Hasse-Weil zeta functions of ${\rm SL}_2$-character varieties of closed orientable hyperbolic $3$-manifolds

TL;DR

This paper proves that the Hasse-Weil zeta functions of certain character varieties of closed hyperbolic 3-manifolds are equal to Dedekind zeta functions of their trace fields, linking algebraic and geometric invariants.

Contribution

It establishes a precise relationship between Hasse-Weil zeta functions of character varieties and Dedekind zeta functions, including special value formulas for arithmetic manifolds.

Findings

Hasse-Weil zeta functions equal Dedekind zeta functions of trace fields

Special value at s=2 relates to hyperbolic volume for arithmetic manifolds

Results connect algebraic geometry with hyperbolic geometry

Abstract

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.