Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen[1]; Thanases Pheidas[2]; Karim Zahidi[3]

  • [1] Mathematisch Instituut Universiteit Utrecht Postbus 80010 3508 TA Utrecht, Nederland
  • [2] Department of Mathematics University of Crete P.O. Box 1470 Herakleio, Crete, Greece
  • [3] Equipe de Logique Mathématique U.F.R. de Mathématiques (case 7012) Université Denis-Diderot Paris 7 2 place Jussieu 75251 Paris Cedex 05, France Adresse actuelle: Departement Wiskunde, Statistiek & Actuariaat Universiteit Amtwerpen Prinsstraat 13 2000 Antwerpen, België

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 3, page 727-735
  • ISSN: 1246-7405

Abstract

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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K ). We relate division-ample sets to arithmetic of abelian varieties.

How to cite

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Cornelissen, Gunther, Pheidas, Thanases, and Zahidi, Karim. "Division-ample sets and the Diophantine problem for rings of integers." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 727-735. <http://eudml.org/doc/249417>.

@article{Cornelissen2005,
abstract = {We prove that Hilbert’s Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over $K$). We relate division-ample sets to arithmetic of abelian varieties.},
affiliation = {Mathematisch Instituut Universiteit Utrecht Postbus 80010 3508 TA Utrecht, Nederland; Department of Mathematics University of Crete P.O. Box 1470 Herakleio, Crete, Greece; Equipe de Logique Mathématique U.F.R. de Mathématiques (case 7012) Université Denis-Diderot Paris 7 2 place Jussieu 75251 Paris Cedex 05, France Adresse actuelle: Departement Wiskunde, Statistiek & Actuariaat Universiteit Amtwerpen Prinsstraat 13 2000 Antwerpen, België},
author = {Cornelissen, Gunther, Pheidas, Thanases, Zahidi, Karim},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {727-735},
publisher = {Université Bordeaux 1},
title = {Division-ample sets and the Diophantine problem for rings of integers},
url = {http://eudml.org/doc/249417},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Cornelissen, Gunther
AU - Pheidas, Thanases
AU - Zahidi, Karim
TI - Division-ample sets and the Diophantine problem for rings of integers
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 727
EP - 735
AB - We prove that Hilbert’s Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over $K$). We relate division-ample sets to arithmetic of abelian varieties.
LA - eng
UR - http://eudml.org/doc/249417
ER -

References

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