Abstract

abstract:

If $V$ is an irreducible algebraic variety over a number field $K$, and $L$ is a field containing $K$, we say that $V$ is {\it 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 $\ell\in S$ and every $n\ge 1$, there are infinitely many cyclic extensions $L/K$ of degree $\ell^n$ for which $V$ is diophantine-stable. We use this result to study the collection of finite extensions of $K$ generated by points in $V(\bar{K})$.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 571-616
Launched on MUSE
2018-05-10
Open Access
No
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