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

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 (Z), or in certain cases, for the Hecke congruence subgroup Γ 0 (N) with N > 1. The short interval is from a large K to K + K 103/135+∈ . 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).