Journal article
Spectral square moments of a resonance sum for Maass forms
Frontiers of Mathematics in China, Vol.12(5), pp.1183-1200
10/2017
DOI: 10.1007/s11464-016-0621-0
Abstract
Let f be a Maass cusp form for Γ0(N) with Fourier coefficients λ f (n) and Laplace eigenvalue $$\frac{1} {4} + k^2 $$ 1 4 + k 2 . For real α ≠ 0 and β > 0, consider the sum S X (f; α, β) = ∑ n λ f (n)e(αn β )ϕ(n/X), where ϕ is a smooth function of compact support. We prove bounds for the second spectral moment of S X (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 $$X^{\tfrac{1} {2} + \varepsilon } $$ X 1 2 + ε , the standard resonance main term for S X (f; $$ \pm 2\sqrt q $$ ± 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 K ε ⩽ L ⩽ K 1−ε . The same bounds can be proved in a similar way for holomorphic cusp forms.
Details
- Title: Subtitle
- Spectral square moments of a resonance sum for Maass forms
- Creators
- Nathan Salazar - Department of Mathematics The University of Iowa Iowa City Iowa 52242-1419 USAYangbo Ye - Department of Mathematics The University of Iowa Iowa City Iowa 52242-1419 USA
- Resource Type
- Journal article
- Publication Details
- Frontiers of Mathematics in China, Vol.12(5), pp.1183-1200
- Publisher
- Higher Education Press; Beijing
- DOI
- 10.1007/s11464-016-0621-0
- ISSN
- 1673-3452
- eISSN
- 1673-3576
- Language
- English
- Date published
- 10/2017
- Academic Unit
- Mathematics
- Record Identifier
- 9983985826502771
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