Given a genus two curve
$X:{y}^{2}={x}^{5}+a{x}^{3}+b{x}^{2}+cx+d$,
we give an explicit parametrization of all other such curves
$Y$
with a specified symplectic isomorphism on three-torsion of Jacobians
$Jac\left(X\right)\left[3\right]\cong Jac\left(Y\right)\left[3\right]$.
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
${C}_{3}\times {Sp}_{4}\left({\mathbb{\mathbb{F}}}_{3}\right)$. 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
${Sp}_{2}\left({\mathbb{\mathbb{F}}}_{3}\right)={SL}_{2}\left({\mathbb{\mathbb{F}}}_{3}\right)$ and
$3$-torsion
on elliptic curves.

Keywords

abelian surfaces, three torsion, Galois representations