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Transactions of the American Mathematical Society

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Improved subconvexity bounds for $ GL(2)\times GL(3)$ and $ GL(3)$ $ L$-functions by weighted stationary phase


Authors: Mark McKee, Haiwei Sun and Yangbo Ye
Journal: Trans. Amer. Math. Soc. 370 (2018), 3745-3769
MSC (2010): Primary 11F66, 11M41, 41A60
DOI: https://doi.org/10.1090/tran/7159
Published electronically: December 14, 2017
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Abstract: Let $ f$ be a fixed self-contragradient Hecke-Maass form for
$ SL(3,\mathbb{Z})$, and let $ u$ be an even Hecke-Maass form for $ SL(2,\mathbb{Z})$ with Laplace eigenvalue $ 1/4+k^2$, $ k\geq 0$. A subconvexity bound $ O\big ((1+k)^{4/3+\varepsilon }\big )$ in the eigenvalue aspect is proved for the central value at $ s=1/2$ of the Rankin-Selberg $ L$-function $ L(s,f\times u)$. Meanwhile, a subconvexity bound $ O\big ((1+\vert t\vert)^{2/3+\varepsilon }\big )$ in the $ t$ aspect is proved for $ L(1/2+it,f)$. These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main techniques in the proofs, other than those used by Li, are $ n$th-order asymptotic expansions of exponential integrals in the cases of the explicit first derivative test, the weighted first derivative test, and the weighted stationary phase integral, for arbitrary $ n\geq 1$. These asymptotic expansions sharpened the classical results for $ n=1$ by Huxley.


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Additional Information

Mark McKee
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
Email: mark.mckee.zoso@gmail.com

Haiwei Sun
Affiliation: School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, People’s Republic of China
Email: hwsun@sdu.edu.cn

Yangbo Ye
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
Email: yangbo-ye@uiowa.edu

DOI: https://doi.org/10.1090/tran/7159
Keywords: $GL(3)$, $GL(3)\times GL(2)$, automorphic $L$-function, Rankin--Selberg $L$-function, subconvexity bound, first derivative test, weighted stationary phase
Received by editor(s): September 6, 2016
Published electronically: December 14, 2017
Additional Notes: These authors contributed equally to this work.
Yangbo Ye is the corresponding author.
The second author was partially supported by the National Natural Science Foundation of China (Grant No. 11601271) and China Postdoctoral Science Foundation Funded Project (Project No. 2016M602125).
Article copyright: © Copyright 2017 American Mathematical Society

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