Uniform bounds for $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions

Abstract

In this paper, we prove uniform bounds for $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$ spectral aspect and the $t$ aspect by a delta method. More precisely, let $\varphi$ be a Hecke--Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ and $f$ a Hecke--Maass cusp form for $\mathrm{SL}(2,\mathbb{Z})$ with the spectral parameter $t_f$. Then for $t\in \mathbb{R}$ and any $\epsilon >0$, we have \[ L(1/2+it,ϕ\times f) \ll_{ϕ,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \] Moreover, we get subconvexity bounds for $L(1/2+it,\varphi \times f)$ whenever $|t|-t_f\gg (|t|+t_f{)}^{3/5+\epsilon }$.

arXiv comment33 pages. Incorporate the referees' comments and corrections. Comments welcome!