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

Abstract

In this paper we prove that for any connected reductive algebraic group G and a large enough prime $l$, there are continuous homomorphisms $$\mathrm{Gal}(\bar\mathbb Q/\mathbb Q) \to G(\bar\mathbb Q_l)$$ with Zariski-dense image, in particular we produce the first such examples for $S{L}_n,S{p}_{2n},Spi{n}_n,E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a mod-$l$ representation of $\mathrm{Gal}(\bar\mathbb Q/\mathbb Q)$ 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.

arXiv admin note: text overlap with arXiv:1507.01294 by other authors