With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Eknath Ghate; Vinayak VatsalOn 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, 11R23Keywords: $\Lambda$-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. Bibliography[Buz03] K. Buzzard, “Analytic continuation of overconvergent eigenforms”, J. Amer. Math. Soc 16 (2003) no. 1, p. 29-55 Article |  MR 1937198 |  Zbl 01832407[BT99] K. Buzzard & R. Taylor, “Companion forms and weight one forms”, Ann. of Math 149 (1999) no. 3, p. 905-919 Article |  MR 1709306 |  Zbl 0965.11019[Col96] R. Coleman, “Classical and overconvergent modular forms”, Invent. Math 124 (1996), p. 215-241 Article |  MR 1369416 |  Zbl 0851.11030[Gha04] E. Ghate, On the local behaviour of ordinary modular Galois representations, Progress in Mathematics, Birkhäuser, 2004, p. 105-124  Zbl 02164178[Gha05] E. Ghate, “Ordinary forms and their local Galois representations”, To appear  Zbl 1085.11029[GV03] R. Greenberg & V. Vatsal, “Iwasawa theory for Artin representations”, To appear [Hid86a] H. Hida, “Iwasawa modules attached to congruences of cusp forms”, Ann. Sci. École Norm. Sup 19 (1986) no. 2, p. 231-273 Numdam |  MR 868300 |  Zbl 0607.10022[Hid86b] H. Hida, “Galois representations into $GL\sb 2({\Bbb Z}\sb p [[X]])$ attached to ordinary cusp forms”, Invent. Math 85 (1986), p. 545-613 Article |  MR 848685 |  Zbl 0612.10021[Hid93] H. Hida, Elementary Theory of $L$-functions and Eisenstein Series, LMSST 26, Cambridge University Press, Cambridge, 1993  MR 1216135 |  Zbl 0942.11024[MT90] B. Mazur & J. Tilouine, “Représentations galoisiennes, différentielles de Kähler et conjectures principales''”, Inst. Hautes Études Sci. Publ. Math 71 (1990), p. 65-103 Numdam |  MR 1079644 |  Zbl 0744.11053[MW86] B. Mazur & A. Wiles, “On $p$-adic analytic families of Galois representations”, Compositio Math. 59 (1986), p. 231-264 Numdam |  MR 860140 |  Zbl 0654.12008[Miy89] T. Miyake, Modular forms, Springer Verlag, 1989  MR 1021004 |  Zbl 05012868[Ser89] J.-P. Serre, Abelian $l$-adic representations and elliptic curves, Advanced Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989  MR 1043865 |  Zbl 0709.14002[Vat05] V. Vatsal, “A remark on the 23-adic representation associated to the Ramanujan Delta function”, Preprint [Wil88] A. Wiles, “On ordinary $\lambda$-adic representations associated to modular forms”, Invent. Math. 94 (1988), p. 529-573 Article |  MR 969243 |  Zbl 0664.10013 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310