An embedding theorem for Eichler orders

Journal of Number Theory

Contreras

2018

We apply the theory of Bruhat-Tits trees to the study of optimal embeddings of two and three dimensional commutative orders into quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in a quaternion algebra yield optimal representations of such orders. This completes the previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as sub-orders. The same technique is used in the second part of this work to compute local embedding numbers, extending previous results by J. Brzezinski.

Section: Introductionmentioning

confidence: 65%

On optimal embeddings and trees

Arenas

Journal of Number Theory

Contreras

2018

“…Since H has maximal rank in L, Example 3.1. Assume A is a split quaternion algebra and H is an order in a maximal unramified subfield L. It is proved in [6] and [9] that there is no selectivity if H embeds non-optimally in D, …”

Section: Proof It Suffices To Prove That If a ∈ H(d T |H) For Infinimentioning

confidence: 99%

“…The main result in [9] and [6] implies that when an order H in a quadratic subfield L ⊆ A is selective, every embedding of H into an order in the genus of D is optimal (in the sense defined in [13]) at any place that is inert for L/K . We show in Example 3.1 bellow that this fails to generalize to arbitrary orders.…”

mentioning

confidence: 99%

Maximal selectivity for orders in fields

Journal of Number Theory

2012

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.

“…If T is omitted, it is assumed that T = ∞(K). Later several authors extended this characterization to Eichler orders [12], [7]. B. Linowitz has given several criteria under which selectivity can be avoided, always assuming Eichler's condition [15].…”

Section: Introductionmentioning

confidence: 99%

Roots of unity in definite quaternion orders

2015

Acta Arith.

Abstract. A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with a type number 3 or larger. The proof extends to a few other closely related orders.