Selectivity in definite quaternion algebras

Abstract

We study the embedding problem for commutative orders into Eichler orders in definite quaternion algebras over the rationals by using methods from the theory of quadratic forms, specifically, Minkowski–Siegel's formula for the representation mass. We characterize when the Gaussian or Eisenstein integers embed into some but not all classes in the genus of an Eichler order. We also give an application to the theory of supersingular elliptic curves.