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Knots and Primes pp 151–159Cite as

Torsions and the Iwasawa Main Conjecture

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Abstract

In this chapter we take up the Iwasawa main conjecture which asserts that the Iwasawa polynomial coincides essentially with the Kubota–Leopoldt p-adic analytic zeta function. According to the analogy between the Iwasawa polynomial and the Alexander polynomial in Chap. 11, we discuss geometric analogues of the Iwasawa main conjecture, namely, some relations between the Reidemeister–Milnor torsion and the Lefschetz or spectral zeta function.

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References

  1. Bayer, P., Neukirch, J.: On values of zeta functions and l-adic Euler characteristics. Invent. Math. 50(1), 35–64 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Coates, J., Perrin-Riou, B.: On p-adic L-functions attached to motives over ℚ. In: Algebraic Number Theory. Advanced Studies in Pure Mathematics, vol. 17, pp. 23–54. Academic Press, Boston (1989)

    Google Scholar 

  3. Coates, J., Sujatha, R.: Cyclotomic Fields and Zeta Values. Springer Monographs in Mathematics. Springer, Berlin (2006)

    MATH  Google Scholar 

  4. Fried, D.: Analytic torsion and closed geodesics on hyperbolic manifolds. Invent. Math. 84(3), 523–540 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Greenberg, R.: Iwasawa theory for p-adic representations. In: Algebraic Number Theory. Advanced Studies in Pure Mathematics, vol. 17, pp. 97–137. Academic Press, Boston (1989)

    Google Scholar 

  6. Hillman, J.: Algebraic Invariants of Links. Series on Knots and Everything, vol. 32. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  7. Kato, K.: Lectures on the approach to Iwasawa theory for Hasse–Weil L-functions via B dR . In: Arithmetic Algebraic Geometry. Lecture Note in Mathematics, vol. 1553, pp. 50–163. Springer, Berlin (1993)

    Chapter  Google Scholar 

  8. Kirk, P., Livingston, C.: Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants. Topology 38(3), 635–661 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kitanao, T., Goda, H., Morifuji, T.: Twisted Alexander Invariants. MSJ Memoir, vol. 5. Math. Soc. Jpn., Tokyo (2006) (in Japanese)

    Google Scholar 

  10. Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39(1), 19–55 (1976)

    MathSciNet  MATH  Google Scholar 

  11. Kubota, T., Leopoldt, H.: Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen. J. Reine Angew. Math. 214/215, 328–339 (1964)

    MathSciNet  Google Scholar 

  12. Lang, S.: Cyclotomic Fields I and II, Combined 2nd edn. Graduate Texts in Mathematics, vol. 121. Springer, New York (1990). With an appendix by Karl Rubin

    Book  MATH  Google Scholar 

  13. Lück, W.: Analytic and topological torsion for manifolds with boundary and symmetry. J. Differ. Geom. 37, 263–322 (1993)

    MATH  Google Scholar 

  14. Mazur, B.: Remarks on the Alexander polynomial. Unpublished Note (1963 or 1964)

    Google Scholar 

  15. Mazur, B., Wiles, A.: Class fields of abelian extensions of ℚ. Invent. Math. 76(2), 179–330 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Milnor, J.: Infinite cyclic coverings. In: Conference on the Topology of Manifolds, Michigan State Univ., E. Lansing, Mich., 1967, pp. 115–133. Prindle, Weber & Schmidt, Boston (1968)

    Google Scholar 

  17. Noguchi, A.: A functional equation for the Lefschetz zeta functions of infinite cyclic coverings with an application to knot theory. Topol. Proc. 29(1), 277–291 (2005). Spring Topology and Dynamical Systems Conference

    MathSciNet  MATH  Google Scholar 

  18. Sugiyama, K.: An analog of the Iwasawa conjecture for a compact hyperbolic threefold. J. Reine Angew. Math. 613, 35–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sugiyama, K.: The geometric Iwasawa conjecture from a viewpoint of the arithmetic topology. In: Proceedings of the Symposium on Algebraic Number Theory and Related Topics. RIMS Kôkyûroku Bessatsu, vol. 4, pp. 235–247. Res. Inst. Math. Sci. (RIMS), Kyoto (2007)

    Google Scholar 

  20. Sugiyama, K.: On geometric analogues of Iwasawa main conjecture for a hyperbolic threefold. In: Noncommutativity and Singularities. Advanced Studies in Pure Mathematics, vol. 55, pp. 117–135. Math. Soc. Jpn., Tokyo (2009)

    Google Scholar 

  21. Washington, L.: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Masanori Morishita

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  1. Masanori Morishita
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Correspondence to Masanori Morishita .

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Morishita, M. (2012). Torsions and the Iwasawa Main Conjecture. In: Knots and Primes. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2158-9_12

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