2010/06/18 by Benjamin Linowitz, Linowitz, Benjamin, Thomas R. Shemanske +1 · 1 citation
Mathematics · #11R54 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA) #math.NT #math.RA #msc:11R54
paper · pdf · doi:10.48550/arxiv.1006.3683
arxiv created 2010/06/18 · arxiv updated 2010/06/21
The question of embedding fields into central simple algebras B over a number field K was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields L of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley \citeChevalley-book which says that with B = Mn(K) the ratio of the number of isomorphism classes of maximal orders in B into which the ring of integers of L can be embedded (to the total number of classes) is [L ∩ \widetilde K : K]-1 where \widetilde K is the Hilbert class field of K. Chinburg and Friedman (\citeChinburg-Friedman) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona \citeArenas-Carmona considers embeddings of the ring of integers into maximal orders in a broad class of higher rank central simple algebras. In this paper, we consider central simple algebras of dimension p2, p an odd prime, and we show that arbitrary commutative orders in a degree p extension of K, embed into none, all or exactly one out of p isomorphism classes of maximal orders. Those commutative orders which are selective in this sense are explicitly characterized; class fields play a pivotal role. A crucial ingredient of Chinberg and Friedman's argument was the structure of the tree of maximal orders for SL2 over a local field. In this work, we generalize Chinburg and Friedman's results replacing the tree by the Bruhat-Tits building for SLp.