A strong splitting of the Frobenius morphism on the algebra of distributions of $SL_2$

Let $p$ be a prime number. Let $Dist(SL_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $SL_2$ over $\mathbb{F}_p$. The Frobenius map $Fr : SL_2 \to SL_2$ induces a map $Fr : Dist(SL_2) \to Dist(SL_2)$ which is, in particular, a surjective algebra homomorphism. In this note, we construct a (unital) section of this map, whenever $p \geq 3$. The main ingredient of this construction is a certain congruence $\operatorname{modulo} p^3$, reminiscent of the congruence${\bigl ( \begin{smallmatrix} np \\ p \end{smallmatrix} \bigr )} \equiv n \operatorname{mod} p^3$.

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Received 3 February 2018

Accepted 30 September 2018

Published 6 March 2020