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arXiv · 1310.7689

Arithmetic invariant theory II

Manjul Bhargava·Benedict H. Gross·Xiaoheng Wang
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Abstract

Let $k$ be a field, let $G$ be a reductive group, and let $V$ be a linear representation of $G$. Let $V//G = Spec(Sym(V^*))^G$ denote the geometric quotient and let $π: V \to V//G$ 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.

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Manjul Bhargava, Benedict H. Gross, Xiaoheng Wang. 2013-10-29. Arithmetic invariant theory II. https://arxiv.org/abs/1310.7689

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