A user's guide to the local arithmetic of hyperelliptic curves

TL;DR

This paper provides a comprehensive, self-contained overview of the recent combinatorial approach using cluster pictures to analyze the local arithmetic of hyperelliptic curves over fields with odd residue characteristic, including numerous examples.

Contribution

It consolidates and summarizes various results on the arithmetic of hyperelliptic curves using cluster pictures into a single, accessible guide with extensive examples.

Findings

Effective computation of arithmetic invariants using cluster pictures

Unified presentation of recent advances in hyperelliptic curve arithmetic

Enhanced understanding through detailed examples

Abstract

A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.