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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 .
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This content will become publicly available on December 1, 2025
The eighth moment of Dirichlet L-functions II
We prove an asymptotic formula for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q, over all moduli q≤Q and with a short average on the critical line. Previously the same result was shown conditionally on the Generalized Riemann Hypothesis by the first two authors.
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- PAR ID:
- 10596298
- Publisher / Repository:
- Duke University Press
- Date Published:
- Journal Name:
- Duke Mathematical Journal
- Volume:
- 173
- Issue:
- 18
- ISSN:
- 0012-7094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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