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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The spinor genus of quaternion orders
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by Gordon L. Nipp PDF
Trans. Amer. Math. Soc. 211 (1975), 299-309 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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