v2
https://www.raco.cat/index.php/CollectaneaMathematica/article/view/56362
2014-03-14T10:41:09Z
Collectanea Mathematica
1997: Vol.: 48 Núm.: 1 -2; 1-30
$p$-adic deformations of cohomology classes of subgroups of $GL(n,\mathbb{Z})$
Ash, Avner
Stevens, G.
1997-01-11 00:00:00
url:https://www.raco.cat/index.php/CollectaneaMathematica/article/view/56362
eng
We construct $p$-adic analytic families of $p$-ordinary cohomology classes in the cohomology of arithmetic subgroups of $GL(n)$ with coefficients in a family of representation spaces for $GL(n)$. 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 $GL(n,\mathbb{Z}_p)$. 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 $\epsilon : T(\mathbb{Z}_p)\longrightarrow\mathbb{Z} ^x_p$ we construct a $\mathbb{Z}_p$-map from $\mathbb{D}$ to $V_{\lambda,\epsilon}$. These maps are equivariant for commuting actions of $T(\mathbb{Z}_p)$ and $\Gamma_\nu$ where $\Gamma_\nu \subseteq GL(n, \mathbb{Z})$ is a congruence subgroup analogous to $\Gamma_0(p^\nu)$ where $p^\nu$ is the conductor of $\epsilon$ . We also make the matrix $\pi := diag(1, p, p^2,\dots , p^{n-1})$ act equivariantly on all these modules. We obtain a $\Lambda := \mathbb{Z}_p[[T(\mathbb{Z}_p)]]$-module structure on $H^\ast(\Gamma,\mathbb{D})$ and Hecke actions on $H^\ast(\Gamma,\mathbb{D})$ and $H^\ast(\Gamma_\nu, V_{\lambda,\epsilon})$ with Hecke equivariant maps $\phi_{\lambda,\epsilon} : H^\ast(\Gamma,\mathbb{D})\longrightarrow H^\ast(\Gamma_\nu, V_{\lambda,\epsilon})$, where $\Gamma$ is a congruence subgroup of $GL(n,\mathbb{Z})$ 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 $\phi^0_ {\lambda,\epsilon}$ denote the map induced by $\phi_{\lambda,\epsilon}$ on the $\Gamma\pi\Gamma$-ordinary part of $H^\ast(\Gamma,\mathbb{D})$. Our main theorem states that the kernel of $\phi^0_{\lambda,\epsilon}$ is $I_{\lambda,\epsilon}H^\ast(\Gamma,\mathbb{D})^0$ where $I_{\lambda,\epsilon}$ is the kernel of the ring homomorphism induced on $\Lambda$ by the character $\lambda$ .