Division-ample sets and the Diophantine problem for rings of integers
Cornelissen, Gunther1 ; Pheidas, Thanases2 ; Zahidi, Karim3
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, Tome 17 (2005) no. 3, pp. 727-735.

Nous demontrons que le dixième problème de Hilbert pour un anneau d’entiers dans un corps de nombres K admet une réponse négative si K satisfait à deux conditions arithmétiques (existence d’un ensemble dit division-ample et d’une courbe elliptique de rang un sur K). Nous lions les ensembles division-ample à l’arithmétique des variétés abéliennes.

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.

DOI : 10.5802/jtnb.516
Cornelissen, Gunther 1 ; Pheidas, Thanases 2 ; Zahidi, Karim 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 
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Cornelissen, Gunther; Pheidas, Thanases; Zahidi, Karim. Division-ample sets and the Diophantine  problem for rings of integers. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 727-735. doi : 10.5802/jtnb.516. https://www.numdam.org/articles/10.5802/jtnb.516/

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