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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Resonance of automorphic forms for $ GL(3)$

Authors: Xiumin Ren and Yangbo Ye
Journal: Trans. Amer. Math. Soc. 367 (2015), 2137-2157
MSC (2010): Primary 11L07, 11F30
Published electronically: August 12, 2014
MathSciNet review: 3286510
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Abstract: Let $ f$ be a Maass form for $ SL_3(\mathbb{Z})$ with Fourier coefficients
$ A_f(m,n)$. A smoothly weighted sum of $ A_f(m,n)$ against an exponential function $ e(\alpha n^\beta )$ of fractional power $ n^\beta $ for $ X\leq n\leq 2X$ is proved to have a main term of size $ X^{2/3}$ when $ \beta =1/3$ and $ \alpha $ is close to $ 3\ell ^{1/3}$ for some integer $ \ell \neq 0$. The sum becomes rapidly decreasing if $ \beta <1/3$. If such a sum is not smoothly weighted, the main term can only be detected under a conjectured bound toward the Ramanujan conjecture. The existence of such a main term manifests the vibration and resonance behavior of individual automorphic forms $ f$ for $ GL(3)$. Applications of these results include a new modularity test on whether a two dimensional array $ a(m,n)$ comes from Fourier coefficients $ A_f(m,n)$ of a Maass form $ f$ for $ SL_3(\mathbb{Z})$. Techniques used in the proof include a Voronoi summation formula, its asymptotic expansion, and the weighted stationary phase.

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Additional Information

Xiumin Ren
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China

Yangbo Ye
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419

Keywords: Automorphic forms for $GL(3)$, resonance, Voronoi summation formula, modularity testing
Received by editor(s): October 4, 2012
Received by editor(s) in revised form: May 6, 2013, and June 5, 2013
Published electronically: August 12, 2014
Article copyright: © Copyright 2014 Xiumin Ren and Yangbo Ye