p-Curvature and Non-Abelian Cohomology

TL;DR

This paper extends Katz's p-curvature conjecture to a non-abelian setting, showing that vanishing p-curvature for isomonodromy foliations implies finite monodromy action on integral characters, with implications for related conjectures.

Contribution

It introduces a non-abelian analogue of Katz's p-curvature conjecture, applying to isomonodromy foliations and integral characters, and proves new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture.

Findings

Vanishing p-curvature implies finite monodromy action on integral characters.

Established a non-abelian version of Katz's formula.

Proved new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture.

Abstract

Let $X\to S$ be a smooth projective morphism. Katz proved the Grothendieck-Katz $p$-curvature conjecture for the Gauss-Manin connection on the $i$-th cohomology of $X/S$: if its $p$-curvature vanishes mod $p$ for infinitely many $p$, then the action of $\pi_1(S,s)$ on $H^i(X_s, \mathbb{Z})$ factors through a finite group. We prove a non-abelian analogue of this statement: if the $p$-curvature of the isomonodromy foliation on the moduli of flat bundles of rank $r$ on $X/S$ vanishes mod $p$ for infinitely many $p$, then the action of $\pi_1(S,s)$ on the rank $r$ integral characters of $\pi_1(X_s)$ factors through a finite group. We deduce many new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture. The proofs rely on a non-abelian version of Katz's formula, and a non-abelian version of the Hodge index theorem.