Eichler orders, trees and Representation Fields

TL;DR

This paper develops a method to compute the representation field for orders in central simple algebras, especially when the order is an intersection of maximal orders like Eichler orders, extending previous work limited to special cases.

Contribution

It introduces a general approach to compute the representation field for intersections of maximal orders, broadening the scope beyond maximal and commutative orders.

Findings

Provides explicit computation methods for Eichler orders

Extends the theory to arbitrary orders H

Includes illustrative examples

Abstract

If H and D are two orders in a central simple algebra A with D of maximal rank, the representation field F(D|H) is a subfield of the spinor class field of the genus of D which determines the set of spinor genera of orders in that genus representing the order H. Previous work have focused on two cases, maximal orders D and commutative orders H. In this work, we show how to compute the representation field F(D|H) when D is the intersection of a finite family of maximal orders, e.g., an Eichler order, and H is arbitrary. Examples are provided.