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

Harada, Shinya

doi:10.1090/S0002-9939-2011-10743-3

Hasse-Weil zeta function of absolutely irreducible $\mathrm {SL}_2$-representations of the figure $8$ knot group
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Proc. Amer. Math. Soc. 139 (2011), 3115-3125 Request permission

Abstract:

Weil-type zeta functions defined by the numbers of absolutely irreducible $\mathrm {SL}_2$-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.

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