Intersection numbers of modular correspondences for genus zero modular curves

Abstract

In this paper, we introduce modular polynomials for the congruence subgroup Γ0(M) when X0(M) has genus zero and therefore the polynomials are defined by a Hauptmodul of X0(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 SL2(Z)-equivalence classes of positive definite binary quadratic forms over 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 Sp2(Z).