Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields

Details Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields

2004-08-19

arxiv.org/abs/math/0408271v1

Mathematics

Number Theory

17 pages

Scientific paper

We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some 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 there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O_{K,S} of S-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 S of density 1 and an infinite Diophantine subset of O_{K,S} 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 S of density 1 such that there exists a Diophantine model of Z over O_{K,S}. This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a non-archimedean topology and questions concerning extensions of Hilbert's Tenth Problem to subrings of number fields.

Affiliated with

Also associated with

No associations

Rating

Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-713691