Soit une forme parabolique primitive de poids au moins et soit la représentation galoisienne -adique associée à . Si est -ordinaire, alors on sait que la restriction de au sous-groupe de décomposition en est “triangulaire supérieure”. Si en plus a multiplication complexe, alors cette représentation est même diagonale. Dans ce travail on étudie la réciproque. Plus précisément, on démontre que la représentation galoisienne locale n’est pas diagonale pour tous les éléments arithmétiques, sauf peut-être un nombre fini, d’une famille de formes -ordinaires n’admettant pas de multiplication complexe. On suppose que est impair et que la représentation galoisienne résiduelle vérifie certaines conditions techniques. On répond aussi à la question analogue pour des formes - ordinaires -adiques, sous des hypothèses similaires.
Let be a primitive cusp form of weight at least 2, and let be the -adic Galois representation attached to . If is -ordinary, then it is known that the restriction of to a decomposition group at is “upper triangular”. If in addition has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members of a non-CM family of -ordinary forms. We assume is odd, and work under some technical conditions on the residual representation. We also settle the analogous question for -ordinary -adic forms, under similar conditions.
Keywords: $\Lambda $-adic forms, $p$-adic families, ordinary primes, Galois representations
Mot clés : formes $\Lambda $-adiques, familles $p$-adiques, premiers ordinaires, représentations galoisiennes
@article{AIF_2004__54_7_2143_0, author = {Ghate, Eknath and Vatsal, Vinayak}, title = {On the local behaviour of ordinary $\Lambda $-adic representations}, journal = {Annales de l'Institut Fourier}, pages = {2143--2162}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {7}, year = {2004}, doi = {10.5802/aif.2077}, mrnumber = {2139691}, zbl = {1131.11341}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2077/} }
TY - JOUR AU - Ghate, Eknath AU - Vatsal, Vinayak TI - On the local behaviour of ordinary $\Lambda $-adic representations JO - Annales de l'Institut Fourier PY - 2004 SP - 2143 EP - 2162 VL - 54 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2077/ DO - 10.5802/aif.2077 LA - en ID - AIF_2004__54_7_2143_0 ER -
%0 Journal Article %A Ghate, Eknath %A Vatsal, Vinayak %T On the local behaviour of ordinary $\Lambda $-adic representations %J Annales de l'Institut Fourier %D 2004 %P 2143-2162 %V 54 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2077/ %R 10.5802/aif.2077 %G en %F AIF_2004__54_7_2143_0
Ghate, Eknath; Vatsal, Vinayak. On the local behaviour of ordinary $\Lambda $-adic representations. Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2143-2162. doi : 10.5802/aif.2077. http://www.numdam.org/articles/10.5802/aif.2077/
[BT99] Companion forms and weight one forms, Ann. of Math, Volume 149 (1999) no. 3, pp. 905-919 | DOI | MR | Zbl
[Buz03] Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc, Volume 16 (2003) no. 1, pp. 29-55 | DOI | MR | Zbl
[Col96] Classical and overconvergent modular forms, Invent. Math, Volume 124 (1996), pp. 215-241 | DOI | MR | Zbl
[Gha04] On the local behaviour of ordinary modular Galois representations, Modular curves and abelian varieties (Progress in Mathematics), Volume volume 224 (2004), pp. 105-124 | Zbl
[Gha05] Ordinary forms and their local Galois representations (To appear) | Zbl
[GV03] Iwasawa theory for Artin representations (To appear)
[Hid86a] Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup, Volume 19 (1986) no. 2, pp. 231-273 | Numdam | MR | Zbl
[Hid86b] Galois representations into attached to ordinary cusp forms, Invent. Math, Volume 85 (1986), pp. 545-613 | DOI | MR | Zbl
[Hid93] Elementary Theory of -functions and Eisenstein Series, LMSST, 26, Cambridge University Press, Cambridge, 1993 | MR | Zbl
[Miy89] Modular forms, Springer Verlag, 1989 | MR | Zbl
[MT90] Représentations galoisiennes, différentielles de Kähler et ``conjectures principales'', Inst. Hautes Études Sci. Publ. Math, Volume 71 (1990), pp. 65-103 | DOI | Numdam | MR | Zbl
[MW86] On -adic analytic families of Galois representations, Compositio Math., Volume 59 (1986), pp. 231-264 | Numdam | MR | Zbl
[Ser89] Abelian -adic representations and elliptic curves, Advanced Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989 | MR | Zbl
[Vat05] A remark on the 23-adic representation associated to the Ramanujan Delta function (Preprint)
[Wil88] On ordinary -adic representations associated to modular forms, Invent. Math., Volume 94 (1988), pp. 529-573 | DOI | MR | Zbl
Cité par Sources :