Hasse-Weil zeta function of absolutely irreducible $\mathrm{SL}_{2}$-representations of the figure $8$ knot group

Shinya Harada
2011 Proceedings of the American Mathematical Society  
Weil-type zeta functions defined by the numbers of absolutely irreducible SL 2 -representations of the figure 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 knot group is also studied. Its central value is written in terms of the Mahler measures of the Alexander polynomial of the figure 8 knot and a certain family of elliptic curves.
doi:10.1090/s0002-9939-2011-10743-3 fatcat:6tw7gzq7rjgw7e73n5j34kpmg4