On the image of $\Lambda $-adic Galois representations

[Sur l’image des représentations galoisiennes $\Lambda $-adiques]

Fischman, Ami

Annales de l'Institut Fourier, Tome 52 (2002), p. 351-378 / Harvested from Numdam

Résumé

Nous explorons la question de détermination de l’image des représentations galoisiennes modulaires $\Lambda $-adiques sans multiplication complexe et montrons que pour un ensemble “générique” de formes modulaires $\Lambda $-adiques (formes propres normalisées sans multiplication complexe), elles ont toutes une image contenant $SL_2(\Lambda )$.

Abstract

We explore the question of how big the image of a Galois representation attached to a $\Lambda $-adic modular form with no complex multiplication is and show that for a “generic” set of $\Lambda $-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

Détail

Publié le : 2002-01-01

MR 1906479 | Zbl 1020.11037

DOI : https://doi.org/10.5802/aif.1890
Classification:  11F80,  11F11,  11F85,  11R23
Mots clés: forme modulaire, famille $p$-adique, représentation galoisienne, forme modulaire $p$-adique

Bibliographie

[DHI98] K. Doi; H. Hida; H. Ishii Discriminant of Hecke fields and twisted adjoint $L$-values for ${G}{L}(2)$, Invent. Math., Tome 134 (1998) no. 3, pp. 547-577 | Article | MR 1660929 | Zbl 0924.11035

[Eis95] D. Eisenbud Commutative algebra with a view toward algebraic geometry, Springer-Verlag, New York (1995) | MR 1322960 | Zbl 0819.13001

[FT93] A. Fröhlich; M. J. Taylor Algebraic number theory, Cambridge University Press, Cambridge (1993) | MR 1215934 | Zbl 0744.11001

[Gou92] F. Q. Gouvêa On the ordinary Hecke algebra, J. Number Theory, Tome 41 (1992) no. 2, pp. 178-198 | Article | MR 1164797 | Zbl 0774.11026

[Hid00] H. Hida Modular forms and Galois cohomology, Cambridge University Press, Cambridge (2000) | MR 1779182 | Zbl 0952.11014

[Hid86a] H. Hida Galois representations into $GL_2({\bf Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math., Tome 85 (1986) no. 3, pp. 545-613 | Article | MR 848685 | Zbl 0612.10021

[Hid86b] H. Hida Hecke algebras for ${\rm GL}_1$ and ${\rm GL}_2$, Séminaire de théorie des nombres, Paris 1984--85, Birkhäuser Boston, Boston, Mass. (1986), pp. 131-163 | Zbl 0648.10020

[Hid86c] H. Hida Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4), Tome 19 (1986) no. 2, pp. 231-273 | Numdam | MR 868300 | Zbl 0607.10022

[Hid87] H. Hida; 2 Galois representations and the theory of p-adic Hecke algebras, Sugaku, in Japanese, Tome 39 (1987), pp. 124-139 | Zbl 0641.10025

[Hid89] H. Hida p-adic hecke algebras and galois representations, Sugaku Expositions 2, (English translation of Hid87), Tome 87 (1989) no. 1, pp. 75-102 | Zbl 0686.10023

[Hid93] H. Hida Elementary theory of L-functions and Eisenstein series, Cambridge University Press, Cambridge (1993) | MR 1216135 | Zbl 0942.11024

[Mom81] F. Momose On the l-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Tome 28 (1981) no. 1, pp. 89-109 | MR 617867 | Zbl 0482.10023

[MW86] B. Mazur; A. Wiles On p-adic analytic families of Galois representations, Compositio Math., Tome 59 (1986) no. 2, pp. 231-264 | Numdam | MR 860140 | Zbl 0654.12008

[Rib85] K. A. Ribet On l-adic representations attached to modular forms II, Glasgow Math. J., Tome 27 (1985), pp. 185-194 | Article | MR 819838 | Zbl 0596.10027

[Ser81] J.-P. Serre Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. (1981) no. 54, pp. 323-401 | Numdam | MR 644559 | Zbl 0496.12011

[Sha94] I. R. Shafarevich Basic algebraic geometry. 2, Springer-Verlag, Berlin, Schemes and complex manifolds. Translated from the Russian edition by Miles Reid (1994) | MR 1328834 | Zbl 0797.14002

[Shi71] G. Shimura; Iwanami Shoten, Publishers Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, Tokyo, Tome No. 11 (1971) | Zbl 0221.10029