Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On certain $ L$-functions for deformations of knot group representations

Authors: Takahiro Kitayama, Masanori Morishita, Ryoto Tange and Yuji Terashima
Journal: Trans. Amer. Math. Soc. 370 (2018), 3171-3195
MSC (2010): Primary 57M25
Published electronically: November 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the twisted knot module for the universal deformation of an $ {\rm SL}_2$-representation of a knot group and introduce an associated $ L$-function, which may be seen as an analogue of the algebraic $ p$-adic $ L$-function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we compute the $ L$-function by some concrete examples for $ 2$-bridge knots.

References [Enhancements On Off] (What's this?)

  • [Ca] Henri Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, $ p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 213-237 (French). MR 1279611,
  • [CS] Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of $ 3$-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109-146. MR 683804,
  • [Fo] Ralph H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547-560. MR 0053938,
  • [FV] Stefan Friedl and Stefano Vidussi, A survey of twisted Alexander polynomials, The mathematics of knots, Contrib. Math. Comput. Sci., vol. 1, Springer, Heidelberg, 2011, pp. 45-94. MR 2777847,
  • [Fu] Takako Fukaya, Hasse zeta functions of non-commutative rings, J. Algebra 208 (1998), no. 1, 304-342. MR 1644015,
  • [G] Ralph Greenberg, Iwasawa theory and $ p$-adic deformations of motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 193-223. MR 1265554
  • [Ha] S. Harada, Modular representations of fundamental groups and associated Weil-type zeta functions, Thesis, Kyushu University, 2008.
  • [H1] Haruzo Hida, Galois representations into $ {\rm GL}_2({\bf Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545-613. MR 848685,
  • [H2] Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 231-273. MR 868300
  • [Hi] Jonathan Hillman, Algebraic invariants of links, 2nd ed., Series on Knots and Everything, vol. 52, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. MR 2931688
  • [I] Kenkichi Iwasawa, On $ {\bf Z}_{l}$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. MR 0349627,
  • [Kt] Kazuya Kato, $ p$-adic Hodge theory and values of zeta functions of modular forms. (English, with English and French summaries), Cohomologies $ p$-adiques et applications arithmétiques. III. Astérisque 295 (2004), ix, 117-290. MR 2104361
  • [Kw] Akio Kawauchi, A survey of knot theory,. translated and revised from the 1990 Japanese original by the author, Birkhäuser Verlag, Basel, 1996. MR 1417494
  • [Ki] Teruaki Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996), no. 2, 431-442. MR 1405595
  • [KL] Paul Kirk and Charles Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999), no. 3, 635-661. MR 1670420,
  • [Ko] K. Kodama, Knot program, available at
  • [Ku] Masato Kurihara, Iwasawa theory and Fitting ideals, J. Reine Angew. Math. 561 (2003), 39-86. MR 1998607,
  • [Le] Le Ty Kuok Tkhang, Varieties of representations and their subvarieties of cohomology jumps for knot groups, Mat. Sb. 184 (1993), no. 2, 57-82 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 187-209. MR 1214944,
  • [LM] Alexander Lubotzky and Andy R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985), no. 336, xi+117. MR 818915,
  • [Ma1] B. Mazur, Remarks on the Alexander Polynomial, available at
  • [Ma2] B. Mazur, Deforming Galois representations, Galois groups over $ {\bf Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385-437. MR 1012172,
  • [Ma3] B. Mazur, The theme of $ p$-adic variation, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 433-459. MR 1754790
  • [MW] B. Mazur and A. Wiles, Class fields of abelian extensions of $ {\bf Q}$, Invent. Math. 76 (1984), no. 2, 179-330. MR 742853,
  • [Mo] Masanori Morishita, Knots and primes, An introduction to arithmetic topology, Universitext, Springer, London, 2012. MR 2905431
  • [MTTU] M. Morishita, Y. Takakura, Y. Terashima, J. Ueki, On the universal deformations for $ {\rm SL}_2$-representations of knot groups, to appear in Tohoku Math. J.MR 3640015
  • [Na] Kazunori Nakamoto, Representation varieties and character varieties, Publ. Res. Inst. Math. Sci. 36 (2000), no. 2, 159-189. MR 1753200,
  • [Ny] Louise Nyssen, Pseudo-représentations, Math. Ann. 306 (1996), no. 2, 257-283 (French). MR 1411348,
  • [O1] Tadashi Ochiai, Control theorem for Greenberg's Selmer groups of Galois deformations, J. Number Theory 88 (2001), no. 1, 59-85. MR 1825991,
  • [O2] Tadashi Ochiai, On the two-variable Iwasawa main conjecture, Compos. Math. 142 (2006), no. 5, 1157-1200. MR 2264660,
  • [Po] Joan Porti, Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc. 128 (1997), no. 612, x+139 (French, with English and French summaries). MR 1396960,
  • [Pr] C. Procesi, The invariant theory of $ n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306-381. MR 0419491,
  • [PS] Józef H. Przytycki and Adam S. Sikora, On skein algebras and $ {\rm Sl}_2({\bf C})$-character varieties, Topology 39 (2000), no. 1, 115-148. MR 1710996,
  • [R] Robert Riley, Nonabelian representations of $ 2$-bridge knot groups, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 138, 191-208. MR 745421,
  • [Sa] Kyoji Saito, Character variety of representations of a finitely generated group in $ {\rm SL}_2$, Topology and Teichmüller spaces (Katinkulta, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 253-264. MR 1659663
  • [Se1] Jean-Pierre Serre, Classes des corps cyclotomiques (d'après K. Iwasawa), Séminaire Bourbaki, Vol. 5, Soc. Math. France, Paris, 1995, Exp. No. 174, pp. 83-93 (French). MR 1603459
  • [Se2] Jean-Pierre Serre, Corps locaux, 1968 (French). Deuxième édition; Publications de l'Université de Nancago, No. VIII, Hermann, Paris. MR 0354618
  • [Se3] Jean-Pierre Serre, Cohomologie galoisienne, with a contribution by Jean-Louis Verdier. Lecture Notes in Mathematics, No. 5, Troisième édition, vol. 1965, Springer-Verlag, Berlin-New York, 1965 (French). MR 0201444
  • [Ta] Richard Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), no. 2, 281-332. MR 1115109,
  • [Ti] Jacques Tilouine, Deformations of Galois representations and Hecke algebras, published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, by Narosa Publishing House, New Delhi, 1996. MR 1643682
  • [Wd] Masaaki Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), no. 2, 241-256. MR 1273784,
  • [Ws] Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M25

Retrieve articles in all journals with MSC (2010): 57M25

Additional Information

Takahiro Kitayama
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
Address at time of publication: Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Masanori Morishita
Affiliation: Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

Ryoto Tange
Affiliation: Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

Yuji Terashima
Affiliation: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan

Keywords: Deformations of representations, character schemes, twisted knot modules, twisted Alexander invariants, $L$-functions
Received by editor(s): December 17, 2015
Received by editor(s) in revised form: February 8, 2016, and July 22, 2016
Published electronically: November 15, 2017
Additional Notes: The first author was partly supported by JSPS Research Fellowships for Young Scientists 26800032
The second author was partly supported by Grants-in-Aid for Scientific Research (B) 24340005
The fourth author was partly supported by Grants-in-Aid for Scientific Research (C) 25400083
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society