Modularity of Galois Representations and Langlands Functoriality

Langlands Functoriality Conjecture

Functoriality conjecture is one of the central and influential subjects of the present day mathematics. Functoriality is the profound lifting problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository article, I describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which a...

Lecture on Langlands Functoriality Conjecture

Modularity lifting theorems for ordinary Galois representations

We generalize the results of [CHT08] and [Tay08] by proving modularity lifting theorems for ordinary l-adic Galois representations of any dimension of a CM or totally real number field F . The main theorems are obtained by establishing an R = T theorem over a Hida family. A key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing l and to determine their i...

First steps towards p-adic Langlands functoriality

— By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type ...

Deformation of Outer Representations of Galois Group

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...