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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits
verfasst von : Manjul Bhargava, Benedict H. Gross, Xiaoheng Wang
Erschienen in: Representations of Reductive Groups
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Let k be a field, let G be a reductive group, and let V be a linear representation of G. Let $$V/\!/G =\mathop{ \mathrm{Spec}}\nolimits ({\mathrm{Sym}}^{{\ast}}(V ^{{\ast}}))^{G}$$ denote the geometric quotient and let $$\pi: V \rightarrow V/\!/G$$ denote the quotient map. Arithmetic invariant theory studies the map π on the level of k-rational points. In this article, which is a continuation of the results of our earlier paper “Arithmetic invariant theory”, we provide necessary and sufficient conditions for a rational element of $$V/\!\!/G$$ to lie in the image of π, assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.