Arithmetic of hyperelliptic curves over local fields

TL;DR

This paper introduces a combinatorial 'cluster picture' to analyze hyperelliptic curves over local fields, linking root p-adic distances to key arithmetic invariants and geometric properties.

Contribution

It presents a novel combinatorial framework that encodes Galois representations, conductors, and semistability of hyperelliptic curves from root configurations.

Findings

Cluster picture determines Galois representation and conductor

It predicts semistability and special fiber structure

Provides explicit formulas for invariants like discriminant

Abstract

We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that this elementary combinatorial object encodes the curve's Galois representation, conductor, whether the curve is semistable, and if so, the special fibre of its minimal regular model, the discriminant of its minimal Weierstrass equation and other invariants.