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Resonance of automorphic forms for GL(3)
Journal article   Open access   Peer reviewed

Resonance of automorphic forms for GL(3)

Xiumin Ren and Yangbo Ye
Transactions of the American Mathematical Society, Vol.367(3), pp.2137-2157
03/01/2015
DOI: 10.1090/s0002-9947-2014-06208-9
url
https://doi.org/10.1090/s0002-9947-2014-06208-9View
Published (Version of record) Open Access

Abstract

Let f be a Maass form for SL3(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 <= n <= 2X is proved to have a main term of size X-2/3 when beta = 1/3 and a is close to 3l(1/3) for some integer l not equal 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 SL3(Z). Techniques used in the proof include a Voronoi summation formula, its asymptotic expansion, and the weighted stationary phase.
 Mathematics Physical Sciences Science & Technology

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