Fine structure of even Galois deformation rings

Source abstract

For every even Galois representation in a family of prime conductor $\ell$, we prove that if $k \in \mathbb{Z}_{\geq 1}$ is maximal such that $$\ell \equiv 1 \bmod 3^k$$ then the even mod $3$ deformation ring $R_S \otimes_{\mathbb{Z}_3} \mathbb{F}_3$ of level $S=\{\ell,3\}$ does not have a quotient isomorphic to $\mathbb{F}_3[T]/(T^{2+3^{k-1}})$. As a corollary, we deduce that the global even deformation rings $R_S$ are finite and flat in the category of $\mathbb{Z}_3$-modules.