Newforms, inner twists, and the inverse Galois problem for projective linear groups
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, p. 395-411

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 2 ) for every prime (3,5(mod8),>3), and PGL(2,𝔽 5 ) for every prime ¬0±1(mod11);>3), are Galois groups over .

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 >3, l’image est aussi grosse que possible. Nous utilisons ce résultat pour prouver que les groupes PGL(2,𝔽 2 )(3,5(mod8),>3) et PGL(2,𝔽 5 )(¬0±1(mod11);>3) sont groupes de Galois sur .

@article{JTNB_2001__13_2_395_0,
     author = {Dieulefait, Luis V.},
     title = {Newforms, inner twists, and the inverse Galois problem for projective linear groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     pages = {395-411},
     zbl = {0996.11042},
     mrnumber = {1879665},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_2_395_0}
}
Dieulefait, Luis V. Newforms, inner twists, and the inverse Galois problem for projective linear groups. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 395-411. http://www.numdam.org/item/JTNB_2001__13_2_395_0/

[AL70] A. Atkin, J. Lehner, Hecke operators on Γ0(m). Math. Ann. 185 (1970), 134-160. | Zbl 0177.34901

[B95] A. Brumer, The rank of J0(N). Astérisque 228 (1995), 41-68. | MR 1330927 | Zbl 0851.11035

[C89] H. Carayol, Sur les representations galoisiennes modulo l attachées aux formes modulaires. Duke Math. J. 59 (1989), 785-801. | MR 1046750 | Zbl 0703.11027

[C92] J. Cremona, Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields. J. London Math. Soc. 45 (1992), 404-416. | MR 1180252 | Zbl 0773.14023

[D71] P. Deligne, Formes modulaires et représentations -adiques. Lecture Notes in Mathematics 179 Springer-Verlag, Berlin-New York, 1971, 139-172. | Numdam | Zbl 0206.49901

[FJ95] G. Faltings, B. Jordan, Crystalline cohomology and GL(2,Q). Israel J. Math. 90 (1995), 1-66. | MR 1336315 | Zbl 0854.14010

[K77] N. Katz, A result on modular forms in characteristic p. Lecture Notes in Math. 601, 53-61, Springer, Berlin, 1977. | MR 463169 | Zbl 0392.10026

[L89] R. Livné, On the conductors of mod Galois representations coming from modular forms. J. Number Theory 31 (1989), 133-141. | MR 987567 | Zbl 0674.10024

[M81] F. Momose, On the -adic representations attached to modular forms. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28:1 (1981), 89-109. | MR 617867 | Zbl 0482.10023

[Q98] J. Quer, La classe de Brauer de l'algèbre d'endomorphismes d'une variété abélienne modulaire. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 227-230. | MR 1650241 | Zbl 0936.14032

[RV95] A. Reverter, N. Vila, Some projective linear groups over finite fields as Galois groups over Q. Contemporary Math. 186 (1995), 51-63. | MR 1352266 | Zbl 0836.12003

[R75] K.A. Ribet, On -adic representations attached to modular forms. Invent. Math. 28 (1975), 245-275. | MR 419358 | Zbl 0302.10027

[R77] K.A. Ribet, Galois representations attached to eigenforms with nebentypus. Lecture Notes in Math. 601, 17-51, Springer, Berlin, 1977. | MR 453647 | Zbl 0363.10015

[R80] K.A. Ribet, Twists of modular forms and endomorphisms of Abelian Varieties, Math. Ann. 253 (1980), 43-62. | MR 594532 | Zbl 0421.14008

[R85] K.A. Ribet, On l-adic representations attached to modular forms II, Glasgow Math. J. 27 (1985), 185-194. | MR 819838 | Zbl 0596.10027

[S71] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Publ. Math. Soc. Japan 11, 199-208, Princeton University Press, Princeton, N.J., 1971. | MR 314766

[S71b] G. Shimura, On elliptic curves with complex multiplication as factors of the jacobian of modular function fields. Nagoya Math. J. 43 (1971), 199-208. | MR 296050 | Zbl 0225.14015

[St] W. Stein, Hecke: The Modular Forms Calculator. Available at: http:// shimura.math. berkeley.edu /~was /Tables /hecke.html.

[S73] H.P.F. Swinnerton-Dyer, On -adic representations and congruences for coefficients of modular forms. Lecture Notes in Math. 350, 1-55, Springer, Berlin, 1973. | MR 406931 | Zbl 0267.10032