Integral embeddings of central simple algebras over number fields
Jiaqi Xie, Fei Xu
TL;DR
This work addresses when integral embeddings of an order $\Xi$ of a degree-$n$ subfield $K$ inside a central simple algebra $\mathcal{A}$ of degree $n$ over a number field $k$ exist inside an $\mathfrak{o}_k$-order $\Gamma$ of $\mathcal{A}$, noting that local–global principles fail for integral embeddings. It develops a criterion using adelic data and Brauer–Manin obstructions: there exist local elements $g_v$ with $\Xi\otimes\mathfrak{o}_{k_v}\subset g_v^{-1}(\Gamma\otimes\mathfrak{o}_{k_v})g_v$ for all $v$ and the idele $(N_v(g_v))_v$ lies in $k^{\times}\cdot N_{K/k}(\mathbb{I}_K)$, equivalently trivial in $\mathrm{Gal}((k^{ab}\cap K)/k)$. The paper introduces class fields attached to orders, analyzes the genus of orders, and uses integral Brauer–Manin obstruction to give a unified, general criterion, recovering and extending classical results (Chevalley, Chinburg–Friedman, Arenas–Carmona) for quaternion and higher-degree algebras. The results provide a systematic framework for counting and locating embeddings via strong approximation and Galois cohomology, with broad implications for integral embedding problems in central simple algebras over number fields.
Abstract
A criterion for determining exactly when an order of a maximal subfield of a central simple algebra over a number field can be embedded into an order of this algebra is given. Various previous results have been generalized and recovered by applying this criterion.