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 $SL_{2}(\mathbb{Z})$, 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).