Global quadratic units and Hecke algebras

Abstract

Let $\{{\rho }_{\mathfrak{p}}{\}}_{\mathfrak{p}}$ be a compatible system of two dimansional $\mathfrak{p}$-adic Galois representations attached to a cusp form of Neben type $\left(\frac{D}{}\right)(D>0)$. We shall give an exact criterion, in terms of the fundamental unit $\epsilon$ of $\mathbb{Q}(\sqrt{D})$, determining primes $\mathfrak{p}$ for which the image of ${\rho }_{\mathfrak{p}}\mathrm{mod}\mathfrak{p}$ is dihedral. Then we shall state a conjecture which gives an explicit description of the universal $p$-ordinary deformation ring of such mod $\mathfrak{p}$ dihedral representations.