Hasse-Weil zeta functions of \rm SL₂-character varieties of closed orientable hyperbolic 3-manifolds

2015/12/24 by Shinya Harada, Harada, Shinya
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1512.07747

openalex publication_date 2015/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is proved that the Hasse-Weil zeta functions of the canonical components of the \rm SL₂ (\rm PSL₂)-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 PSL₂-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.

Related

• Hasse-Weil zeta functions of SL₂-character varieties of arithmetic two bridge link complements
• Pure L-functions from algebraic geometry over finite fields
• Derived equivalence, Albanese varieties, and the zeta functions of\n 3-dimensional varieties
• Zeta Functions for the Adjoint Action of GL(n) and density of residues of Dedekind zeta functions
• Zeta functions in higher Teichmuller theory