RESONANCE OF AUTOMORPHIC FORMS FOR GL(3)

作者：Ren Xiumin^*; Ye Yangbo

来源：Transactions of the American Mathematical Society, 2015, 367(3): 2137-2157.

• Automorphic forms for GL(3)
• resonance
• Voronoi summation formula
• modularity testing

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.

• 出版日期2015-3
• 单位山东大学