The nonabelian reciprocity law for local fields (Q2756713)

This note reports on the local Langlands correspondence for \(\text{GL}_n\) over a \(p\)-adic field, which was proved by \textit{M. Harris} and \textit{R. Taylor} in 1998 [see ``The geometry and cohomology of some simple Shimura varieties'', Annals of Mathematics Studies. 151. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)]. A second proof was given by \textit{G. Henniart} shortly thereafter [see Invent. Math. 139, 439-455 (2000; Zbl 1048.11092)]. The local Langlands correspondence is a non-Abelian generalization of the reciprocity law of local class field theory. It gives a remarkable relationship between non-Abelian Galois groups and infinite-dimensional representation theory. Posed as an open problem thirty years ago in [\textit{R. Langlands}, Lect. Notes Math. 170, 18-61 (1970; Zbl 0225.14022)], its proof in full generality represents a milestone in algebraic number theory. The goal of this note, aimed at the nonspecialist, is to state the main result with some motivations and necessary definitions. NEWLINENEWLINENEWLINEAn excellent exposition of one of the most outstanding results in algebraic number theory!