×

The Langlands conjecture for \(\mathrm{GL}_2\) of a local field. (English) Zbl 0469.22013

Let \(F\) be any \(p\)-field and \(\mathfrak{A}_{2}(F)\) denote the set of equivalence classes of 2-dimensional semi-simple continuous complex representations of the absolute Weil group \(W(F)\) of \(G\). Let \(\mathfrak{A}(\mathrm{GL}_2(F))\) be the set of irreducible admissible non-special representations of \(\mathrm{GL}_{2}(F)\). The author proves the following result first conjectured by Langlands: there is a bijection \(\sigma \rightarrow \pi(\sigma)\) of \(\mathfrak{A}_{2}(F)\) with \(\mathfrak{A}(\mathrm{GL}_{2}(F))\) such that
(i)
\(\pi(\sigma \otimes \chi) = \pi(\sigma) \otimes \chi \) for any character \(\chi\) of \(F^{\times}\) viewed as a character of \(W(F)\),
(ii)
\(\operatorname{det}\sigma\) should be the central character of \(\pi(\sigma)\) and
(iii)
\(\sigma\) and \(\pi(\sigma)\) have the same ‘local factors’ \(L\) as well as \(\varepsilon\) with respect to a fixed character of \(F^{+}\).
The proof rests on the construction of supercuspidal representations given in his earlier papers.
Reviewer: S. Raghavan

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F80 Galois representations
11S37 Langlands-Weil conjectures, nonabelian class field theory