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

Fischman, Ami

doi:10.5802/aif.1890

On the image of

Λ

-adic Galois representations
[Sur l’image des représentations galoisiennes

Λ

-adiques]

Fischman, Ami ¹

¹ 517 N 137th Street, Seattle WA 98133 (USA)

Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 351-378.

Résumé
Abstract

Nous explorons la question de détermination de l’image des représentations galoisiennes modulaires $Λ$ -adiques sans multiplication complexe et montrons que pour un ensemble “générique” de formes modulaires $Λ$ -adiques (formes propres normalisées sans multiplication complexe), elles ont toutes une image contenant $S L_{2} (Λ)$ .

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

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1890

Classification : 11F80, 11F11, 11F85, 11R23
Keywords: modular form, $p$-adic family, Galois representation, $p$-adic modular form
Mot clés : forme modulaire, famille $p$-adique, représentation galoisienne, forme modulaire $p$-adique

Affiliations des auteurs :

Fischman, Ami ¹

¹ 517 N 137th Street, Seattle WA 98133 (USA)

@article{AIF_2002__52_2_351_0,
     author = {Fischman, Ami},
     title = {On the image of $\Lambda $-adic {Galois} representations},
     journal = {Annales de l'Institut Fourier},
     pages = {351--378},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     doi = {10.5802/aif.1890},
     mrnumber = {1906479},
     zbl = {1020.11037},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1890/}
}

TY  - JOUR
AU  - Fischman, Ami
TI  - On the image of $\Lambda $-adic Galois representations
JO  - Annales de l'Institut Fourier
PY  - 2002
SP  - 351
EP  - 378
VL  - 52
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1890/
DO  - 10.5802/aif.1890
LA  - en
ID  - AIF_2002__52_2_351_0
ER  -

%0 Journal Article
%A Fischman, Ami
%T On the image of $\Lambda $-adic Galois representations
%J Annales de l'Institut Fourier
%D 2002
%P 351-378
%V 52
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1890/
%R 10.5802/aif.1890
%G en
%F AIF_2002__52_2_351_0

Fischman, Ami. On the image of $\Lambda $-adic Galois representations. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 351-378. doi : 10.5802/aif.1890. http://www.numdam.org/articles/10.5802/aif.1890/

Bibliographie
Cité par

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

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

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

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

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

[Hid86a] H. Hida Galois representations into $G L_{2} (𝐙_{p} [[X]])$ attached to ordinary cusp forms, Invent. Math., Volume 85 (1986) no. 3, pp. 545-613 | DOI | MR | Zbl

[Hid86b] H. Hida Hecke algebras for ${GL}_{1}$ and ${GL}_{2}$ , Séminaire de théorie des nombres, Paris 1984--85 (1986), pp. 131-163 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

[Shi71] G. Shimura; Iwanami Shoten, Publishers Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Volume No. 11 (1971) | Zbl

Cité par Sources :