Spectral square moments of a resonance sum for Maass forms

作     者：Nathan SALAZAR Yangbo YE 

作者机构：Department of Mathematics The University of Iowa Iowa City Iowa 52242-1419 USA 

出 版 物：《Frontiers of Mathematics in China》 (中国高等学校学术文摘·数学（英文）)

年 卷 期：2017年第12卷第5期

页      面：1183-1200页

核心收录：

学科分类：08[工学] 080502[工学-材料学] 0805[工学-材料科学与工程（可授工学、理学学位）] 0701[理学-数学] 080101[工学-一般力学与力学基础] 0801[工学-力学（可授工学、理学学位）] 

主　　题：Cusp form Maass form Fourier coefficient of cusp form Kuznetsovtrace formula resonance sum first derivative test weighted stationary phase 

摘      要：Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.