Let $T_E=W^2$ be a rank $2$ crystalline $G_{\mathbb{Q}_p}$-representation of weights $[0,1]$ with non-ordinary reduction where $W$ is the ring of integers of some extension of $\mathbb{Q}_p$, and let $\bar{T}_E$ be its residual representation. Suppose $l\ge 2$ and fix some big enough $N$ which only depends on $T_E$. We show that the group $\mathrm{Ext}^1_{cr, [0,l-1]}(\bar{T}_E, \bar{T}_E)$ (Definition 2.30) of extensions with crystalline liftings of weights $[0,l-1]$, which are themselves extensions of $G_{\mathbb{Q}_p}$-representations which are congruent to $T_E \pmod {p^N}$, is isomorphic to the group of finite flat extensions $\mathrm{Ext}^1_{fl}(\bar{T}_E, \bar{T}_E)$ ([18, Chapter 1.1]). In addition, we construct a certain functor $\mathcal{D}$ of deformations of $\bar{T}_E$ with liftings of certain type and weights $[0,l-1]$, satisfying certain congruences with $T_E$, show $\mathcal{D}$ has a representable hull, and demonstrate some evidence that $t_{\mathcal{D}} \subset \mathrm{Ext}^1_{cr, [0,l-1]}(\bar{T}_E, \bar{T}_E)$ and $ V_{\mathbf{T}_{\mathfrak{m}}} \otimes W/\mathfrak{m}_W \in {\mathcal{D}}(\mathbf{T}_{\mathfrak{m}} \otimes W/\mathfrak{m}_W)$ where $\mathbf{T}$ is the Hecke algebra $\mathbf{T}_l(\Gamma _1(M))$, $\mathbf{m}$ is its maximal ideal given by a weight $l$ eigenform of level $\Gamma _1(M)$ whose Galois representation is congruent modulo $p^N$ to $T_E$, and $V_{\mathbf{T}_{\mathfrak{m}}}$ is its associated Galois representation.
Soit $W$ l’anneau des entiers d’une extension de $\mathbb{Q}_p$, et soit $T_E = W^2$ une représentation cristalline de rang 2 de $G_{\mathbb{Q}_p}$ à poids de Hodge–Tate $[0,1]$ ayant réduction non ordinaire. On note $\bar{T}_E$ la représentation résiduelle de $T_E.$ Soit $l \ge 2$ et soit $N$ un entier fixé suffisamment grand, qui ne dépend que de $T_E$. Nous montrons que le groupe $\mathrm{Ext}^1_{cr, [0,l-1]}(\bar{T}_E, \bar{T}_E)$ d’extensions admettant des relèvements cristallins de poids $[0,l-1]$ qui sont eux-mêmes extensions de représentations de $G_{\mathbb{Q}_p}$ congrues à $T_E$ modulo ${p^N}$ (cf. Définition 2.30), est isomorphe au groupe d’extensions finies plates $\mathrm{Ext}^1_{fl}(\bar{T}_E, \bar{T}_E)$ (cf. [18, Chapitre 1.1]). En outre, nous construisons le foncteur $\mathcal{D}$ des déformations de $\bar{T}_E$ de poids $[0,l-1]$ ayant relèvements d’un certain type et satisfaisant certaines congruences avec $T_E$ et montrons que $\mathcal{D}$ admet une enveloppe représentable. Nous conjecturons que $t_{\mathcal{D}} \subset \mathrm{Ext}^1_{cr, [0,l-1]}(\bar{T}_E, \bar{T}_E)$ et $V_{\mathbf{T}_{\mathfrak{m}}} \otimes W/\mathfrak{m}_W \in {\mathcal{D}}(\mathbf{T}_{\mathfrak{m}} \otimes W/\mathfrak{m}_W)$, où $\mathbf{T}$ est l’algèbre de Hecke $\mathbf{T}_l(\Gamma _1(M))$, $\mathbf{m}$ son idéal maximal donné par une forme propre de poids $l$ et de niveau $\Gamma _1(M)$ dont la représentation galoisienne est congrue à $T_E$ modulo $p^N$, et $V_{\mathbf{T}_{\mathfrak{m}}}$ la représentation galoisienne associée. Enfin, nous donnons des résultats à l’appui de cette conjecture.
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Keywords: Local conditions of adjoint representations, Gorenstein Hecke algebras, patching, congruences of $p$-adic $L$-functions
Byoung Du (BD) Kim 1
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@article{JTNB_2025__37_1_325_0,
author = {Byoung Du (BD) Kim},
title = {Local conditions of adjoint representations with supersingular reduction, and representable functors of deformations with higher weight liftings},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {325--355},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {1},
doi = {10.5802/jtnb.1323},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1323/}
}
TY - JOUR AU - Byoung Du (BD) Kim TI - Local conditions of adjoint representations with supersingular reduction, and representable functors of deformations with higher weight liftings JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 325 EP - 355 VL - 37 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1323/ DO - 10.5802/jtnb.1323 LA - en ID - JTNB_2025__37_1_325_0 ER -
%0 Journal Article %A Byoung Du (BD) Kim %T Local conditions of adjoint representations with supersingular reduction, and representable functors of deformations with higher weight liftings %J Journal de théorie des nombres de Bordeaux %D 2025 %P 325-355 %V 37 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1323/ %R 10.5802/jtnb.1323 %G en %F JTNB_2025__37_1_325_0
Byoung Du (BD) Kim. Local conditions of adjoint representations with supersingular reduction, and representable functors of deformations with higher weight liftings. Journal de théorie des nombres de Bordeaux, Volume 37 (2025) no. 1, pp. 325-355. doi: 10.5802/jtnb.1323
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