A Strong Splitting of the Frobenius Morphism on the Algebra of Distributions of $SL_2$

TL;DR

This paper constructs a splitting of the Frobenius map on the algebra of distributions of SL_2 over finite fields for primes p ≥ 3, revealing new algebraic structure related to congruences modulo p^3.

Contribution

It provides an explicit section of the Frobenius map on the distribution algebra of SL_2 over finite fields for primes p ≥ 3, using novel congruence techniques.

Findings

Constructed a section of the Frobenius map for p ≥ 3

Established a congruence modulo p^3 related to binomial coefficients

Enhanced understanding of algebraic structures in positive characteristic

Abstract

Let $p$ be a prime number, and 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 section of this map, whenever $p\geq 3$. The main ingredient of this construction is a certain congruence modulo $p^3$, reminiscent of the congruence $\binom{np}{p}\equiv n\mod p^3$.