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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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  • [1] H. Bateman, Higher Transcendental Functions, vol.1, McGraw-Hill, 1953, New York.
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, with Based, in part, on notes left by Harry Bateman, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. MR 0065685
  • [3] Valentin Blomer, Rizwanur Khan, and Matthew Young, Distribution of mass of holomorphic cusp forms, Duke Math. J. 162 (2013), no. 14, 2609-2644. MR 3127809, https://doi.org/10.1215/00127094-2380967
  • [4] Daniel Bump, Automorphic forms on $ {\rm GL}(3,{\bf R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698
  • [5] Daniel Bump, The Rankin-Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 49-109. MR 993311
  • [6] J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic $ L$-functions, Ann. of Math. (2) 151 (2000), no. 3, 1175-1216. MR 1779567, https://doi.org/10.2307/121132
  • [7] Amit Ghosh and Peter Sarnak, Real zeros of holomorphic Hecke cusp forms, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 465-487. MR 2881302, https://doi.org/10.4171/JEMS/308
  • [8] Dorian Goldfeld, Automorphic forms and $ L$-functions for the group $ {\rm GL}(n,\mathbf{R})$, with With an appendix by Kevin A. Broughan, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. MR 2254662
  • [9] D. R. Heath-Brown, The twelfth power moment of the Riemann-function, Quart. J. Math. Oxford Ser. (2) 29 (1978), no. 116, 443-462. MR 517737, https://doi.org/10.1093/qmath/29.4.443
  • [10] Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and the Siegel zero, with With an appendix by Dorian Goldfeld, Hoffstein, and Daniel Lieman, Ann. of Math. (2) 140 (1994), no. 1, 161-181. MR 1289494, https://doi.org/10.2307/2118543
  • [11] M. N. Huxley, Area, lattice points, and exponential sums, London Mathematical Society Monographs. New Series, vol. 13, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. MR 1420620
  • [12] Aleksandar Ivić, The Riemann zeta-function: The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. MR 792089
  • [13] H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of $ L$-functions, Geom. Funct. Anal. Special Volume (2000), 705-741. GAFA 2000 (Tel Aviv, 1999). MR 1826269, https://doi.org/10.1007/978-3-0346-0425-3_6
  • [14] Hervé Jacquet and Joseph Shalika, Exterior square $ L$-functions, Automorphic forms, Shimura varieties, and $ L$-functions, Vol.II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143-226. MR 1044830
  • [15] Matti Jutila and Yoichi Motohashi, Uniform bound for Hecke $ L$-functions, Acta Math. 195 (2005), 61-115. MR 2233686, https://doi.org/10.1007/BF02588051
  • [16] N. V. Kuznetsov, Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture, Sums of Kloosterman sums, Math. USSR Sbornik 29 (1981), 299-342.
  • [17] Erez M. Lapid, On the nonnegativity of Rankin-Selberg $ L$-functions at the center of symmetry, Int. Math. Res. Not. 2 (2003), 65-75. MR 1936579, https://doi.org/10.1155/S1073792803204013
  • [18] Yuk-Kam Lau, Jianya Liu, and Yangbo Ye, A new bound $ k^{2/3+\epsilon}$ for Rankin-Selberg $ L$-functions for Hecke congruence subgroups, IMRP Int. Math. Res. Pap. (2006), Art. ID 35090, 78. MR 2235495
  • [19] Xiaoqing Li, The central value of the Rankin-Selberg $ L$-functions, Geom. Funct. Anal. 18 (2009), no. 5, 1660-1695. MR 2481739, https://doi.org/10.1007/s00039-008-0692-5
  • [20] Xiaoqing Li, Bounds for $ {\rm GL}(3)\times{\rm GL}(2)$ $ L$-functions and $ {\rm GL}(3)$ $ L$-functions, Ann. of Math. (2) 173 (2011), no. 1, 301-336. MR 2753605, https://doi.org/10.4007/annals.2011.173.1.8
  • [21] Jianya Liu and Yangbo Ye, Subconvexity for Rankin-Selberg $ L$-functions of Maass forms, Geom. Funct. Anal. 12 (2002), no. 6, 1296-1323. MR 1952930, https://doi.org/10.1007/s00039-002-1296-0
  • [22] Jianya Liu and Yangbo Ye, Petersson and Kuznetsov trace formulas, Lie groups and automorphic forms, AMS/IP Stud. Adv. Math., vol. 37, Amer. Math. Soc., Providence, RI, 2006, pp. 147-168. MR 2272921
  • [23] Qing Lu, Bounds for the spectral mean value of central values of $ L$-functions, J. Number Theory 132 (2012), no. 5, 1016-1037. MR 2890524, https://doi.org/10.1016/j.jnt.2011.12.008
  • [24] Wenzhi Luo and Peter Sarnak, Quantum variance for Hecke eigenforms, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 769-799 (English, with English and French summaries). MR 2103474, https://doi.org/10.1016/j.ansens.2004.08.001
  • [25] Mark McKee, Haiwei Sun, and Yangbo Ye, Weighted stationary phase of higher orders, Front. Math. China 12 (2017), no. 3, 675-702. MR 3630423, https://doi.org/10.1007/s11464-016-0615-y
  • [26] Philippe Michel, Analytic number theory and families of automorphic $ L$-functions, Automorphic forms and applications, IAS/Park City Math. Ser., vol. 12, Amer. Math. Soc., Providence, RI, 2007, pp. 181-295. MR 2331346
  • [27] Stephen D. Miller and Wilfried Schmid, Automorphic distributions, $ L$-functions, and Voronoi summation for $ {\rm GL}(3)$, Ann. of Math. (2) 164 (2006), no. 2, 423-488. MR 2247965, https://doi.org/10.4007/annals.2006.164.423
  • [28] Zhuangzhuang Peng, Zeros and central values of automorphic L-functions, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-Princeton University. MR 2701928
  • [29] Xiumin Ren and Yangbo Ye, Asymptotic Voronoi's summation formulas and their duality for $ SL_3(\mathbb{Z})$, Number theory--arithmetic in Shangri-La, Ser. Number Theory Appl., vol. 8, World Sci. Publ., Hackensack, NJ, 2013, pp. 213-236. MR 3089018, https://doi.org/10.1142/9789814452458_0012
  • [30] XiuMin Ren and YangBo Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $ \mathrm{GL}_m(\mathbb{Z})$, Sci. China Math. 58 (2015), no. 10, 2105-2124. MR 3400638, https://doi.org/10.1007/s11425-014-4955-3
  • [31] Peter Sarnak, Estimates for Rankin-Selberg $ L$-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), no. 2, 419-453. MR 1851004, https://doi.org/10.1006/jfan.2001.3783
  • [32] Yangbo Ye, The fourth power moment of automorphic $ L$-functions for $ {\rm GL}(2)$ over a short interval, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2259-2268. MR 2197443, https://doi.org/10.1090/S0002-9947-05-03831-6
  • [33] Yangbo Ye and Deyu Zhang, Zero density for automorphic $ L$-functions, J. Number Theory 133 (2013), no. 11, 3877-3901. MR 3084304, https://doi.org/10.1016/j.jnt.2013.05.012

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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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