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We study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when   $\textbf {k}$   is a number field, a finite   $p$  -subgroup of the group of polynomial automorphisms of   $\textbf {k}^d$   is isomorphic to a subgroup of   $\textrm {GL}_d(\textbf {k})$  . In the case of infinite nilpotent group actions, we show that a finitely generated nilpotent group   $H$   acting on a complex quasiprojective variety   $X$   of dimension   $d$   can be embedded in a   $p$  -adic Lie group that acts faithfully and analytically on   $\textbf {Z}_p^d$  . As a consequence, we show that the virtual derived length of   $H$   is at most the dimension of   $X$  .

Actions of Nilpotent Groups on Complex Algebraic Varieties