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

Eknath Ghate^{[1]}; Vinayak Vatsal

- [1] Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400005 (India), University of British Columbia, Department of Mathematics, Vancouver (Canada)

Annales de l'Institut Fourier (2004)

- Volume: 54, Issue: 7, page 2143-2162
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topGhate, Eknath, and Vatsal, Vinayak. "On the local behaviour of ordinary $\Lambda $-adic representations." Annales de l'Institut Fourier 54.7 (2004): 2143-2162. <http://eudml.org/doc/116170>.

@article{Ghate2004,

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.},

affiliation = {Tata Institute of Fundamental Research, School of Mathematics, Homi Bhabha Road, Mumbai 400005 (India), University of British Columbia, Department of Mathematics, Vancouver (Canada)},

author = {Ghate, Eknath, Vatsal, Vinayak},

journal = {Annales de l'Institut Fourier},

keywords = {$\Lambda $-adic forms; $p$-adic families; ordinary primes; Galois representations; -adic forms; -adic families},

language = {eng},

number = {7},

pages = {2143-2162},

publisher = {Association des Annales de l'Institut Fourier},

title = {On the local behaviour of ordinary $\Lambda $-adic representations},

url = {http://eudml.org/doc/116170},

volume = {54},

year = {2004},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 54

IS - 7

SP - 2143

EP - 2162

AB - 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.

LA - eng

KW - $\Lambda $-adic forms; $p$-adic families; ordinary primes; Galois representations; -adic forms; -adic families

UR - http://eudml.org/doc/116170

ER -

## References

top- K. Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc 16 (2003), 29-55 Zbl1076.11029MR1937198
- K. Buzzard, R. Taylor, Companion forms and weight one forms, Ann. of Math 149 (1999), 905-919 Zbl0965.11019MR1709306
- R. Coleman, Classical and overconvergent modular forms, Invent. Math 124 (1996), 215-241 Zbl0851.11030MR1369416
- E. Ghate, On the local behaviour of ordinary modular Galois representations, Modular curves and abelian varieties volume 224 (2004), 105-124, Birkhäuser Zbl1166.11330
- E. Ghate, Ordinary forms and their local Galois representations Zbl1085.11029
- R. Greenberg, V. Vatsal, Iwasawa theory for Artin representations Zbl1032.11046
- H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup 19 (1986), 231-273 Zbl0607.10022MR868300
- H. Hida, Galois representations into $G{L}_{2}\left({\mathbb{Z}}_{p}\left[\left[X\right]\right]\right)$ attached to ordinary cusp forms, Invent. Math 85 (1986), 545-613 Zbl0612.10021MR848685
- H. Hida, Elementary Theory of $L$-functions and Eisenstein Series, 26 (1993), Cambridge University Press, Cambridge Zbl0942.11024MR1216135
- B. Mazur, J. Tilouine, Représentations galoisiennes, différentielles de Kähler et ``conjectures principales'', Inst. Hautes Études Sci. Publ. Math 71 (1990), 65-103 Zbl0744.11053MR1079644
- B. Mazur, A. Wiles, On $p$-adic analytic families of Galois representations, Compositio Math. 59 (1986), 231-264 Zbl0654.12008MR860140
- T. Miyake, Modular forms, (1989), Springer Verlag Zbl1159.11014MR1021004
- J.-P. Serre, Abelian $l$-adic representations and elliptic curves, (1989), Addison-Wesley Publishing Company, Redwood City, CA Zbl0709.14002MR1043865
- V. Vatsal, A remark on the 23-adic representation associated to the Ramanujan Delta function
- A. Wiles, On ordinary $\lambda $-adic representations associated to modular forms, Invent. Math. 94 (1988), 529-573 Zbl0664.10013MR969243

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.