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

Abstract

Let $f$ be a Maass cusp form for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+\hat{I}{¼}_f^2$, $\hat{I}{¼}_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$. In this paper, we establish the following uniform subconvexity bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ $$L\left(1/2+it,f\otimes g\right)\ll (\hat{I}{¼}_f+|t|{)}^{9/10+\epsilon },$$ where the implied constant depends only on $\epsilon$ and $g$.

28 pages. Comments are welcome! arXiv admin note: substantial text overlap with arXiv:2108.09410