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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On certain $L$-functions for deformations of knot group representations
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by Takahiro Kitayama, Masanori Morishita, Ryoto Tange and Yuji Terashima PDF
Trans. Amer. Math. Soc. 370 (2018), 3171-3195 Request permission

Abstract:

We study the twisted knot module for the universal deformation of an $\textrm {SL}_2$-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.
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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
  • MR Author ID: 880899
  • Email: kitayama@ms.u-tokyo.ac.jp
  • Masanori Morishita
  • Affiliation: Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
  • MR Author ID: 261276
  • Email: morisita@math.kyushu-u.ac.jp
  • Ryoto Tange
  • Affiliation: Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
  • Email: rtange.math@gmail.com
  • Yuji Terashima
  • Affiliation: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
  • Email: tera@is.titech.ac.jp
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3171-3195
  • MSC (2010): Primary 57M25
  • DOI: https://doi.org/10.1090/tran/7037
  • MathSciNet review: 3766846