Maximality of Galois actions for abelian and hyperkahler varieties

Abstract

Let $\{{\rho }_{\ell }{\}}_{\ell }$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete smooth variety $X$ defined over a number field $K$. Let ${\Gamma }_{\ell }$ and ${\mathbf{G}}_{\ell }$ be respectively the image and the algebraic monodromy group of ${\rho }_{\ell }$. We prove that the reductive quotient of ${\mathbf{G}}_{\ell }^{\circ }$ is unramified over every degree 12 totally ramified extension of ${\mathbb{Q}}_{\ell }$ for all sufficiently large $\ell$. We give a necessary and sufficient condition $(\ast )$ on $\{{\rho }_{\ell }{\}}_{\ell }$ such that for all sufficiently large $\ell$, the subgroup ${\Gamma }_{\ell }$ is in some sense maximal compact in ${\mathbf{G}}_{\ell }({\mathbb{Q}}_{\ell })$. This is used to deduce Galois maximality results for $\ell$-adic representations arising from abelian varieties (for all $i$) and hyperkähler varieties ($i=2$) defined over finitely generated fields over $\mathbb{Q}$.

Accepted in Duke Math J

References in corpus (3)

• Surjectivity Criteria for p-adic Representations, Part I
• Adelic openness without the Mumford-Tate conjecture
• l-independence for Compatible Systems of (mod l) Representations

Cited by in corpus (1)

• On the Frobenius fields of abelian varieties over number fields