Level compatibility in Sharifi's conjecture

Sharifi has constructed a map from the first homology of the modular curve $X_1(M)$ to the $K$-group $K_2(\mathbf{Z}[\zeta_M, \frac{1}{M}])$, where $\zeta_M$ is a primitive $M$th root of unity. We study how these maps relate when $M$ varies. Our method relies on the techniques developed by Sharifi and Venkatesh.