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

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 $j^{\mathrm{qt}}(f)$.

21 pages

Cited by in corpus (1)

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