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Groupes p-divisibles et corps gauches. (p-divisible groups and skew fields). (French) Zbl 0586.14036
The author extends the Lubin-Tate construction of formal groups with complex multiplication by considering the following situation (in fact a more general one). Let K be a finite extension of \({\mathbb{Q}}_ p\), A its ring of integers, \(\bar K\) its algebraic closure, and put \(G=Gal(\bar K/K)\). Given a skew field D of dimension \(n^ 2\) over K and center K, the author constructs a connected p-divisible group R over A of dimension n and height \([D:{\mathbb{Q}}_ p]\) whose Tate module \(V_ R\) satisfies \(End_ G(V_ R\otimes {\mathbb{Q}}_ p)=D\). The algebraic envelope of G, that is the smallest algebraic subgroup defined over \({\mathbb{Q}}_ p\) of \(Aut_{{\mathbb{Q}}_ p}(V_ R\otimes {\mathbb{Q}}_ p)\) containing the image of G, is then proven to be isomorphic to the multiplicative group \(D^*\) of D.
Reviewer: F.Baldassarri
14L05 Formal groups, \(p\)-divisible groups
Full Text: Numdam EuDML
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