\(p\)-adic deformations of cohomology classes of subgroups of \(\text{GL}(N,\mathbb{Z})\)
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 \(I_{\lambda,\varepsilon}H^*(\Gamma,\mathbb{D})^0\) where \(I_{\lambda,\varepsilon}\) is the kernel of the ring homomorphism induced on \(\Lambda\) by the character \(\lambda\varepsilon\).
arithmetic subgroups of \(\text{GL}(n)\)
scientific article; zbMATH DE number 937449
scientific article