Abstract: Let $k$ be any field and let $V$ be a linear and faithful representation of a finite group $G$. The Noether problem asks whether $V/G$ is a (stably) rational variety over k. It is known that if $p = char(k) > 0$ and $G$ is a $p$-group, then $V/G$ is always rational. On the other hand, Saltman and later Bogomolov constructed many examples of $p$-groups such that $V/G$ is not stably rational over the complex numbers.

The aim of the talk is to study what happens over dvr of mixed characteristic $(0,p)$. We show for instance that for all the examples found by Saltaman and Bogomolov, there cannot exist a smooth projective scheme over $R$ whose special resp. generic fibre are stably birational to $V/G$ (and, hence, that $P^n_R/G$ never admits a relative resolution of singularities over $R$). The proof combines integral $p$-adic Hodge theory and the study of differential forms in positive characteristic.