Resonance of automorphic forms for $GL(3)$
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- by Xiumin Ren and Yangbo Ye PDF
- Trans. Amer. Math. Soc. 367 (2015), 2137-2157
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.References
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Additional Information
- Xiumin Ren
- Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong 250100, Peopleโs Republic of China
- Email: xmren@sdu.edu.cn
- Yangbo Ye
- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 261621
- Email: yangbo-ye@uiowa.edu
- 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
- © Copyright 2014 Xiumin Ren and Yangbo Ye
- Journal: Trans. Amer. Math. Soc. 367 (2015), 2137-2157
- MSC (2010): Primary 11L07, 11F30
- DOI: https://doi.org/10.1090/S0002-9947-2014-06208-9
- MathSciNet review: 3286510