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 $\mathbb {Q}$
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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.

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