Subconvexity bound for $GL(2)$ L-functions: \lowercase{t}-aspect

TL;DR

This paper establishes a subconvexity bound for $GL(2)$ $L$-functions in the $t$-aspect, demonstrating that the $L$-function grows at most on the order of $|t|^{1/3+ ext{epsilon}}$, which improves previous bounds.

Contribution

The paper proves the Weyl exponent bound for $GL(2)$ $L$-functions using the circle method, achieving a significant subconvexity result in the $t$-aspect.

Findings

Proves $L(1/2 + it, f) \, \ll_{f, \epsilon} \, (2 + |t|)^{1/3 + \epsilon}$.

Establishes the Weyl bound for $GL(2)$ $L$-functions in the $t$-aspect.

Uses circle method to derive the subconvexity bound.

Abstract

Let $f $ be a holomorphic Hecke eigenform 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 \[ L \left( \frac{1}{2} + it, f \right) \ll_{f, \epsilon} \left( 2 + |t|\right)^{1/3 + \epsilon}, \] for any $\epsilon > 0.$