Deformations with respect to an algebraic group

Abstract

Let $\mathcal{G}$ be a smooth linear algebraic group over the ring of Witt vectors of a finite field $k$. In this paper, we study deformations of representations of a profinite group into the points $\mathcal{G}(k)$ of $\mathcal{G}$ over $k$. We show that the $\mathcal{G}$-deformation functor has a versal deformation ring, and we generalize criteria of Tilouine concerning when this ring is universal. If $\mathcal{G}$ is an algebraic subgroup of $\mathrm{GL}_n$, we study when the $\mathcal{G}$-deformation functor is a subfunctor of the $\mathrm{GL}_n$-deformation functor studied by Mazur. When $\mathcal{G}$ is an orthogonal group, this leads to studying versal versions of results of Serre and Fröhlich about the connection between Stiefel-Whitney classes, spinor norms and Hasse-Witt invariants of orthogonal Galois representations.