Embedding Orders Into Central Simple Algebras

TL;DR

This paper investigates how commutative orders in degree p extensions embed into maximal orders of central simple algebras of dimension p^2, revealing selectivity phenomena characterized by class fields and generalizing previous results using Bruhat-Tits buildings.

Contribution

It extends embedding results to central simple algebras of dimension p^2 for odd primes p, characterizing selective orders and generalizing the structure from trees to Bruhat-Tits buildings.

Findings

Embeddings depend on class field theory and are either none, all, or exactly one of p classes.

Explicit characterization of selective orders in central simple algebras of dimension p^2.

Generalization of the structure of maximal order trees to Bruhat-Tits buildings for SL_p.

Abstract

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 $SL_p$.