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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of $\mathbb {Q}$
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by Bjorn Poonen PDF
J. Amer. Math. Soc. 16 (2003), 981-990 Request permission


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
  • MR Author ID: 250625
  • ORCID: 0000-0002-8593-2792
  • Email:
  • 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:
  • MathSciNet review: 1992832