Quantum Drinfeld Modules I: Quantum Modular Invariant and Hilbert Class Fields

TL;DR

This paper introduces a quantum modular invariant function in positive characteristic and demonstrates its role in generating Hilbert class fields for real quadratic extensions, advancing solutions to Hilbert's 12th problem in this setting.

Contribution

It presents the first construction of a quantum modular invariant function that generates Hilbert class fields in positive characteristic, addressing Manin's Real Multiplication program.

Findings

The quantum modular invariant $j^{qt}$ is multivalued and modular.

Hilbert class fields are generated by values of $j^{qt}$ at quadratic units.

The approach provides a new method for explicit class field theory in positive characteristic.

Abstract

This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of $\mathbb{Q}$ -- in positive characteristic, using quantum analogs of the exponential function and the modular invariant. In this first paper, we treat the problem of Hilbert class field generation. If $k=\mathbb{F}_{q}(T)$ and $k_{\infty}$ is the analytic completion of $k$, we introduce the quantum modular invariant \[ j^{\rm qt}: k_{\infty}\multimap k_{\infty}\] as a multivalued, modular invariant function. Then if $K=k(f)\subset k_{\infty}$ is a real quadratic extension of $k$ where $f$ is a quadratic unit, we show that the Hilbert class field $H_{\mathcal{O}_{K}}$ (associated to $\mathcal{O}_{K}=$ integral closure of $\mathbb{F}_{q}[T]$ in $K$) is generated over $K$ by the product of the multivalues of $j^{\rm qt}(f)$.