Type numbers of locally tiled orders in central simple algebras

TL;DR

This paper studies the classification of locally tiled orders in central simple algebras over number fields, providing geometric descriptions, explicit formulas, and algorithms for their type numbers, extending previous work to higher degrees.

Contribution

It introduces a geometric approach using Bruhat-Tits buildings to analyze local normalizers and derives explicit formulas and algorithms for computing type numbers of orders.

Findings

Strong approximation holds for certain central simple algebras.

Provides a geometric description of local normalizers using Bruhat-Tits buildings.

Develops explicit formulas and algorithms for type number computation.

Abstract

Let $A$ be a central simple algebra over a number field $K$ with ring of integers $\mathcal{O}_K$, such that either the degree of the algebra $n \ge 3$, or $n=2$ and $A$ is not a totally definite quaternion algebra. Then strong approximation holds in $A$, which allows us to describe the genus of an $\mathcal{O}_K$-order $\Gamma \subset A$ in terms of idelic quotients of the field $K$. We consider orders $\Gamma$ that are tiled at every finite place $\nu$ of $K$ and use the Bruhat-Tits building for $SL_n(K_\nu)$ to give a geometric description for the local normalizers of $\Gamma$. We also give explicit formulas and algorithms to compute the type number of $\Gamma$. Our results generalize work of Vign\'{e}ras for orders in higher degree central simple algebras.