Optimal spinor selectivity for quaternion orders

Abstract

Let D be a quaternion algebra over a number field F, and G be an arbitrary genus of OF-orders of full rank in D. Let K be a quadratic field extension of F that embeds into D, and B be an OF-order in K that can be optimally embedded into some member of G. We provide a necessary and sufficient condition for B to be optimally spinor selective for the genus G, which generalizes previous existing optimal selectivity criterions for Eichler orders as given by Arenas, Arenas-Carmona and Contreras, and by Voight independently. This allows us to obtain a refinement of the classical trace formula for optimal embeddings, which will be called the spinor trace formula. When G is a genus of Eichler orders, we extend Maclachlan's relative conductor formula for optimal selectivity from Eichler orders of square-free levels to all Eichler orders.