A Universal Genus-Two Curve from Siegel Modular Forms

Abstract

Let p be any point in the moduli space of genus-two curves M2 and K its field of moduli. We provide a universal equation of a genus-two curve Cα,β defined over K(α,β), corresponding to p, where α and β satisfy a quadratic α²+bβ²=c such that b and c are given in terms of ratios of Siegel modular forms. The curve Cα,β is defined over the field of moduli K if and only if the quadratic has a K-rational point (α,β). We discover some interesting symmetries of the Weierstrass equation of Cα,β. This extends previous work of Mestre and others.