Lie invariant Frobenius lifts on linear algebraic groups

TL;DR

The paper investigates the existence of Lie invariant Frobenius lifts on the $p$-adic completions of linear algebraic groups over number fields, showing such lifts are rare for non-torus groups at almost all primes.

Contribution

It establishes that non-torus linear algebraic groups over number fields generally lack Lie invariant Frobenius lifts in their $p$-adic completions for all but finitely many primes.

Findings

Frobenius lifts are not Lie invariant mod p for most primes p.

Non-torus groups rarely admit Lie invariant Frobenius lifts.

Contrast with elliptic curves where such lifts are more common.

Abstract

We show that if $G$ is a linear algebraic group over a number field and if $G$ is not a torus then for all but finitely many primes $p$ the $p$-adic completion of $G$ does not possess a Frobenius lift that is "Lie invariant mod $p$" (in the sense of \cite{alie1}). This is in contrast with the situation of elliptic curves studied in \cite{alie1}.