  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.
