Exponential sums involving automorphic forms for GL(3) over arithmetic progressions

doi:10.1007/s11464-018-0732-x

Front. Math. China

2018, Vol. 13

Issue (6) : 1355-1368 https://doi.org/10.1007/s11464-018-0732-x

RESEARCH ARTICLE

Exponential sums involving automorphic forms for GL(3) over arithmetic progressions

Xiaoguang HE(

)

School of Mathematics, Shandong University, Jinan 250100, China

Download: PDF(322 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Let f be a Hecke-Maass cusp form for SL(3; $ℤ$ ) with Fourier coefficients A_f(m; n); and let $ϕ$ (x) be a $C ∞$ -function supported on [1; 2] with derivatives bounded by $ϕ (j) (x) ≪ j$ 1. We prove an asymptotic formula for the nonlinear exponential sum $Σ n ≡ l ⁢ mod ⁢ q ⁡ A f (m, n) φ (n / X) e (3 (k n)) 1 / 3 / q$ , where $e (z) = e 2 π i z$ and $k ∈ ℤ + .$

Keywords Automorphic forms for GL(3) exponential sum, arithmetic progression

Corresponding Author(s): Xiaoguang HE

Issue Date: 02 January 2019

Cite this article:

Xiaoguang HE. Exponential sums involving automorphic forms for GL(3) over arithmetic progressions[J]. Front. Math. China, 2018, 13(6): 1355-1368.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0732-x
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I6/1355

1	DGoldfeld. Automorphic forms and L-functions for the group GL(n;ℝ): Cambridge Stud Adv Math, Vol 99. Cambridge: Cambridge Univ Press, 2006
2	DGoldfeld, X QLi. Voronoi formulas on GL(n): Int Math Res Not IMRN, 2006, Art ID: 86295 (25pp)
3	DGoldfeld, X QLi. The Voronoi formula for GL(n;ℝ): Int Math Res Not IMRN, 2008, (9): rnm144 (39pp)
4	HIwaniec, W ZLuo, PSarnak. Low lying zeros of families of L-functions. Inst Hautes Études Sci Publ Math, 2000, 91: 55–131
5	JKaczorowski, A Perelli. On the structure of the Selberg class VI: non-linear twists. Acta Arith, 2005, 116: 315–341
6	HKim. Functoriality for the exterior square of GL4 and the symmetric fourth of GL2: J Amer Math Soc, 2003, 16: 139–183
7	X QLi. The central value of the Rankin-Selberg L-functions. Geom Funct Anal, 2009, 18(5): 1660–1695
8	S DMiller, W.SchmidAutomorphic distributions, L-functions, and Voronoi summation for GL(3): Ann of Math (2), 2006, 164(2): 423–488
9	X MRen, Y BYe. Resonance between automorphic forms and exponential functions. Sci China Math, 2010, 53: 2463–2472
10	X MRen, Y B.YeAsymptotic Voronoi's summation formulas and their duality for SL3( ℤ): In: Kanemitsu S, Li H-Z, Liu J-Y, eds. Number Theory: Arithmetic in Shangri-La. Proceedings of the 6th China-Japan Seminar. Series on Number Theory and Its Applications, Vol 8. Singapore: World Scientific, 2013, 213–236
11	X MRen, Y B.YeResonance of automorphic forms for GL(3): Trans Amer Math Soc, 2015, 367(3): 2137–2157
12	P.SarnakNotes on the generalized Ramanujan conjectures. In: Arthur J, Ellwood D, Kottwitz R, eds. Harmonic Analysis, the Trace Formula, and Shimura Varieties. Clay Math Proc, Vol 4. Providence: Amer Math Soc, 2005, 659–685
13	Q FSun, Y YWu. Exponential sums involving Maass forms. Front Math China, 2014, 9(6): 1349–1366
14	X FYan. On some exponential sums involving Maass forms over arithmetic progressions. J Number Theory, 2016, 160: 44–59

Viewed

Full text

Abstract

Cited

Shared

Discussed