ETD

• non-abelian cohomology,moduli stacks,p-curvature

This thesis presents some recent developments on the p-curvature conjecture in the context of connections arising from cohomology, which generalize to a non-abelian setting the classical result for the Gauss–Manin connection. This connection expresses the variation of cohomology classes of the fibers of a smooth projective family, and Katz’s formula is a central result relating the action of the Gauss–Manin connection on the Hodge filtration in cohomology, defined in characteristic zero, to the arithmetic properties that appear after reduction to positive characteristic, namely the action of the p-curvature on the conjugate filtration.
In the first part of the work, we present the classical proof of this formula, recalling the necessary definitions and the main constructions concerning algebraic de Rham cohomology, the Gauss–Manin connection, the Kodaira–Spencer map, and p-curvature. In the second part, we explain how one passes from these classical objects to their counterparts in non-abelian cohomology, where cohomology groups are replaced by moduli spaces of flat vector bundles. We present the non-abelian form of the p-curvature conjecture, in the setting of foliations on moduli stacks, and we discuss Lam and Litt’s proof of Katz’s formula in the non-abelian setting, concerning the isomonodromy foliation. The aim of the thesis is to explain how the classical technique leads to the contemporary reformulation in terms of moduli spaces of local systems.