An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel

doi:10.5802/jtnb.877

Rafael von Känel ¹

¹ IHÉS 35 Route de Chartres 91440 Bures-sur-Yvette France

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 507-530.

Résumé
Abstract

Soit $C$ une courbe hyperelliptique de genre $g \geq 1$ 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 $g \geq 1$ , 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 $g \geq 1$ 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 $g \geq 1$ one can in principle determine the set of $K$ -isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$ .

MR

DOI : 10.5802/jtnb.877

Affiliations des auteurs :

Rafael von Känel ¹

¹ IHÉS 35 Route de Chartres 91440 Bures-sur-Yvette France

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     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},
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     year = {2014},
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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/articles/10.5802/jtnb.877/

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