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

2011
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Proceedings of the American Mathematical Society
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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