Patching and the $p$-adic local Langlands correspondence

THE LOCAL LANGLANDS CORRESPONDENCE FOR GLn OVER A p-ADIC FIELD

The Local Langlands Conjecture posits the existence of a bijection between certain classes of irreducible representations of a reductive group G, such as GLnpKq, and certain ndimensional representations of an additive extension of the Weil group WK (closely related to the Galois group). Here we focus solely on the case G “ GLn and K is a p-adic field (such as Qp). The Local Langlands conjecture...

The Jacquet-Langlands correspondence via `-adic uniformization

On the Local Langlands Correspondence

The local Langlands correspondence for GL(n) of a non-Archimedean local field F parametrizes irreducible admissible representations of GL(n, F ) in terms of representations of the Weil-Deligne group WDF of F . The correspondence, whose existence for p-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of s...

An introduction to the local Langlands correspondence

In these notes, based on my lectures at the FRG workshop on “Characters, Liftings, and Types” at American University in June 2012, I give an introduction to the conjectural Local Langlands Correspondence (LLC), for split semisimple groups over a nonarchimedean local field. This conjecture has been evolving over the past 45 years (with roots going back much further) to the point that today’s sta...

8 On a p - adic extension of the Jacquet - Langlands correspondence to weight 1

In this paper, we consider a novel version of the classical Jacquet-Langlands correspondence, explore a p-adic extension of the correspondence , and as an explicit example we find an overconvergent automorphic form of weight 1 which corresponds to a classical modular form of weight 1, using both experimental and theoretical methods.