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

Yunqing Tang

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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, ∞}.

Original languageEnglish (US)
Pages (from-to)1427-1457
Number of pages31
JournalInternational Journal of Number Theory
Issue number5
StatePublished - Jun 1 2018

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • Grothendieck-Katz p-curvature conjecture
  • algebraization theorems


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