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Chandrashekhar Khare; Michael Larsen; Gordan Savin
Functoriality and the Inverse Galois problem II: groups of type $B_n$ and $G_2$
Annales de la faculté des sciences de Toulouse Sér. 6, 19 no. 1 (2010), p. 37-70, doi: 10.5802/afst.1235
Article PDF | Analyses MR 2597780 | Zbl 1194.11063

Résumé - Abstract

Cet article donne une application du principe de fonctorialité de Langlands au problème classique suivant  : quels groupes finis, en particulier quels groupes simples, apparaissent comme groupes de Galois sur $\mathbb{Q}$  ? Soit $\ell $ une nombre premier et $t$ un entier positif. Nous montrons que les groupes finis simples de type de Lie $B_{n}(\ell ^{k})=3DSO_{2n+1}({\mathbb{F}}_{\ell ^{k}})^{der}$ lorsque $\ell \equiv 3,5\hspace{4.44443pt}(\@mod \; 8)$ et $G_{2}(\ell ^{k})$ sont des groupes de Galois sur $\mathbb{Q}$ pour un entier $k$ divisant $t$. En particulier, pour chacun de ces deux types de Lie et pour un entier $\ell $ fixé, nous construisons une infinité de groupes de Galois, mais nous n’avons pas de contrôle précis sur $k$.

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