Sharedwww / tables / genus2reduction / WARNINGSOpen in CoCalc
Author: William A. Stein
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In the output of genus2reduction, the group of components
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$\Phi$ of the Neron model is the group over the algebraic
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closure of F_p. The set of rational points of $\Phi$
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can be computed using Theorem 1.17 in
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S. Bosch and Q. Liu "Rational points of the group of
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components of a N\'eron model", Manuscripta Math. 98
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(1999), 275-293.
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Be carefull that the formula
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valuation of the naive minimal discriminant
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= f + n - 1 + 11c(X)
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(Q. Liu : "Conducteur et discriminant minimal de courbes de
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genre 2", Compositio Math. 94 (1994) 51-79, Theoreme 2)
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is valid only if the residual field is algebraically
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closed as stated in the paper. So this equality does not
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hold in general over Q_p. The fact is that the minimal
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discriminant may change after unramified extension.
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One can show however that, at worst, the change will
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stabilize after a quadratic unramified extension
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(Q. Liu : "Modeles entiers de courbes hyperelliptiques sur un
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corps de valuation discrete", Trans. AMS 348 (1996), 4577-4610,
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\S 7.2, Proposition 4).
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