Algebraicity and integrality of solutions to differential equations

Abstract

We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point. For linear differential equations, this conjecture is a strengthening of the Grothendieck-Katz $p$-curvature conjecture. We prove the conjecture for many differential equations and initial conditions of algebro-geometric interest. For linear differential equations, we prove it for Picard-Fuchs equations at initial conditions corresponding to cycle classes, among other cases. For non-linear differential equations, we prove it for isomonodromy differential equations, such as the Painlev\'e VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs.