On the Cohomology of GL(N) and Adjoint Selmer Groups

Abstract

Abstract We prove under certain conditions (local-global compatibility and vanishing of modulo $p$ cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of $\operatorname{\textsf{GL}}(N)$ over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well.