\(p\)-adic deformations of cohomology classes of subgroups of \(\text{GL}(N,\mathbb{Z})\)
type: Item
instance of: scholarly article
title: \(p\)-adic deformations of cohomology classes of subgroups of \(\text{GL}(N,\mathbb{Z})\)
zbMATH Open document ID: 0866.20038
author: Avner Ash
author: Glenn Stevens
published in: Collectanea Mathematica
publication date: 1996-11-21T00:00:00Z
full work available at URL: https://eudml.org/doc/40485
review text: We construct \(p\)-adic analytic families of \(p\)-ordinary cohomology classes in the cohomology of arithmetic subgroups of \(\text{GL}(n)\) with coefficients in a family of representation spaces for \(\text{GL}(n)\). These analytic families are parametrized by the highest weights of the coefficient modules. More precisely, we consider the cohomology of a compact \(\mathbb{Z}_p\)-module \(\mathbb{D}\) of \(p\)-adic measures on a certain homogeneous space of \(\text{GL}(n,\mathbb{Z}_p)\). For any dominant weight \(\lambda\) with respect to a fixed choice \((B,T)\) of a Borel subgroup \(B\) and a maximal split torus \(T\subseteq B\) and for any finite ``nebentype'' character \(\varepsilon:T(\mathbb{Z}_p)\to\mathbb{Z}_p^\times\) we construct a \(\mathbb{Z}_p\)-map from \(\mathbb{D}\) to \(V_{\lambda,\varepsilon}\). These maps are equivariant for commuting actions of \(T(\mathbb{Z}_p)\) and \(\Gamma_\nu\) where \(\Gamma_\nu\subseteq\text{GL}(n,\mathbb{Z})\) is a congruence subgroup analogous to \(\Gamma_0(p^\nu)\) where \(p^\nu\) is the conductor of \(\varepsilon\). We also make the matrix \(\pi:=\text{diag}(1,p,p^2,\dots,p^{n-1})\) act equivariantly on all these modules. We obtain a \(\Lambda:=\mathbb{Z}_p[[T(\mathbb{Z}_p)]]\)-module structure on \(H^*(\Gamma,\mathbb{D})\) and Hecke actions on \(H^*(\Gamma,\mathbb{D})\) and \(H^*(\Gamma_\nu,V_{\lambda,\varepsilon})\) with Hecke equivariant maps \(\varphi_{\lambda,\varepsilon}:H^*(\Gamma,\mathbb{D})\to H^*(\Gamma_\nu,V_{\lambda,\varepsilon})\), where \(\Gamma\) is a congruence subgroup of \(\text{GL}(N,\mathbb{Z})\) of level prime to \(p\) and \(\Gamma_\nu\) is one of a certain family of congruence subgroups of \(\Gamma\) with \(p\) in their level. Let \(\varphi^0_{\lambda,\varepsilon}\) denote the map induced by \(\varphi_{\lambda,\varepsilon}\) on the \(\Gamma\pi\Gamma\)-ordinary part of \(H^*(\Gamma,\mathbb{D})\). Our main theorem states that the kernel of \(\varphi^0_{\lambda,\varepsilon}\) is \(I_{\lambda,\varepsil…
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 937449
zbMATH Keywords: \(p\)-adic analytic families
zbMATH Keywords: \(p\)-ordinary cohomology classes
zbMATH Keywords: arithmetic subgroups of \(\text{GL}(n)\)
zbMATH Keywords: representation spaces
zbMATH Keywords: highest weights
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: dominant weights
zbMATH Keywords: Borel subgroups
zbMATH Keywords: maximal split torus
zbMATH Keywords: congruence subgroups
zbMATH Keywords: Hecke actions
zbMATH Keywords: equivariant maps
MaRDI profile type: Publication
Recommended article: Rigidity of p -adic cohomology classes of congruence subgroups of GL(n , ℤ)
Recommended article: Q3138807
Recommended article: On the cohomology groups of $\mathfrak{p}$-adic number fields
Recommended article: \(p\)-adic deformation of algebraic cycle classes
Recommended article: \(p\)-adic deformation of motivic Chow groups
Recommended article: Q2702133
Recommended article: Q5854553
Recommended article: Deforming 𝑙-adic representations of the fundamental group of a smooth variety
Recommended article: The \(\text{mod-}p\) cohomology of \(\text{GL}(2p-2,\mathbb{Z})\)
Recommended article: A constructive approach to p-adic deformations of arithmetic homology
description: scientific article; zbMATH DE number 937449