An effective proof of the hyperelliptic Shafarevich conjecture
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 507-530.

Soit C une courbe hyperelliptique de genre g1 sur un corps de nombres K avec bonne réduction en dehors d’un ensemble fini S de places de K. Nous démontrons que C possède un modèle de Weierstrass sur l’anneau des entiers de K avec hauteur effectivement bornée en termes de g, S et K. En particulier, nous démontrons que pour tout corps de nombres K, tout ensemble fini S de places de K et tout entier g1, on peut déterminer en principe l’ensemble des classes d’isomorphisme de courbes hyperelliptiques de genre g sur K avec bonne réduction en dehors de S.

Let C be a hyperelliptic curve of genus g1 over a number field K with good reduction outside a finite set of places S of K. We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g, S and K. In particular, we obtain that for any given number field K, finite set of places S of K and integer g1 one can in principle determine the set of K-isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside S.

Reçu le : 2012-12-20
Accepté le : 2013-05-20
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.877
@article{JTNB_2014__26_2_507_0,
     author = {Rafael von K\"anel},
     title = {An effective proof of the hyperelliptic Shafarevich conjecture},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {507--530},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.877},
     mrnumber = {3320490},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_507_0/}
}
Rafael von Känel. An effective proof of the hyperelliptic Shafarevich conjecture. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 507-530. doi : 10.5802/jtnb.877. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_507_0/

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