Siegel’s theorem and the Shafarevich conjecture
Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 705-727.

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus g would imply an effective version of Siegel’s theorem for integral points on hyperelliptic curves of genus g.

Il est connu que dans le cas des courbes hyperelliptiques la conjecture de Shafarevich peut être rendue effective, c’est à dire, pour tout corps de nombres k et tout ensemble fini de places S de k, on peut effectivement calculer l’ensemble des classes d’isomorphisme des courbes hyperelliptiques sur k ayant bonne réduction en dehors de S. Nous montrons ici qu’une extension de ce résultat à une version effective de la conjecture de Shafarevich pour les Jacobiennes de courbes hyperelliptiques de genre g impliquerait une version effective du théorème de Siegel pour les points entiers sur les courbes hyperelliptiques de genre g.

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DOI: 10.5802/jtnb.818
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Aaron Levin. Siegel’s theorem and the Shafarevich conjecture. Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 705-727. doi : 10.5802/jtnb.818. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.818/

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