Deformation theory of Galois representations and automorphic forms

The Bloch-Kato Selmer group is a group attached to a p-adic Galois representation which generalises the group of rational points on an elliptic curve. The Bloch-Kato conjecture relates the rank of this group to special values of L-functions. My aim is to prove a very special case of this conjecture: The Bloch-Kato Selmer group of the adjoint representation of an automorphic Galois representation vanishes.

This has been done in the polarizable case by P. Allen using the Taylor-Wiles method and results of Kisin on the generic fibre of p-adic Galois deformation rings. To extend this to the non-polarizable case one can use the modularity lifting method of Calegari-Geraghty joined with the 'Khare-Thorne trick' and the construction of Galois representations as in the 10 author paper. However, there are (at least) 2 technical obstacles.

Firstly, when constructing these non-polarizable Galois representations, they take values in a Hecke algebra modulo a certain nilpotent ideal. This causes problems in the Calegari-Geraghty method.
Secondly, to use Kisin's results on Galois deformation rings we need to know that these Galois representations satisfy a local-global compatibility at p, e.g. are crystalline at p with distinct HT weights. This is also proven in the 10 author paper but is another source of nilpotent ideals.

I have addressed the question of nilpotent ideals by finding a new patching lemma that uses the smoothness of generic fibres of certain local deformation rings as observed by Allen. Now I am working on weakening the local-global compatibility assumption in order to remove conditions on p.

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