2022/09/20 by Taïbi, Olivier
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2209.09744
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 ℚ. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over ℚℓ with coefficients in a complex of continuous ℓ-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous ℓ-adic representations''. This is motivated by Pink's formula computing the derived direct image of an ℓ-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.