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Free, publicly-accessible full text available May 1, 2024
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Let ρ ¯ : G Q → GSp 4 ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the 3 3 -torsion of a principally polarized abelian surface A / Q A/\mathbf {Q} . We prove that the moduli space A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces B / Q B/\mathbf {Q} admitting a symplectic isomorphism B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over Q \mathbf {Q} when ρ ¯ \overline {\rho } is surjective, even though it is both rational over C \mathbf {C} and unirational over Q \mathbf {Q} via a map of degree 6 6 .more » « less
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Abstract We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$ A over ${\mathbf {Q}}$ Q with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.more » « less