Derived Galois deformation rings

TL;DR

This paper introduces a derived version of Mazur's Galois deformation ring, a pro-simplicial ring that generalizes classical deformation rings and connects to the Langlands program and homology actions.

Contribution

It defines a derived Galois deformation ring that extends classical deformation rings to a homotopical setting, providing new tools for understanding Galois representations.

Findings

Derived deformation rings recover classical rings at zeroth homotopy.

Evidence suggests these rings act on the homology of arithmetic groups.

Taylor--Wiles method can upgrade actions to graded actions of the entire ring spectrum.

Abstract

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $\pi_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $\pi_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $\pi_* \mathcal{R}$ on the homology.