The modular Cauchy kernel for the Hilbert modular surface

Abstract

In this paper we construct the modular Cauchy kernel on the Hilbert modular surface ${\Xi }_{\mathrm{Hil},m}(z)(z_2-\overline{z_2})$, i.e. the function of two variables, $(z_1,z_2)\in \mathbb{H}\times \mathbb{H}$, which is invariant under the action of the Hilbert modular group, with the first order pole on the Hirzebruch-Zagier divisors. The derivative of this function with respect to $\overline{z_2}$ is the function ${\omega }_m(z_1,z_2)$ introduced by Don Zagier in \cite{Za1}. We consider the question of the convergence and the Fourier expansion of the kernel function. The paper generalizes the first part of the results obtained in the preprint \cite{Sa}