  In the present article, we study the conjecture of Sharifi on the
surjectivity of the map $\varpi_{\theta}$. Here $\theta$ is a primitive even
Dirichlet character of conductor $Np$, which is exceptional in the sense of
Ohta. After localizing at the prime ideal $\mathfrak{p}$ of the Iwasawa algebra
related to the trivial zero of the Kubota\textendash Leopoldt $p$-adic
$L$-function $L_p(s,\theta^{-1}\omega^2)$, we compute the image of
$\varpi_{\theta,\mathfrak{p}}$ 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.
