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Working paper

Modular Cauchy kernel for the Hilbert modular surface

arxiv.org. math. Cornell University, 2018. No. 1802.08661.
In this paper we construct the modular Cauchy kernel on the Hilbert modular surface ΞHil,m(z)(z2−z2¯), i.e. the function of two variables, (z1,z2)∈H×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 z2¯ is the function ωm(z1,z2) 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}