ON SHARIFI'S CONJECTURE: EXCEPTIONAL CASE

In the present article, we study the conjecture of Sharifi on the surjectivity of the map ωθ. Here θ is a primitive even Dirichlet character of conductor Np, which is exceptional in the sense of Ohta. After localizing at the prime ideal p of the Iwasawa algebra related to the trivial zero of the Kubota- Leopoldt p-adic L-function Lp(s, θ-1ω2), we compute the image of ωθ,p in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization.