Reduction modulo $p$ of the Noether problem

TL;DR

This paper investigates the relationship between the stable rationality of special fibers and torsion properties of certain cohomology groups in mixed characteristic, applying advanced p-adic Hodge theory to the Noether problem.

Contribution

It establishes a link between stable rationality in characteristic p and torsion-freeness of $H^3$ in mixed characteristic, using integral p-adic Hodge theory.

Findings

If $X_k$ is stably rational, then $H^3(X_K, \\mathbb{Z}_p)$ is torsion-free.

Application to the Noether problem for finite p-groups.

Uses integral p-adic Hodge theory of Bhatt-Morrow-Scholze.

Abstract

Let $R$ be a complete valuation ring of mixed characteristic $(0,p)$ with algebraically closed fraction field $K$ and residue field $k$. Let $X/R$ be a smooth projective morphism. We show that if $X_k$ is stably rational, then $H^3(X_K, \mathbb Z_p)$ is torsion-free. The proof uses integral $p$-adic Hodge theory of Bhatt-Morrow-Scholze and the study of differential forms in positive characteristic. We then apply this result to study the Noether problem for finite $p$-groups.