Minimal Weierstrass equations for genus 2 curves

TL;DR

This paper investigates methods to find minimal Weierstrass equations for genus 2 curves over rings of integers, utilizing reduction theory and Julia invariants, especially when curves have extra automorphisms or are in standard form.

Contribution

It introduces a systematic approach to compute minimal Weierstrass equations for genus 2 curves, highlighting simplifications when automorphisms are present or curves are in standard form.

Findings

Reduction theory and Julia invariants facilitate minimal models.

Extra automorphisms simplify the computation process.

Standard form $y^2=f(x^2)$ with specific conditions is shown to be reduced.

Abstract

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we show that when the curve is given in the standard form $y^2=f(x^2)$, where $f(x)$ is a monic polynomial, $f(0)=1$ which is defined over $\mathcal O_{\mathbb F}$ then this form is reduced.