An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel[1]

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

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 2, page 507-530
  • ISSN: 1246-7405

Abstract

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Let C be a hyperelliptic curve of genus g ≥ 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 ≥ 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 .

How to cite

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von Känel, Rafael. "An effective proof of the hyperelliptic Shafarevich conjecture." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 507-530. <http://eudml.org/doc/275799>.

@article{vonKänel2014,
abstract = {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 $S$.},
affiliation = {IHÉS 35 Route de Chartres 91440 Bures-sur-Yvette France},
author = {von Känel, Rafael},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {10},
number = {2},
pages = {507-530},
publisher = {Société Arithmétique de Bordeaux},
title = {An effective proof of the hyperelliptic Shafarevich conjecture},
url = {http://eudml.org/doc/275799},
volume = {26},
year = {2014},
}

TY - JOUR
AU - von Känel, Rafael
TI - An effective proof of the hyperelliptic Shafarevich conjecture
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/10//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 2
SP - 507
EP - 530
AB - 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 $S$.
LA - eng
UR - http://eudml.org/doc/275799
ER -

References

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