Optimal spinor selectivity for quaternion Bass orders

Abstract

Let A be a quaternion algebra over a number field F , and 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 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 [Optimal embeddings in quaternion algebras.

On superspecial abelian surfaces over finite fields III

“…In fact, L/F is the unique unramified quadratic field extension if e(O) = −1, and it is a ramified quadratic field extension if e(O) = 0. In the latter case, the ramified quadratic extension L/F can be arbitrary if n(O) = 2 according to[4, (3.14)]; and it is uniquely determined by O if n(O) ≥ 3 and F is nondyadic according to[17, Lemma 3.5].2 From the proof of [4, Theorem 3.3 and 3.10], any two embeddings of O L into O are conjugate by an element of the normalizer of O, thus expression (3.4) does not depend on the choice of the embedding O L ֒→ O.…”