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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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Moduli Spaces of Quadratic Maps: Arithmetic and Geometry
Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $$n$$th Gleason polynomial $$G_{n}\in{\mathbb{Q}}[c]$$ comprise the $$0$$-dimensional moduli space of quadratic polynomials with an $$n$$-periodic critical point. $$\operatorname{Per}_{n}(0)$$ is the $$1$$-dimensional moduli space of quadratic rational maps on $${\mathbb{P}}^{1}$$ with an $$n$$-periodic critical point. We show that if $$G_{n}$$ is irreducible over $${\mathbb{Q}}$$, then $$\operatorname{Per}_{n}(0)$$ is irreducible over $${\mathbb{C}}$$. To do this, we exhibit a $${\mathbb{Q}}$$-rational smooth point on a projective completion of $$\operatorname{Per}_{n}(0)$$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $$n$$, $$\operatorname{Per}_{n}(0)$$ itself has no $${\mathbb{Q}}$$-rational points.
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- Award ID(s):
- 1928930
- PAR ID:
- 10529232
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnae126
