Arithmetic invariant theory II

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 $\pi: V \to V//G$ denote the quotient map. Arithmetic invariant theory studies the map $\pi$ 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 $\pi$, assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.