nLab Galois deformation ring

Idea

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.

Definitions

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.

Definition

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 $\overline{\rho}$.

Definition

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.

Definition

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 $A$.

Theorem

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.

Definition

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 $A$

Definition

We say that $\overline{\rho}$ is Schur if $\mathrm{End}_{\mathbb{F}[\mathrm{Gal}(\overline{F}/F)]}\overline{\rho}=\mathbb{F}$.

Theorem

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.

Tangent spaces and Galois cohomology

(…)

Derived version

A version of Galois deformation rings in derived geometry has been developed by Soren Galatius and Akshay Venkatesh.

Related

• deformation theory

References

• Toby Gee, Modularity Lifting Theorems: Notes for Arizona Winter School, pdf

• Soren Galatius and Akshay Venkatesh, Derived Galois Deformation Rings, arxiv:1608.07236