A Local-Global Study of Obstructed Deformation Problems II

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 $H^0(G_p,{\bar{\epsilon }}_{\mathrm{\ell }}\otimes {\mathrm{a}\mathrm{d}}^0\bar{\rho })$ for every inertial type arising at primes $p$ with $p^2\mid N$ and $p\mathrm{\ne }\mathrm{\ell }$. For each type and in each of three arithmetic cases ($p\equiv ̸\pm 1$, $p\equiv -1$, and $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell })$), 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 ($\tau \simeq {\tau }_{\mathrm{S}\mathrm{t},p}\otimes {\epsilon }_p$), the principal series cases, and the non-exceptional supercuspidal cases, including the full family of types at $p=3$.