A Kronecker congruence relation for modular functions of higher level and genus

Abstract

Let j be the elliptic modular function, a weakly holomorphic modular function for ${\mathrm{SL}}_2\left(Z\right)$. Weber showed that for each prime p the modular polynomial ${\Phi }_p(x,y)$ of j satisfies what is known as the Kronecker congruence relation ${\Phi }_p(x,y)\equiv (x^p-y)(x-y^p)\left(\text{mod}pZ\right[x,y\left]\right)$. We give a generalization of this congruence applicable to certain weakly holomorphic modular functions of higher level in terms of integrality over $Z\left[j\right]$.