Singular moduli of higher level and special cycles

TL;DR

This paper studies the values of modular functions at CM points for $ ext{Gamma}_0(N)$, relating them to special cycles on CM elliptic curves and providing explicit formulas and prime ideal factorizations.

Contribution

It introduces explicit formulas for CM values of modular functions for $ ext{Gamma}_0(N)$ in terms of special cycles, extending previous results to higher levels and Borcherds products.

Findings

Explicit formulas for CM values in terms of special cycles

Prime ideal factorizations of CM values obtained

Application to Borcherds products of weight 0

Abstract

We describe the complex multiplication (CM) values of modular functions for $\Gamma_0(N)$ whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the stack of CM elliptic curves. In particular, our results apply to Borcherds products of weight $0$ for $\Gamma_0(N)$. By working out explicit formulas for the special cycles, we obtain the prime ideal factorizations of such CM values.