Extended modular functions and definite form class groups

For a positive integer N, we define an extended modular function of level N motivated by physics and investigate its fundamental properties. Let K be an imaginary quadratic field, and let $O$ be an order in K of discriminant D. Let $K_{O,N}$ denote the ray class field of $O$ modulo $NO$. For $N\ge 3$, we provide an explicit description of the Galois group $\mathrm{Gal}(K_{O,N}/Q)$ using special values of extended modular functions of level N and the definite form class group of discriminant D and level N.