Solvable points on genus one curves over local fields

TL;DR

This paper investigates the existence of rational points on genus one curves over local fields, proving solvability under certain conditions and exploring the limitations of solvable extensions for reduction properties.

Contribution

It establishes conditions for the existence of solvable points on genus one curves over local fields and constructs examples where solvable extensions do not achieve semi-stable reduction.

Findings

Genus one curves with degree a power of p have solvable points over F.

Existence of fields where no solvable extension yields unramified composite extensions.

Examples of curves that do not acquire semi-stable reduction over any solvable extension.

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 $F$.