t-Aspect Subconvexity Bound for $GL(2)$ L-functions

Abstract

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it \right) \ll_{f, \epsilon} \left( 1 + |t|\right)^{1/3 + \epsilon}, \] for any $\epsilon > 0.$