Local selectivity of orders in central simple algebras

Abstract

Let [Formula: see text] be a central simple algebra of degree [Formula: see text] over a number field [Formula: see text], and [Formula: see text] be a strictly maximal subfield. We say that the ring of integers [Formula: see text] is selective if there exists an isomorphism class of maximal orders in [Formula: see text] no element of which contains [Formula: see text]. 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 [Formula: see text] by leveraging the theory of affine buildings for [Formula: see text] where [Formula: see text] 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 [Formula: see text], and distinguish those which are guaranteed to contain [Formula: see text]. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra.