Models of Bihyperelliptic Curves

TL;DR

This paper provides an explicit combinatorial method to describe the minimal regular model of bihyperelliptic curves with semistable reduction over local fields, extending cluster picture techniques.

Contribution

It generalizes the cluster picture approach to bihyperelliptic curves, enabling explicit determination of their minimal regular models and Frobenius action.

Findings

Introduces chromatic cluster picture for bihyperelliptic curves

Determines minimal regular model from combinatorial data

Describes Frobenius action explicitly

Abstract

We give an explicit description of the minimal regular model of bihyperelliptic curves with semistable reduction over a local field of odd residue characteristic. We do this using a generalisation of the cluster picture; a completely combinatorial object attached to a hyperelliptic curve $y^2 = f(x)$ over $K$ which contains the data of the $p$-adic distances between the roots of $f$. We add some information, resulting in a chromatic cluster picture, and show that this determines the minimal regular model of $Y$ with the action of Frobenius.