RESEARCH ARTICLE

Spectral square moments of a resonance sum for Maass forms

  • Nathan SALAZAR ,
  • Yangbo YE
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  • Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419, USA

Received date: 28 Jan 2016

Accepted date: 25 Nov 2016

Published date: 30 Sep 2017

Copyright

2016 Higher Education Press and Springer-Verlag GmbH Germany

Abstract

Let fbe a Maass cusp form for Γ0(N) with Fourier coefficients λf (n)and Laplace eigenvalue 14+k2.For real α0 and β>0,consider the sum SX(f; α, β) =nλf(n)e(αnβ)ϕ(n/X),where φis 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 fhas its eigenvalue beyond X12+ε,the standard resonance main term for SX(f;±2q, 1/2), q+,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 KεLK1ε. The same bounds can be proved in a similar way for holomorphic cusp forms.

Cite this article

Nathan SALAZAR , Yangbo YE . Spectral square moments of a resonance sum for Maass forms[J]. Frontiers of Mathematics in China, 2017 , 12(5) : 1183 -1200 . DOI: 10.1007/s11464-016-0621-0

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