Title: Analytic and arithmetic properties of the $(Γ,χ)$-automorphic reproducing kernel function,
Abstract: We consider the reproducing kernel function of the theta Bargmann-Fock
Hilbert space associated to given full-rank lattice and pseudo-character, and
we deal with some of its analytical and arithmetical properties. Specially, the
distribution and discreteness of its zeros are examined and analytic sets
inside a product of fundamental cells is characterized and shown to be finite
and of cardinal less or equal to the dimension of the theta Bargmann-Fock
Hilbert space. Moreover, we obtain some remarkable lattice sums by evaluating
the so-called complex Hermite-Taylor coefficients. Some of them generalize some
of the arithmetic identities established by Perelomov in the framework of
coherent states for the specific case of von Neumann lattice. Such complex
Hermite-Taylor coefficients are nontrivial examples of the so-called lattice's
functions according the Serre terminology. The perfect use of the basic
properties of the complex Hermite polynomials is crucial in this framework.