On representations of spinor genera

Chan, Wai Kiu; Xu, Fei

doi:10.1112/s0010437x03000484

Cited by 29 publications

(14 citation statements)

References 11 publications

(25 reference statements)

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“…When L is a maximal commutative sub-algebra, the conditions on A and H for which selectivity can occur were described completely by T. Chinburg and E. Friedman in [9]. These results where extended to Eichler orders D in [10] and [8]. The second author of the present work gave a generalization to representations of an arbitrary suborder H into into finite intersections of maximal orders [5].…”

Section: Introductionmentioning

confidence: 65%

On optimal embeddings and trees

Arenas

Arenas-Carmona

Contreras

2018

Journal of Number Theory

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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.

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Section: Introductionmentioning

confidence: 65%

On optimal embeddings and trees

Arenas

Arenas-Carmona

Contreras

2018

Journal of Number Theory

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“…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: 97%

Maximal selectivity for orders in fields

Arenas-Carmona

2012

Journal of Number Theory

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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.

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“…The corresponding problem when the embedded order is commutative was studied in [5,6], and [1]. When n = 2, i.e., A is a quaternion algebra, the problem can be treated in terms of representation of quadratic forms as pointed out in [4], but the study of algebras of higher dimension requires the general theory developed in [1].…”

Section: Introductionmentioning

confidence: 99%