Grothendieck–Katz p-curvature conjecture

The Grothendieck–Katz p-curvature conjecture predicts that a linear differential equation over a number field has finite monodromy if its p-curvature vanishes modulo almost every prime of good reduction.

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p-curvature: An endomorphism-valued invariant of a connection in characteristic p, defined by comparing its p-th covariant derivative with the derivative associated to a p-th power vector field. Its vanishing at almost all reductions is the conjecture’s arithmetic test.
Number field: A finite-degree field extension of the rational numbers. The equation’s coefficients lie in this arithmetic setting before reduction modulo primes.
Alexander Grothendieck: A French mathematician whose work transformed algebraic geometry and introduced foundational ideas in modern arithmetic geometry. His proposed link between p-curvature and finite monodromy gives the conjecture its first name.
Gauss hypergeometric equation: A second-order linear differential equation with three regular singular points and parameters controlling its solutions. Its parameter families provide explicit settings for studying finite monodromy and p-curvature.
Nilpotent p-curvature conjecture: A related conjecture predicting quasi-unipotent monodromy when p-curvature is nilpotent for almost all primes. It replaces vanishing by nilpotence and predicts a weaker restriction on monodromy.
Monodromy representation: A representation of the fundamental group obtained by analytically continuing solutions of a differential equation around loops. Finite monodromy is the predicted consequence of vanishing p-curvature.
Reduction modulo a prime: The passage from arithmetic data over a number field to data over a finite residue field at a prime. It turns the original connection into one where p-curvature is defined.
Nicholas Katz: An American mathematician known for work in algebraic geometry, number theory, and differential equations. His study of the p-curvature criterion developed and publicized the conjecture.
Schwarz list: A classification of parameter choices for the Gauss hypergeometric equation that yield algebraic solutions. It gives concrete finite-monodromy cases against which arithmetic criteria can be tested.
Global nilpotence: The property that a differential equation has nilpotent p-curvature for almost every prime of good reduction. It is a broader arithmetic condition whose consequences are studied alongside the vanishing criterion.