Patching and the -adic Langlands program for

Prerequisites for the Langlands Program∗

For learning about the Langlands program, knowledge of Lie-group structure theory, algebraic number theory, algebraic geometry, modular forms, and infinite-dimensional representation theory is appropriate. This article describes how one can get a glimpse of the program with less background than this.

The Jacquet-Langlands correspondence via `-adic uniformization

Weak local-global compatibility in the p-adic Langlands program for U(2)

We study the completed cohomology Ĥ of a definite unitary group G in two variables associated with a CM-extension K/F . When the prime p splits, we prove that (under technical asumptions) the p-adic local Langlands correspondence for GL2(Qp) occurs in Ĥ. As an application, we obtain a result towards the Fontaine-Mazur conjecture over K: if x is a point on the eigenvariety such that ρx is geomet...

Strong local-global compatibility in the p-adic Langlands program for U(2)

For certain mod p Galois representations ρ̄, arising from modular forms on definite unitary groups in two variables, we express the ρ̄-part of completed cohomology Ĥ ρ̄ (away from Σ = Σp ∪Σ0) as a tensor product Πp⊗ΠΣ0 . Here Πp is attached to the universal deformation ρ via the p-adic local Langlands correspondence for GL2(Qp), and ΠΣ0 is given by the local Langlands correspondence in families, o...

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 ...