#### Vol. 13, No. 6, 2019

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

### Shiang Tang

Vol. 13 (2019), No. 6, 1353–1394
DOI: 10.2140/ant.2019.13.1353
##### Abstract

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

$Gal\left(\stackrel{̄}{ℚ}∕ℚ\right)\to G\left({\stackrel{̄}{ℚ}}_{l}\right)$

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 $Gal\left(\stackrel{̄}{ℚ}∕ℚ\right)$ 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.

##### Keywords
Galois representation, Galois deformation theory
Primary: 11F80