Functoriality and the Inverse Galois problem II: groups of type $B_n$ and $G_2$

Khare, Chandrashekhar; Larsen, Michael; Savin, Gordan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 1, pp. 37-70.

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 $B_n\left({\ell }^k\right)=3DS{O}_{2n+1}{\left({𝔽}_{{\ell }^k}\right)}^{der}$ lorsque $\ell \equiv 3,5(mod8)$ et $G_2\left({\ell }^k\right)$ 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 $k$.

This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over $\mathbb{Q}$? Let $\ell$ be a prime and $t$ a positive integer. We show that that the finite simple groups of Lie type $B_n\left({\ell }^k\right)=3DS{O}_{2n+1}{\left({𝔽}_{{\ell }^k}\right)}^{der}$ if $\ell \equiv 3,5(mod8)$ and $G_2\left({\ell }^k\right)$ appear as Galois groups over $\mathbb{Q}$, for some $k$ divisible by $t$. In particular, for each of the two Lie types and fixed $\ell$ we construct infinitely many Galois groups but we do not have a precise control of $k$.

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