Solvable points on genus one curves over local fields

Abstract

Let $F$ be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic $p>0$. We prove that every smooth, projective, geometrically irreducible curve of genus one defined over $F$ with a non-zero divisor of degree a power of $p$ has a solvable point over $F$. We also show that there is a field $F$ complete with respect to a discrete valuation whose residue field is perfect and there is a finite Galois extension $K|F$ such that there is no solvable extension $L|F$ such that the extension $KL|K$ is unramified, where $KL$ is the composite of $K$ and $L$. As an application we deduce that that there is a field $F$ as above and there is a smooth, projective, geometrically irreducible curve over $F$ which does not acquire semi-stable reduction over any solvable extension of $F$.