Modular Cauchy kernel corresponding to the Hecke curve

I’ll talk about the construction of the 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. Using the Rankin-Selberg method, Don Zagier proved that the Hecke operator $T_k(m)$ on the space of cusp forms of weight $k>2$ can be defined by a kernel $\omega_m(z_1,\bar{z_2}, k)$. Firstly, we prove generalization of the Zagier theorem 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 $\omega_{m, N}(z_1,\bar{z_2}, 2)$. 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 $\Gamma_0(N)$. https://arxiv.org/pdf/1802.03299.pdf