Universal Deformations and p-adic L-functions

Abstract

In this thesis we study deformations of certain $2$-dimensional reducible representations whose image is in the Borel subgroup of $GL_2(\F)$. Our method of understanding the universal deformation ring is via the Jordan-H\"older factors of the residual representation. Using the vanishing of cup products of appropriate cohomology classes we can compute the tangent space of the universal deformation ring and some obstruction classes to lifting representations. In this process, we can also explicitly construct certain big meta-abelian extensions inside the fixed field of the kernel of the universal representation. We give an explicit example of our construction of an unramified extension in the case of elliptic curves of conductor $11$. We also give an Iwasawa theoretic description of various fields that are cut out by the universal representation. The Galois theoretic description of the constructed meta-abelian unramified extension is then later used as an ingredient for the isomorphism criterion in the modularity lifting results. When the isomorphism criterion is satisfied, we could prove some modularity lifting results allowing us to recover some results of Skinner-Wiles and prove a conjecture of Wake in this special case. We also show that the representations considered by Skinner-Wiles have big image inside the universal deformation ring.