There are no abelian varieties over \(\mathbb Z\).
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.
scientific article; zbMATH DE number 3989548