# Semistable types of hyperelliptic curves

## TL;DR

This paper develops combinatorial descriptions and classifications of semistable hyperelliptic curves over local fields, linking their geometric, algebraic, and Galois-theoretic properties to better understand their local invariants.

## Contribution

It introduces explicit correspondences between dual graphs, quotient trees, and cluster pictures for hyperelliptic curves, extending the classification framework beyond elliptic curves.

## Key findings

- Established bijections between combinatorial models
- Classified semistable types with Frobenius action
- Enhanced understanding of local invariants like Tamagawa numbers

## Abstract

In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. This is the higher genus analogue of the distinction between good, split and non-split multiplicative reduction for elliptic curves. Our motivation is to understand $L$-factors, Galois representations, conductors, Tamagawa numbers and other local invariants of hyperelliptic curves and their Jacobians.