Newforms, inner twists, and the inverse Galois problem for projective linear groups

Dieulefait, Luis V.

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 395-411.

Nous reformulons de manière plus explicite les résultats de Momose, Ribet et Papier sur les images des représentations galoisiennes attachées à des newforms sans multiplication complexe, en nous restreignant aux formes de poids $2$ et de caractère trivial. Nous calculons deux tels exemples de newforms, possédant une unique tordue conjuguée à la forme, et nous démontrons que pour tout nombre premier $\ell >3$, l’image est aussi grosse que possible. Nous utilisons ce résultat pour prouver que les groupes $\text{PGL}(2,{𝔽}_{{\ell }^2})(\ell \equiv 3,5\left(\text{mod}8\right),\ell >3)$ et $\text{PGL}(2,{𝔽}_{{\ell }^5})(\ell \neg \equiv 0\pm 1\left(\text{mod}11\right);\ell >3)$ sont groupes de Galois sur $\mathbb{Q}$.

We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight $2$ and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than $3$ the image is as large as possible. As a consequence, we prove that the groups $\text{PGL}(2,{𝔽}_{{\ell }^2})$ for every prime $(\ell \equiv 3,5(\text{mod}8),\ell >3)$, and $\text{PGL}(2,{𝔽}_{{\ell }^5})$ for every prime $\ell \neg \equiv 0\pm 1\left(\text{mod}11\right);\ell >3)$, are Galois groups over $\mathbb{Q}$.

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