigamma functions, SIAM J. Numer. Anal. 31 (1994), no. 3, 931-944. MR 1275121 (95g:33002)

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Additional Information:

Andrew Knightly
Affiliation: Department of Mathematics and Statistics, Neville Hall, University of Maine, Orono, Maine 04469-5752

Charles Li
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

DOI: 10.1090/S0002-9947-09-04923-X
PII: S 0002-9947(09)04923-X
Received by editor(s): January 31, 2008
Posted: September 25, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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