Embedding orders into central simple algebras

作者：Linowitz Benjamin^*; Shemanske Thomas R

来源：Journal de Theorie des Nombres de Bordeaux, 2012, 24(2): 405-424.

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 [6] which says that with B = M-n (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 boolean AND (K) over tilde : K](-1) where (K) over tilde is the Hilbert class field of K. Chinburg and Friedman ([7]) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona [2] 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 p(2), 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 Chinburg 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.

• 出版日期2012