A Galois deformation ring is the ring that represents the deformation functor $\mathrm{Def}_{\overline{\rho}}$ that assigns to a complete Noetherian local $W(\mathbb{F})$-algebra $A$ the set of deformations (equivalence classes of lifts) of a fixed residual representation $\overline{\rho}:\Gal(\overline{F}/F)\to GL_{n}(\mathbb{F})$ to $A$.
They were first introduced by Mazur, and find application in modularity, i.e. showing that a Galois representation comes from a modular form. As such they are instrumental in the proof of the Taniyama-Shimura conjecture.
Let $F$ be a number field, let $\mathbb{F}$ be a finite field, with ring of Witt vectors $W(\mathbb{F})$, and let $\overline{\rho} \colon \Gal(\overline{F}/F)\to GL_{n}(\mathbb{F})$ be a linear representation. In reference to $\mathbb{F}$ being the residue field of $W(\mathbb{F})$ we refer to $\overline{\rho}$ as a residual representation.
Let $A$ be a complete Noetherian $W(\mathbb{F})$-algebra. A lift, or framed deformation, of $\overline{\rho}$ is a Galois representation $\rho \colon \Gal(\overline{F}/F)\to GL_{n}(A)$ such that reduction of $\rho$ by the unique maximal ideal $\mathfrak{m}$ of $A$ gives back $\overline{\rho}$.
A deformation is an equivalence class of lifts, where two lifts are considered equivalent if they are conjugate by an element of the kernel of the reduction map.
The framed deformation functor $\mathrm{Def}_{\overline{\rho}}^{\Box}$ is the functor assigns to a complete Noetherian local algebra $A$ the set of lifts of $\overline{\rho}$ to $A$.
The framed deformation functor $\mathrm{Def}_{\overline{\rho}}^{\Box}$ (Def. ) is represented by a lift $\rho(R_{\overline{\rho}}^{\Box})$. We refer to the ring $R_{\overline{\rho}}^{\Box}$ as the universal framed deformation ring.
The deformation functor $\mathrm{Def}_{\overline{\rho}}^{\Box}$ is the functor assigns to a complete Noetherian local algebra $A$ the set of lifts of $\overline{\rho}$ to $A$
We say that $\overline{\rho}$ is Schur if $\mathrm{End}_{\mathbb{F}[\mathrm{Gal}(\overline{F}/F)]}\overline{\rho}=\mathbb{F}$.
Let $\overline{\rho}$ be Schur. Then the deformation functor $\mathrm{Def}_{\overline{\rho}}$ is represented by a deformation $\rho(R_{\overline{\rho}}^{\Box})$. We refer to the ring $R_{\overline{\rho}}$ as the universal deformation ring.
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A version of Galois deformation rings in derived geometry has been developed by Soren Galatius and Akshay Venkatesh.
Toby Gee, Modularity Lifting Theorems: Notes for Arizona Winter School, pdf
Soren Galatius and Akshay Venkatesh, Derived Galois Deformation Rings, arxiv:1608.07236
Last revised on July 3, 2022 at 01:44:52. See the history of this page for a list of all contributions to it.