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

Nous démontrons 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 ens...

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Published in:Journal de theorie des nombres de bordeaux 2005-01, Vol.17 (3), p.727-735
Main Authors: CORNELISSEN, Gunther, PHEIDAS, Thanases, ZAHIDI, Karim
Format: Article
Language:English
Subjects:
Algebra
Algebraic geometry
Arithmetic
Computer algebra systems
Diophantine sets
Exact sciences and technology
Geometry
Integers
Mathematical relations
Mathematical rings
Mathematics
Number theory
Numbers
Sciences and techniques of general use
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PHEIDAS, Thanases
ZAHIDI, Karim
description Nous démontrons 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 divisionample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.
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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 divisionample set of integers and of an elliptic curve of rank one over K). 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subjects Algebra
Algebraic geometry
Arithmetic
Computer algebra systems
Diophantine sets
Exact sciences and technology
Geometry
Integers
Mathematical relations
Mathematical rings
Mathematics
Number theory
Numbers
Sciences and techniques of general use
title Division-ample sets and the Diophantine problem for rings of integers
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