Solutions of algebraic differential equations are conjectured to be algebraic exactly when their Taylor coefficients have almost no primes in their denominators, and the conjecture is proved for Picard-Fuchs equations and for isomonodromy equations such as Painlevé VI.
Eisenstein’s last theorem
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Algebraicity and integrality of solutions to differential equations
Solutions of algebraic differential equations are conjectured to be algebraic exactly when their Taylor coefficients have almost no primes in their denominators, and the conjecture is proved for Picard-Fuchs equations and for isomonodromy equations such as Painlevé VI.