On the infinite fern of Galois representations of unitary type

Gaëtan Chenevier

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 6, page 963-1019
  • ISSN: 0012-9593

Abstract

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Let E be a CM number field, p an odd prime totally split in  E , and let  X be the p -adic analytic space parameterizing the isomorphism classes of  3 -dimensional semisimple p -adic representations of  Gal ( E ¯ / E ) satisfying a selfduality condition “of type U ( 3 ) ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in  X has dimension at least 3 [ E : ℚ ] . As important steps, and in any rank, we prove that any first order deformation of a generic enough crystalline representation of  Gal ( ℚ ¯ p / ℚ p ) is a linear combination of trianguline deformations, and that unitary eigenvarieties are étale over weight space at the non-critical classical points. As another application, we give a surjectivity criterion for the localization at  p of theadjoint ' Selmer group (Pronounce “adjoint primed Selmer group.”) of a p -adic Galois representation attached to a cuspidal cohomological automorphic representation of  GL n ( 𝔸 E ) of type U ( n ) (for any  n ).

How to cite

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Chenevier, Gaëtan. "On the infinite fern of Galois representations of unitary type." Annales scientifiques de l'École Normale Supérieure 44.6 (2011): 963-1019. <http://eudml.org/doc/272123>.

@article{Chenevier2011,
abstract = {Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\{\rm Gal\}(\overline\{E\}/E)$ satisfying a selfduality condition “of type $\{\rm U\}(3)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb \{Q\}]$. As important steps, and in any rank, we prove that any first order deformation of a generic enough crystalline representation of $\{\rm Gal\}(\overline\{\mathbb \{Q\}\}_p/\mathbb \{Q\}_p)$ is a linear combination of trianguline deformations, and that unitary eigenvarieties are étale over weight space at the non-critical classical points. As another application, we give a surjectivity criterion for the localization at $p$ of theadjoint$^\{\prime \}$Selmer group (Pronounce “adjoint primed Selmer group.”) of a $p$-adic Galois representation attached to a cuspidal cohomological automorphic representation of $\{\rm GL\}_n(\mathbb \{A\}_E)$ of type $\{\rm U\}(n)$ (for any $n$).},
author = {Chenevier, Gaëtan},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Galois representation; automorphic form; unitary group; trianguline; infinite fern; eigenvariety; Selmer group},
language = {eng},
number = {6},
pages = {963-1019},
publisher = {Société mathématique de France},
title = {On the infinite fern of Galois representations of unitary type},
url = {http://eudml.org/doc/272123},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Chenevier, Gaëtan
TI - On the infinite fern of Galois representations of unitary type
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 6
SP - 963
EP - 1019
AB - Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of ${\rm Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type ${\rm U}(3)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb {Q}]$. As important steps, and in any rank, we prove that any first order deformation of a generic enough crystalline representation of ${\rm Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)$ is a linear combination of trianguline deformations, and that unitary eigenvarieties are étale over weight space at the non-critical classical points. As another application, we give a surjectivity criterion for the localization at $p$ of theadjoint$^{\prime }$Selmer group (Pronounce “adjoint primed Selmer group.”) of a $p$-adic Galois representation attached to a cuspidal cohomological automorphic representation of ${\rm GL}_n(\mathbb {A}_E)$ of type ${\rm U}(n)$ (for any $n$).
LA - eng
KW - Galois representation; automorphic form; unitary group; trianguline; infinite fern; eigenvariety; Selmer group
UR - http://eudml.org/doc/272123
ER -

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