A Universal Genus-Two Curve from Siegel Modular Forms

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 $\mathcal C_{α, β}$ defined over $K(α, β)$, corresponding to $\mathfrak p$, where $α$ and $β$ satisfy a quadratic $α^2+ b β^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{α, β}$ 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 $\mathcal C_{α, β}$. This extends previous work of Mestre and others.

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