Abstract
Abstract We establish the Bernstein-centre type of results for the category of mod p representations of $\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p)$ . We treat all the remaining open cases, which occur when p is $2$ or $3$ . Our arguments carry over for all primes p. This allows us to remove the restrictions on the residual representation at p in Lue Pan’s recent proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of $\operatorname {\mathrm {Gal}}(\overline {\mathbb Q}/\mathbb {Q})$ with equal Hodge–Tate weights.
Highlights
Our arguments carry over for all primes p. This allows us to remove the restrictions on the residual representation at p in Lue Pan’s recent proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of Gal(Q/Q) with equal Hodge–Tate weights
Lue Pan gave a new proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of Gal(Q/Q) with equal Hodge–Tate weights [38]
We show in Corollary 6.16 that Vinduces an antiequivalence of categories between BanaGd,mζ (L)Πfl and the category of modules of finite length over Rρζ ε, the universal deformation ring of ρ parameterising deformations of ρ with determinant equal to ζ ε to local Artinian L-algebras
Summary
Lue Pan gave a new proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of Gal(Q/Q) with equal Hodge–Tate weights [38]. We show in Corollary 6.16 that Vinduces an antiequivalence of categories between BanaGd,mζ (L)Πfl and the category of modules of finite length over Rρζ ε, the universal deformation ring of ρ parameterising deformations of ρ with determinant equal to ζ ε to local Artinian L-algebras Such results were known before for p ≥ 5 [39], and under assumptions on the reduction modulo p of a G-invariant lattice in Π if p = 2 or p = 3 [41]. We showed that Rtprsρ, ̄ζ ε [1/p] is normal, ZB [1/p] is reduced, and map (1) induces a bijection on maximal spectra m-Spec ZB [1/p] → m-Spec Rtprsρ, ̄ζ ε [1/p] and an isomorphism of the residue fields This is replaced by a different argument in the final version, which proves the first part of Theorem 1.4. We expect the rings Rρ [1/p] to be normal for any d-dimensional representation ρof GF , where F is a finite extension of Qp
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