The modular Cauchy kernel for the Hilbert modular surface

TL;DR

This paper constructs a modular Cauchy kernel on the Hilbert modular surface, analyzing its properties, convergence, and Fourier expansion, extending previous work by Zagier and others.

Contribution

It introduces a new modular Cauchy kernel function on the Hilbert modular surface with specific invariance and pole properties, generalizing prior results.

Findings

Constructed the modular Cauchy kernel with specified invariance and pole structure.

Analyzed the convergence and Fourier expansion of the kernel.

Extended previous results on related modular functions.

Abstract

In this paper we construct the modular Cauchy kernel on the Hilbert modular surface $\Xi_{\mathrm{Hil},m}(z)(z_2-\bar{z_2})$, 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 $\bar{z_2}$ 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}