Explicit non-Gorenstein R = T via rank bounds I:deformation theory

Ribet has proven remarkable results about non-optimal levels of residually reducibleGalois representations. We focus on a non-optimal levelNthat is the product oftwo distinct primes and where the Galois deformation ring is not expected to beGorenstein. We prove a Galois-theoretic criterion for the deformation ring to be assmall as possible-that is, for there to be auniquenewform of level N with reducible residual representation. When this criterion is satisfied, we deduce an R = T theorem.