Hasse-Weil zeta function of absolutely irreducible 𝑆𝐿₂-representations of the figure 8 knot group

Abstract

Weil-type zeta functions defined by the numbers of absolutely irreducible S L 2 \mathrm {SL}_2 -representations of the figure 8 8 knot group over finite fields are computed explicitly. They are expressed in terms of the congruence zeta functions of reductions of a certain elliptic curve defined over the rational number field. Then the Hasse-Weil type zeta function of the figure 8 8 knot group is also studied. Its central value is written in terms of the Mahler measures of the Alexander polynomial of the figure 8 8 knot and a certain family of elliptic curves.