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A new bound k2/3+ε for Rankin-Selberg L-functions for Hecke congruence subgroups
Journal article   Peer reviewed

A new bound k2/3+ε for Rankin-Selberg L-functions for Hecke congruence subgroups

Yuk Kam Lau, Jianya Liu and Yangbo Ye
International mathematics research papers, Vol.2006, 35090
2006
DOI: 10.1155/IMRP/2006/35090

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Abstract

Let be a holomorphic Hecke eigenform for Γ ( ) of weight , or a Maass eigenform for Γ ( ) with Laplace eigenvalue 1/4 + . Let be a fixed holomorphic or Maass cusp form for Γ ( ). A subconvexity bound for central values of the Rankin-Selberg -function ( , ⊗ ) is proved in the -aspect: (1/2 + , ⊗ ) ≪ , while a convexity bound is only ≪ . The dependence of the implied constant on and the level is polynomial. This new bound improves earlier subconvexity bounds for these Rankin-Selberg -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 > −1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.

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