Optimal spinor selectivity for quaternion orders

TL;DR

This paper establishes a comprehensive criterion for optimal spinor selectivity of quadratic orders in quaternion algebras, extending classical results and refining the trace formula for optimal embeddings.

Contribution

It generalizes existing selectivity criteria for Eichler orders and introduces the spinor trace formula, broadening the understanding of optimal embeddings in quaternion orders.

Findings

Provides a necessary and sufficient condition for optimal spinor selectivity.

Refines the classical trace formula to the spinor trace formula.

Extends Maclachlan's conductor formula to all Eichler orders.

Abstract

Let $D$ be a quaternion algebra over a number field $F$, and $\mathscr{G}$ be an arbitrary genus of $O_F$-orders of full rank in $D$. Let $K$ be a quadratic field extension of $F$ that embeds into $D$, and $B$ be an $O_F$-order in $K$ that can be optimally embedded into some member of $\mathscr{G}$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus $\mathscr{G}$, 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 $\mathscr{G}$ 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.