Optimal spinor selectivity for quaternion Bass orders

TL;DR

This paper establishes a precise criterion for when a quadratic order can be optimally embedded into a quaternion Bass order, extending previous results to more general settings with well-behaved dyadic primes.

Contribution

It provides a necessary and sufficient condition for optimal spinor selectivity of quadratic orders in quaternion Bass orders, generalizing earlier work on Eichler orders of various levels.

Findings

Characterization of optimal spinor selectivity for Bass orders

Extension of selectivity criteria beyond Eichler orders

Partial generalization of previous selectivity results

Abstract

Let $A$ be a quaternion algebra over a number field $F$, and $\mathcal{O}$ be an $O_F$-order of full rank in $A$. Let $K$ be a quadratic field extension of $F$ that embeds into $A$, and $B$ be an $O_F$-order in $K$. Suppose that $\mathcal{O}$ is a Bass order that is well-behaved at all the dyadic primes of $F$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus of $\mathcal{O}$. This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan [Optimal embeddings in quaternion algebras. J. Number Theory, 128(10):2852-2860, 2008] for Eichler orders of square-free levels, and independently by M. Arenas et al. [On optimal embeddings and trees. J. Number Theory, 193:91-117, 2018] and by J. Voight [Chapter 31, Quaternion algebras, volume 288 of Graduate Texts in Mathematics. Springer-Verlag, 2021] for Eichler orders of arbitrary levels.