Rigid $G$-connections and nilpotency of $p$-curvatures

TL;DR

This paper extends the nilpotency result of $p$-curvatures from rigid irreducible connections to integrable $G$-connections, supporting the broader understanding of their motivic nature.

Contribution

It generalizes the nilpotency of $p$-curvatures to integrable $G$-connections, broadening the scope beyond previous cases.

Findings

Nilpotency of $p$-curvatures for integrable $G$-connections

Extension of Esnault and Groechenig's results

Supports conjectures on motivicity of rigid connections

Abstract

Motivated by Simpson's conjecture on the motivicity of rigid irreducible connections, Esnault and Groechenig demonstrated that the mod-$p$ reductions of such connections on smooth projective varieties have nilpotent $p$-curvatures. In this paper, we extend their result to integrable $G$-connections.