Representation field for orders of small ranks

TL;DR

This paper investigates the existence of representation fields for orders in central simple algebras, proving their existence for ranks up to 7 and in certain cases, while providing a counterexample at rank 8.

Contribution

It establishes the existence of representation fields for all orders of rank 7 or less and for orders in algebras with commutative semi-simple reduction, extending previous results.

Findings

Representation field exists for any order of rank ≤7.

Representation field exists for orders in algebras with commutative semi-simple reduction.

Counterexample showing non-existence of representation field at rank 8.

Abstract

A representation field for a non-maximal order H in a central simple algebra A is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. In this article, we prove that the representation field is defined for any order H of rank 7 or lower. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order whose representation field is not defined.