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