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We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. We then show that triple quadratic (resp. cubic) residue symbols of primes in the rational (resp. Eisenstein) number field are expressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobenius elements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramified covering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded as a higher analog of the dilogarithmic function, for the gerbe associated to the mod $l$ Heisenberg group, and we study the monodromy transformations of certain functions on ${\cal D}^{(l)}$ along the pro-$l$ longitudes of Frobenius elements for $l=2,3$., Comment: 33 pages, 1 figure

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and R\'{e}dei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to R\'{e}dei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology., Comment: 42 pages. The previous version entitled "Arithmetic Milnor invariants and multiple power residue symbols in number fields" by F. Amano and M. Morishita contained some mistakes concerning Theorem 2.8. This is a new version for the Eisenstein number field, which revises largely the previous one, with the collaboration of Y. Mizusawa. To appear in Research in Number Theory

We study the asymptotic distribution of S-integral points on affine homogeneous spaces in the light of the Hardy–Littlewood property introduced by Borovoi and Rudnick. We introduce the S-Hardy–Littlewood property for affine homogeneous spaces defined over an algebraic number field and a finite set S of places of the base field. We work with the adelic harmonic analysis on affine algebraic groups over a number field to determine the asymptotic density of S-integral points under congruence conditions. We give some new examples of strongly or relatively S-Hardy–Littlewood homogeneous spaces over number fields. As an application, we prove certain asymptotically uniform distribution property of integral points on an ellipsoid defined by a totally positive definite tenary quadratic form over a totally real number field.

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension of a number field. We give their cohomological interpretation in terms of Massey products in Galois cohomology., Comment: 38 pages, to appear in J. of Algebra, changed title from "p-Johnson homomorphisms in non-Abelian Iwasawa theory", added references in section 5, corrected typos

In this chapter we recollect the preliminary materials from topology and number theory. In particular, we give a summary about fundamental groups and Galois theory for topological spaces and arithmetic rings, together with the basic concepts and examples in 3-dimensional topology and number fields. We also review class field theory as arithmetic duality theorems in Galois, etale cohomology groups.

In this chapter we discuss close parallels between Alexander-Fox theory and Iwasawa theory such as the Alexander polynomial and the Iwasawa polynomial, based on the analogy between the infinite cyclic covering of a knot exterior and the cyclotomic ℤ p -extension of a number field. In particular, we give asymptotic formulas on the orders of the 1st homology groups (p-ideal class groups) of cyclic ramified coverings (extensions).

In this chapter we review Hilbert theory which deals with, in a group-theoretic manner, the decomposition of a prime in a finite Galois extension of number fields. Based on the analogies in Chap. 3, we give a topological analogue of Hilbert theory for a finite Galois covering of 3-manifolds.

The purpose of this paper is to investigate analogues for 3-manifolds after the model of capitulation theorems for number fields. This will give one of the foundational analogies between 3-dimensional topology and algebraic number theory.

Following the conceptual analogies between knots and primes, 3-manifolds and number fields, we discuss an analogue in knot theory after the model of the arithmetical theory of genera initiated by Gauss. We present an analog for cyclic coverings of links following along the line of Iyanaga-Tamagawa's genus theory for cyclic extentions over the rational number field. We also give examples of $\mathbf{Z} / 2\mathbf{Z} \times \mathbf{Z} / 2\mathbf{Z}$-coverings of links for which the principal genus theorem does not hold.