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The canonical lifting of an ordinary Jacobian variety need not be a Jacobian variety. (English) Zbl 0605.14031
An abelian variety X over a perfect field k of characteristic $$p>0$$ is called ordinary if $$X(\bar k)$$ has $$p^ g$$-torsion points, where $$g=\dim X.$$
It is well known that an ordinary polarized abelian variety $$(X,\lambda_ 0)$$ can be lifted to a polarized abelian scheme ($${\mathcal X},\lambda)$$ over Spec W(k), where W(k) denotes the ring of Witt vectors of k. The authors show by a counterexample that if $$(X,\lambda_ 0)$$ is the Jacobian of a smooth curve over k, ($${\mathcal X},\lambda)$$ is in general not a Jacobian. A deformation argument then shows that this holds even generically if $$p\geq 5$$ and $$g\geq 2(p-1)$$. The curve which is used for the counterexample is a cyclic Galois covering of $${\mathbb{P}}_ 1$$ of order p.
Reviewer: F.Herrlich

##### MSC:
 14H40 Jacobians, Prym varieties 14G15 Finite ground fields in algebraic geometry 14K15 Arithmetic ground fields for abelian varieties
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