On certain 𝐿-functions for deformations of knot group representations

Kitayama, Takahiro; Morishita, Masanori; Tange, Ryoto; Terashima, Yuji

doi:10.1090/tran/7037

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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