Holmstrom Langlands program global

arXiv:0907.4527](http://front.math.ucdavis.edu/0907.4527) Lecture on Langlands Functoriality Conjecture from arXiv Front: math.AG by Jae-Hyun Yang This is a survey lecture note on the applications of Langlands functoriality which were obtained recently by some people at the Langalnds school. This lecture was delivered at the Department of Mathematics, Kyoto University, Japan on June 30 (Tuesday), 2009.

See Lafforgues webpage for several papers on his recent ideas.

The FE and beyond endoscopy

http://mathoverflow.net/questions/1252/where-stands-functoriality-in-2009

http://mathoverflow.net/questions/104397/base-change-and-automorphic-induction-for-gl-1

http://mathoverflow.net/questions/10578/what-is-the-current-status-of-the-function-fields-langlands-conjectures

http://mathoverflow.net/questions/31538/non-abelian-class-field-theory-and-fundamental-groups

http://mathoverflow.net/questions/1972/langlands-dual-groups

http://mathoverflow.net/questions/71939/what-is-the-strongest-most-natural-conjectural-form-of-langlands

http://mathoverflow.net/questions/74472/what-makes-langlands-for-n2-easier-than-langlands-for-n2

http://mathoverflow.net/questions/33269/fontaine-mazur-for-gl-1

http://mathoverflow.net/questions/72319/how-does-one-see-hecke-operators-as-helping-to-generalize-quadratic-reciprocity

http://mathoverflow.net/questions/75335/what-is-the-reason-for-modularity-results

http://mathoverflow.net/questions/71743/what-is-the-precise-relationship-between-langlands-and-tannakian-formalism

http://mathoverflow.net/questions/70833/does-the-ramanujan-conjecture-imply-some-cases-of-the-riemann-hypothesis

arXiv:1003.4578 Formule des Traces et Fonctorialité: le Début d’un Programme from arXiv Front: math.AG by Edward Frenkel, Robert Langlands, Ngo Bao Chau We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues are expressed in terms of the L-functions of automorphic representations and begin the analysis of their traces using the orbital side of the stable trace formula. We show that the most interesting part, corresponding to regular conjugacy classes, is nothing but a sum over a finite-dimensional vector space over the global field, which we call the Steinberg-Hitchin base. Therefore it may be analyzed using the Poisson summation formula. Our main result is that the leading term of the dual sum (the value at 0) is precisely the dominant term of the trace formula (the contribution of the trivial representation). This gives us hope that the full Poisson summation formula would reveal the patterns predicted by functoriality.

Buzzard email: Let me finish by explaining what will be happening in the Langlands study group. Langlands made some conjectures, perhaps slightly vague at first but gradually becoming more precise in the 80s, relating the L-functions of Shimura varieties to the L-functions attached to automorphic forms. These conjectures vastly generalise the work of Eichler-Shimura/Ihara which shows that the L-function of a modular curve can be expressed in terms of the L-functions attached to modular forms. Definitive steps towards their proof were made in the 80s (Langlands, Rapoport; Langlands gave a completely new proof of Eichler-Shimura/Ihara) and 90s (Kottwitz, Reimann, Milne). My initial plan was to go through Kottwitz’ paper but I am still at the stage where I break out in a sweat whenever I open it. My current plan is to start off slowly, with Langlands’ proof of the modular curve case (which is vastly different to Eichler-Shimura’s proof but more amenable to generalisation), and then to move on to principally polarized abelian varieties, where Eichler-Shimura breaks down to a large extent but Langlands’ methods do not. We’ll see how far we get.

nLab page on Langlands program global