The p-curvature conjecture and monodromy about simple closed loops

Abstract

The Grothendieck-Katz p-curvature curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its p-curvature vanishes modulo p, for almost all primes p. We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.