Local Solubility of Hyperelliptic Curves

TL;DR

This paper provides a criterion for the local solvability of hyperelliptic curves over local fields, based on their cluster picture and reduction properties, especially when the residue field is large and the curve has tame semistable reduction.

Contribution

It introduces a new solvability criterion for hyperelliptic curves using cluster pictures, linking local solubility to reduction behavior and residue field size.

Findings

Criterion for local solubility based on cluster picture

Applicable when curve has tame semistable reduction

Residue field size influences solvability condition

Abstract

We give a condition for a hyperelliptic curve $C$ over a local field $K$ to be locally soluble, on the condition that $C$ obtains semistable reduction after a tame extension of $K$, and that the residue field $k$ is sufficiently large relative to the genus of the curve. The condition is presented in terms of the cluster picture of $C$, a combinatorial object which determines much of the local arithmetic of $C$.