Narrow Results

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of $S$-integers of $F$, where $S$ is a finite set of finite primes of a number field $F$. As an application, we give the asymptotic growth of twisted homology groups.

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient $\mathbb{Z}/n\mathbb{Z}$$(n\ge 2)$ associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient $\mathbb{Z}/2\mathbb{Z}$ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient $\mathbb{Z}/n\mathbb{Z}$ in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient $\mathbb{Z}/n\mathbb{Z}$) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542].

In this note, we would like to discuss leisurely analogies between knot theory and number theory, focused on the variation of representations of the knot and prime groups. More precisely, we will discuss two things: First, we recall

1. the basic analogies between knots and primes and, based on them, secondly we will discuss some analogies between

2. the deformations of hyperbolic structures on a knot complement and of modular Galois representations

The latter analogy was first pointed out by Kazuhiro Fujiwara ([6]. See also [14]). We will make his observation more precise and discuss intriguing analogous features between SL₂(ℂ) Chern-Simons invariants and p-adic modular L-functions for GL₂.