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

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- by Mark McKee, Haiwei Sun and Yangbo Ye PDF
- Trans. Amer. Math. Soc.
**370**(2018), 3745-3769 Request permission

## 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 $O\big ((1+|t|)^{2/3+\varepsilon }\big )$ 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.## References

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## Additional Information

**Mark McKee**- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 867414
- Email: mark.mckee.zoso@gmail.com
**Haiwei Sun**- Affiliation: School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, People’s Republic of China
- MR Author ID: 856910
- Email: hwsun@sdu.edu.cn
**Yangbo Ye**- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 261621
- Email: yangbo-ye@uiowa.edu
- 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). - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 3745-3769 - MSC (2010): Primary 11F66, 11M41, 41A60
- DOI: https://doi.org/10.1090/tran/7159
- MathSciNet review: 3766865