Nakamura, Tetsuo \(p\)-divisible groups with complex multiplication over \(W(k)\). (English) Zbl 0758.14030 Compos. Math. 80, No. 2, 229-234 (1991). Fix a prime number \(p\). Let \(k\) be an algebraically closed field of characteristic \(p\) and \(W=W(k)\) the ring of Witt vectors over \(k\). Let \(G\) be a \(p\)-divisible group over \(W\) of finite height \(h\). For an extension \(E\) of \(\mathbb{Q}_ p\) of degree \(h\), we say that \(G\) has CM (complex multiplication) by \(E\) if there is given a homomorphism of \(E\) into \(\mathbb{Q}_ p\otimes\text{End}(G)\). The action of \(E\) on the tangent space to \(G\) has character \(\Sigma_ \Phi\tau\) for some subset \(\Phi\) of \(\text{Hom}(E,\overline E)\). We say \(G\) has type \((E,\Phi)\). We denote by \(K_ h\) the unramified extension over \(\mathbb{Q}_ p\) of degree \(h\) and by \(W_ h\) its maximal order. The following results are proved:(i) A \(p\)-divisible group \(G\) over \(W\) with CM of height \(h\) is elementary if and only if \(\text{End}(G)\cong W_ h\).(ii) A \(p\)-divisible group over \(W\) with CM is isomorphic to a product of several copies of an elementary group over \(W\).(iii) Any two \(p\)-divisible groups over \(W\) of the same type \((K_ h,\Phi)\) are isomorphic over \(W\). Reviewer: T.Nakamura (Sendai) MSC: 14L05 Formal groups, \(p\)-divisible groups Keywords:characteristic \(p\); \(p\)-divisible group; complex multiplication; Witt vectors PDFBibTeX XMLCite \textit{T. Nakamura}, Compos. Math. 80, No. 2, 229--234 (1991; Zbl 0758.14030) Full Text: Numdam EuDML References: [1] J.-M. Fontaine , Groupes p-divisible sur les corps locaux . Astérisque 47-48, Soc. Math. de France, 1977. · Zbl 0377.14009 [2] J.-M. Fontaine , Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate , in Journées de Géométrie Algébrique de Renne , Astérisque 65, Soc. Math. de France, 1979, pp. 3-80. · Zbl 0429.14016 [3] J.-M. Fontaine , Sur certains types de représentations p-adiques du groupe de Galois d’un corps local: construction d’un anneau de Barsotti-Tate , Ann Math. 115 (1982) 529-577. · Zbl 0544.14016 [4] T. Honda , On the theory of commutative formal groups , J. Math. Soc. Japan 22 (1970) 213-246. · Zbl 0202.03101 [5] Yu. I. Manin , The theory of commutative formal groups over fields of finite characteristic , Russian Math. Surveys 18 (1963) 1-83. · Zbl 0128.15603 [6] J.-P. Serre , Abelian l-adic Representations and Elliptic Curves , Benjamin, New York, 1968. · Zbl 0186.25701 [7] W. Waterhouse , A classification of almost full formal groups , Proc. Amer. Math. Soc. 20 (1969) 426-428. · Zbl 0176.30303 [8] W. Waterhouse , On p-divisible groups over complete valuation rings , Ann. Math. 95 (1972) 55-65. · Zbl 0227.14018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.