Diophantine stability

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(K̄).