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.

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.

@article{JTNB_2005__17_3_727_0,
     author = {Gunther Cornelissen and Thanases Pheidas and Karim Zahidi},
     title = {Division-ample sets and the Diophantine  problem for rings of integers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {727--735},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.516},
     zbl = {05016583},
     mrnumber = {2212121},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/}
}
Gunther Cornelissen; Thanases Pheidas; Karim Zahidi. 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://jtnb.centre-mersenne.org/articles/10.5802/jtnb.516/

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