Jacobi theta functions over number fields (Q1881647)

Eichler and Zagier developed a theory of holomorphic Jacobi forms and showed that Jacobi theta functions corresponding to positive definite quadratic forms are examples of such forms. N.-P. Skoruppa introduced the notion of skew-holomorphic Jacobi forms and presented examples using Jacobi theta functions corresponding to indefinite quadratic forms with signature \((1,n-1)\). The authors use Jacobi theta functions to create examples of Jacobi forms and skew-holomorphic Jacobi forms over number fields. That is, they define \(\Theta_{Q,R,w}^{(K)}(\tau,z)\), a Jacobi theta function attached to an arbitrary quadratic form \(Q\) defined over \(K\). If \(K=\mathbb Q\), then \(\Theta_{Q,R,w}^{(\mathbb Q)}(\tau,z)\) is the usual Jacobi theta function. If, in addition, \(Q\) is of type \((1,n-1)\), then \(\Theta_{Q,R,w}^{(\mathbb Q)}(\tau,z)\) is a skew-holomorphic Jacobi form in the sense of Skoruppa. They determine the behavior of \(\Theta_{Q,R,w}^{(K)}(\tau,z)\) under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions. In addition, they show how sums of these Jacobi theta functions appear as a single coefficient of a symplectic theta function. (revised version, September 2007)