A Local-Global Study of Obstructed Deformation Problems -- I

Abstract

We study obstructed deformation problems for two-dimensional residual Galois representations arising from weight~$2$ newforms of level~$N$. Using Poitou-Tate duality, we isolate local and global sources of obstructions and give concrete criteria for when they occur. In several cases we also describe the resulting universal deformation ring explicitly.