Taiwanese Journal of Mathematics

Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$

Xiumin Ren and Yangbo Ye

Full-text: Open access

Abstract

Hyper-Kloosterman sums of different moduli appear naturally in Voronoi's summation formula for cusp forms for $\operatorname{GL}_m(\mathbb{Z})$. In this paper their square moment is evaluated and their bounds are proved in the case of consecutively dividing moduli. As an application, smooth sums of Fourier coefficients of a Maass form for $\operatorname{SL}_m(\mathbb{Z})$ against an exponential function $e(\alpha n)$ are estimated. These sums are proved to have rapid decay when $\alpha$ is a fixed rational number or a transcendental number with approximation exponent $\tau(\alpha) \gt m$. Non-trivial bounds are proved for these sums when $\tau(\alpha) \gt (m+1)/2$.

Article information

Source
Taiwanese J. Math., Volume 20, Number 6 (2016), 1251-1274.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874530

Digital Object Identifier
doi:10.11650/tjm.20.2016.7389

Mathematical Reviews number (MathSciNet)
MR3580294

Zentralblatt MATH identifier
1357.11068

Subjects
Primary: 11L05: Gauss and Kloosterman sums; generalizations 11L07: Estimates on exponential sums 11F30: Fourier coefficients of automorphic forms

Keywords
hyper-Kloosterman sum Maass form for $\operatorname{SL}_m(\mathbb{Z})$ oscillation of Maass form Voronoi summation formula

Citation

Ren, Xiumin; Ye, Yangbo. Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$. Taiwanese J. Math. 20 (2016), no. 6, 1251--1274. doi:10.11650/tjm.20.2016.7389. https://projecteuclid.org/euclid.twjm/1498874530


Export citation

References

  • T. Cochrane, M.-C. Liu and Z. Zheng, Upper bounds on $n$-dimensional Kloosterman sums, J. Number Theory 106 (2004), no. 1, 259–274.
  • K. Czarnecki, Resonance sums for Rankin-Selberg products of $\orn{SL}_m(\mb{Z})$ Maass cusp forms, J. Number Theory 163 (2016), 359–374.
  • P. Deligne, Applications de la formule des traces aux sommes trigonométrigues, in Cohomologie Etale, 168–232, Lecture Notes in Mathematics 569, Springer, New York, 1977.
  • A.-M. Ernvall-Hytönen, On certain exponential sums related to $\orn{GL}(3)$ cusp forms, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 5–8.
  • A.-M. Ernvall-Hytönen, J. Jääsaari and E. V. Vesalainen, Resonances and $\Omega$-results for exponential sums related to Maass forms for $\orn{SL}(n,\mb{Z})$, J. Number Theory 153 (2015), 135–157.
  • D. Goldfeld and X. Li, Voronoi formulas on $\orn{GL}(n)$, Int. Math. Res. Not. 2006, Art. ID 86295, 25 pp.
  • ––––, The Voronoi formula for $\orn{GL}(n,\mb{R})$, Int. Math. Res. Not. IMRN 2008, no. 2, Art. ID rnm144, 39 pp.
  • ––––, Addendum to: “The Voronoi formula for $\orn{GL}(n,\mb{R})$", [Int. Math. Res. Not. IMRN 2008, no. 2, Art. ID rnm144, 39 pp.; MR2418857], Int. Math. Res. Not. IMRN 2008, Art. ID rnn123, 1 p.
  • J. L. Hafner, Some remarks on odd Maass wave forms (and a correction to: “Zeros of $L$-functions attached to Maass forms”, [Math. Z. 190 (1985), no. 1, 113–128] by Hafner, C. Epstein and P. Sarnak), Math. Z. 196 (1987), no. 1, 129–132.
  • J. L. Hafner and A. Ivić, On sums of Fourier coefficients of cusp forms, Enseign. Math. (2) 35 (1989), no. 3-4, 375–382.
  • H. Hasse, Theorei der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper, J. Reine Angew. Math. 172 (1934), 37–54.
  • C. Hooley, An asymptotic formula in the theory of numbers, Proc. London. Math. Soc. (3) 7 (1957), 396–413.
  • N. M. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Annals of Mathematics Studies 116, Princeton University Press, Princeton, NJ, 1988.
  • H. D. Kloosterman, On the representations of a number in the form $ax^2+by^2+cz^2+dt^2$, Acta. Math. 49 (1926), 407–464.
  • X. Li, Bounds for $\orn{GL}(3) \times \orn{GL}(2)$ $L$-functions and $\orn{GL}(3)$ $L$-functions, Ann. of Math. (2) 173 (2011), no. 1, 301–336.
  • S. D. Miller, Cancellation in additively twisted sums on $\orn{GL}(n)$, Amer. J. Math. 128 (2006), no. 3, 699–729.
  • S. D. Miller and W. Schmid, A general Voronoi summation formula for $\orn{GL}(n,\mb{Z})$, in Geometry and Analysis, No. 2, 173–224, Adv. Lect. Math. 18, Int. Press, Somerville, 2011.
  • L. J. Mordell, On a special polynomial congruence and exponential sum, in 1963 Calcutta Math. Soc. Golden Jubilee Commemoration Vol. pp. 29–32, Calcutta Math. Soc., Calcutta.
  • X. Ren and Y. Ye, Asymptotic Voronoi's summation formulas and their duality for $\orn{SL}_3(\mb{Z})$, in Number Theory: Arithmetic in Shangri-La, 213–236, Ser. Number Theory Appl. 8, World Sci., Hackensack, NJ, 2013.
  • ––––, Sums of Fourier coefficients of a Maass form for $\orn{SL}_3(\mb{Z})$ twisted by exponential functions, Forum Math. 26 (2014), no. 1, 221–238.
  • ––––, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $\orn{GL}_m(Z)$, Sci. China Math. 58 (2015), no. 10, 2105–2124.
  • ––––, Resonance of automorphic forms for $\orn{GL}(3)$, Trans. Amer. Math. Soc. 367 (2015), no. 3, 2137–2157.
  • R. A. Smith, On $n$-dimensional Kloosterman sums, J. Number Theory 11 (1979), no. 3, 324–343.
  • A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207.
  • Y. Ye, Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith. 86 (1998), no. 3, 255–267.
  • ––––, Estimation of exponential sums of polynomials of higher degrees II, Acta Arith. 93 (2000), no. 3, 221–235.