Algebraic solutions of differential equations over ℙ¹ - {0, 1, ∞}

The Grothendieck-Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for almost all p has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on ℙ¹ -{0, 1, ∞}. We prove a variant of this conjecture for ℙ¹ -{0, 1, ∞}, which asserts that if the equation satisfies a certain convergence condition for all p, then its monodromy is trivial. For those p for which the p-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of p-curvatures and certain local monodromy groups. We also prove similar variants of the p-curvature conjecture for an elliptic curve with j-invariant 1728 minus its identity and for ℙ¹ -{±1, ±i, ∞}.