Article contents
UNIFORM BOUNDS FOR
$\operatorname {GL}(3)\times \operatorname {GL}(2)$ L-FUNCTIONS
Published online by Cambridge University Press: 17 October 2023
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
$$\begin{align*}L(1/2+it,\phi\times f) \ll_{\phi,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \end{align*}$$
$L(1/2+it,\phi \times f)$ whenever
$|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon }$.
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline688.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline689.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline690.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline691.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline692.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline693.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline694.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline695.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline696.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline697.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline698.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline699.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline700.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline701.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline702.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline703.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline704.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline705.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline706.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline707.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline708.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline709.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline710.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline711.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline712.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline713.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline714.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline715.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline716.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline717.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline718.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline719.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline720.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline721.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline722.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline723.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline724.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline725.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline726.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline727.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline728.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline729.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline730.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline731.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline732.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline733.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline734.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline735.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline736.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline737.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline738.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231016130844414-0590:S1474748023000348:S1474748023000348_inline739.png?pub-status=live)
- 1
- Cited by