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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

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