2025/06/06 by Tiso, Mattia
#13A35 #14F10 #14G15 #14L30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2506.06246
In this dissertation we study the Hodge-Witt cohomology of the d-dimensional Drinfeld's upper half space X ⊂ ℙkd over a finite field k. We consider the natural action of the k-rational points G of the linear group GLd+1 on H0(X,WnΩℙkdi), making them natural Wn(k)[G]-modules. To study these representations, we introduce a theory of differential operators over the Witt vectors for smooth k-schemes X, through a quasi-coherent sheaf of Wn(k)-algebras DWn(X). We apply this theory to equip suitable local cohomology groups arising from H0(X,WnOℙkd) with a Γ(ℙkd,DWn(ℙkd))-module structure. Those local cohomology groups are naturally modules over some parabolic subgroup of GLd+1(k), and we prove that they are finitely generated Γ(ℙkd,DWn(ℙkd))-modules.