On certain birational invariants of the Fermat curves
type: Item
instance of: scholarly article
title: On certain birational invariants of the Fermat curves
zbMATH Open document ID: 1129.14028
author: E. Kunz
author: R. Waldi
published in: Journal of Pure and Applied Algebra
publication date: 2007-04-26T00:00:00Z
review text: Let \(K_{m} = {\mathbb Q}(x, y),\) \(x^{m}+y^{m}=1\) (\(m > 2)\) be the field of rational functions of the Fermat curve over \({\mathbb Q}\), and let \(V\) be the set of all discrete valuation rings \((R, m_{R})\) with \(Q(R)=K_{m}\) which are essentially of finite type over \({\mathbb Z}\). Moreover, let \(V_{s}=\{R\in V: R \text{ is smooth over}\, {\mathbb Z}\}\). The authors study, in the case of Fermat fields, the groups of integral differentials introduced by Kähler and Bost for arithmetic function fields. They derive some properties of these birational invariants of the Fermat curves and compute them for small \(m;\) describe the elements of \(V_{s}\) and their modules of differentials explicitly, which is done in section 3. This section is based on the explicit description of the discrete valuation rings of the rational functions field \(\mathbb Q(x)\), which are essentially of finite type and smooth over \(\mathbb Z\), and of their modules of differentials.
reviewed by: V. V. Chueshev
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 5146860
zbMATH Keywords: Fermat curve over \({\mathbb{Q}}\)
zbMATH Keywords: integral differentials
zbMATH Keywords: birational invariants
zbMATH Keywords: discrete valuation rings
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.08.016
OpenAlex ID: W2083974485
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DOI: 10.1016/J.JPAA.2006.08.016
description: scientific article; zbMATH DE number 5146860