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.

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.