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UNIFORM BOUNDS FOR $\operatorname {GL}(3)\times \operatorname {GL}(2)$ L-FUNCTIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  17 October 2023

Bingrong Huang Open the ORCID record for Bingrong Huang [Opens in a new window]
Bingrong Huang*
Affiliation:
Data Science Institute and School of Mathematics, Shandong University, Jinan, Shandong 250100, China
*
brhuang@sdu.edu.cn
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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 $|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon }$.

Keywords

uniformsubconvexityL-functionGL(3) × GL(2)spectraldelta method

MSC classification

Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 44
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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