Diophantine stability

TL;DR

This paper investigates conditions under which certain algebraic varieties over number fields remain rational points stable across infinite extensions, revealing that for specific varieties, there are infinitely many cyclic extensions where stability persists.

Contribution

It proves the existence of infinitely many cyclic extensions of prime power degree where diophantine stability holds for certain varieties, extending understanding of rational points over field extensions.

Findings

Existence of positive density set of primes with stability properties

Infinite cyclic extensions of prime power degree where stability persists

Application to the structure of fields generated by rational points

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 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 $V(\bar{K})$.