Analytic and arithmetic properties of the $$(\Gamma ,\chi )$$ ( Γ , χ ) -automorphic reproducing kernel function and associated Hermite–Gauss series

Abstract

We consider the reproducing kernel function of the theta Bargmann–Fock Hilbert space associated with given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and the discreteness of its zeros are examined. The analytic sets of zeros of the theta Bargmann–Fock space inside a given fundamental cell is characterized and shown to be finite and of cardinal less or equal to its dimension. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite–Gauss coefficients. Some of them generalize some of the arithmetic identities given by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite–Gauss 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.