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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The fourth power moment of automorphic $ L$-functions for $ GL(2)$ over a short interval

Author: Yangbo Ye
Journal: Trans. Amer. Math. Soc. 358 (2006), 2259-2268
MSC (2000): Primary 11F66
Published electronically: October 31, 2005
MathSciNet review: 2197443
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Abstract: In this paper we will prove bounds for the fourth power moment in the $ t$ aspect over a short interval of automorphic $ L$-functions $ L(s,g)$ for $ GL(2)$ on the central critical line Re$ s=1/2$. Here $ g$ is a fixed holomorphic or Maass Hecke eigenform for the modular group $ SL_{2}(\mathbb{Z})$, or in certain cases, for the Hecke congruence subgroup $ \Gamma _{0}({\mathcal{N}})$ with $ \mathcal{N}>1$. The short interval is from a large $ K$ to $ K+K^{103/135+\varepsilon }$. The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg $ L$-function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).

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Yangbo Ye
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419

Received by editor(s): March 26, 2004
Received by editor(s) in revised form: July 27, 2004
Published electronically: October 31, 2005
Additional Notes: This project was sponsored by the National Security Agency under Grant Number MDA904-03-1-0066. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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