Open AccessHasse-Weil zeta function of absolutely irreducible 𝑆𝐿₂-representations of the figure 8 knot group
Author(s) -
Shigeru Harada
Publication year - 2011
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/s0002-9939-2011-10743-3
Subject(s) - mathematics , knot (papermaking) , pure mathematics , elliptic curve , combinatorics , riemann zeta function , algebra over a field , engineering , chemical engineering
Weil-type zeta functions defined by the numbers of absolutely irreducible S L 2 \mathrm {SL}_2 -representations of the figure 8 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 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 8 knot and a certain family of elliptic curves.