Modular Cauchy kernel corresponding to the Hecke curve

Abstract

In this paper we construct the modular Cauchy kernel ${\Xi }_N(z_1,z_2)$, i.e. the modular invariant function of two variables, $(z_1,z_2)\in \mathbb{H}\times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1,z_2)\in \mathbb{H}\times \mathbb{H}|z_2=\gamma z_1,\gamma \in {\Gamma }_0(N)\right\}.$$ The function ${\Xi }_N(z_1,z_2)$ is used in two cases and for two different purposes. Firstly, we prove generalization of the Zagier theorem ([La], [Za3]) for the Hecke subgroups ${\Gamma }_0(N)$ of genus $g>0$. Namely, we obtain a kind of "kernel function" for the Hecke operator $T_N(m)$ on the space of the weight 2 cusp forms for ${\Gamma }_0(N)$, which is the analogue of the Zagier series ${\omega }_{m,N}(z_1,\overline{z_2},2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, $J_{{\Gamma }_0(N)}(z_1)-J_{{\Gamma }_0(N)}(z_2)$, for genus zero congruence subgroup ${\Gamma }_0(N)$.