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An introduction to the Langlands correspondence. (English) Zbl 1380.11063

Kerr, Matt (ed.) et al., Recent advances in Hodge theory. Period domains, algebraic cycles, and arithmetic. Proceedings of the summer school and conference, University of British Columbia, Vancouver, Canada, June 10–20, 2013. Cambridge: Cambridge University Press (ISBN 978-1-107-54629-5/pbk; 978-1-316-38788-7/ebook). London Mathematical Society Lecture Note Series 427, 333-367 (2016).
“For us the Langlands correspondence is a family of conjectured correspondences which can be summarized by the following fundamental triangle: \[ \{\text{algebraic automorphic representations}\} \]
\[ \swarrow \nearrow\qquad \qquad\nwarrow\searrow \]
\[ \begin{matrix} \{\text{geometric Galois}\\ \text{representations}\} \end{matrix}\longleftrightarrow \{\text{pure motives}\}. \] As such, it should be understood from the outset that the Langlands correspondence means something not only due to Langlands, but originating with Langlands and then developed by many people, including Fontaine-Mazur, Grothendieck, Deligne, Serre, Clozel and Buzzard-Gee.”
“This article is divided into two parts, followed by one appendix. The first part, comprised of §§2,3 is motivation for the general Langlands correspondence. The second part which consists of §4 is about the Langlands correspondence proper. Appendix A touches briefly on functoriality and its relationship with the Langlands correspondence.”
“ §4.4. states some of the main remaining open problems regarding the correspondence.”
For the entire collection see [Zbl 1348.14004].

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F80 Galois representations
14D24 Geometric Langlands program (algebro-geometric aspects)
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