Journal article
Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(\mathbb Z)
Science China. Mathematics, Vol.58(10), pp.1-20
11/21/2014
DOI: 10.1007/s11425-014-4955-3
Abstract
Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a smoothly weighted average of $A_f(n,1,...,1)$ against $e(\alpha|n|^\beta)$ is proved to be rapidly decayed when $0<\beta<1/m$. When $\beta=1/m$ and $\alpha$ equals or approaches $\pm mq^{1/m}$ for a positive integer $q$, this smooth average has a main term of the size of $|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients $A_f(n,1,...,1)$. Similar estimate is also proved for a sharp-cut sum.
Details
- Title: Subtitle
- Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(\mathbb Z)
- Creators
- Xiumin RenYangbo Ye
- Resource Type
- Journal article
- Publication Details
- Science China. Mathematics, Vol.58(10), pp.1-20
- DOI
- 10.1007/s11425-014-4955-3
- ISSN
- 1674-7283
- eISSN
- 1869-1862
- Language
- English
- Date published
- 11/21/2014
- Academic Unit
- Mathematics
- Record Identifier
- 9983985859102771
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