Grothendieck–Katz p-curvature conjecture for flat bundles

Establish the equivalence for a flat vector bundle (E, ∇) on a smooth R-scheme S, where R is a finitely generated subalgebra of C, between (i) the existence of a finite étale cover S′ of S_C that trivializes (E, ∇) and (ii) vanishing p-curvature of (E, ∇) modulo p for almost all primes p.

Background

The Grothendieck–Katz p-curvature conjecture is a central open problem connecting arithmetic and differential equations in algebraic geometry. It predicts that a flat connection has finite monodromy after a finite étale cover over C if and only if its p-curvatures vanish for almost all primes.

In this paper the authors prove non-abelian analogues for isomonodromy foliations and use them to deduce new cases of related conjectures; they recall the classical conjecture here as context and motivation.