A Universal Genus-Two Curve from Siegel Modular Forms

TL;DR

This paper constructs a universal genus-two curve over the moduli space using Siegel modular forms, providing explicit equations and conditions for defining the curve over the field of moduli, extending prior results.

Contribution

It introduces a universal equation for genus-two curves over the moduli space using Siegel modular forms, with explicit conditions for the field of definition.

Findings

Explicit universal equation for genus-two curves over moduli space

Conditions for defining the curve over the field of moduli

Discovery of symmetries in the Weierstrass equation

Abstract

Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{\alpha, \beta}$ defined over $K(\alpha, \beta)$, corresponding to $\mathfrak p$, where $\alpha $ and $\beta$ satisfy a quadratic $\alpha^2+ b \beta^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{\alpha, \beta}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(\alpha, \beta)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{\alpha, \beta}$. This extends previous work of Mestre and others.