Commentarii Mathematici Helvetici


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Volume 87, Issue 4, 2012, pp. 825–859
DOI: 10.4171/CMH/270

Published online: 2012-10-10

Jacobi forms over complex quadratic fields via the cubic Casimir operators

Kathrin Bringmann[1], Charles H. Conley[2] and Olav K. Richter[3]

(1) Universität Köln, Germany
(2) University of North Texas, Denton, USA
(3) University of North Texas, Denton, USA

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 Kohnen's plus space for modular forms of half-integral weight over $K=\mathbb{Q}(i)$, and provide a lift from it to the space of Jacobi forms over $K=\mathbb{Q}(i)$.

Keywords: Complex quadratic fields: Jacobi forms, Kohnen's plus space, invariant differential operators

Bringmann Kathrin, Conley Charles, Richter Olav: Jacobi forms over complex quadratic fields via the cubic Casimir operators. Comment. Math. Helv. 87 (2012), 825-859. doi: 10.4171/CMH/270