Kutzko, Philip The Langlands conjecture for \(\mathrm{GL}_2\) of a local field. (English) Zbl 0469.22013 Ann. Math. (2) 112, 381-412 (1980). 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 6 ReviewsCited in 41 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 11F80 Galois representations 11S37 Langlands-Weil conjectures, nonabelian class field theory Keywords:2-dimensional semi-simple continuous complex representations; absolute Weil group; irreducible admissible non-special representations; central character; supercuspidal representations × Cite Format Result Cite Review PDF Full Text: DOI Euclid