Algebraic solutions of differential equations over the projective line minus three points

TL;DR

This paper proves a variant of the Grothendieck--Katz $p$-curvature conjecture for differential equations on the projective line minus three points, linking $p$-curvature vanishing to trivial monodromy and describing the differential Galois group.

Contribution

It establishes a new version of the $p$-curvature conjecture for specific algebraic curves, connecting $p$-curvature conditions to monodromy and Galois groups.

Findings

Proves a variant of the $p$-curvature conjecture for $ ext{P}^1 - ext{points}$.

Describes the differential Galois group in terms of $p$-curvatures.

Extends results to elliptic curves and other punctured projective lines.

Abstract

The Grothendieck--Katz $p$-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo $p$ has vanishing $p$-curvatures for {\em almost all} $p,$ has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on $\mathbb{P}^{1}-\{0,1,\infty\}.$ We prove a variant of this conjecture for $\mathbb{P}^{1}-\{0,1,\infty\},$ which asserts that if the equation satisfies a certain convergence condition for {\em 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 the elliptic curve with $j$-invariant $1728$ minus its identity and for $\mathbb{P}^1-\{\pm 1,\pm i,\infty\}$.