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Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series

Rafail Psyroukis

Abstract

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 $4 \mid n$, 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 $\mathbb{C}$ as a corollary. In the case of the $E_8$ lattice, we are able to further deduce a precise functional equation for the Dirichlet series.

Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series

Abstract

We investigate the analytic properties of a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms for orthogonal groups of signature . 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 -dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally , we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree . We obtain, in this way, the meromorphic continuation of the Dirichlet series to as a corollary. In the case of the lattice, we are able to further deduce a precise functional equation for the Dirichlet series.

Paper Structure

This paper contains 9 sections, 24 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier-Jacobi Forms
  4. Dirichlet Series and Integral Representation
  5. Manipulation of the Eisenstein Series
  6. Theta Series and Transformation Properties
  7. Differential Operators
  8. Theta Correspondence
  9. The $E_8$ lattice

Key Result

Proposition 4.4

For $W \in \mathcal{H}_S$ and $s \in \mathbb{C}$ with $\textup{Re}(s) > n+2$, we have where $\langle \textup{ }, \textup{ }\rangle$ denotes the inner product of Definition inner_product_orthogonal and which is a finite group.

Theorems & Definitions (62)

  • Proposition 4.4
  • Proposition 5.3
  • Theorem 8.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • ...and 52 more