Bloch cycles and the weak two-variable Chinburg conjecture

Let $-f<0$ be a fundamental discriminant, $w$ the number of roots of unity in $\mathbb{Q}(\sqrt{-f})$, and ${\chi }_{-f}$ the associated quadratic character. We prove that there is a polynomial $R_f\in \mathbb{Q}[x,y]$ such that $m(R_f)=4w{L}^{\mathrm{\prime }}({\chi }_{-f},-1)$. Hence the weak two-variable Chinburg conjecture holds.