  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
$\mathbb{P}^1-\{\pm 1,\pm i,\infty\}$.
