Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of ℚ

Poonen, Bjorn

doi:10.1090/S0894-0347-03-00433-8

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
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Abstract: We give the first examples of infinite sets of primes such that Hilbert's Tenth Problem over $\mathbb{Z}[S^{-1}]$ has a negative answer. In fact, we can take to be a density 1 set of primes. We show also that for some such there is a punctured elliptic curve 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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