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

Bjorn Poonen·Alexandra Shlapentokh
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Abstract

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.

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BibTeXRIS

Bjorn Poonen, Alexandra Shlapentokh. 2004-08-19. Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields. https://doi.org/10.1515/crll.2005.2005.588.27

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