Global quadratic units and Hecke algebras (Q1271412)

Summary: Let \(\{\rho_{\mathfrak p}\}_{\mathfrak p}\) be a compatible system of two dimensional \({\mathfrak p}\)--adic Galois representations attached to a cusp form of nebentype \(({D\over })\) (\(D>0\)). The author gives an exact criterion, in terms of the fundamental unit \(\varepsilon\) of \({\mathbb Q}(\sqrt{D})\), determining primes \({\mathfrak p}\) for which the image of \(\rho_{\mathfrak p}\mod{\mathfrak p}\) is dihedral. Then he states a conjecture which gives an explicit description of the universal \(p\)-ordinary deformation ring of such mod \({\mathfrak p}\) dihedral representations.