Some explicit constructions of integral structures in quaternion algebras

TL;DR

This paper provides explicit constructions and embeddings of Eichler orders in quaternion algebras over Q, including bases for chains of such orders and analysis of their intersections.

Contribution

It offers explicit methods for constructing and embedding Eichler orders in quaternion algebras, extending prior characterizations and analyzing their intersections.

Findings

Explicit embeddings of Eichler orders in local matrix rings

Bases for chains of Eichler orders in quaternion algebras

Results on intersections of Eichler orders

Abstract

Let B be an undefined quaternion algebra over Q. Following the explicit chacterization of some Eichler orders in B given by Hashimoto, we define explicit embeddings of these orders in some local rings of matrices; we describe the two natural inclusions of an Eichler order of leven Nq in an Eichler order of level N. Moreover we provide a basis for a chain of Eichler orders in B and prove results about their intersection.