A Fourier-Jacobi Dirichlet series attached to modular forms of $SO(2,4)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Bouganis, Thanasis Psyroukis, Rafail |
| author_facet | Bouganis, Thanasis Psyroukis, Rafail |
| contents | We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a Jacobi-Poincaré series, our main result gives that $D(F,G;s)$ is equal to the standard $L$-function attached to $F$, up to an explicit constant. To establish this, we use a correspondence between binary Hermitian forms and ideals of quaternion algebras, as established by Latimer, together with the fact that the even Clifford algebra of a three-dimensional definite quadratic space can be identified with a quaternion division algebra. Our work should be seen as a generalisation of a work of Kohnen and Skoruppa, whose result corresponds to the case of the orthogonal group of real signature $(2,3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Fourier-Jacobi Dirichlet series attached to modular forms of $SO(2,4)$ Bouganis, Thanasis Psyroukis, Rafail Number Theory 11F55, 11E88, 11E41 (Primary) 11F66, 11M41 (Secondary) We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a Jacobi-Poincaré series, our main result gives that $D(F,G;s)$ is equal to the standard $L$-function attached to $F$, up to an explicit constant. To establish this, we use a correspondence between binary Hermitian forms and ideals of quaternion algebras, as established by Latimer, together with the fact that the even Clifford algebra of a three-dimensional definite quadratic space can be identified with a quaternion division algebra. Our work should be seen as a generalisation of a work of Kohnen and Skoruppa, whose result corresponds to the case of the orthogonal group of real signature $(2,3)$. |
| title | A Fourier-Jacobi Dirichlet series attached to modular forms of $SO(2,4)$ |
| topic | Number Theory 11F55, 11E88, 11E41 (Primary) 11F66, 11M41 (Secondary) |
| url | https://arxiv.org/abs/2510.26403 |