| Let f be a normalized eigenform of level Nℓalpha for some positive integer alpha and some odd prime ℓ satisfying gcd(ℓ, n) = 1. A construction of Deligne, Shimura, et al., attaches an ℓ-adic continuous two-dimensional Galois representation to f. The Refined Conjecture of Serre states that such a representation should in fact arise from a normalized eigenform of level prime to ℓ.;In this thesis we present a proof of Ribet which allows us to "strip" these powers of ℓ from the level while still retaining the original Galois representation, i.e., the residual of our new representation arising from level N will remain isomorphic to the residual of our original representation arising from level Nℓalpha. |
