Reference: A conjecture in the arithmetic theory of differntial equations by Katz
Might be useful:
Nilpotent connections and the monodromy theorem
Algebraic solutions of differntial equations
Let \(R\) be an \(\mathbb{F}_p\)-algebra and \(X\) a smooth \(R\)-scheme, and a vector bundle (locally free sheaf) with integrable connection \((M,\nabla)\).
Recall that the derivation \(\mathrm{Der}_R(\mathcal{O}_X)\) is a sheaf and has Lie-algebra structure because vector fields (or derivations) are closed under the Lie bracket. When we are in characteristic \(p\) however, there is a new operation that makes the derivation closed which is the \(p\)-th power.
\[ D \mapsto D^p \]
as we can see by Leibniz rule
\[ D^p(fg) = \sum_{i=0}^p \binom{p}{i} D^i(f) D^{p-i}(g) = f D^p(g) + g D^p(f). \]
Therefore, just like we have integrable requirement that connection commutes with Lie bracket \(\nabla_{[D_1,D_2]} = [\nabla_{D_1},\nabla_{D_2}]\), in characteristic \(p\) we can have another requirement that connection commutes with \(p\)-th power,
\[ \nabla_{D^p} = (\nabla_D)^p \in \mathrm{End}_R(M). \]
The failure of this commutativity is measured by the p-adic curvature \(\psi_p(D)\)
\[\psi_p(D) := (\nabla_D)^p - \nabla_{D^p} \in \mathrm{End}_R(M).\]
\(D\mapsto \psi_p(D)\) is \(p\)-linear, this means
\[\psi_p(a_1D_1 + a_2D_2) = a_1^p\psi_p(D_1) + a_2^p\psi_p(D_2)\]
Theorem (Cartier) \(p\)-adic curvature is the obstruction to the existence of enough horizontal sections.