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

Abstract

Let f be a Hecke-Maass cusp form for SL(3;\mathrm{\mathbb{Z}}) with Fourier coefficients A_f(m; n); and let \varphi (x) be a C^{\infty } -function supported on [1; 2] with derivatives bounded by {\varphi }^{(j)}(x)\ll j 1. We prove an asymptotic formula for the nonlinear exponential sum {\mathrm{\Sigma }}_{n\equiv l\mathrm{mod}q}{A}_f(m,n)\phi (n/X)e{\left(3\left(kn\right)\right)}^{1/3}/q, where e(z)={\mathrm{e}}^{\mathrm{2}\pi iz} and k\in {\mathrm{\mathbb{Z}}}^{+}.

Key words： Automorphic forms for GL(3); exponential sum, arithmetic progression

References

1 Goldfeld D. Automorphic forms and L-functions for the group GL(n;ℝ): Cambridge Stud Adv Math, Vol 99. Cambridge: Cambridge Univ Press, 2006

2 Goldfeld D, Li X Q. Voronoi formulas on GL(n): Int Math Res Not IMRN, 2006, Art ID: 86295 (25pp)

3 Goldfeld D, Li X Q. The Voronoi formula for GL(n;ℝ): Int Math Res Not IMRN, 2008, (9): rnm144 (39pp)

4 Iwaniec H, Luo W Z, Sarnak P. Low lying zeros of families of L-functions. Inst Hautes Études Sci Publ Math, 2000, 91: 55–131

5 Kaczorowski J, Perelli A. On the structure of the Selberg class VI: non-linear twists. Acta Arith, 2005, 116: 315–341

6 Kim H. Functoriality for the exterior square of GL₄ and the symmetric fourth of GL₂: J Amer Math Soc, 2003, 16: 139–183

7 Li X Q. The central value of the Rankin-Selberg L-functions. Geom Funct Anal, 2009, 18(5): 1660–1695

8 Miller S D, Schmid W.Automorphic distributions, L-functions, and Voronoi summation for GL(3): Ann of Math (2), 2006, 164(2): 423–488

9 Ren X M, Ye Y B. Resonance between automorphic forms and exponential functions. Sci China Math, 2010, 53: 2463–2472

10 Ren X M, Ye Y B.Asymptotic 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 Ren X M, Ye Y B.Resonance of automorphic forms for GL(3): Trans Amer Math Soc, 2015, 367(3): 2137–2157

12 Sarnak P.Notes 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 Sun Q F, Wu Y Y. Exponential sums involving Maass forms. Front Math China, 2014, 9(6): 1349–1366

14 Yan X F. On some exponential sums involving Maass forms over arithmetic progressions. J Number Theory, 2016, 160: 44–59