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arXiv · 1608.07236

Derived Galois deformation rings

Soren Galatius·Akshay Venkatesh
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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.

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Soren Galatius, Akshay Venkatesh. 2016-08-25. Derived Galois deformation rings. https://doi.org/10.1016/j.aim.2017.08.016

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