Let k be a perfect field of characteristic p\(>0\), \(A=W(k)\) the ring of Witt vectors over k, K the quotient field of A and \(G=Gal(\bar K/K)\) the Galois group of the algebraic closure \(\bar K\) of K. Let C be the category of all finite commutative group schemes over A which are killed by the multiplication by p, and let MG(k) be the category of finite commutative G-modules which are killed by the multiplication by p. Then one has the functor \(F:\quad C\to MG(k)\) associating to every \(E\in Ob(C)\) the G-module \(E(\bar K)\) of \(\bar K-\)points.
The aim of the paper under review is to give necessary and sufficient conditions for a G-module \(H\in Ob(MG(k))\) in order to belong to the image of F. - As applications one proves the non-existence of abelian schemes over the rings of integers of the following number fields: \({\mathbb{Q}}\), \({\mathbb{Q}}(\sqrt{-1})\), \({\mathbb{Q}}(\sqrt{\pm 2})\), \({\mathbb{Q}}(\sqrt{-3})\), \({\mathbb{Q}}(\sqrt{-7})\) and \({\mathbb{Q}}(^ 5\sqrt{1})\).