Effect of increasing the ramification on pseudo-deformation rings

Shaunak V. Deo

doi:10.5802/jtnb.1198

Shaunak V. Deo ¹

¹ Department of Mathematics Indian Institute of Science Bangalore 560012, India

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 189-236.

Abstract
Résumé

Given a continuous, odd, semi-simple $2$ -dimensional representation of $G_{ℚ, N p}$ over a finite field of odd characteristic $p$ and a prime $ℓ$ not dividing $N p$ , we study the relation between the universal deformation rings of the corresponding pseudo-representations for the groups $G_{ℚ, N ℓ p}$ and $G_{ℚ, N p}$ . As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and Böckle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and $p$ does not divide $ℓ^{2} - 1$ . When the pseudo-representation is unobstructed and $p$ divides $ℓ + 1$ , we prove that the universal deformation rings in characteristic $0$ and $p$ of the pseudo-representation for $G_{ℚ, N ℓ p}$ are not local complete intersection rings. As an application of our main results, we prove a big $R = 𝕋$ theorem.

Etant donnée une représentation continue, impaire et semi-simple de dimension $2$ de $G_{ℚ, N p}$ sur un corps fini de caractéristique impaire $p$ et un nombre premier $ℓ$ ne divisant pas $N p$ , nous étudions la relation entre les anneaux de déformation universels des pseudo-représentations correspondantes pour les groupes $G_{ℚ, N ℓ p}$ et $G_{ℚ, N p}$ . Nous nous intéressons aussi au problème connexe de savoir si la pseudo-représentation universelle provient d’une véritable représentation sur l’anneau de déformation universel. Sous certaines hypothèses, nous prouvons des analogues des théorèmes de Boston et Böckle pour les anneaux de pseudo-déformation réduits. Nous améliorons ces résultats dans le cas où la pseudo-représentation est non obstruée et $p$ ne divise pas $ℓ^{2} - 1$ . Lorsque la pseudo-représentation est non obstruée et $p$ divise $ℓ + 1$ , nous prouvons que les anneaux de déformation universels de la pseudo-représentation de $G_{ℚ, N ℓ p}$ en caractéristique $0$ et $p$ ne sont pas des anneaux locaux d’intersection complète. Comme application de nos résultats principaux, nous prouvons un théorème $R = 𝕋$ pour les algèbres de Hecke élargies et les anneaux de pseudo-représentations.

Received: 2020-09-14
Revised: 2021-12-08
Accepted: 2022-01-30
Published online: 2022-07-07

DOI: 10.5802/jtnb.1198

Classification: 11F80, 11F70, 11F33, 13H10
Mots-clés : pseudo-representations, deformation of Galois representations, structure of deformation rings

Author's affiliations:

Shaunak V. Deo ¹

¹ Department of Mathematics Indian Institute of Science Bangalore 560012, India

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Shaunak V. Deo. Effect of increasing the ramification on pseudo-deformation rings. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 189-236. doi : 10.5802/jtnb.1198. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1198/

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