Reduction modulo p of Noether’s problem

Ambrosi, Emiliano; Valloni, Domenico

doi:10.2140/ant.2025.19.2359

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Reduction modulo $p$ of Noether's problem

Emiliano Ambrosi and Domenico Valloni

Vol. 19 (2025), No. 12, 2359–2367

DOI: 10.2140/ant.2025.19.2359

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 $H_{ét}^{3} (X_{K}, \hat{ℤ})$ is torsion-free. The proof uses the integral $p$ -adic Hodge theory of Bhatt, Morrow and Scholze and the study of differential forms in positive characteristic. We then apply this result to study the Noether problem for finite $p$ -groups.

Keywords

Noether's problem, crystalline cohomology, Artin–Mumford invariant

Mathematical Subject Classification

Primary: 14E08, 14F30, 14M20

Milestones

Received: 2 April 2023

Revised: 20 November 2024

Accepted: 23 December 2024

Published: 20 October 2025

Authors

Emiliano Ambrosi
IRMA (Université de Strasbourg) 67084 Strasbourg France

Domenico Valloni
EPFL 1015 Lausanne Switzerland

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 12, 2025