Solvable points on genus one curves over local fields

2012/02/12 by Ambrus Pal, Pal, Ambrus · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1202.2548

JIMJ, recommended for publication

arxiv created 2012/02/12 · arxiv updated 2012/02/14

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.

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