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

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

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; [email protected]
YONGXIAO LIN
Affiliation:
EPFL/MATH/TAN, Station 8, CH-1015Lausanne, Switzerland; [email protected], [email protected]
PHILIPPE MICHEL
Affiliation:
EPFL/MATH/TAN, Station 8, CH-1015Lausanne, Switzerland; [email protected], [email protected]
WILL SAWIN
Affiliation:
Columbia University, USA; [email protected]

Abstract

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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
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

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