对应于Hecke曲线的模Cauchy核
Modular Cauchy kernel corresponding to the Hecke curve
摘要：
本文构造了模柯西核$Ξ_N(z_1, z_2)$，即二元模不变函数$(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$，其一阶极点在曲线$$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=γz_1, ~γ\in Γ_0(N) \right\}.$$上。函数$Ξ_N(z_1, z_2)$用于两种情况和两种不同的目的。首先证明了Zagier定理（[La]，[Za 3]）对亏格$g>0$的Hecke子群$Γ_0(N)$的推广.即在$Γ_0(N)$的权为2的尖点形式空间上得到了Hecke算子$T_N(m)$的一种“核函数“，它是Zagier级数$ω_{m, N}(z_1,\bar{z_2}, 2)$的类似.其次，我们考虑亏格零同余子群$Γ_0(N)$的两个正规化Hauptmoduls之差的无穷Borcherds积公式$J_{Γ_0(N)}(z_1)-J_{Γ_0(N)}(z_2)$的初等证明。
英文摘要：
In this paper we construct the modular Cauchy kernel $Ξ_N(z_1, z_2)$, 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 $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=γz_1, ~γ\in Γ_0(N) \right\}.$$ The function $Ξ_N(z_1, z_2)$ is used in two cases and for two different purposes. Firstly, we prove generalization of the Zagier theorem ([La], [Za3]) for the Hecke subgroups $Γ_0(N)$ 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 $Γ_0(N)$, which is the analogue of the Zagier series $ω_{m, N}(z_1,\bar{z_2}, 2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, $J_{Γ_0(N)}(z_1)-J_{Γ_0(N)}(z_2)$, for genus zero congruence subgroup $Γ_0(N)$.
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