Global Weierstrass equations of hyperelliptic curves

TL;DR

This paper establishes conditions under which hyperelliptic curves over number fields can be globally represented by Weierstrass equations, especially when they have good reduction and certain class number properties.

Contribution

It provides necessary and sufficient conditions for hyperelliptic curves to have global Weierstrass models, confirming a conjecture for cases with prime-to-2(2g+1) class number.

Findings

Hyperelliptic curves with good reduction can be globally modeled by Weierstrass equations under specific class number conditions.

Confirmed Sadek's conjecture for cases where the class number is prime to 2(2g+1).

Identified criteria linking local models to global Weierstrass equations.

Abstract

Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathscr{C}$ of $C$ over the ring of integers ${\mathcal O}_K$ (i.e. the hyperelliptic involution of $C$ extends to $\mathscr{C}$ and the quotient is a smooth model of ${\mathbb P}^1_K$ over ${\mathcal O}_K$), we give necessary and sometimes sufficient conditions for $\mathscr{C}$ to be defined by a global Weierstrass equation. In particular, if $C$ has everywhere good reduction, we prove that it is defined by a global Weierstrass equation with invertible discriminant if the class number $h_K$ is prime to $2(2g+1)$, confirming a conjecture of M. Sadek.