Intersection numbers of modular correspondences for genus zero modular curves

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 $\mathrm{Sp}_2(\mathbb{Z})$.