Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908874568630272 |
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| author | Psyroukis, Rafail |
| author_facet | Psyroukis, Rafail |
| contents | We investigate the analytic properties of a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms for orthogonal groups of signature $(2,n+2)$. Using an orthogonal Eisenstein series of Klingen type, we obtain an integral representation for this Dirichlet series. In the case when the corresponding lattice has only one $1$-dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally $4 \mid n$, we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree $2$. We obtain, in this way, the meromorphic continuation of the Dirichlet series to $\mathbb{C}$ as a corollary. In the case of the $E_8$ lattice, we are able to further deduce a precise functional equation for the Dirichlet series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series Psyroukis, Rafail Number Theory 11F55, 11F66 (Primary) 11F60, 11F50 (Secondary) We investigate the analytic properties of a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms for orthogonal groups of signature $(2,n+2)$. Using an orthogonal Eisenstein series of Klingen type, we obtain an integral representation for this Dirichlet series. In the case when the corresponding lattice has only one $1$-dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally $4 \mid n$, we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree $2$. We obtain, in this way, the meromorphic continuation of the Dirichlet series to $\mathbb{C}$ as a corollary. In the case of the $E_8$ lattice, we are able to further deduce a precise functional equation for the Dirichlet series. |
| title | Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series |
| topic | Number Theory 11F55, 11F66 (Primary) 11F60, 11F50 (Secondary) |
| url | https://arxiv.org/abs/2411.15956 |