Lie invariant Frobenius lifts

Abstract

We begin with the observation that the p-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are “Lie invariant mod p” in the sense that the “normalized” action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be “infinitesimally deformed” in the sense that the above mod p result has a mod p2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: 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.