Finiteness of unramified deformation rings

Abstract

We prove that the universal unramified deformation ring $R^{\mathrm{unr}}$ of a continuous Galois representation $\overline{\rho }:G_{F^{+}}\to {\mathrm{GL}}_n(k)$ (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 $R^{\mathrm{loc}}$.

To appear in Algebra & Number Theory

References in corpus (2)

• On the automorphy of $l$-adic Galois representations with small residual image
• Adequate subgroups