2024/11/24 by Psyroukis, Rafail
#11F50 (Secondary) #11F55 #11F66 (Primary) 11F60 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2411.15956
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 | 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 ℂ as a corollary. In the case of the E8 lattice, we are able to further deduce a precise functional equation for the Dirichlet series.