Sur les représentations automorphes non ramifiées des groupes linéaires sur Q de petits rangs.

Abstract : In this these we study the different algebraic automorphic representations discovered by Chenevier-Renard. We focus more particularly on their Satake parameters. To do so, we use Arthur's theory, which enables us to see these representations through discrete automorphic representations for the special orthogonal group of well chosen lattices. Afterwards, we can compute some properties of Hecke operators acting on these lattices. This gives us a lot of information on these Satake parameters. In particular, we can determine the trace in the standard representation for many of these algebraic representations, which weight can be arbitrarily high. These results also enable us to compute many Hecke operators, connected to the notion of neighbourhood developed by Kneser, seen as linear operators acting on the classes of isomorphism of even lattices with determinant 2 in dimension 23 or 25.
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Thomas Mégarbané. Sur les représentations automorphes non ramifiées des groupes linéaires sur Q de petits rangs.. Théorie des nombres [math.NT]. Université Paris-Saclay, 2016. Français. ⟨NNT : 2016SACLX046⟩. ⟨tel-01495417⟩

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