Uniform bounds for $\rm GL(3) \times GL(2)$ $L$-functions

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