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research
Selectivity in Quaternion Algebras
Authors
Benjamin Linowitz
Publication date
11 February 2012
Publisher
Doi
Cite
View
on
arXiv
Abstract
We prove an integral version of the classical Albert-Brauer-Hasse-Noether theorem regarding quaternion algebras over number fields. Let
A
\mathfrak A
A
be a quaternion algebra over a number field
K
K
K
and assume that
A
\mathfrak A
A
satisfies the Eichler condition; that is, there exists an archimedean prime of
K
K
K
which does not ramify in
A
\mathfrak A
A
. Let
Ī©
\Omega
Ī©
be a commutative, quadratic
O
K
\mathcal{O}_K
O
K
ā
-order and let
R
ā
A
\mathcal{R}\subset \mathfrak A
R
ā
A
be an order of full rank. Assume that there exists an embedding of
Ī©
\Omega
Ī©
into
R
\mathcal R
R
. We describe a number of criteria which, if satisfied, imply that every order in the genus of
R
\mathcal R
R
admits an embedding of
Ī©
\Omega
Ī©
. In the case that the relative discriminant ideal of
Ī©
\Omega
Ī©
is coprime to the level of
R
\mathcal R
R
and the level of
R
\mathcal R
R
is coprime to the discriminant of
A
\mathfrak A
A
, we give necessary and sufficient conditions for an order in the genus of
R
\mathcal R
R
to admit an embedding of
Ī©
\Omega
Ī©
. We explicitly parameterize the isomorphism classes of orders in the genus of
R
\mathcal R
R
which admit an embedding of
Ī©
\Omega
Ī©
. In particular, we show that the proportion of the genus of
R
\mathcal{R}
R
admitting an embedding of
Ī©
\Omega
Ī©
is either 0, 1/2 or 1. Analogous statements are proven for optimal embeddings.Comment: Final version; to appear in the Journal of Number Theor
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Elsevier - Publisher Connector
See this paper in CORE
Go to the repository landing page
Download from data provider
Last time updated on 06/05/2017