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

Publication date: 24 December 2015

Abstract: It is proved that the Hasse-Weil zeta functions of the canonical components of the

{m S L}_{2}

(

{m P S L}_{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

{m P S L}_{2}

-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.

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