  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.

Number Theory;Algebraic Geometry;Representation Theory