On the local behaviour of ordinary $\Lambda$-adic representations

Ghate, Eknath; Vatsal, Vinayak

Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2143-2162.

Soit $f$ une forme parabolique primitive de poids au moins $2$ et soit ${\rho }_f$ la représentation galoisienne $p$-adique associée à $f$. Si $f$ est $p$-ordinaire, alors on sait que la restriction de ${\rho }_f$ au sous-groupe de décomposition en $p$ est “triangulaire supérieure”. Si en plus $f$ 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 $p$-ordinaires n’admettant pas de multiplication complexe. On suppose que $p$ est impair et que la représentation galoisienne résiduelle vérifie certaines conditions techniques. On répond aussi à la question analogue pour des formes $p$- ordinaires $\Lambda$-adiques, sous des hypothèses similaires.

Let $f$ be a primitive cusp form of weight at least 2, and let ${\rho }_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of ${\rho }_f$ to a decomposition group at $p$ is “upper triangular”. If in addition $f$ 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 $p$-ordinary forms. We assume $p$ is odd, and work under some technical conditions on the residual representation. We also settle the analogous question for $p$-ordinary $\Lambda$-adic forms, under similar conditions.

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