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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/tran/7037
Published electronically: November 15, 2017
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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.


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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
Email: kitayama@ms.u-tokyo.ac.jp

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

DOI: https://doi.org/10.1090/tran/7037
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

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