Representation fields for commutative orders

A representation field for a non-maximal order \Ha in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of \Ha. Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.Comment: Annales de l'institut Fourier, vol 61, 201

Roots of unity in definite quaternion orders

A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with a type number 3 or larger. The proof extends to a few other closely related orders