In this paper, we introduce modular polynomials for the congruence subgroup $\Gamma_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respect to $\mathrm{Sp}_2(\mathbb{Z})$.

中文翻译：

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属零模曲线的模对应的交点数

在本文中，我们为同余子群 $\Gamma_0(M)$ 引入模多项式，当 $ X_0(M) $ 具有零属时，因此多项式由 $ X_0(M) $ 的 Hauptmodul 定义。我们证明了由两个模多项式定义的两条曲线的交点数可以表示为 $\mathrm{SL}_2(\mathbb{Z})$ - 正定二元二次型形式的 $\mathrm{SL}_2(\mathbb{Z})$-等价类的数目之和\mathbb{Z}$。我们还表明，交集数也可以由 Siegel Eisenstein 级数 2、权重 2 相对于 $\mathrm{Sp}_2(\mathbb{Z})$ 的傅立叶系数组合写入。