Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

Abstract

We compute the dimension of the tangent space to, and the Krull dimension of, the prorepresentable hull of two deformation functors. The first one is the "algebraic" deformation functor of an ordinary curve $X$ over a field of positive characteristic with prescribed action of a finite group $G$, and the data are computed in terms of the ramification behaviour of $X\to G\backslash X$. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group $N$ in ${\rm PGL}(2, K)$ over a nonarchimedean-valued field $K$, and the data are computed in terms of the Bass-Serre representation of $N$ via a graph of groups. Finally, if $\Gamma$ is a free subgroup of $N$ such that $N$ is contained in the normalizer of $\Gamma$ in ${\rm PGL}(2, K)$, then the Mumford curve associated to $\Gamma$ becomes equipped with an action of $N/\Gamma$, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of $N$.