Selmer group actions on p-adic automorphic cohomology

Abstract

This thesis studies connections between Galois representations and automorphic forms, two central objects in modern number theory. In contrast to the classical setting of modular forms, the Hecke eigenspaces in automorphic cohomology of the general linear group over a CM field are spread out across multiple cohomological degrees. In the p-adic setting and under technical hypotheses, we explain this phenomenon by equipping such eigenspaces with actions of their associated dual adjoint Bloch-Kato Selmer groups, thereby establishing new cases of a conjecture of Venkatesh. Appended to the thesis are two articles: In Article I we treat the special case of PGL_2 with p a totally split prime. Under various hypotheses, we construct an action using the p-adic local Langlands correspondence for GL_2(Q_p) and completed homology. The main result is a `big R=T' theorem in characteristic 0, proved via a Taylor-Wiles patching argument, from which the Selmer group action is obtained in a natural way. In Article II we consider the general case of \GL_n. Under mild assumptions, we define a Selmer group action using derived Galois deformation rings and the extension of the Taylor-Wiles method, building on previous work of Galatius--Venkatesh. We also establish an equivalence between the derived deformation ring of rho and the completion at the point corresponding to rho of the derived deformation ring of its mod p reduction, thus generalising frequently cited results of Kisin.