Let (rho) over bar : G(Q) -> GSp(4)(F-3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider (rho) over bar to be the Galois representation associated to the 3-torsion of a principally polarized abelian surface A/Q. We prove that the moduli space A(2)((rho) over bar) of principally polarized abelian surfaces B/Q admitting a symplectic isomorphism B3 similar or equal to (rho) over bar of Galois representations is never rational over Q when (rho) over bar is surjective, even though it is both rational over C and unirational over Q via a map of degree 6.
展开▼
