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.

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 |