  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.

Number Theory