Moduli of flat connections in positive characteristic

Exploiting the description of rings of differential operators as Azumaya algebras on cotangent bundles, we show that the moduli stack of flat connections on a curve (allowed to acquire orbifold points), defined over an algebraically closed field of positive characteristic is étale locally equivalent to a moduli stack of Higgs bundles over the Hitchin base. We then study the interplay with stability and generalise a result of Laszlo–Pauly, concerning properness of the Hitchin map. Using Arinkin’s autoduality of compactified Jacobians we extend the main result of Bezrukavnikov–Braverman, the Langlands correspondence for $D$-modules in positive characteristic for smooth spectral curves, to the locus of integral spectral curves. We prove that Arinkin’s autoduality satisfies an analogue of the Hecke eigenproperty.

Full Text (PDF format)

Accepted 29 February 2016

Published 16 September 2016