A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups

TL;DR

This paper introduces a new Dirichlet series involving Fourier-Jacobi coefficients of cusp forms on orthogonal groups, establishing connections with standard L-functions and deriving explicit Euler product formulas for special cases.

Contribution

It constructs a Fourier-Jacobi Dirichlet series for orthogonal cusp forms, linking it to L-functions and providing explicit Euler product formulas for certain orthogonal groups.

Findings

Connection with standard L-functions when F is a Hecke eigenform and G is a Maass lift

Explicit Euler product formulas for specific orthogonal groups

Recovery of classical results for Siegel modular forms and new examples for other modular forms

Abstract

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms $F,G$ for orthogonal groups of signature $(2,n+2)$. In the case when $F$ is a Hecke eigenform and $G$ is a Maass lift of a Poincar\'e series, we establish a connection with the standard $L$-function attached to $F$. What is more, we find explicit choices of orthogonal groups, for which we obtain a clear-cut Euler product expression for this Dirichlet series. Through our considerations, we recover a classical result for Siegel modular forms, first introduced by Kohnen and Skoruppa, but also provide a range of new examples, which can be related to other kinds of modular forms, such as paramodular, Hermitian, and quaternionic.