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Zbl 0703.11062
Henniart, Guy
Formes de Maass et représentations galoisiennes [d'après Blasius, Clozel, Harris, Ramakrishnan et Taylor]. (Maass forms and Galois representations). (French)
[A] Sémin. Bourbaki, Vol. 1988/89, 41e année, Exp. No.711, Astérisque 177-178, 277-302 (1989).

Given an irreducible 2-dimensional complex representation $\sigma$ of the Galois group of ${\bbfQ}$, it is conjectured that there exists a cuspidal automorphic representation $\pi$ of GL(2,${\bbfA})$ with the same L- function and this has been proved when $\sigma$ is not of icosahedral type. In the case $\sigma$ is odd, the infinity component of $\pi$ is known (and $\pi$ corresponds to a holomorphic modular form of weight 1) and given $\pi$ with the right infinity component, Deligne and Serre constructed $\sigma$ with the same L-function as $\pi$ [see {\it P. Deligne} and {\it J.-P. Serre}, Ann. Sci. Ec. Norm. Super, IV. Ser. 7, 507-530 (1975; Zbl 0321.10026)]. \par The present Bourbaki lecture is concerned with the case $\sigma$ even. There are now two possibilities for the infinity type of $\pi$ (and these correspond to certain non-holomorphic modular forms) and given such $\pi$ one wants to construct $\sigma$. A particular case was treated in: [{\it J.-P. Labesse} and {\it R. P. Langlands}, Can. J. Math. 31, 726-785 (1979; Zbl 0421.12014)]. Excluding this case the method is described. It is shown how one attaches to $\pi$ a cuspidal automorphic representation of GSp(4,${\bbfA})$ (corresponding to a holomorphic Siegel modular form; here a conjecture which has not yet been proved comes in) and how the method of Deligne-Serre can be imitated. \par There is in particular a result on the algebraicity of the eigenvalues of Hecke operators. Some remarks are made on the case of representations $\sigma$ of degree $>2$ and the case of number fields.
[J.G.M.Mars]

MSC 2000:: *11R39 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods in automorphic theory
11F37 Forms of half-integer weight, etc.

Keywords: Langlands-Weil conjecture; cuspidal automorphic representation; algebraicity; eigenvalues of Hecke operators

Citations: Zbl 0321.10026; Zbl 0421.12014

Cited in: Zbl 0748.11057

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