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

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 SL2( Z), or in certain cases, for the Hecke congruence subgroup Gamma(0)(N) with N > 1. The short interval is from a large K to K + K-103/(135+epsilon). 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).