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

Abstract

Given a finite dimensional representation $\ddot{I}€$ of a finitely generated group $G=⟨g_1,\dots ,g_n⟩$, the associated characteristic polynomial is defined as $Q_{\ddot{I}}€(z):=\det (z_{0}I+z_1\ddot{I}€(g_1)+\cdots +z_n\ddot{I}€(g_n))$, and it is known to contain a good amount of structural information about $G$ and $\ddot{I}€$. This paper is a part of an ongoing project to investigate the number-theoretic properties of the algebraic varieties (called {\em eigensurfaces}) $\{z\in {\mathbb{C}}^{n+1}:Q_{\ddot{I}}€(z)=0\}$. Its focus is the distribution of prime triples in the eigensurface $S:=\{z\in {\mathbb{C}}^3:(z_0+z_1+z_2{)}^2+z_0{z}_1=0\}$ associated with the braid group ${\mathbf{B}}_3$ 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.

24 pages, 1 figure