Contents

# Contents

## 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.

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## Derived version

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