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 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 the image is as large as possible. As a consequence, we prove that the groups for every prime , and for every prime , 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 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’image est aussi grosse que possible. Nous utilisons ce résultat pour prouver que les groupes et 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}, pages = {395--411}, publisher = {Universit\'e Bordeaux I}, volume = {13}, number = {2}, year = {2001}, mrnumber = {1879665}, zbl = {0996.11042}, language = {en}, url = {http://archive.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://archive.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://archive.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, Volume 13 (2001) no. 2, pp. 395-411. http://archive.numdam.org/item/JTNB_2001__13_2_395_0/
[AL70] Hecke operators on Γ0(m). Math. Ann. 185 (1970), 134-160. | Zbl
, ,[B95] The rank of J0(N). Astérisque 228 (1995), 41-68. | MR | Zbl
,[C89] Sur les representations galoisiennes modulo l attachées aux formes modulaires. Duke Math. J. 59 (1989), 785-801. | MR | Zbl
,[C92] 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] Formes modulaires et représentations -adiques. Lecture Notes in Mathematics 179 Springer-Verlag, Berlin-New York, 1971, 139-172. | Numdam | Zbl
,[FJ95] Crystalline cohomology and GL(2,Q). Israel J. Math. 90 (1995), 1-66. | MR | Zbl
, ,[K77] A result on modular forms in characteristic p. Lecture Notes in Math. 601, 53-61, Springer, Berlin, 1977. | MR | Zbl
,[L89] On the conductors of mod Galois representations coming from modular forms. J. Number Theory 31 (1989), 133-141. | MR | Zbl
,[M81] On the -adic representations attached to modular forms. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28:1 (1981), 89-109. | MR | Zbl
,[Q98] 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] Some projective linear groups over finite fields as Galois groups over Q. Contemporary Math. 186 (1995), 51-63. | MR | Zbl
, ,[R75] On -adic representations attached to modular forms. Invent. Math. 28 (1975), 245-275. | MR | Zbl
,[R77] Galois representations attached to eigenforms with nebentypus. Lecture Notes in Math. 601, 17-51, Springer, Berlin, 1977. | MR | Zbl
,[R80] Twists of modular forms and endomorphisms of Abelian Varieties, Math. Ann. 253 (1980), 43-62. | MR | Zbl
,[R85] On l-adic representations attached to modular forms II, Glasgow Math. J. 27 (1985), 185-194. | MR | Zbl
,[S71] Introduction to the Arithmetic Theory of Automorphic Functions. Publ. Math. Soc. Japan 11, 199-208, Princeton University Press, Princeton, N.J., 1971. | MR
,[S71b] 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] Hecke: The Modular Forms Calculator. Available at: http:// shimura.math. berkeley.edu /~was /Tables /hecke.html.
,[S73] On -adic representations and congruences for coefficients of modular forms. Lecture Notes in Math. 350, 1-55, Springer, Berlin, 1973. | MR | Zbl
,