Neutral representations in dimension $\leq 3$ and fields of moduli

Abstract

A representation $V$ of an algebraic group $G$ induces a vector bundle $[V/G] \to BG$. The representation $V$ of $G$ is neutral if, for every twisted form $\mathcal{V} \to \mathcal{G}$ of $[V/G] \to BG$ over a field $k$, we have $\mathcal{G}(k) \neq \emptyset$. Twisted forms of representations arise in many ways, for instance as cohomology of families of varieties on residual gerbes of moduli spaces, and from quotient singularities. Moreover, every Tannakian category is the category of vector bundles on some gerbe. Because of this, studying neutral representations yields numerous applications, especially to problems about fields of moduli. The present article has three main results. First, we completely classify neutral, faithful representations of finite groups in dimension $\leq 3$. Second, we give a very general, computation-friendly result for proving that representations of finite abelian groups are neutral, in arbitrary dimensions. Third, we develop the abstract concept of the normalizer $\mathcal{G} \to \mathcal{N} \to \mathcal{H}$ of a morphism of gerbes $\mathcal{G} \to \mathcal{H}$ on an arbitrary site (twisted representations correspond to morphisms of gerbes $\mathcal{G} \to B\mathrm{GL}_{n}$), and show that the normalizer $\mathcal{N}$ only depends on the geometric type of $\mathcal{G} \to \mathcal{H}$.