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

Thanasis Bouganis, Rafail Psyroukis

Abstract

We consider a Dirichlet series $D_{F,G}(s)$ attached to two modular 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 Poincaré 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 $(2,3)$.

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

Abstract

We consider a Dirichlet series attached to two modular forms and of an orthogonal group of real signature , involving their Fourier--Jacobi coefficients. When is a Hecke eigenform and a Poincaré series, our main result gives that is equal to the standard -function attached to , 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 .

Paper Structure

This paper contains 9 sections, 27 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modular Forms on $\textup{SO}(2,n)$
  4. Equivalence of Quadratic Spaces
  5. The Dirichlet Series $D_{F,\xi_i, \mathcal{P}}(s)$
  6. Quaternion Algebras
  7. Representation Numbers of Binary Hermitian Forms
  8. Partial Zeta functions and Latimer's work
  9. Main Theorem

Key Result

Theorem 1.1

Assume $K$ has class number $1$ and it is not $\mathbb{Q}(\sqrt{-2})$. Choose $\ell$ so that it satisfies the conditions of Proposition stabilisers of lattices. Assume $F$ is a Hecke eigenform for the corresponding Hecke algebra. Then, outside a finite set of primes $\mathcal{P}$, we have where $L(F;s)$ is the standard $L$-function attached to $F$, and $A(\xi)$ is the $\xi^{th}$ Fourier coefficie

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 42 more