Bloch 循环与弱二元 Chinburg 猜想

中文摘要 AI 辅助

设 $-f<0$ 为基本判别式，$w$ 为 $\mathbb{Q}(\sqrt{-f})$ 中单位根的个数，$\chi_{-f}$ 为相应的二次特征。我们证明存在多项式 $R_f\in\mathbb{Q}[x,y]$ 使得 $m(R_f)=4wL'(\chi_{-f},-1)$。因此，弱二元 Chinburg 猜想成立。

英文摘要

For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ 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 $χ_{-f}$ 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 $\mathbb{Q}(\sqrt{-f})$.

发表机构

• Nanjing University（南京大学）
• Hefei University of Technology（合肥工业大学）

机构由 AI 辅助整理，请以论文原文为准。