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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hasse-Weil zeta function of absolutely irreducible $ \mathrm{SL}_2$-representations of the figure $ 8$ knot group


Author: Shinya Harada
Journal: Proc. Amer. Math. Soc. 139 (2011), 3115-3125
MSC (2010): Primary 11S40; Secondary 14G10, 57M27
Published electronically: January 21, 2011
MathSciNet review: 2811266
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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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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: http://dx.doi.org/10.1090/S0002-9939-2011-10743-3
PII: S 0002-9939(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.