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UNIFORM BOUNDS FOR
$\operatorname {GL}(3)\times \operatorname {GL}(2)$ L-FUNCTIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 4 / July 2024
- Published online by Cambridge University Press:
- 17 October 2023, pp. 1607-1650
- Print publication:
- July 2024
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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 Moreover, we get subconvexity bounds for
$$\begin{align*}L(1/2+it,\phi\times f) \ll_{\phi,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \end{align*}$$
$L(1/2+it,\phi \times f)$ whenever 
$|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon }$.
