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

Abstract

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\mathrm{SL}}_2$ (${\mathrm{PSL}}_2$)-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 ${\mathrm{PSL}}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.

25 pages, revision: Content of Section 3.1 (Corollary 3.2, Lemma 3.3) has been revised