Algebraicity and integrality of solutions to differential equations

TL;DR

This paper proposes a conjecture linking algebraic solutions of differential equations to prime factors in their Taylor coefficients, extending known conjectures and proving it in various algebraic and geometric contexts.

Contribution

It formulates a new conjecture on algebraic solutions of differential equations and proves it for many linear and non-linear cases, including Picard-Fuchs and isomonodromy equations.

Findings

Conjecture relates algebraic solutions to prime factors in coefficients.

Proved the conjecture for many linear differential equations.

Established results for non-linear equations like Painlevé VI.

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.