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Canonical liftings of Jacobians. (English) Zbl 0603.14029
The Jacobians of Fermat curves are of CM type. Is there a systematic way to construct other curves with this property ? The authors have this problem in mind and ask if the canonical lifting, due to Serre and Tate, of the Jacobian of an ordinary curve over a perfect field k of characteristic \(p>0\) is again the Jacobian of some curve. In this paper it is shown that, when p is odd and the genus is \(\geq 4\), the answer is ”no” for most curves, even if one works \(mod p^ 2.\) The same problem is independently treated by F. Oort and T. Sekiguchi [J. Math. Soc. Japan 38, 427-437 (1986; Zbl 0605.14031)], and the results considerably overlap in both works. But the general ideas of the arguments are quite different from each other. Our authors proceed by ”pure thought”, while the others follow a very concrete way.
Reviewer: S.Koizumi

MSC:
14K30 Picard schemes, higher Jacobians
14H40 Jacobians, Prym varieties
14H10 Families, moduli of curves (algebraic)
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References:
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