Local conditions of adjoint representations with supersingular reduction, and representable functors of deformations with higher weight liftings

Byoung Du (BD) Kim

doi:10.5802/jtnb.1323

Byoung Du (BD) Kim ¹

¹ School of Mathematics and Statistics Victoria University of Wellington Wellington, New Zealand

Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 325-355.

Abstract (VO)
Résumé

Let $T_{E} = W^{2}$ be a rank $2$ crystalline $G_{Q_{p}}$ -representation of weights $[0, 1]$ with non-ordinary reduction where $W$ is the ring of integers of some extension of $Q_{p}$ , and let ${\bar{T}}_{E}$ be its residual representation. Suppose $l \geq 2$ and fix some big enough $N$ which only depends on $T_{E}$ . We show that the group ${Ext}_{c r, [0, l - 1]}^{1} ({\bar{T}}_{E}, {\bar{T}}_{E})$ (Definition 2.30) of extensions with crystalline liftings of weights $[0, l - 1]$ , which are themselves extensions of $G_{Q_{p}}$ -representations which are congruent to $T_{E} (\mod p^{N})$ , is isomorphic to the group of finite flat extensions ${Ext}_{f l}^{1} ({\bar{T}}_{E}, {\bar{T}}_{E})$ ([18, Chapter 1.1]). In addition, we construct a certain functor $D$ of deformations of ${\bar{T}}_{E}$ with liftings of certain type and weights $[0, l - 1]$ , satisfying certain congruences with $T_{E}$ , show $D$ has a representable hull, and demonstrate some evidence that $t_{D} \subset {Ext}_{c r, [0, l - 1]}^{1} ({\bar{T}}_{E}, {\bar{T}}_{E})$ and $V_{T_{m}} \otimes W / m_{W} \in D (T_{m} \otimes W / m_{W})$ where $T$ is the Hecke algebra $T_{l} (Γ_{1} (M))$ , $m$ is its maximal ideal given by a weight $l$ eigenform of level $Γ_{1} (M)$ whose Galois representation is congruent modulo $p^{N}$ to $T_{E}$ , and $V_{T_{m}}$ is its associated Galois representation.

Soit $W$ l’anneau des entiers d’une extension de $Q_{p}$ , et soit $T_{E} = W^{2}$ une représentation cristalline de rang 2 de $G_{Q_{p}}$ à poids de Hodge–Tate $[0, 1]$ ayant réduction non ordinaire. On note ${\bar{T}}_{E}$ la représentation résiduelle de $T_{E} .$ Soit $l \geq 2$ et soit $N$ un entier fixé suffisamment grand, qui ne dépend que de $T_{E}$ . Nous montrons que le groupe ${Ext}_{c r, [0, l - 1]}^{1} ({\bar{T}}_{E}, {\bar{T}}_{E})$ d’extensions admettant des relèvements cristallins de poids $[0, l - 1]$ qui sont eux-mêmes extensions de représentations de $G_{Q_{p}}$ congrues à $T_{E}$ modulo $p^{N}$ (cf. Définition 2.30), est isomorphe au groupe d’extensions finies plates ${Ext}_{f l}^{1} ({\bar{T}}_{E}, {\bar{T}}_{E})$ (cf. [18, Chapitre 1.1]). En outre, nous construisons le foncteur $D$ 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 $D$ admet une enveloppe représentable. Nous conjecturons que $t_{D} \subset {Ext}_{c r, [0, l - 1]}^{1} ({\bar{T}}_{E}, {\bar{T}}_{E})$ et $V_{T_{m}} \otimes W / m_{W} \in D (T_{m} \otimes W / m_{W})$ , où $T$ est l’algèbre de Hecke $T_{l} (Γ_{1} (M))$ , $m$ son idéal maximal donné par une forme propre de poids $l$ et de niveau $Γ_{1} (M)$ dont la représentation galoisienne est congrue à $T_{E}$ modulo $p^{N}$ , et $V_{T_{m}}$ la représentation galoisienne associée. Enfin, nous donnons des résultats à l’appui de cette conjecture.

Reçu le : 2023-11-24
Révisé le : 2024-11-12
Accepté le : 2024-12-28
Publié le : 2025-06-23

DOI : 10.5802/jtnb.1323

Classification : 11S25, 11R34, 12G05, 11F85, 11G99
Keywords: Local conditions of adjoint representations, Gorenstein Hecke algebras, patching, congruences of

p

-adic

L

-functions

Affiliations des auteurs :

Byoung Du (BD) Kim ¹

¹ School of Mathematics and Statistics Victoria University of Wellington Wellington, New Zealand

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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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, Tome 37 (2025) no. 1, pp. 325-355. doi : 10.5802/jtnb.1323. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1323/

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