Subconvexity for Symmetric Square L-functions off-centre

TL;DR

This paper establishes a subconvex bound for symmetric square L-functions of modular forms, improving understanding of their growth in both level and spectral parameters, which is significant for analytic number theory.

Contribution

The paper proves a new subconvexity bound for symmetric square L-functions, combining level and spectral aspects in a novel way.

Findings

Bound $L( ext{sym}^2f, 1/2 + it)$ by $p^{1/2+ ext{epsilon}} t^{3/4 - 1/12 + ext{epsilon}}$

Achieves subconvexity in the $t$-aspect and nearly convex in the level aspect

Advances the understanding of L-function bounds in multiple aspects simultaneously.

Abstract

Let $p$ be a prime. Let $f$ be a holomorphic modular form of level $p$ with trivial nebentypus. We prove the bound $L\left(\text{sym}^2f, \frac{1}{2} + it\right) \ll_{f,\epsilon} p^{1/2+\epsilon}t^{3/4-1/12 + \epsilon}$. This bound is subconvex in the $t$-aspect and almost convex in the level aspect simultaneously.