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Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of $\mathbb{Q}$

Author: Bjorn Poonen
Journal: J. Amer. Math. Soc. 16 (2003), 981-990
MSC (2000): Primary 11U05; Secondary 11G05
Published electronically: July 8, 2003
MathSciNet review: 1992832
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Abstract: We give the first examples of infinite sets of primes $S$such that Hilbert's Tenth Problem over $\mathbb{Z}[S^{-1}]$ has a negative answer. In fact, we can take $S$ to be a density 1 set of primes. We show also that for some such $S$ there is a punctured elliptic curve $E'$ over $\mathbb{Z}[S^{-1}]$such that the topological closure of $E'(\mathbb{Z}[S^{-1}])$ in $E'(\mathbb{R})$has infinitely many connected components.

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Additional Information

Bjorn Poonen
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840

Keywords: Hilbert's Tenth Problem, elliptic curve, Mazur's Conjecture, diophantine definition
Received by editor(s): December 8, 2002
Published electronically: July 8, 2003
Additional Notes: This research was supported by NSF grant DMS-0301280 and a Packard Fellowship.
Article copyright: © Copyright 2003 American Mathematical Society