Note on Hermitian Jacobi forms

数理解析研究所講究録, 1992

1993

For any positive integer g ∈ Z+, we let Hg := { Z ∈ C | t Z = Z , Im Z > 0 } be the Siegel upper half plane of degree g and 0g := Sp(g,Z) be the Siegel modular group of degree g. Let ρ be an irreducible finite dimensional representation of GL(g,C) and M be a symmetric half integral positive definite matrix of degree h. It is known ([Z] Theorem 1.8) that the vector space Jρ,M(0g) of all Jacobi forms of index M with respect to ρ on 0g is finite dimensional. For the precise definition of Jρ,M(0g), we refer to Definition 2.2. It is a natural question to ask under which conditions the vector space Jρ,M(0g) vanishes. In this paper, the author gives a vanishing theorem on Jρ,M(0g). In section 2, we establish the notations and review some properties of Jacobi forms. In section 3, we define the Siegel-Jacobi operator and give the relation between the corank of a Jacobi form and the corank of ρ. In the final section, we establish the Shimura isomorphism and give a vanishing theorem on Jaco...

Mathematische Zeitschrift, 2009

In this paper, we discuss the dimension formula for Jacobi forms via the Selberg trace formula, an explicit dimension formula for the space of Jacobi cusp forms of degree 2 is given as an application.

Journal of Number Theory, 1996

In this paper we describe the space of Jacobi forms on H_C n . This type of Jacobi forms appears in the Fourier Jacobi expansions of all kinds of modular forms of several variables. We can characterize the space of Jacobi forms of weight k by vector valued elliptic modular forms of weight k&nÂ2. Then we use the Jordan theoretic language in order to describe modular forms on the orthogonal group O(2, n+2), whose Fourier Jacobi expansions also yield Jacobi forms of our type. Finally we determine a Maa? space as the isomorphic image of a particular space of Jacobi forms. Especially our procedure guarantees the existence of certain nontrivial singular modular forms.

Transactions of American Math. Soc., Vol. 347 (1995), n. 6, 2041-2049

We introduce the differential operator Mgj,^ characterizing singular Jacobi forms. We also characterize singular Jacobi forms by the weight of the associated rational representation of the general linear group. And we provide eigenfunctions of the differential operator Mgj,^ .

Rocky Mountain Journal of Mathematics, 2009

We present a combinatorial characterization of Jacobi forms a la Henri Cohen.

Kyungpook Math. J. 53 (2013), 49-86

This article is a continuation of the paper . In this paper we deal with Maass-Jacobi forms on the Siegel-Jacobi space H × C m , where H denotes the Poincaré upper half plane and m is any positive integer.

Manuscripta Mathematica, 2001

This paper deals with Jacobi forms Φ on ?×ℂ. The Rankin–Selberg doubling method is employed to study properties of the standard L-function of Hecke–Jacobi eigenforms. It is shown that every analytic Klingen–Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke–Jacobi cusp eigenforms of weight k > 4 and k≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved.

arXiv: Number Theory, 2007

In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations among Fourier coefficients. We prove that this space can also be characterized by duality properties of certain distinguished embedded Hecke operators. We then show that this space is Hecke invariant with respect to all good Hecke operators. As explicit examples we give Eisenstein series. Conversely we show the existence of forms that are not contained in this space.