The p-curvature conjecture for the non-abelian Gauss-Manin connection

Abstract

Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz in [9], the p-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known. In [8], Katz proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. This dissertation addresses the non-abelian analogue of Katz’ theorem, in the sense of Simpson’s non-abelian Hodge theory, surveyed in [15], and later developed in characteristic p in [11]. Specifically, there is a canonical non-abelian Gauss-Manin connection on MdR, the stack of vector bundles with integrable connection, which is the appropriate definition of non-abelian de Rham cohomology. In this dissertation, I introduce this connection and its p-curvature; this requires the generalization of the p-curvature conjecture due to Bost, Ekedahl and Shepherd-Barron. Then, I prove that the analogue of the main technical result of Katz’ theorem holds for this connection.