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

Author:
Shinya Harada

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3115-3125

MSC (2010):
Primary 11S40; Secondary 14G10, 57M27

DOI:
https://doi.org/10.1090/S0002-9939-2011-10743-3

Published electronically:
January 21, 2011

MathSciNet review:
2811266

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Abstract: Weil-type zeta functions defined by the numbers of absolutely irreducible -representations of the figure 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 knot group is also studied. Its central value is written in terms of the Mahler measures of the Alexander polynomial of the figure knot and a certain family of elliptic curves.

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Additional Information

**Shinya Harada**

Affiliation:
School of Mathematics, Korea Institute for Advanced Study (KIAS), 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130-722, Republic of Korea

Email:
harada@kias.re.kr

DOI:
https://doi.org/10.1090/S0002-9939-2011-10743-3

Keywords:
Hasse-Weil zeta function,
modular representation,
topological invariant,
figure $8$ knot

Received by editor(s):
June 22, 2010

Received by editor(s) in revised form:
August 13, 2010

Published electronically:
January 21, 2011

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.