Siegel’s theorem and the Shafarevich conjecture

Levin, Aaron

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 705-727.

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 $g$.

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 $g$.

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