论文标题
Siegel模块化属的曲折属的合理性$ 2 $和级别$ 3 $
Rationality of twists of the Siegel modular variety of genus $2$ and level $3$
论文作者
论文摘要
令$ \ OVERLINEC：g _ {\ MATHBF {Q}} \ rightarrow {\ rm gsp} _4（\ Mathbf {f} _3）$是连续的galois代表，与环形类似物的角色 - 或与$ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3相关的环形层面的galois代表。 $ a/\ mathbf {q} $。我们证明了Moduli空间$ \ MATHCAL {a} _2（\overlineρ）$主要两极分化的Abelian表面$ B/\ Mathbf {Q} $承认交响素异态$ b [3]即使$ \ mathbf {c} $和$ \ mathbf {q} $通过度量$ 6 $的$ \ mathbf {c} $都合理，而在$ \ mathbf {q} $上都是合理的。
Let $\overlineρ: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ 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 $\mathcal{A}_2(\overlineρ)$ of principally polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism $B[3] \simeq \overlineρ$ of Galois representations is never rational over $\mathbf{Q}$ when $\overlineρ$ is surjective, even though it is both rational over $\mathbf{C}$ and unirational over $\mathbf{Q}$ via a map of degree $6$.
