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Article overview
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Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(mathbb Z)$ | | Xiumin Ren
; Yangbo Ye
; | | Date: |
21 Nov 2014 | | 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
$e(alpha|n|^eta)$ is proved to be rapidly decayed when $0<eta<1/m$. When
$eta=1/m$ 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. | | Source: | arXiv, 1411.6022 | | Services: | Forum | Review | PDF | Favorites | |
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