Maximal selectivity for orders in fields

TL;DR

This paper investigates the conditions under which certain orders in central simple algebras exhibit maximal selectivity in representing orders within their genus, extending known results to broader cases.

Contribution

It characterizes orders that are selective for at least one genus of maximal rank orders in central simple algebras, generalizing previous selectivity conditions.

Findings

Proportion of spinor genera representing H is r/p when the representation field exists.

Selectivity occurs when the order H is contained in a maximal subfield and the algebra dimension is p^2.

Provides criteria for selectivity beyond previously known cases.

Abstract

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.