We show relations between rigidity of connections in characteristic 0 and nilpotency of their $p$-curvatures (a consequence of a conjecture by Simpson and of a generalization of Grothendieck's $p$-curvature conjecture).
Work in progress with Michael Groechenig.

Joint with Moritz Kerz. We study arithmetic subvarieties of the character variety of normal complex varieties defined over a field of finite type.