Algebraicity and integrality of solutions to differential equations
Yeuk Hay Joshua Lam, Daniel Litt
TL;DR
The paper introduces a unifying conjecture characterizing when solutions to algebraic (possibly nonlinear) differential equations are algebraic, by linking the denominators of Taylor-series coefficients to primes across almost all reductions. It proves this conjecture in key nonlinear settings obtained from isomonodromy (including Painlevé VI and Schlesinger systems) at Picard–Fuchs initial conditions, and in linear cases via Picard–Fuchs equations and cycle-class initial data. The approach blends arithmetic geometry (p-curvature, crystalline cohomology, Fontaine–Laffaille theory) with non-abelian Hodge theory (Ogus–Vologodsky theory, Higgs–de Rham flow) to extend the Hodge filtration along isomonodromic deformations and deduce algebraicity of leaves. The results illuminate a nonlinear analogue of Katz’s p-curvature theorem, and imply algebro-geometric consequences and connections to the relative Fontaine–Mazur conjecture and variational motivic principles. Collectively, the work offers a concrete framework for predicting and proving when algebraic leaves and algebraic solutions arise in families of differential equations, with broad implications for the arithmetic of differential equations and the geometry of moduli of flat bundles.
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 Painlevé VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs.