Local Selectivity of Orders in Central Simple Algebras

TL;DR

This paper investigates the local structure of maximal orders in central simple algebras over number fields, characterizing when they contain the ring of integers of a maximal subfield, and applies this to compute representatives and understand selectivity.

Contribution

It introduces a local characterization of maximal orders containing the ring of integers using affine buildings and applies this to global selectivity problems.

Findings

Characterization of maximal orders containing $\\mathcal{O}_L$ via affine buildings.

Method to compute representatives of maximal orders containing $\\mathcal{O}_L$.

Insights into the role of partial ramification in selectivity.

Abstract

Let $B$ be a central simple algebra of degree $n$ over a number field $K$, and $L\subset B$ a strictly maximal subfield. We say that the ring of integers $\mathcal O_L$ is "selective" if there exists an isomorphism class of maximal orders in $B$ no element of which contains $\mathcal O_L$. Many authors have worked to characterize the degree to which selectivity occurs, first in quaternion algebras, and more recently in higher-rank algebras. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers $\mathcal O_L$ by leveraging the theory of affine buildings for $SL_r(D)$ where $D$ is a local central division algebra. Then as an application, we use the local result and a local-global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in $B$, and distinguish those which are guaranteed to contain $\mathcal O_L$. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result mentioned above, and give examples which clarify the interesting role of partial ramification in the algebra.