Local Selectivity of Orders in Central Simple Algebras

الأصل · EN

Let B be a central simple algebra of degree n over a number field K, and L⊂ B a strictly maximal subfield. We say that the ring of integers Oₗ is "selective" if there exists an isomorphism class of maximal orders in B no element of which contains Oₗ. 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 Oₗ by leveraging the theory of affine buildings for SLᵣ(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 Oₗ. 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.