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.
Categories: math.NT math.GT