Rationality of twists of the Siegel modular variety of genus $2$ and level $3$

Abstract:

Let $\overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3)$ be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider $\overline {\rho }$ to be the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf {Q}$. We prove that the moduli space $\mathcal {A}_2(\overline {\rho })$ of principally polarized abelian surfaces $B/\mathbf {Q}$ admitting a symplectic isomorphism $B[3] \simeq \overline {\rho }$ of Galois representations is never rational over $\mathbf {Q}$ when $\overline {\rho }$ is surjective, even though it is both rational over $\mathbf {C}$ and unirational over $\mathbf {Q}$ via a map of degree $6$.