A Universal Genus-Two Curve from Siegel Modular Forms

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.

References in corpus (3)

• K3 surfaces, modular forms, and non-geometric heterotic compactifications
• The Satake sextic in elliptic fibrations on K3
• Rational points in the moduli space of genus two

Cited by in corpus (4)

• Six line configurations and string dualities
• On isogenies among certain abelian surfaces
• On the mixed-twist construction and monodromy of associated Picard-Fuchs systems
• Isogenies of certain K3 surfaces of rank 18