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Journal für die reine und angewandte Mathematik

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Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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

Bjorn Poonen / Alexandra Shlapentokh

Citation Information: Journal für die reine und angewandte Mathematik. Volume 2005, Issue 588, Pages 27–47, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crll.2005.2005.588.27, December 2005

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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 
-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.

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