Abstract
We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$ . This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$ , we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$ , with Hodge–Tate weights in $[0,p]$ , are potentially diagonalizable.
Highlights
We establish new instances of potential diagonalizability, a notion introduced in [1], which concerns the geometry of crystalline deformation rings
If K /Qp is a finite extension, a p-adic representation of G K = Gal(K /K ) is potentially diagonalizable if, after possibly restricting to a finite extension of K, it lies on the same irreducible component of an appropriate crystalline deformation ring as a direct sum of characters
This goes part of the way to proving potential diagonalizability of ρ. It unclear whether or not ρ and ρ lie in the same irreducible component of a crystalline deformation ring. We address this by proving that two such crystalline representations ρ and ρ, with Hodge–Tate weights contained in [0, p], are contained in the same irreducible component of a crystalline deformation ring if the Breuil–Kisin modules associated with ρ and ρ are congruent modulo p
Summary
The starting point of this paper is the results of Gee–Liu–Savitt [16] They consider, for K /Qp unramified, a lattice ρ : G K → GL2(Zp) inside a crystalline representation with Hodge–Tate weights in [0, p]. We address this by proving that two such crystalline representations ρ and ρ , with Hodge–Tate weights contained in [0, p], are contained in the same irreducible component of a crystalline deformation ring if the Breuil–Kisin modules associated with ρ and ρ are congruent modulo p This is deduced from the following theorem, which summarizes the key new results in this paper. Part (2) implies any crystalline deformation x of VF to O with Hodge–Tate weights contained in [0, p], where O is the ring of integers inside a finite extension E/Qp, corresponds to an O-valued point of Spec Rcryps, which factors uniquely through Lcryps.
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