Global Quadratic Units and Hecke Algebras

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

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