Coherent State Transforms and Abelian Varieties

We extend the coherent state transform (CST) of Hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group K=U(1)g. We show that this transform, applied to appropriate distributions on K, gives all classical theta functions, and that, by defining on this space of theta functions an inner product related to the K-averaged heat kernel, the unitarity of the CST transform is still preserved.