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.
Classification : 11F80, 11F33, 11R23
Mots clés : formes -adiques, familles -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}, zbl = {1131.11341}, mrnumber = {2139691}, 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 DA - 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/ UR - https://zbmath.org/?q=an%3A1131.11341 UR - https://www.ams.org/mathscinet-getitem?mr=2139691 UR - https://doi.org/10.5802/aif.2077 DO - 10.5802/aif.2077 LA - en ID - AIF_2004__54_7_2143_0 ER -
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/
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