UNIFORM BOUNDS FOR L-FUNCTIONS

Abstract

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