Embedding Orders Into Central Simple Algebras

Details Embedding Orders Into Central Simple Algebras Embedding Orders Into Central Simple Algebras

2010-06-18

arxiv.org/abs/1006.3683v1

Mathematics

Number Theory

Scientific paper

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 \cite{Chevalley-book} 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 $[L \cap \widetilde K : K]^{-1}$ where $\widetilde K$ is the Hilbert class field of $K$. Chinburg and Friedman (\cite{Chinburg-Friedman}) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona \cite{Arenas-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 $p^2$, $p$ 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 Chinberg and Friedman's argument was the structure of the tree of maximal orders for $SL_2$ over a local field. In this work, we generalize Chinburg and Friedman's results replacing the tree by the Bruhat-Tits building for $SL_p$.

Affiliated with

Also associated with

No associations

Rating

Embedding Orders Into Central Simple Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Embedding Orders Into Central Simple Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Embedding Orders Into Central Simple Algebras will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-260577