$t$-aspect subconvexity for $GL(2)$ $L$-functions

TL;DR

This paper proves a subconvexity bound for $GL(2)$ $L$-functions, specifically for holomorphic cusp forms, improving the known bounds on their growth at the critical line.

Contribution

The paper establishes a $t$-aspect subconvexity bound for $GL(2)$ $L$-functions using methods inspired by Munshi, advancing the understanding of their size.

Findings

Proves the Burgess bound for $L(1/2+it,f)$ with explicit exponent

Improves the known bounds on the growth of $GL(2)$ $L$-functions in the $t$-aspect

Provides a new approach inspired by Munshi's techniques

Abstract

Let $f$ be a holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k>1$. In these notes, we follow Munshi to prove the Burgess bound $$ L(1/2+it,f)\ll_{f,\varepsilon} (1+|t|)^{1/2-1/8+\varepsilon}. $$