The Langlands Conjecture for Gl 2 of a Local Field

The local Langlands conjecture for GL(n) over a p-adic field

The local Langlands conjecture for GL(n) over a p-adic field, n < p.

MAXIMAL VARIETIES AND THE LOCAL LANGLANDS CORRESPONDENCE FOR GL(n)

The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for GL(n) over a nonarchimedean local field. In this article we make progress towards a purely local proof of this fact. To wit, we find a family of formal schemes V such that the generic fiber of V is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomolo...

Lecture Note on Local Langlands Conjecture

The Weight Part of Serre’s Conjecture for Gl(2)

Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call “pseudo-Barsotti–Tate representations”, over arbitrary finite extensions of Qp. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for GL(2) over arbitrary totally real fields. 2010 ...

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...