Minimal Weierstrass equations for genus 2 curves

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.