$p$-integrability

TL;DR

This paper establishes the equivalence between the Grothendieck-Katz $p$-curvature Conjecture and Conjecture F in Ekedahl-Shepherd-Barron-Taylor, linking two important conjectures in algebraic geometry.

Contribution

It proves the equivalence of the $p$-curvature Conjecture with Conjecture F, clarifying their relationship in the context of foliations and integrable connections.

Findings

Conjecture F implies the $p$-curvature Conjecture.

The $p$-curvature Conjecture implies Conjecture F for certain foliations.

The equivalence links two major conjectures in algebraic geometry.

Abstract

In this short note, we prove the equivalence of Grothendieck-Katz $p$-curvature Conjecture with Conjecture F in Ekedahl-Shepherd-Barron-Taylor. More precisely, we show that Conjecture F implies the $p$-curvature conjecture, and that the $p$-curvature Conjecture implies Conjecture F for the foliation attached to a vector bundle with integrable connection.