Olav K. Richter

Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition

Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers’s PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers’s μ-functions to establish a theta-like decomposition.

Holomorphic projections and Ramanujan’s mock theta functions

Significance Mock theta functions were introduced by Ramanujan in 1920. They have become a vivid area of research, and they continue to play important roles in different parts of mathematics and physics. In this paper, we extend the concept of holomorphic projection, which allows us to prove identities for the Fourier series coefficients of Ramanujan’s mock theta functions. We use spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan’s mock theta functions.

The action of the heat operator on Jacobi forms

We investigate the action of the heat operator on Jacobi forms. In particular, we present two explicit characterizations of this action on Jacobi forms of index 1. Furthermore, we study congruences and filtrations of Jacobi forms. As an application, we determine when an analog of Atkins U-operator applied to a Jacobi form is nonzero modulo a prime.

RAMANUJAN CONGRUENCES FOR SIEGEL MODULAR FORMS

We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.

Theta functions of indefinite quadratic forms over real number fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

On congruences of Jacobi forms

We consider congruences and filtrations of Jacobi forms. More specifically, we extend Tates theory of theta cycles to Jacobi forms, which allows us to prove a criterion for an analog of Atkins U-operator applied to a Jacobi form to be nonzero modulo a prime.

A remark on the behavior of theta series of degree n under modular transformations

A. Andrianov and G. Maloletkin [3] , [4] and Andrianov [1] , [2] investigate transformation properties of theta series corresponding to quadratic forms. Let F be a symmetric, integral matrix of rankm with even diagonal entries, and let q be the level of F; that is, qF is integral and qF has even diagonal entries. Suppose that F is of type (k, l), and let H be a majorant of F; that is,HFH = F and H = H > 0. For Z in the Siegel upper half-plane, H = {Z ∈ Mn,n(C) | Z = Z and Im(Z) > 0}, and for ζ+, ζ− ∈Mm,n(C) (withm > n), Andrianov and Maloletkin [4] define the theta series

Jacobi forms over complex quadratic fields via the cubic Casimir operators

We prove that the center of the algebra of differential operators invariant under the action of the Jacobi group over a complex quadratic field is generated by two cubic Casimir operators, which we compute explicitly. In the spirit of Borel, we consider Jacobi forms over complex quadratic fields that are also eigenfunctions of these Casimir operators, a new approach in the complex case. Theta functions and Eisenstein series provide standard examples. In addition, we introduce an analog of Kohnens plus space for modular forms of half-integral weight over K D Q.i/, and provide a lift from it to the space of Jacobi forms over K D Q.i/. Mathematics Subject Classification (2010). Primary 11F50; Secondary 43A85.

Classification of the space spanned by theta series and applications

We determine a class of functions spanned by theta series of higher degree. We give two applications: A simple proof of the inversion formula of such theta series and a classification of skew-holomorphic Jacobi forms.

On Rankin-Cohen brackets for Siegel modular forms

We determine an explicit formula for a Rankin-Cohen bracket for Siegel modular forms of degree n on a certain subgroup of the symplectic group. Moreover, we lift that bracket via a Poincare series to a Siegel cusp form on the full symplectic group.