Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic ?-adic representations of $textup {Gal}(overline {mathbb{Q}}/mathbb{Q})$ into exceptional groups of type E ₇,E ₈ and G ₂ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(mathbb{F}_{ell})$ are Galois groups over $mathbb{Q}$ for large enough primes ?.