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 O K , such that either the degree of the algebra n ≥ 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 O K -order Γ ⊂ A in terms of idelic quotients of the field K. We consider orders Γ that are tiled at every finite place ν of K and use the Bruhat-Tits building for SL n (K ν ) to give a geometric description for the local normalizers of Γ. We also give explicit formulas and algorithms to compute the type number of Γ. Our results generalize work of Vignéras [25] for orders in higher degree central simple algebras.

Elasticities of Orders in Central Simple Algebras

“…They have seen some study in recent years [Bab20;Bab19;She10]. If n = 1 then A p = D p is a division ring.…”