Grothendieck–Katz p-curvature conjecture for flat bundles

Establish the equivalence for a flat vector bundle (E, ∇) on a smooth R-scheme S, where R is a finitely generated subalgebra of C, between (i) the existence of a finite étale cover S′ of S_C that trivializes (E, ∇) and (ii) vanishing p-curvature of (E, ∇) modulo p for almost all primes p.

Background

The Grothendieck–Katz p-curvature conjecture is a central open problem connecting arithmetic and differential equations in algebraic geometry. It predicts that a flat connection has finite monodromy after a finite étale cover over C if and only if its p-curvatures vanish for almost all primes.

In this paper the authors prove non-abelian analogues for isomonodromy foliations and use them to deduce new cases of related conjectures; they recall the classical conjecture here as context and motivation.

References

Conjecture [Grothendieck-Katz] Let R\subset \mathbb{C} be a finitely-generated \mathbb{Z}-algebra, and S a smooth R-scheme. Let (\mathscr{E}, \nabla) be a flat vector bundle on S/R. There exists a finite etale cover S' of S_{\mathbb{C}} trivializing (\mathscr{E}, \nabla) if and only if (\mathscr{E}, \nabla) mod p has vanishing p-curvature for almost all primes p.

— p-Curvature and Non-Abelian Cohomology  (2601.07933 - Lam et al., 12 Jan 2026) in Introduction, Subsection “Main results (applications)”, Conjecture [Grothendieck–Katz]

A well-known (and still wildly open) conjecture of Grothendieck and Katz predicts that the existence of algebraic solutions to a differential equation in characteristic $0$ is encoded in the $p$-curvature of its mod $p$ reduction across different primes $p$; we refer the readers to Lecture 2 for a nice introduction to this topic and for a more detailed overview.

— p-curvature in non-commutative Hodge theory and the Kontsevich-Soibelman operad  (2609.26765 - Chen, 22 Sep 2026) in Section 1, subsection “Methods of proof,” subsection “p-curvature”