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La résolution des conjectures d’Artin dans les cas $${\mathfrak A}_ 4$$ et $${\mathfrak S}_ 4$$ par les méthodes de Langlands. (French) Zbl 0532.12011
Sémin. Théor. Nombres, Univ. Bordeaux I 1982-1983, Exp. No. 8, 12 p. (1983).
This paper gives a clear and lucid exposition of some parts of the proof - by Langlands and Tunnell - of the Artin conjecture for degree two representations of the Galois group $$G_ F$$ of a global field F, whose image in PGL(2,$${\mathbb{C}})$$ is isomorphic to $${\mathfrak A}_ 4$$ or $${\mathfrak S}_ 4$$. Artin conjectured that the L-function of an irreducible representation $$\sigma$$ of $$G_ F$$ is entire. Langlands conjectured further that to $$\sigma$$ is attached an automorphic cuspidal representation $$\pi$$ of $$GL(n,{\mathbb{A}}_ F)$$ (where n is the degree of $$\sigma$$ and $${\mathbb{A}}_ F$$ the adèle group of F), such that for all but a finite number of places v of F, the component $$\pi_ v$$ of $$\pi$$ at v is the unramified principal series representation of $$GL(2,F_ v)$$ corresponding to the component $$\sigma_ v$$ of $$\sigma$$ at v, which is a sum of unramified characters; in particular we would have $$L(\sigma,s)=L(\pi,s)$$. Since L-functions of cuspidal representations are entire, this would imply and explain the Artin conjecture.
This strong form of the Artin conjecture is known when $$\sigma$$ is monomial, of degree 2 or 3. When $$\sigma$$ is of degree 2 and when its image in PGL(2,$${\mathbb{C}})$$ is isomorphic to $${\mathfrak A}_ 4$$ or $${\mathfrak S}_ 4$$, $$\sigma$$ is not monomial; however, going to a suitable cubic extension E of F yields a restriction $$\sigma_ E$$ which is monomial, hence a corresponding cuspidal representation $$\pi_ E$$ of $$GL(2,{\mathbb{A}}_ E)$$. The problem is then to construct $$\pi$$ from $$\pi_ E$$ (this is called the base change problem) and to verify that $$\pi$$ has the right properties. Base change for cyclic E/F (due to Langlands) enabled him to treat the $${\mathfrak A}_ 4$$-type, whereas for $${\mathfrak S}_ 4$$-type $$\sigma$$, Tunnell had to use base change for non-cyclic E/F (due to Jacquet-Piatetskij-Shapiro-Shalika Tunnell uses also the cyclic case). That $$\pi$$ has the right properties comes from Gelbart and Jacquet’s results on functoriality from GL(2) to GL(3) (i.e. the relationship between automorphic representations of $$GL(2,{\mathbb{A}}_ F)$$ and $$GL(3,{\mathbb{A}}_ F)$$ corresponding to the passage, on the Galois side, from $$\sigma$$ to Ad$${\mathbb{O}}\sigma$$, where Ad is the adjoint representation of GL(2,$${\mathbb{C}}).$$
Granting all technical results on the side of cuspidal representations, this paper explains elegantly and carefully the ”dévissage” necessary to get the strong Artin conjecture.
Reviewer: G.Henniart
##### MSC:
 11R39 Langlands-Weil conjectures, nonabelian class field theory 11R42 Zeta functions and $$L$$-functions of number fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11S15 Ramification and extension theory
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