Lie invariant Frobenius lifts on linear algebraic groups

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}.