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If V is an irreducible algebraic variety over a number field K, and L is a field containing K, we say that V is diophantine-stable for L/K if V (L) =V (K). We prove that if V is either a simple abelian variety, or a curve of genus at least one, then under mild hypotheses there is a set S of rational primes with positive density such that for every S and every n 1, there are infinitely many cyclic extensions L/K of degree ⁿ for which V is diophantine-stable. We use this result to study the collection of finite extensions of K generated by points in V (K).
Mazur et al. (Mon,) studied this question.