Mathematical Research Letters
Volume 23 (2016)
Moduli of flat connections in positive characteristic
Pages: 989 – 1047
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.