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There are no abelian varieties over \(\mathbb Z\). (Il n’y a pas de variété abélienne sur \(\mathbb Z\).) (French) Zbl 0612.14043
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.
Reviewer: S.Koizumi

14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
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