On representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $\mathrm{Aut}(\hat{F}_d)$

Frauke M. Bleher, Ted Chinburg, Alexander Lubotzky

Published 2020-04-19Version 1

It is well known that $G=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the absolute Galois group of the field of rational numbers $\mathbb{Q}$, is embedded as an (infinite index) subgroup in $A=\mathrm{Aut}(\hat{F}_2)$, the automorphism group of the free profinite group on 2 generators $\hat{F}_2$. In this note, we prove a "super-rigidity" result for some classical Galois representations of $G$. Namely, let $X$ be a smooth projective irreducible curve over $\overline{\mathbb{Q}}$, realized as a cover $\lambda:X\to \mathbb{P}^1_{\overline{\mathbb{Q}}}$ that is unramified outside $\{0,1,\infty\}$, and let $\ell$ is a prime number. Then the action of a natural finite index subgroup of $G$ on the $\ell$-adic Tate module of the generalized Jacobian of $X$ with respect to the ramification locus of $\lambda$, can be extended, up to a finite index subgroup, to an action of a finite index subgroup of $A$. Moreover, this holds even for the product of the Tate modules of the generalized Jacobian of $X$ over all primes $\ell$.