Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields

Abstract

We prove that some infinite

-adically discrete sets have Diophantine definitions in large subrings of number fields. First, if K  is a totally real number field or a totally complex degree-2 extension of a totally real number field, then for every prime  of K  there exists a set of K-primes  of density arbitrarily close to 1 such that there is an infinite -adically discrete set that is Diophantine over the ring  of -integers in K. Second, if K  is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes of density 1 and an infinite Diophantine subset of that is v-adically discrete for every place v of K. Third, if K  is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes of density 1 such that there exists a Diophantine model of ℤ over . This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a nonarchimedean topology and questions concerning extensions of Hilbert’s Tenth Problem to subrings of number fields.