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Eknath Ghate; Vinayak Vatsal
On the local behaviour of ordinary $\Lambda$-adic representations
(Sur le comportement local des représentations ordinaires $\Lambda$-adiques)
Annales de l'institut Fourier, 54 no. 7 (2004), p. 2143-2162, doi: 10.5802/aif.2077
Article PDF | Reviews MR 2139691 | Zbl 1131.11341 | 1 citation in Cedram
Class. Math.: 11F80, 11F33, 11R23
Keywords: $\Lambda$-adic forms, $p$-adic families, ordinary primes, Galois representations

Résumé - Abstract

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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