On the local behaviour of ordinary Λ -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

Abstract

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Let f be a primitive cusp form of weight at least 2, and let ρ f be the p -adic Galois representation attached to f . If f is p -ordinary, then it is known that the restriction of ρ 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 Λ -adic forms, under similar conditions.

How to cite

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Ghate, 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

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  15. A. Wiles, On ordinary λ -adic representations associated to modular forms, Invent. Math. 94 (1988), 529-573 Zbl0664.10013MR969243

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