Number Theory Seminar: The $p$-curvature conjecture for privileged local systems (Joint with Michael Groechenig)

We define the notion of privileged local systems on a smooth complex quasi-projective variety. It generalizes the notion of physically rigid local systems on an open of the projective line, due to Nick Katz. We prove the $p$-curvature conjecture for  privileged local systems in any dimension. In particular it yields a new proof  of Katz’ theorem, not based on classification,  that the rigid local systems satisfy the $p$-curvature conjecture  on an open of the projective line. Our proof raises natural questions in general which, if time permits, I hope to discuss as well.