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.

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 .

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 .

@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},
     pages = {395--411},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     mrnumber = {1879665},
     zbl = {0996.11042},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_2_395_0/}
}
TY  - JOUR
AU  - Dieulefait, Luis V.
TI  - Newforms, inner twists, and the inverse Galois problem for projective linear groups
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2001
SP  - 395
EP  - 411
VL  - 13
IS  - 2
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_2001__13_2_395_0/
LA  - en
ID  - JTNB_2001__13_2_395_0
ER  - 
%0 Journal Article
%A Dieulefait, Luis V.
%T Newforms, inner twists, and the inverse Galois problem for projective linear groups
%J Journal de théorie des nombres de Bordeaux
%D 2001
%P 395-411
%V 13
%N 2
%I Université Bordeaux I
%U http://www.numdam.org/item/JTNB_2001__13_2_395_0/
%G en
%F 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, Tome 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

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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

[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 | Zbl

[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 | Zbl