On the local behaviour of ordinary \(\Lambda\)-adic representations. (English) Zbl 1131.11341

Summary: 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.


11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11R23 Iwasawa theory
Full Text: DOI Numdam EuDML


[1] Companion forms and weight one forms, Ann. of Math, 149, 3, 905-919 (1999) · Zbl 0965.11019 · doi:10.2307/121076
[2] Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc, 16, 1, 29-55 (2003) · Zbl 1076.11029 · doi:10.1090/S0894-0347-02-00405-8
[3] Classical and overconvergent modular forms, Invent. Math, 124, 215-241 (1996) · Zbl 0851.11030 · doi:10.1007/s002220050051
[4] On the local behaviour of ordinary modular Galois representations, Modular curves and abelian varieties, volume 224, 105-124 (2004) · Zbl 1166.11330
[5] Ordinary forms and their local Galois representations · Zbl 1085.11029
[6] Iwasawa theory for Artin representations · Zbl 1480.11047
[7] Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup, 19, 2, 231-273 (1986) · Zbl 0607.10022
[8] Galois representations into \(GL\sb 2({\Bbb Z}\sb p [[X]])\) attached to ordinary cusp forms, Invent. Math, 85, 545-613 (1986) · Zbl 0612.10021 · doi:10.1007/BF01390329
[9] Elementary Theory of \(L\)-functions and Eisenstein Series, 26 (1993) · Zbl 0942.11024
[10] Modular forms (1989) · Zbl 1159.11014
[11] Représentations galoisiennes, différentielles de Kähler et “conjectures principales”, Inst. Hautes Études Sci. Publ. Math, 71, 65-103 (1990) · Zbl 0744.11053 · doi:10.1007/BF02699878
[12] On \(p\)-adic analytic families of Galois representations, Compositio Math., 59, 231-264 (1986) · Zbl 0654.12008
[13] Abelian \(l\)-adic representations and elliptic curves (1989) · Zbl 0709.14002
[14] A remark on the 23-adic representation associated to the Ramanujan Delta function
[15] On ordinary \(\lambda \)-adic representations associated to modular forms, Invent. Math., 94, 529-573 (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.