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.References
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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