On a conjecture of Sharifi and Mazur's Eisenstein ideal

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur's Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf{Z}_p)$ for the complex conjugation. We give a conjectural explicit description of the group $I\cdot H_+/I^2\cdot H_+$ in terms of the second $K$-group of the cyclotomic field $\mathbf{Q}(\zeta_N)$. We prove that this conjecture follows from a conjecture of Sharifi about some Eisenstein ideal of level $\Gamma_1(N)$. Following the work of Fukaya--Kato, we prove partial results on Sharifi's conjecture. This allows us to prove partial results on our conjecture.