Sharedwww / Tables / genus2reduction / WARNINGSOpen in CoCalc
Author: William A. Stein
1In the output of genus2reduction, the group of components
2$\Phi$ of the Neron model is the group over the algebraic
3closure of F_p. The set of rational points of $\Phi$
4can be computed using Theorem 1.17 in
5S. Bosch and Q. Liu "Rational points of the group of
6components of a N\'eron model", Manuscripta Math. 98
7(1999), 275-293.
8
9Be carefull that the formula
10	valuation of the naive minimal discriminant
11	= f + n - 1 + 11c(X)
12(Q. Liu : "Conducteur et discriminant minimal de courbes de
13genre 2", Compositio Math. 94 (1994) 51-79, Theoreme 2)
14is valid only if the residual field is algebraically
15closed as stated in the paper. So this equality does not
16hold in general over Q_p. The fact is that the minimal
17discriminant may change after unramified extension.
18One can show however that, at worst, the change will
19stabilize after a quadratic unramified extension
20(Q. Liu : "Modeles entiers de courbes hyperelliptiques sur un
21corps de valuation discrete", Trans. AMS 348 (1996), 4577-4610,
22\S 7.2, Proposition 4).
23