### Abstract

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 ℙ^{1} -{0, 1, ∞}. We prove a variant of this conjecture for ℙ^{1} -{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 ℙ^{1} -{±1, ±i, ∞}.

Original language | English (US) |
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Pages (from-to) | 1427-1457 |

Number of pages | 31 |

Journal | International Journal of Number Theory |

Volume | 14 |

Issue number | 5 |

DOIs | |

State | Published - Jun 1 2018 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Keywords

- Grothendieck-Katz p-curvature conjecture
- algebraization theorems

## Fingerprint Dive into the research topics of 'Algebraic solutions of differential equations over ℙ<sup>1</sup> - {0, 1, ∞}'. Together they form a unique fingerprint.

## Cite this

^{1}- {0, 1, ∞}.

*International Journal of Number Theory*,

*14*(5), 1427-1457. https://doi.org/10.1142/S1793042118500884