Intersection numbers of modular correspondences for genus zero modular curves

TL;DR

This paper introduces modular polynomials for genus zero modular curves and expresses their intersection numbers through quadratic forms and Fourier coefficients of Siegel Eisenstein series, linking algebraic and automorphic perspectives.

Contribution

It defines modular polynomials for $ ext{X}_0(M)$ with genus zero and relates their intersection numbers to quadratic forms and Siegel Eisenstein series.

Findings

Intersection numbers expressed via quadratic forms.

Intersection numbers related to Fourier coefficients of Eisenstein series.

Provides a new algebraic and automorphic connection for modular correspondences.

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