The weak Chinburg conjecture on Mahler measures

Abstract

For every negative fundamental discriminant $-f$ and every $k\ge 1$, we construct a rational function $R_{f,2k}\in \mathbb{Q}(x_1,\dots ,x_{2k})$ and a constant $r_{f,2k}\in {\mathbb{Q}}^{\times }$ such that \[ m(R_{f,2k})=r_{f,2k}L'(χ_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and ${\chi }_{-f}$ is the quadratic Dirichlet character associated with $-f$. This proves the weak Chinburg conjecture. An independent construction using Bloch cycles yields a stronger result in two variables: there exists a nonzero polynomial $R_f\in \mathbb{Q}[x,y]$ satisfying \[ m(R_f)=4wL'(χ_{-f},-1), \] where $w$ is the number of roots of unity in $\mathbb{Q}(\sqrt{-f})$.