Explicit Class Field Theory for Orders in Global Function Fields

TL;DR

This paper advances explicit class field theory for orders in global function fields, extending Hayes theory to rank 1 and exploring rank 2 cases using quantum invariants for generating class fields.

Contribution

It develops explicit class field theory for rank 1 and rank 2 orders in global function fields, including an orders version of Shimura's Main Theorem and applications to real quadratic fields.

Findings

Extended Hayes theory to rank 1 orders in global function fields.

Used quantum modular invariants to generate Hilbert class fields of rank 2 orders.

Provided explicit descriptions of class fields for orders in 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 $\mathbb{F}_{q}[T]$.