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

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- by Bjorn Poonen
- J. Amer. Math. Soc.
**16**(2003), 981-990 - DOI: https://doi.org/10.1090/S0894-0347-03-00433-8
- Published electronically: July 8, 2003
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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.## References

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## Bibliographic Information

**Bjorn Poonen**- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
- MR Author ID: 250625
- ORCID: 0000-0002-8593-2792
- Email: poonen@math.berkeley.edu
- 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.
- © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 981-990 - MSC (2000): Primary 11U05; Secondary 11G05
- DOI: https://doi.org/10.1090/S0894-0347-03-00433-8
- MathSciNet review: 1992832