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

Gunther Cornelissen·Thanases Pheidas·Karim Zahidi
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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.

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Gunther Cornelissen, Thanases Pheidas, Karim Zahidi. 2003-12-19. Division-ample sets and the Diophantine problem for rings of integers. https://arxiv.org/abs/math/0312382

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