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- W4379914511 abstract "In this paper, we first give a new proof of the Voronoi summation formula for $mathrm{GL}_n$ over a number field by means of the $pi$-Poisson summation formula on $mathrm{GL}_1$ for any irreducible cuspidal automorphic representation $pi$ of $mathrm{GL}_n$. The duality on both sides of the Voronoi formula is related by the $pi$-Fourier transform. Then we introduce the notion of the Godement-Jacquet kernels $H_{pi,s}$ and their dual kernels $K_{pi,s}$ for any irreducible cuspidal automorphic representation $pi$ of $mathrm{GL}_n$ and show that $H_{pi,s}$ and $K_{pi,1-s}$ are related by the nonlinear $pi_infty$-Fourier transform if and only if $sinmathbb{C}$ is a zero of $L_f(s,pi_f)=0$, the finite part of the standard automorphic $L$-function $L(s,pi)$, which are the $(mathrm{GL}_n,pi)$-versions of Clozel's Theorem that is for the Tate kernel with $n=1$ and $pi$ the trivial character." @default.
- W4379914511 created "2023-06-09" @default.
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- W4379914511 date "2023-06-04" @default.
- W4379914511 modified "2023-09-26" @default.
- W4379914511 title "The Voronoi Summation Formula for $mathrm{GL}_n$ and the Godement-Jacquet Kernels" @default.
- W4379914511 doi "https://doi.org/10.48550/arxiv.2306.02554" @default.
- W4379914511 hasPublicationYear "2023" @default.
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