On Sharifi's conjecture: exceptional case

Abstract

In the present article, we study the conjecture of Sharifi on the surjectivity of the map ̟ θ . Here θ is a primitive even Dirichlet character of conductor N p, 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 Kupota-Leopoldt p-adic L-function Lp(s, θ −1 ω 2 ), we compute the image of ̟ θ,p in local Galois cohomology groups 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. Sharifi's ConjectureThroughout this paper, we will denote by N a positive integer, denote by p ≥ 5 a prime number not dividing N φ(N ). The goal of this section is to review Sharifi's conjecture on the map ̟ following [13] and to state Theorem 1.1. We refer the reader to loc. cit. for more details on Sharifi's conjecture.