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Preprint Number 2235

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2235. Barry Mazur, Karl Rubin, and Alexandra Shlapentokh
Existential definability and diophantine stability

Submission date: 21 August 2022


Let K be a number field, let L be an algebraic (possibly infinite degree) extension of K, and let O_K ⊂ O_L be their rings of integers. Suppose A is an abelian variety defined over K such that A(K) is infinite and A(L)/A(K) is a torsion group. If at least one of the following conditions is satisfied:
1. L is a number field, 2. L is totally real, 3. L is a quadratic extension of a totally real field,
then O_K has a diophantine definition over O_L.

Mathematics Subject Classification: 11U05 (Primary) 11G05, 11G10, 03D35 (Secondary)

Keywords and phrases:

Full text arXiv 2208.09963: pdf, ps.

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