Finiteness of unramified deformation rings

TL;DR

This paper proves that certain universal unramified deformation rings of Galois representations are finite over their base rings in many cases, advancing understanding of deformation theory in number theory.

Contribution

It establishes finiteness of universal unramified deformation rings and their relation to local deformation rings under specific conditions.

Findings

Universal unramified deformation rings are finite over W(k) in many cases.

Universal deformation rings are finite over local deformation rings.

Results apply to Galois representations over totally real fields.

Abstract

We prove that the universal unramified deformation ring $R^{\mathrm{unr}}$ of a continuous Galois representation $\overline{\rho}: G_{F^{+}} \rightarrow \mathrm{GL}_n(k)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O} = W(k)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{\mathrm{univ}}$ is finite over the local deformation ring $R^{\mathrm{loc}}$.