A Fourier-Jacobi Dirichlet series attached to modular forms of $SO(2,4)$

TL;DR

This paper studies a special Dirichlet series linked to automorphic forms on the orthogonal group SO(2,4), showing it equals the standard L-function for certain forms, extending previous work on lower-dimensional cases.

Contribution

It establishes a new explicit relation between a Fourier-Jacobi Dirichlet series and the standard L-function for automorphic forms on SO(2,4), generalizing earlier results for SO(2,3).

Findings

Proves the Dirichlet series equals the standard L-function up to a constant.

Uses quaternion algebra correspondence to establish the result.

Extends known results from SO(2,3) to SO(2,4).

Abstract

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-Poincar\'e 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 $(2,3)$.