Cohomologically rigid complex local systems with finite determinant and quasi-unipotent monodromies at infinity are integral

Joint work with Michael Groechenig.

If we dropped ‘Cohomologically’ from the title, this would be a complete (positive) answer to Simpson’s conjecture. 
We prove the conjecture under the (perhaps ?) stronger assumption that the local systems are cohomologically rigid. Our proof can’t be extended to the case of rigid connections, and I shall explain why.
Initially, in the projective case, our proof consisted in going to the de Rham side, showing that the restriction of the connection on the p-adic varieties is a Frobenius isocrystal, descend mod p, considering the ℓ-adic companions (the existence of which has been proven recently by Abe and myself) and going back to the complex numbers by showing that the induced local systems stemming from the companions are still cohomologically rigid. This last step requires the study of weights and the L-functions of the companions.
The new proof is purely Betti-ℓ-adic. 

We also  prove that the  overconvergent isocrystals with a Frobenius structure coming from cohomologically rigid connectionshave a full package of companions. There are also mild consequences for the p-curvature conjecture.