Maximal selectivity for orders in fields

Abstract

If H ⊆ D are two orders in a central simple algebra A with D of maximal rank, the theory of representation fields describes the set of spinor genera of orders in the genus of D representing the order H. When H is contained in a maximal subfield of A and the dimension of A is the square of a prime p, the proportion of spinor genera representing H has the form r/p. In fact, when the representation field exists, this proportion is either 1 or 1/p. In the later case the order H is said to be selective for the genus. The condition for selectivity is known when D is maximal and also when p = 2 and D is an Eichler order. In this work we describe the orders H that are selective for at least one genus of orders of maximal rank in A.

On optimal embeddings and trees

“…It was proved in [3] that, when L ∼ = K( √ d) is a field, F max (A|H) is the largest field of the form F (D|H), where D runs over the set of full orders in A containing H, and it can be computed as follows:…”

Optimal spinor selectivity for quaternion Bass orders

“…Arenas-Carmona [5] and Linowitz [18] obtained selectivity theorems for more general classes of orders. More broadly, selectivity results in the context of A being a central simple F -algebra have been obtained by Linowitz and Shemanske [19] and Arenas-Carmona [2,3,4].…”

“…Generally, the (optimal) selectivity question admits a satisfactory answer only if A satisfies the Eichler condition, that is, A is split at an infinite place of F . Indeed, almost all literature [1,3,5,10,13,15,18,20,25] on (optimal) selectivity assumes the Eichler condition.…”

Selectivity on division algebras

“…It is immediate from the definition of H(D|H), that we have the contention F ⊆ L∩Σ for any maximal field L containing H. This is the only bound needed in the sequel. A stronger bound of the form F ⊆ F 0 (A|H) ∩ Σ, was used in [4] to prove that an order contained in a quadratic extension is spinor selective for some genus in exactly one quaternion algebra. The field F 0 (A|H) is the class field corresponding to a class group K * H(H) whose definition is analogous to (1).…”