Motives with exceptional Galois groups and the inverse Galois problem

Details Motives with exceptional Galois groups and the inverse Galois problem Motives with exceptional Galois groups and the inverse Galois problem

2011-12-12

arxiv.org/abs/1112.2434v1

Mathematics

Number Theory

40 pages

Scientific paper

We construct motivic $\ell$-adic representations of $\GQ$ 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 uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\FF_{\ell})$ are Galois groups over $\QQ$ for large enough primes $\ell$.

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