Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL_m(Z)  

作　　者：REN XiuMin YE YangBo 

机构地区：[1]Department of Mathematics, Shandong University [2]Department of Mathematics, The University of Iowa

出　　处：《Science China Mathematics》2015年第10期2105-2124,共20页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.10971119);Program for Changjiang Scolars and Innovative Research Team in University(Grant No.1264)

摘　　要：Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of this formula, a smoothly weighted average of Af(n, 1,..., 1) against e(α|n|β) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of |Af(1,..., 1, q) + Af(1,..., 1,-q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af(n, 1,..., 1). Similar estimate is also proved for a sharp-cut sum.Let f be a full-level cusp form for GLm（Z） with Fourier coefficients Af（n1,..., nm-1）. In this paper,an asymptotic expansion of Voronoi’s summation formula for Af（n1,..., nm-1） is established. As applications of this formula, a smoothly weighted average of Af（n, 1,..., 1） against e（α｜n｜β） is proved to be rapidly decayed when 0 〈 β 〈 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of ｜Af（1,..., 1, q） ＋ Af（1,..., 1,-q）｜X1/（2m）＋1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af（n, 1,..., 1）. Similar estimate is also proved for a sharp-cut sum.

关 键 词：cusp form for GLm（Z） Voronoi’s summation formula Fourier coefficient of cusp forms RESONANCE 

分 类 号：O174.2[理学—数学] TG143.5[理学—基础数学]