Explicit Class Field Theory for Orders in Global Function Fields

Abstract: This paper develops explicit class field theory for orders: of rank 1 in any global function field -- Hayes theory -- and of rank 2 in real quadratic function fields -- Real Multiplication. The essential ingredient in the development of the Hayes Theory is an orders version of Shimura's Main Theorem on Complex Multiplication. The section on Real Multiplication for orders uses values of the quantum modular invariant to generate the Hilbert class field of a rank 2 order contained in the integral closure of $\mathbb{F}_{q}[T]$.