COLENGTH ONE DEFORMATION RINGS

Daniel Le; Bao V. Le Hung; Stefano Morra; Chol Park; Zicheng Qian

doi:10.1090/tran/9191

COLENGTH ONE DEFORMATION RINGS

Daniel Le, Bao V. Le Hung, Stefano Morra, Chol Park, Zicheng Qian

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let K/Qp be a finite unramified extension, ρ : Gal(Q_p/K) → GLn(Fp) a continuous representation, and τ a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on τ, the potentially crystalline deformation ring R_ρ^η,τ in parallel Hodge–Tate weights η = (n − 1, · · ·, 1, 0) and inertial type τ when the shape of ρ with respect to τ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].

Original language	English (US)
Pages (from-to)	5749-5786
Number of pages	38
Journal	Transactions of the American Mathematical Society
Volume	377
Issue number	8
DOIs	https://doi.org/10.1090/tran/9191
State	Published - Aug 2024

Funding

The first author was supported by the National Science Foundation under agreements Nos. DMS-1128155 and DMS-1703182 and an AMS-Simons travel grant. The second author acknowledges support from the National Science Foundation under grant Nos. DMS-1128155, DMS-1802037 and the Alfred P. Sloan Foundation. The third author was supported by the ANR-18-CE40-0026 (CLap CLap) and the Institut Universitaire de France. The fourth author was supported by Samsung Science and Technology Foundation under Project Number SSTFBA2001-02. Part of the work was carried out during a visit at the Universit\u00E0 degli Studi di Padova (2019), which we would like to heartily thank for the excellent working conditions which provided to us. The first author was supported by the National Science Foundation under agreements Nos. DMS-1128155 and DMS-1703182 and an AMS-Simons travel grant. The second author acknowledges support from the National Science Foundation under grant Nos. DMS-1128155, DMS-1802037 and the Alfred P. Sloan Foundation. The third author was supported by the ANR-18-CE40-0026 (CLap CLap) and the Institut Universitaire de France. The fourth author was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2001-02.

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/tran/9191

Cite this

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title = "COLENGTH ONE DEFORMATION RINGS",

abstract = "Let K/Qp be a finite unramified extension, ρ : Gal(Qp/K) → GLn(Fp) a continuous representation, and τ a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on τ, the potentially crystalline deformation ring Rρη,τ in parallel Hodge–Tate weights η = (n − 1, · · ·, 1, 0) and inertial type τ when the shape of ρ with respect to τ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre{\textquoteright}s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [M{\'e}m. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].",

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AU - Le Hung, Bao V.

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N2 - Let K/Qp be a finite unramified extension, ρ : Gal(Qp/K) → GLn(Fp) a continuous representation, and τ a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on τ, the potentially crystalline deformation ring Rρη,τ in parallel Hodge–Tate weights η = (n − 1, · · ·, 1, 0) and inertial type τ when the shape of ρ with respect to τ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].

AB - Let K/Qp be a finite unramified extension, ρ : Gal(Qp/K) → GLn(Fp) a continuous representation, and τ a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on τ, the potentially crystalline deformation ring Rρη,τ in parallel Hodge–Tate weights η = (n − 1, · · ·, 1, 0) and inertial type τ when the shape of ρ with respect to τ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].

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COLENGTH ONE DEFORMATION RINGS

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