There are no abelian varieties over \(\mathbb Z\).
type: Item
instance of: scholarly article
title: There are no abelian varieties over \(\mathbb Z\).
zbMATH Open document ID: 0612.14043
DOI: 10.1007/BF01388584
author: Jean-Marc Fontaine
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143270
review text: Over the field \(E={\mathbb Q}\), \({\mathbb Q}(\sqrt{-1})\), \({\mathbb Q}(\sqrt{-3})\), or \({\mathbb Q}(\sqrt{-5})\), there are no abelian varieties of dimension \(\geq 1\) having good reduction everywhere. This is the final result of the paper. In particular, in the case \(E={\mathbb Q}\), this assertion, which is also the title of this paper, has been known as a conjecture of Shafarevich. Now, a little more generally, let \(J\) be a finite group scheme killed by \(p\) over the ring of integers of a number field \(E\) and \(F\) the field generated by the algebraic points of \(J\) over \(E\). A majoration of the discriminant of the field \(F\) over \(E\), which is discussed in this paper, a minoration of it given by the method of Odlyzko-Poitou-Serre, and the theory of Néron models of abelian varieties are the keys to the proof of the final result. The local version of the majoration of the discriminant, i.e., in the case of a complete valuation field \(K\) of characteristic 0, instead of the number field \(E\), was the starting point in this paper.
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G30
zbMATH DE Number: 3989548
zbMATH Keywords: nonexistence of abelian varieties
zbMATH Keywords: good reduction
zbMATH Keywords: conjecture of Shafarevich
zbMATH Keywords: Néron models
reviewed by: Shoji Koizumi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3989548