Type numbers of locally tiled orders in central simple algebras

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 $S{L}_n(K_{\nu })$ 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 Vignéras for orders in higher degree central simple algebras.

23 pages, 4 figures. The paper incorporates material from author's PhD thesis

Cited by in corpus (1)

• Elasticities of Orders in Central Simple Algebras