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

DOI:
https://doi.org/10.1090/S0894-0347-03-00433-8

Published electronically:
July 8, 2003

MathSciNet review:
1992832

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
250625

ORCID:
0000-0002-8593-2792

Email:
poonen@math.berkeley.edu

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