A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting

A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type.