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

Bjorn Poonen
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Abstract

We give the first examples of infinite sets of primes S such that Hilbert's Tenth Problem over 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 Z[S^{-1}] such that the topological closure of E'(Z[S^{-1}]) in E'(R) has infinitely many connected components.

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Bjorn Poonen. 2003-06-18. Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of Q. https://doi.org/10.1090/s0894-0347-03-00433-8

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