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

TL;DR

This paper constructs new examples of Galois representations with Zariski-dense images in various algebraic groups for large primes, advancing understanding of algebraic monodromy groups of l-adic Galois representations.

Contribution

It provides the first known examples of Galois representations with Zariski-dense images in groups like SL_n, Sp_{2n}, and E_7^{sc} for large primes, using deformation techniques.

Findings

Constructed Galois representations with dense images in classical and exceptional groups.

Extended deformation methods to produce characteristic zero representations from mod-l data.

Established existence results for monodromy groups of l-adic representations for large primes.

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 $SL_n, Sp_{2n}, Spin_n, E_6^{sc}$ 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.