On the pro-étale cohomology of quotient stacks of Drinfeld spaces

2025/10/03 by Yi, Zecheng
#11S25 #11S31 #14F30 (Primary) #22E41 (Secondary) #22E50 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2510.02699

Abstract

Let H^n-1_K denote the (n-1)-dimensional Drinfeld space over a p-adic field K. We give an explicit description of the ℓ-adic and p-adic pro-étale cohomology of quotient stacks [H^n-1_K/GL_n(O_K)] and [H^n-1_K/GL_n(K)], which are moduli stacks of special formal O_D-modules. The computation makes use of the isomorphism between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze--Weinstein, as well as the p-adic pro-étale cohomology of the Drinfeld spaces computed by Colmez--Dospinescu--Niziol. As an application, we also compute the continuous group cohomology of GL_n(ℚ_p) over duals of generalized Steinberg representations over ℚ_p.

Citations

• Arithmetic duality for p-adic pro-étale cohomology of analytic curves
• Solid locally analytic representations of p-adic Lie groups
• On the p-adic pro-étale cohomology of Drinfeld symmetric spaces
• Continuous Group Cohomology and Ext-Groups
• Cohomology of p-adic Stein spaces
• On the cohomology of the affine space
• Moduli of p-divisible groups
• L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld : demonstration du resultat principal
• On Extensions of generalized Steinberg Representations

Related

• The pro-étale cohomology of Drinfeld's upper half space
• On the p-adic pro-étale cohomology of Drinfeld symmetric spaces
• Factorization de la cohomologie étale p-adique de la tour de Drinfeld
• Cohomology of p-adic Stein spaces
• Cohomologie p-adique de la tour de Drinfeld: le cas de la dimension 1