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


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.