Intersection numbers of modular correspondences for genus zero modular curves

Abstract

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

27 pages, 2 tables