Maximal Orders in Central Simple Algebras and Bruhat–Tits Buildings
✍ Scribed by Anne R. Schwartz; Thomas Shemanske
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 1001 KB
- Volume
- 56
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
We study the affine building for SL n over a local field and give a characterization of distance involving Hecke operators. For n=3 we give an explicitly computable distance formula. We use this local information to show that the class number of a maximal order in a central simple algebra of dimension n 2 over a number field K is equal to the number of orbits of a group of isometries (related to the unit group of the maximal order) acting on a Bruhat Tits building for SL n (K). This generalizes results of Serre and Vigne ras who considered the quaternion case in which the Bruhat Tits building is a tree.