Diophantine analysis and the Braid group ${\bf B}_3$

Given a finite dimensional representation $\pi$ of a finitely generated group $G=\langle g_1, \ldots, g_n\rangle$, the associated characteristic polynomial is defined as $Q_\pi(z):=\det(z_0I+z_1\pi(g_1)+\cdots +z_n\pi(g_n))$, and it is known to contain a good amount of structural information about $G$ and $\pi$.

This paper is a part of an ongoing project to investigate the number-theoretic properties of the algebraic varieties (called {\em eigensurfaces}) $\{z\in \mathbb{C}^{n+1}: Q_\pi(z)=0\}$.

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.