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\(p\)-divisible groups with complex multiplication over \(W(k)\). (English) Zbl 0758.14030
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 \(W\).
MSC:
14L05 Formal groups, \(p\)-divisible groups
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References:
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