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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The spinor genus of quaternion orders


Author: Gordon L. Nipp
Journal: Trans. Amer. Math. Soc. 211 (1975), 299-309
MSC: Primary 10C05; Secondary 16A18
DOI: https://doi.org/10.1090/S0002-9947-1975-0376526-6
MathSciNet review: 0376526
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Abstract: Let D be a global domain whose quotient field F does not have characteristic 2, let $ \mathfrak{A}$ be a quaternion algebra over F, and let $ \mathfrak{D}$ be an order on $ \mathfrak{A}$ over D. A right $ \mathfrak{D}$-module M which is simultaneously a lattice on $ \mathfrak{A}$ over D is said to be right $ \mathfrak{D}$-generic if there exists $ \alpha \in \mathfrak{A},N(\alpha ) \ne 0$, such that $ {\alpha ^{ - 1}}M \in {\operatorname{gen}}\;\mathfrak{D}$. Our main result is that every right $ \mathfrak{D}$-generic module is cyclic if and only if every class in the spinor genus of $ \mathfrak{D}$ represents a unit in D. One consequence is that $ \mathfrak{D}$ is in a spinor genus of one class if and only if $ \mathfrak{D}$-generic modules are cyclic and $ \mathfrak{D}$ represents every unit represented by its spinor genus. In addition, it is shown that a necessary and sufficient condition that an integral ternary lattice L be in a spinor genus of one class is that every right $ {\mathfrak{D}_L}$-generic pair be equivalent to a two-sided $ {\mathfrak{D}_L}$-generic pair, where $ {\mathfrak{D}_L}$ is the quaternion order associated with L.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0376526-6
Article copyright: © Copyright 1975 American Mathematical Society