  We study nilpotent groups acting faithfully on complex algebraic varieties.
We use a method of base change. For finite p-groups, we go from $k$, a number
field, to a finite field in order to use counting lemmas. We show that a finite
$p$-group of polynomial automorphisms of $k^d$ is isomorphic to a subgroup of
GL$_d(k)$. For infinite groups, we go from $\mathbb{C}$ to $\mathbb{Z}_p$ and
use p-adic analytic tools and the theory of p-adic Lie groups. We show that a
finitely generated nilpotent group $H$ acting faithfully on a complex
quasiprojective variety $X$ of dimension $d$ can be embedded into a $p$-adic
Lie group acting faithfully and analytically on $\mathbb{Z}_p^d$; we deduce
that $d$ is larger than the virtual derived length of $H$.
