Classical Theta Functions and Quantum Tori

The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schr\"odinger equation in which the role of time is played by the deformation parameter $\hbar$ and the role of the hamiltonian by a Poisson structure. At least in some special cases, the kernel can be written as a sum of products of single-variable theta functions.