Intersection numbers of modular correspondences for genus zero modular curves
Let \(j(E)\) denote the invariant of the elliptic curve \(E\) over \(\mathbb{C}\). For \(N\ge1\) define \(\Phi_N:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}\) by \[ \Phi_N(j(E),j(E'))=\prod_{(f,E_1')}(j(E)-j(E_1')), \] where the product ranges over the isomorphism classes of pairs \((f,E_1')\), with \(E_1'\) an elliptic curve over \(\mathbb{C}\) and \(f:E_1'\rightarrow E'\) an isogeny of degree \(N\). It is known that the function \(\Phi_N\) is given by a symmetric polynomial with coefficients in \(\mathbb{Z}\). Let \(T_N\) denote the affine plane curve defined by \(\Phi_N(X,Y)=0\). In [Math. Ann. 88, 26--52 (1922; JFM 48.1164.04)] \textit{A. Hurwitz} showed that \(T_{N_1}\) and \(T_{N_2}\) intersect properly if and only if \(N_1N_2\) is not a square. In this case he computed the intersection number \(T_{N_1}\cdot T_{N_2}\) in terms of equivalence classes of positive definite binary quadratic forms. The paper under review generalizes Hurwitz's formula by replacing the affine \(j\)-line \(Y_0(1)\) with a modular curve of the form \(Y_0(M)\) for any \(M\ge1\) for which the compactification \(X_0(M)\) has genus 0. Given such an \(M\) the author finds a parameter \(t\) on \(X_0(M)\) which is holomorphic on \(X_0(M)\smallsetminus\{\infty\}\), has a simple pole at \(\infty\) with residue 1, and whose Fourier expansion at the cusp \(\infty\) has coefficients in \(\mathbb{Z}\). For \(N\) relatively prime to \(M\) this allows him to define the modular polynomial \(\Phi_N^{\Gamma_0(M)}(X,Y)\) of level \(N\) for the congruence subgroup \(\Gamma_0(M)\), with \(t\) playing the role of \(j\) in the definition of \(\Phi_N(X,Y)\). Once again, \(\Phi_N^{\Gamma_0(M)}(X,Y)\) is a symmetric polynomial with coefficients in \(\mathbb{Z}\). Let \(T_N^{\Gamma_0(M)}\) denote the curve in \(Y_0(M)\times Y_0(M)\) defined by \(\Phi_N^{\Gamma_0(M)}(X,Y)=0\). A point on \(T_N^{\Gamma_0(M)}\) corresponds to two pairs \((E,C)\), \((E',C')\), each consisting of an elliptic curve over \(\mathbb{C}\) and a cyclic subgroup of order \(M\) such that there is an isogeny \(f:E\rightarrow E'\) of degree \(N\) with \(f(C)=C'\). Let \(N_1,N_2\) be positive integers which are relatively prime to \(M\). The author shows that \(T_{N_1}^{\Gamma_0(M)}\) and \(T_{N_2}^{\Gamma_0(M)}\) intersect properly if and only if \(N_1N_2\) is not a square. When this holds he gives a formula for the intersection number \(T_{N_1}^{\Gamma_0(M)}\cdot T_{N_2}^{\Gamma_0(M)}\) in terms of equivalence classes of positive definite binary quadratic forms. In the case where \(M\) is prime he gives a version of the intersection number formula which expresses \(T_{N_1}^{\Gamma_0(M)}\cdot T_{N_2}^{\Gamma_0(M)}\) in terms of coefficients of a Siegel Eisenstein series.
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