Maximal selectivity for orders in fields

If H and D are two orders in a central simple algebra A with D of maximal rank and containing H, 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.

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