Uniform subconvex bounds for Rankin-Selberg $L$-functions

TL;DR

This paper proves a uniform subconvexity bound for Rankin-Selberg L-functions involving Maass cusp forms, improving understanding of their growth and distribution in the critical strip.

Contribution

It establishes a new uniform subconvexity bound for Rankin-Selberg L-functions for Maass forms, extending previous bounds to a broader class of automorphic forms.

Findings

Proves the bound $L(1/2+it,f\otimes g) \ll (\mu_f+|t|)^{9/10+\varepsilon}$.

The implied constant depends only on $\varepsilon$ and $g$.

Advances the understanding of L-function behavior in the critical strip.

Abstract

Let $f$ be a Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+\mu_f^2$, $\mu_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$. In this paper, we establish the following uniform subconvexity bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ $$ L\left(1/2+it,f\otimes g\right)\ll (\mu_f+|t|)^{9/10+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$ and $g$.