Modular Cauchy kernel corresponding to the Hecke curve

TL;DR

This paper constructs a modular Cauchy kernel for Hecke curves, generalizes Zagier's theorem for higher genus subgroups, and provides an elementary proof for Borcherds product formulas related to Hauptmoduls.

Contribution

It introduces a new modular invariant kernel function for Hecke subgroups and extends key theorems to higher genus cases, also simplifying Borcherds product proofs.

Findings

Generalized Zagier's theorem to genus > 0 Hecke subgroups

Constructed a kernel function for Hecke operators on cusp forms

Provided an elementary proof for Borcherds product formula

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 $\omega_{m, N}(z_1,\bar{z_2}, 2)$. 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 $\Gamma_0(N)$.