Let $\overline{ \rho}: G_{ \mathbf{ Q}} \rightarrow { \rm GSp}_4(\mathbf{ F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, 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 $6$.
Categories: math.NT