Hilbert’s Tenth Problem and Mazur’s Conjecture for Large Subrings of Q

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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@inproceedings{Poonen2003HilbertsTP, title={Hilbert’s Tenth Problem and Mazur’s Conjecture for Large Subrings of Q}, author={Bjorn Poonen}, year={2003} }