G-Valued Galois Deformation Rings when $\ell \neq p$

TL;DR

This paper investigates the structure of G-valued Galois deformation rings over local fields where the residue characteristic differs from the prime p, revealing their regularity and equidimensionality properties.

Contribution

It establishes the regularity and equidimensionality of the generic fiber of G-valued Galois deformation rings for certain smooth group schemes over extensions of Z_p.

Findings

The generic fiber of the deformation ring admits a regular dense open locus.

The deformation ring is equidimensional of dimension equal to that of G.

The results apply to Galois representations over local fields with residue characteristic not equal to p.

Abstract

For a smooth group scheme $G$ over an extension of $\mathbf{Z}_p$ such that the generic fiber of $G$ is reductive, we study the generic fiber of the Galois deformation ring for a $G$-valued mod $p$ representation of the absolute Galois group of a finite extension of $\mathbf{Q}_\ell$ with $\ell \neq p$. In particular, we show it admits a regular dense open locus, and that it is equidimensional of dimension $\dim G$.