The Hilbert reciprocity law on 3-manifolds

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law for a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.Comment: 8 page

The pp-adic limits of class numbers in Zp\mathbb{Z}_p-towers

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class numbers in a $\mathbb{Z}_p$-extension of a global field and a similar result in a $\mathbb{Z}_p$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the $p$-adic limit of the $p$-power-th cyclic resultants of a polynomial using roots of unity of orders prime to $p$, the $p$-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with $p$-adic limits being in $\mathbb{Z}$ and find the cases such that the base $p$-class numbers are small and $\nu$'s are arbitrarily large.Comment: 28 pages. new results on lim in Z and large nu in v2. minor corrections in later version

The Iwasawa invariants of Zp d\mathbb{Z}_p^{\,d}-covers of links

Let $p$ be a prime number and let $d\in \mathbb{Z}_{>0}$. In this paper, following the analogy between knots and primes, we study the $p$-torsion growth in a compatible system of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of 3-manifolds and establish several analogues of Cuoco--Monsky's multivariable versions of Iwasawa's class number formula. Our main goal is to establish the Cuoco--Monsky type formula for branched covers of links in rational homology 3-spheres. In addition, we prove the precise formula over integral homology 3-spheres prompted by Greenberg's conjecture. We also obtain results on reduced Alexander polynomials and on the Betti number periodicity. Furthermore, we investigate the twisted Whitehead links in $S^3$ and point out that the Iwasawa $\mu$-invariant of a $\mathbb{Z}_p^{\,2}$-cover can be an arbitrary non-negative integer. We also calculate the Iwasawa $\mu$ and $\lambda$-invariants of the Alexander polynomials of all links in Rolfsen's table.Comment: 56 pages, 2 figures, minor corrections in v