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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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