Genus field

Philosophy

Algebraic number

Pure mathematics

Biology

Discrete mathematics

Statistics

Algebraic number field

Variety (cybernetics)

Genus

Abelian variety

Field (mathematics)

Abelian group

Stability (learning theory)

Degree (music)

Bar (unit)

Computer science

Botany

Acoustics

Simple (philosophy)

Physics

Elementary abelian group

Mathematics

Epistemology

Mathematical analysis

Diophantine equation

Combinatorics

Finite field

Meteorology

Machine learning

American Journal of Mathematics

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 $V(\bar{K})$.

Diophantine stability