# The Quantum j-Invariant and Hilbert Class Fields of Real Quadratic
Extensions in Positive Characteristic

Research paper by **L. Demangos, T. M. Gendron**

Indexed on: **08 Jul '16**Published on: **08 Jul '16**Published in: **Mathematics - Number Theory**

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#### Abstract

Let $k=\mathbb{F}_{q}(T)$ where $\mathbb{F}_{q}$ is the field with $q$
elements and let $k_{\infty}$ be its completion with respect to the valuation
$v_{\infty}(f)=-{\rm deg}_{T}(f)$. The quantum $j$-invariant is introduced as a
modular-invariant, discontinuous and multi-valued function \[ j^{\rm
qt}:k_{\infty}\cup \{\infty\} \multimap k_{\infty}\cup \{\infty\}.\] If
$K\subset k_{\infty}$ is a quadratic extension of $k$ and $\mathcal{O}_{K}$ is
the integral closure of the ring $\mathbb{F}_{q}[T]$ in $K$, we prove that
there exists $f\in K-k$ such that the Hilbert class field $H_{\mathcal{O}_{K}}$
associated to $\mathcal{O}_{K}$ is generated over $K$ by the product of the
elements of the set $j^{\rm qt}(f)$. This result can be viewed as a solution in
positive characteristic to the Hilbert class field part of Manin's Real
Multiplication program.