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 $j^{ \rm qt}(f)$.
Categories: math.NT