Deformations of Galois representations and exceptional monodromy

TL;DR

This paper constructs new geometric $bla$-adic Galois representations with exceptional algebraic monodromy groups, including types F4 and E6, by extending Ramakrishna's lifting techniques to general reductive groups.

Contribution

It introduces a method to produce geometric Galois representations with exceptional monodromy groups, filling a gap in known examples for types F4 and E6.

Findings

First examples of such representations in types F4 and E6

Extension of Ramakrishna's lifting techniques to reductive groups

Construction of geometric Galois representations with specified monodromy groups

Abstract

For any simple algebraic group $G$ of exceptional type, we construct geometric $\ell$-adic Galois representations with algebraic monodromy group equal to $G$, in particular producing the first such examples in types $\mathrm{F}_4$ and $\mathrm{E}_6$. To do this, we extend to general reductive groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois representations to geometric $\ell$-adic representations.