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

Cornelissen, Gunther; Pheidas, Thanases; Zahidi, Karim

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.

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