A Local-Global Study of Obstructed Deformation Problems II

Abstract: We continue the local-global study of obstructed deformation problems for two-dimensional residual Galois representations arising from weight $2$ newforms of level $N$, initiated in \cite{Bin26}. Using the Greenberg-Wiles formula and the explicit classification of inertial Weil-Deligne types due to Dembélé-Freitas-Voight \cite{DFV22}, we systematically compute the local obstruction groups $H^0(G_p, \bar{\varepsilon}\ell \otimes \mathrm{ad}^0\barρ)$ for every inertial type arising at primes $p$ with $p^2 \mid N$ and $p \neq \ell$. For each type and in each of three arithmetic cases ($p \not\equiv \pm 1$, $p \equiv -1$, and $p \equiv 1 \pmod{\ell}$), we give the dimension of the local obstruction group and an explicit presentation of the universal deformation ring as a power series ring over the Witt vectors modulo explicit relations. We treat in detail the twisted Steinberg case ($τ\simeq τ{\mathrm{St},p} \otimes \varepsilon_p$), the principal series cases, and the non-exceptional supercuspidal cases, including the full family of types at $p = 3$.