On Fourier–Jacobi Dirichlet Series for Hermitian and Orthogonal Modular Forms

Abstract

This thesis is concerned with the study of analytic and arithmetic properties of Dirichlet series involving Fourier-Jacobi coefficients of Hermitian and orthogonal modular forms. It is naturally divided into two main parts.

In the first part, motivated by a work of B. Heim, we consider a Dirichlet series associated with three Hermitian cuspidal eigenforms of degrees and over and study its -factor for every rational prime . Using factorisation methods in parabolic Hecke rings, we show that for inert primes, this factor can be identified with the -twist of the degree Euler factor attached to a Hermitian modular form of degree by Gritsenko. For split primes, we obtain a rational expression for the local factor, allowing us to show that the Dirichlet series has an Euler product. Moreover, we show that this Dirichlet series arises as part of a Rankin-Selberg integral representation.

In the second part, we consider, in the spirit of Kohnen and Skoruppa, a Dirichlet series involving the Fourier-Jacobi coefficients of a pair of orthogonal modular forms of real signature . First, we obtain an integral representation of Rankin-Selberg type and use theta correspondence to deduce its analytic properties for certain orthogonal groups. Next, using results of Sugano and Shimura, we obtain, for certain orthogonal groups, an Euler product for the Dirichlet series and relate it to the standard -function for .