Algebraic monodromy groups of $l$-adic representations of Gal$(\overline{\mathbb{Q}} /\mathbb{Q})$

Abstract

In this paper we prove that for any connected reductive algebraic group $G$ and a large enough prime $l$, there are continuous homomorphisms

$$Gal\left(\bar{\mathbb{Q}}∕\mathbb{Q}\right)\to G\left({\bar{\mathbb{Q}}}_l\right)$$

with Zariski-dense image, in particular we produce the first such examples for ${SL}_n,{Sp}_{2n},{Spin}_n,E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a mod-$l$ representation of $Gal\left(\bar{\mathbb{Q}}∕\mathbb{Q}\right)$ related to the Weyl group of $G$ and use a variation of Stefan Patrikis’ generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.