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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.


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