An effective proof of the hyperelliptic Shafarevich conjecture

von Känel, Rafael

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

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

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