Level compatibility in Sharifi’s conjecture

Abstract Romyar Sharifi has constructed a map $\varpi _M$ from the first homology of modular curve $X_1(M)$ to K -group $K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M+\zeta _M^{-1}, \frac {1}{M}]) \otimes \operatorname {Z}}}[1/2]$ , where $\zeta is primitive M th root unity. conjectured that annihilated by certain Eisenstein ideal. Fukaya and Kato proved this conjecture after tensoring with $\operatorname {Z}}}_p$ for prime $p\geq 5$ dividing . More recently, Venkatesh Hecke operators away In note, we prove two main results. First, give relation between _{M'}$ when $M' \mid M$ Our method relies on techniques developed Venkatesh. We then use result in combination results order get property index