Let X be a smooth irreducible variety over a field of characteristic zero, and let F⊆ T_X be an algebraic foliation of rank r admitting reductions modulo almost all primes of good reduction. The conjecture asserts that if the p-curvature of the reduction F_p vanishes for almost all primes p, equivalently if F_p is closed under the p-th power operation on vector fields, then F is algebraically integrable: the field of rational first integrals f∈ K(X):D(f)=0 for every local section D of F has transcendence degree dim X-r over the base field.

Generalized Grothendieck–Katz p-curvature conjecture

Let $X$ be a smooth irreducible variety over a field of characteristic zero, and let $\mathcal{F}\subseteq T_X$ be an algebraic foliation of rank $r$ admitting reductions modulo almost all primes of good reduction. The conjecture asserts that if the $p$ -curvature of the reduction $\mathcal{F}_p$ vanishes for almost all primes $p$ , equivalently if $\mathcal{F}_p$ is closed under the $p$ -th power operation on vector fields, then $\mathcal{F}$ is algebraically integrable: the field of rational first integrals $\{f\in K(X):D(f)=0\text{ for every local section }D\text{ of }\mathcal{F}\}$ has transcendence degree $\dim X-r$ over the base field.

References

Primary source

arXiv

Additional references

On Algebraic Integrability of Vector Fields with Rational Function Coefficients that Separate Variables — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new September 2026 result settles a structured special case, but the full conjecture remains open.

The conjecture, originating with Alexander Grothendieck in the late 1960s and later formulated precisely by Katz, predicts that vanishing $p$ -curvature for almost all primes forces sufficiently many algebraic solutions. Its generalized form concerns algebraic integrability of foliations; the full conjecture remains unresolved.

Known results

Katz proved the conjecture for Picard–Fuchs equations.
Farb and Kisin treated a broad class of cases.
Rank-two connections on generic curves were handled in a theorem covering arbitrary genus and punctures.
A 2025 result established equivalence with the Ekedahl–Shepherd-Barron–Taylor conjecture in the stated setting.

September 2026 special-case verification

A newly reported preprint proves a necessary-and-sufficient integrability criterion for rational vector fields with separated variables, gives explicit integrable fields and first integrals, and claims verification of the generalized conjecture for the resulting foliations. This is substantial progress for that structured family, not a solution of the general conjecture, and the claim remains unverified.

Current status (as of September 2026): the separated-variable family is claimed to be completely treated, while the generalized Grothendieck–Katz conjecture in full remains open.

Sources

No solutions have been posted yet.