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

Luis V. Dieulefait

Luis V. Dieulefait

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

Résumé
Abstract

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 $ℓ > 3$ , l’image est aussi grosse que possible. Nous utilisons ce résultat pour prouver que les groupes $PGL (2, 𝔽_{ℓ^{2}}) (ℓ \equiv 3, 5 (mod 8), ℓ > 3)$ et $PGL (2, 𝔽_{ℓ^{5}}) (ℓ \neg \equiv 0 \pm 1 (mod 11); ℓ > 3)$ sont groupes de Galois sur $ℚ$ .

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 $PGL (2, 𝔽_{ℓ^{2}})$ for every prime $(ℓ \equiv 3, 5 (mod 8), ℓ > 3)$ , and $PGL (2, 𝔽_{ℓ^{5}})$ for every prime $ℓ \neg \equiv 0 \pm 1 (mod 11); ℓ > 3)$ , are Galois groups over $ℚ$ .

MR Zbl

@article{JTNB_2001__13_2_395_0,
     author = {Luis V. Dieulefait},
     title = {Newforms, inner twists, and the inverse {Galois} problem for projective linear groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {395--411},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     zbl = {0996.11042},
     mrnumber = {1879665},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2001__13_2_395_0/}
}

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Luis V. Dieulefait. Newforms, inner twists, and the inverse Galois problem for projective linear groups. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 395-411. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_2_395_0/

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