Hostname: page-component-7f64f4797f-m2b9p Total loading time: 0 Render date: 2025-11-04T08:38:55.332Z Has data issue: false hasContentIssue false
  • English
  • Français

PERIODIC TWISTS OF $\operatorname{GL}_{3}$-AUTOMORPHIC FORMS

Part of: Finite fields and commutative rings (number-theoretic aspects) Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory Exponential sums and character sums

Published online by Cambridge University Press:  12 March 2020

EMMANUEL KOWALSKI ,
YONGXIAO LIN ,
PHILIPPE MICHEL  and
WILL SAWIN
EMMANUEL KOWALSKI
Affiliation:
ETHZ, Rämistrasse 101, 8092Zürich, Switzerland; kowalski@math.ethz.ch
YONGXIAO LIN
Affiliation:
EPFL/MATH/TAN, Station 8, CH-1015Lausanne, Switzerland; yongxiao.lin@epfl.ch, philippe.michel@epfl.ch
PHILIPPE MICHEL
Affiliation:
EPFL/MATH/TAN, Station 8, CH-1015Lausanne, Switzerland; yongxiao.lin@epfl.ch, philippe.michel@epfl.ch
WILL SAWIN
Affiliation:
Columbia University, USA; sawin@math.columbia.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.

MSC classification

Primary: 11F55: Other groups and their modular and automorphic forms (several variables) 11M41: Other Dirichlet series and zeta functions 11L07: Estimates on exponential sums
Secondary: 11T23: Exponential sums 32N10: Automorphic forms

Information

Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e15
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Aggarwal, K., Holowinsky, R., Lin, Y. and Sun, Q., ‘The Burgess bound via a trivial delta method’, Ramanujan J. (2018), (to appear) arXiv:1803.00542.Google Scholar
Blomer, V., ‘Subconvexity for twisted L-functions on GL(3)’, Amer. J. Math. 134(5) (2012), 1385–1421.CrossRefGoogle Scholar
Conrey, J. B. and Iwaniec, H., ‘The cubic moment of central values of automorphic L-functions’, Ann. of Math. (2) 151(3) (2000), 1175–1216.CrossRefGoogle Scholar
Fouvry, É., Kowalski, E. and Michel, Ph., ‘Algebraic trace functions over the primes’, Duke Math. J. 163(9) (2014), 1683–1736.CrossRefGoogle Scholar
Fouvry, É., Kowalski, E. and Michel, Ph., Trace Functions Over Finite Fields and Their Applications, Colloquium De Giorgi 2013 and 2014, Colloquia, 5 (Ed. Norm., Pisa, 2014), 7–35.Google Scholar
Fouvry, É., Kowalski, E. and Michel, Ph., ‘Algebraic twists of modular forms and Hecke orbits’, GAFA 25(2) (2015), 580–657.Google Scholar
Fouvry, É., Kowalski, E. and Michel, Ph., ‘On the exponent of distribution of the ternary divisor function’, Mathematika 61(1) (2015), 121–144.CrossRefGoogle Scholar
Fouvry, É., Kowalski, E., Michel, Ph., Raju, C., Rivat, J. and Soundararajan, K., ‘On short sums of trace functions’, Ann. Inst. Fourier 67 (2017), 423–449.CrossRefGoogle Scholar
Friedlander, J. B. and Iwaniec, H., ‘Incomplete Kloosterman sums and a divisor problem’, Ann. of Math. (2) 121(2) (1985), 319–350. (with an appendix by B. J. Birch and E. Bombieri).CrossRefGoogle Scholar
Goldfeld, D., Automorphic Forms and L-Functions for the Group GL(n, R), Cambridge Studies in Advanced Mathematics, 99 (Cambridge University Press, Cambridge, 2006), With an appendix by Kevin A. Broughan.Google Scholar
Holowinsky, R., Munshi, R. and Qi, Z., ‘Character sums of composite moduli and hybrid subconvexity’, inAdvances in the Theory of Automorphic Forms and Their L-functions, Contemporary Mathematics, 664 (American Mathematical Society, Providence, RI, 2016), 135–148.CrossRefGoogle Scholar
Holowinsky, R. and Nelson, P., ‘Subconvex bounds on GL(3) via degeneration to frequency zero’, Math. Ann. 372(1–2) (2018), 299–319.CrossRefGoogle Scholar
Iwaniec, H., Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
Katz, N. M., Exponential Sums and Differential Equations, Annals of Mathematics Studies, 124 (Princeton University Press, Princeton, NJ, 1990).CrossRefGoogle Scholar
Kim, H. and Sarnak, P., ‘Refined estimates towards the Ramanujan and Selberg conjectures’, J. Amer. Math. Soc. 16(1) (2003), 139–183. Appendix to H. Kim, Functoriality for the exterior square of $\text{GL}_{4}$ and the symmetric fourth of $\text{GL}_{2}$.CrossRefGoogle Scholar
Kowalski, E., Michel, Ph. and Sawin, W., Stratification and averaging for exponential sums: Bilinear forms with generalized Kloosterman sums, Ann. Sc. Norm. Super. Pisa, (to appear), Preprint, 2018, arXiv:1802.09849.CrossRefGoogle Scholar
Lin, Y., ‘Bounds for twists of $\text{GL}(3)$$L$-functions’, Preprint, 2018, arXiv:1802.05111.Google Scholar
Miller, S. D., ‘Cancellation in additively twisted sums on GL(n)’, Amer. J. Math. 128(3) (2006), 699–729.CrossRefGoogle Scholar
Molteni, G., ‘Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product’, Duke Math. J. 111(1) (2002), 133–158.CrossRefGoogle Scholar
Munshi, R., ‘The circle method and bounds for L-functions—IV: Subconvexity for twists of GL(3) L-functions’, Ann. of Math. (2) 182(2) (2015), 617–672.CrossRefGoogle Scholar
Munshi, R., ‘Twists of $\operatorname{GL}(3)$$L$-functions’, Preprint, 2016, arXiv:1604.08000.Google Scholar
Petrow, I. and Young, M., ‘The Weyl bound for Dirichlet $L$-functions of cube-free conductor’, Preprint, 2018, arXiv:1811.02452.Google Scholar
Sun, Q. and Zhao, R., ‘Bounds for GL3 L-functions in depth aspect’, Forum Math. 31(2) (2019), 303–318.CrossRefGoogle Scholar
Zacharias, R., ‘Simultaneous non-vanishing for Dirichlet L-functions’, Ann. Inst. Fourier 69(4) (2019), 1459–1524.CrossRefGoogle Scholar
Zhou, F., ‘The Voronoi formula on $GL(3)$ with ramification’, Preprint, 2018,arXiv:1806.10786.Google Scholar