Tiled orders are a class of orders in matrix algebras over a non-Archimedean local field generalizing maximal and hereditary orders. Normalizers of tiled orders contain valuable information for finding type numbers of associated global orders. We describe an algorithm for computing normalizers of tiled orders in matrix algebras.MSC2010: 11H06, 11S45.

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.

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