Quantum Drinfeld Modules II: Quantum Exponential and Ray Class Fields

Abstract

This is the second in a series of two papers presenting a solution to Manin's Real Multiplication program \cite{Man} in positive characteristic. If $K$ is a quadratic and real extension of $\mathbb{F}_{q}(T)$ and $\mathcal{O}_{K}$ is the integral closure of $\mathbb{F}_{q}[T]$ in $K$, we associate to each modulus $\mathfrak{M}\subset \mathcal{O}_{K}$ the {\it unit narrow ray class field} $K^{\mathfrak{M}}$: a class field containing the narrow ray class field, whose class group contains an additional contribution coming from $\mathcal{O}^{\times}_{K}$. For $f\in K$ a fundamental unit, we introduce the associated {\it quantum Drinfeld module} $\rho^{\rm qt}_{f}$ of $f$: a generalization of Drinfeld module whose elements are multi-points. The main theorem of the paper is that $$K^{\mathfrak{M}}=H_{\mathcal{O}_{K}} ( {\sf Tr}(\rho^{\rm qt}_{f}[\mathfrak{M}]), {\sf Tr}(\rho^{\rm qt}_{f^{-1}}[\mathfrak{M}]))$$ where $H_{\mathcal{O}_{K}}$ is the Hilbert class field of $\mathcal{O}_{K}$ and ${\sf Tr}(\rho^{\rm qt}_{f}[\mathfrak{M}])$, ${\sf Tr}(\rho^{\rm qt}_{f^{-1}}[\mathfrak{M}])$ are the groups of traces of $\mathfrak{M}$ torsion points of $\rho^{\rm qt}_{f}$, $\rho^{\rm qt}_{f^{-1}}$.Comment: 41 page