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 $SL_n(K_\nu)$ 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.