Resonance of automorphic forms for 𝐺𝐿(3)

Abstract

Letffbe a Maass form forSL3(Z)SL_3(\mathbb Z)with Fourier coefficientsAf(m,n)A_f(m,n). A smoothly weighted sum ofAf(m,n)A_f(m,n)against an exponential functione(αnβ)e(\alpha n^\beta )of fractional powernβn^\betaforX≤n≤2XX\leq n\leq 2Xis proved to have a main term of sizeX2/3X^{2/3}whenβ=1/3\beta =1/3andα\alphais close to3ℓ1/33\ell ^{1/3}for some integerℓ≠0\ell \neq 0. The sum becomes rapidly decreasing ifβ>1/3\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 formsffforGL(3)GL(3). Applications of these results include a new modularity test on whether a two dimensional arraya(m,n)a(m,n)comes from Fourier coefficientsAf(m,n)A_f(m,n)of a Maass formffforSL3(Z)SL_3(\mathbb Z). Techniques used in the proof include a Voronoi summation formula, its asymptotic expansion, and the weighted stationary phase.