A note on crystalline liftings in the $\protect \mathbb{Q}_p$ case

Gao, Hui

doi:10.24033/bsmf.2754

A note on crystalline liftings in the

ℚ_{p}

case
[Note sur les élévations cristallines dans le cas

ℚ_{p}

]

Gao, Hui ¹

¹ Department of Mathematics and Statistics, FIN-00014 University of Helsinki, Finland

Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 1, pp. 141-153

Abstract (VO)
Résumé

Let $p > 2$ be a prime. Let $ρ$ be a crystalline representation of $G_{ℚ_{p}}$ with distinct Hodge-Tate weights in $[0, p]$ , such that its reduction $\bar{ρ}$ is upper triangular. Under certain conditions, we prove that $\bar{ρ}$ has an upper triangular crystalline lift $ρ^{'}$ such that $HT (ρ^{'}) = HT (ρ)$ . The method is based on the author’s previous work, combined with an inspiration from the work of Breuil-Herzig.

Soit $p > 2$ un premier. Soit $ρ$ une représentation cristalline de $G_{ℚ_{p}}$ avec des poids distincts de Hodge-Tate dans $[0, p]$ , de telle sorte que sa réduction $\bar{ρ}$ soit triangulaire supérieure. Dans certaines conditions, nous prouvons que $\bar{ρ}$ a une élévation cristalline triangulaire supérieure $ρ^{'}$ telle que $HT (ρ^{'}) = HT (ρ)$ . La méthode est basée sur le travail antérieur de l’auteur, combiné avec une inspiration de l’oeuvre de Breuil-Herzig.

Reçu le : 2015-06-25
Accepté le : 2017-07-09
Publié le : 2018-07-21

MR Zbl

DOI : 10.24033/bsmf.2754

Classification : 11F80, 11F33
Keywords: Kisin modules, crystalline representations, modules de Kisin, représentations cristallines
Mots-clés : WARNING

Affiliations des auteurs :

Gao, Hui ¹

¹ Department of Mathematics and Statistics, FIN-00014 University of Helsinki, Finland

@article{BSMF_2018__146_1_141_0,
     author = {Gao, Hui},
     title = {A note on crystalline liftings in the $\protect \mathbb{Q}_p$ case},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {141--153},
     year = {2018},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {146},
     number = {1},
     doi = {10.24033/bsmf.2754},
     mrnumber = {3864872},
     zbl = {1446.11102},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2754/}
}

TY  - JOUR
AU  - Gao, Hui
TI  - A note on crystalline liftings in the $\protect \mathbb{Q}_p$ case
JO  - Bulletin de la Société Mathématique de France
PY  - 2018
SP  - 141
EP  - 153
VL  - 146
IS  - 1
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2754/
DO  - 10.24033/bsmf.2754
LA  - en
ID  - BSMF_2018__146_1_141_0
ER  -

%0 Journal Article
%A Gao, Hui
%T A note on crystalline liftings in the $\protect \mathbb{Q}_p$ case
%J Bulletin de la Société Mathématique de France
%D 2018
%P 141-153
%V 146
%N 1
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2754/
%R 10.24033/bsmf.2754
%G en
%F BSMF_2018__146_1_141_0

Gao, Hui. A note on crystalline liftings in the $\protect \mathbb{Q}_p$ case. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 1, pp. 141-153. doi: 10.24033/bsmf.2754

Bibliographie
Cité par

Breuil, Christophe; Herzig, Florian Ordinary representations of $G (ℚ_{p})$ and fundamental algebraic representations, Duke Math. J., Volume 164 (2015), pp. 1271-1352 | DOI | MR | Zbl

Caruso, Xavier; Liu, Tong Some bounds for ramification of $p^{n}$ -torsion semi-stable representations, J. Algebra, Volume 325 (2011), pp. 70-96 | DOI | MR | Zbl

Gao, Hui Crystalline liftings and the weight part of Serre’s conjecture, Israel J. Math., Volume 221 (2017), pp. 117-164 | DOI | MR | Zbl

Gee, Toby; Geraghty, David Companion forms for unitary and symplectic groups, Duke Math. J., Volume 161 (2012), pp. 247-303 | DOI | MR | Zbl

Gee, Toby; Liu, Tong; Savitt, David The Buzzard-Diamond-Jarvis conjecture for unitary groups, J. Amer. Math. Soc., Volume 27 (2014), pp. 389-435 | DOI | MR | Zbl

Gee, Toby; Liu, Tong; Savitt, David The weight part of Serre’s conjecture for $GL (2)$ , Forum Math. Pi, Volume 3 (2015), p. e2 | DOI | MR | Zbl

Gao, Hui; Sorensen, Claus Locally algebraic vectors in the Breuil-Herzig ordinary part, Manuscripta Math., Volume 151 (2016), pp. 113-131 | DOI | MR | Zbl

Kisin, Mark Crystalline representations and $F$ -crystals, Algebraic geometry and number theory (Progr. Math.), Volume 253, Birkhäuser, 2006, pp. 459-496 | DOI | MR | Zbl

Levin, Brandon $G$ -valued crystalline representations with minuscule $p$ -adic Hodge type, Algebra Number Theory, Volume 9 (2015), pp. 1741-1792 | DOI | MR | Zbl

Liu, Tong A note on lattices in semi-stable representations, Math. Ann., Volume 346 (2010), pp. 117-138 | DOI | MR | Zbl

Cité par Sources :