AI 中文总结
研究辫群${\bf B}_3$及其约化Burau表示相关特征曲面中素数三元组分布，通过定义特征多项式，证明该曲面中素数三元组出现频率高于周围格点，揭示群表示理论与解析数论的联系。
AI 中文摘要
给定有限生成群$G=\langle g_1, \ldots, g_n\rangle$的有限维表示$\pi$，其相关特征多项式定义为$Q_\pi(z):=\det(z_0I+z_1\pi(g_1)+\cdots +z_n\pi(g_n))$。本文是研究代数簇（称为“特征曲面”）$\{z\in \mathbb{C}^{n+1}: Q_\pi(z)=0\}$数论性质项目的一部分。重点研究与辫群${\bf B}_3$及其约化Burau表示相关的特征曲面$S:=\{z\in \mathbb{C}^3: (z_0+z_1+z_2)^2+z_0z_1=0\}$中素数三元组的分布。证明了这些三元组在$S$上出现的频率高于在周围格点中，揭示了群表示理论与解析数论之间意外的联系。
英文摘要
Given a finite dimensional representation $π$ of a finitely generated group $G=\langle g_1, \ldots, g_n\rangle$, the associated characteristic polynomial is defined as $Q_π(z):=\det(z_0I+z_1π(g_1)+\cdots +z_nπ(g_n))$, and it is known to contain a good amount of structural information about $G$ and $π$. This paper is a part of an ongoing project to investigate the number-theoretic properties of the algebraic varieties (called {\em eigensurfaces}) $\{z\in \mathbb{C}^{n+1}: Q_π(z)=0\}$. Its focus is the distribution of prime triples in the eigensurface $S:=\{z\in \mathbb{C}^3: (z_0+z_1+z_2)^2+z_0z_1=0\}$ associated with the braid group ${\bf B}_3$ and its reduced Burau representation. We prove that such triples occur with higher frequency on $S$ than in the ambient lattice, revealing an unexpected connection between group representation theory and analytic number theory.
Comments24 pages, 1 figure
