Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits

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→V//G denotethequotientmap.Arithmeticinvarianttheorystudiesthe map 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.