Symmetry, Integrability and Geometry: Methods and Applications
National Academy of Science of Ukraine
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.
Malmendier, Andreas and Shaska, Tony, "A Universal Genus-Two Curve from Siegel Modular Forms" (2017). Mathematics and Statistics Faculty Publications. Paper 221.