A Strong Splitting of the Frobenius Morphism on the Algebra of Distributions of $S{L}_2$

Abstract

Let $p$ be a prime number, and let $Dist(S{L}_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $S{L}_2$ over ${\mathbb{F}}_p$. The Frobenius map $Fr:S{L}_2\to S{L}_2$ induces a map $Fr:Dist(S{L}_2)\to Dist(S{L}_2)$ which is in particular a surjective algebra homomorphism. In this note, we construct a section of this map, whenever $p\ge 3$. The main ingredient of this construction is a certain congruence modulo $p^3$, reminiscent of the congruence $\left(\genfrac{}{}{0ex}{}{np}{p}\right)\equiv n\mathrm{mod}{p}^3$.