Resonance between automorphic forms and exponential functions  

作　　者：REN XiuMin1,& YE YangBo1,2 1School of Mathematics,Shandong University,Jinan 250100,China 2Department of Mathematics,The University of Iowa,Iowa City,Iowa 52242-1419,USA 

出　　处：《Science China Mathematics》2010年第9期2463-2472,共10页中国科学：数学（英文版）

基　　金：supported in part by National Basic Research Program of China (973-Program) (Grant No.);National Natural Science Foundation of China (Grant No.10971119)

摘　　要：Let f be a holomorphic cusp form of weight k for SL2(Z) and λf(n) its n-th Fourier coefficient.In this paper,the exponential sum X【n 2X λf(n)e(αnβ) twisted by Fourier coefficients λf(n) is proved toh ave a main term of size |λf(q)|X3/4 when β = 1/2 and α is close to ±2√q,q ∈ Z,and is smaller otherwise for β 【 3/4.This is a manifestation of the resonance spectrum of automorphic forms for SL2(Z).Let f be a holomorphic cusp form of weight k for SL2(Z) and λf(n) its n-th Fourier coefficient.In this paper,the exponential sum X<n 2X λf(n)e(αnβ) twisted by Fourier coefficients λf(n) is proved toh ave a main term of size |λf(q)|X3/4 when β = 1/2 and α is close to ±2√q,q ∈ Z,and is smaller otherwise for β < 3/4.This is a manifestation of the resonance spectrum of automorphic forms for SL2(Z).

关 键 词：automorphic FORM EXPONENTIAL SUM RESONANCE 

分 类 号：O174[理学—数学]