Abelian surfaces with fixed $3$-torsion

Abstract

Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong \mbox{Jac}(Y)[3]$. It is known that under certain conditions modularity of $X$ implies modularity of infinitely many of the $Y$, and we explain how our formulas render this transfer of modularity explicit. Our method centers on the invariant theory of the complex reflection group $C_3 \times \operatorname{Sp}_4(\mathbf{F}_3)$. We discuss other examples where complex reflection groups are related to moduli spaces of curves, and in particular motivate our main computation with an exposition of the simpler case of the group $\operatorname{Sp}_2(\mathbf{F}_3) = \mathrm{SL}_2(\mathbf{F}_3)$ and $3$-torsion on elliptic curves.