Optimal spinor selectivity for quaternion Bass orders

Abstract

Let A be a quaternion algebra over a number field F, and O be an OF-order of full rank in A. Let K be a quadratic field extension of F that embeds into A, and B be an OF-order in K. Suppose that 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 O. This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan (2008) [21] for Eichler orders of square-free levels, and independently by M. Arenas et al. (2018) [1] and by J. Voight (2021) [27, Chapter 31] for Eichler orders of arbitrary levels.