Effect of increasing the ramification on pseudo-deformation rings

Deo, Shaunak V.

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

Etant donnée une représentation continue, impaire et semi-simple de dimension $2$ de $G_{\mathbb{Q},Np}$ sur un corps fini de caractéristique impaire $p$ et un nombre premier $\ell$ ne divisant pas $Np$, nous étudions la relation entre les anneaux de déformation universels des pseudo-représentations correspondantes pour les groupes $G_{\mathbb{Q},N\ell p}$ et $G_{\mathbb{Q},Np}$. 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 ${\ell }^2-1$. Lorsque la pseudo-représentation est non obstruée et $p$ divise $\ell +1$, nous prouvons que les anneaux de déformation universels de la pseudo-représentation de $G_{\mathbb{Q},N\ell p}$ 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.

Given a continuous, odd, semi-simple $2$-dimensional representation of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$ and a prime $\ell$ not dividing $Np$, we study the relation between the universal deformation rings of the corresponding pseudo-representations for the groups $G_{\mathbb{Q},N\ell p}$ and $G_{\mathbb{Q},Np}$. 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 ${\ell }^2-1$. When the pseudo-representation is unobstructed and $p$ divides $\ell +1$, we prove that the universal deformation rings in characteristic $0$ and $p$ of the pseudo-representation for $G_{\mathbb{Q},N\ell p}$ are not local complete intersection rings. As an application of our main results, we prove a big $R=𝕋$ theorem.

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

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