On Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2

Let M be a number field. Let W be a set of non-archimedean primes of M . Let OM,W = {x ∈ M | ordpx ≥ 0∀p 6∈ W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particular, the following results are proved. Let M be a totally real field or a totally complex extension of degree 2 of a totally real field. Then for any ε > 0 there exists a set WM of primes of M whose density is bigger than 1 − [M : Q]−1 − ε and such that Z has a Diophantine definition over OM,WM . (Thus, Hilbert’s Tenth Problem is undecidable in OM,WM .) Let M be as above and let ε > 0 be given. Let SQ be the set of all rational primes splitting in M . (If the extension is Galois but not cyclic, SQ contains all the rational primes.) Then there exists a set of M -primes WM such that the set of rational primes WQ below WM differs from SQ by a set contained in a set of density less than ε and such that Z has a Diophantine definition over OM,WM . (Again this will imply that Hilbert’s Tenth Problem is undecidable in OM,WM .)