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

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

2003-12-19

arxiv.org/abs/math/0312382v1

Mathematics

Number Theory

9 pages, uses a4paper, calrsfs

Scientific paper

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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