Author: Eiko

Tags: p-adic, differential equations, p-adic curvature

Time: 2024-11-27 08:02:44 - 2024-11-27 08:02:44 (UTC)

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

p-adic Curvature

Let $R$ be an ${\mathbb{F}}_p$-algebra and $X$ a smooth $R$-scheme, and a vector bundle (locally free sheaf) with integrable connection $(M,\mathrm{\nabla })$.

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

$$D^p(fg)=\sum _{i=0}^p(\genfrac{}{}{0ex}{}{p}{i})D^i(f)D^{p-i}(g)=f{D}^p(g)+g{D}^p(f).$$

Therefore, just like we have integrable requirement that connection commutes with Lie bracket ${\mathrm{\nabla }}_{[D_1,D_2]}=[{\mathrm{\nabla }}_{D_1},{\mathrm{\nabla }}_{D_2}]$, in characteristic $p$ we can have another requirement that connection commutes with $p$-th power,

$${\mathrm{\nabla }}_{D^p}=({\mathrm{\nabla }}_D{)}^p\in {\mathrm{End}}_R(M).$$

The failure of this commutativity is measured by the p-adic curvature ${\psi }_p(D)$

$${\psi }_p(D):=({\mathrm{\nabla }}_D{)}^p-{\mathrm{\nabla }}_{D^p}\in {\mathrm{End}}_R(M).$$

• $D\mapsto {\psi }_p(D)$ is $p$-linear, this means

  $${\psi }_p(a_1{D}_1+a_2{D}_2)=a_1^p{\psi }_p(D_1)+a_2^p{\psi }_p(D_2)$$

Theorem (Cartier) $p$-adic curvature is the obstruction to the existence of enough horizontal sections.