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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

Ihara initiated to study a certain Galois representation which may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further, following after some issues and their inter-relations in the theory of braids and links such as Milnor invariants, Johnson homomorphisms, Magnus-Gassner cocycles and Alexander invariants, and study relations with arithmetic in Ihara theory., Comment: 66pages, to appear in Publ. Res. Inst. Math. Sci., corrected typos

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Redei's triple symbol.

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

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol $[a_1,a_2, p]$ describes the decomposition law of a prime number $p$ in a certain dihedral extension over $\mathbb{Q}$ of degree 8 determined by $a_1$ and $a_2$. In this paper, we show that the triple symbol $[-p_1,p_2, p_3]$ for certain prime numbers $p_1, p_2$ and $p_3$ can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.

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 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.

In this chapter we introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. We then discuss analogous constructions for pro-finite (pro-l) groups to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for primes and the exact sequence relationg the complete Alexander module with a Galois (Iwasawa) module.

In this chapter we discuss the analogy between Galois groups with restricted ramification and link groups. In particular, we shall see the close analogy between Milnor’s theorem on the structure of a link group and a theorem by H. Koch on the structure of a pro-l Galois group over the rational number field with restricted ramification.

In this final chapter, as an extension of the analogies in Chap. 13 to 2-dimensional representations, we discuss intriguing analogies between the family of holonomy representations associated to deformations of hyperbolic structures (due to W. Thurston) and the family of Galois representations associated to deformations of p-adic ordinary modular forms (due to H. Hida and B. Mazur).

In this chapter, in view of the analogy between a knot group and a prime group, we discuss some analogies between the moduli spaces of 1-dimensional representations of knot and prime groups, and the associated higher Alexander and Iwasawa invariants.

Following the analogies between knots and primes, we introduce arithmetic analogues of higher linking matrices for prime numbers by using the arithmetic Milnor numbers. As an application, we describe the Galois module structure of the l-class group of a cyclic extension of ℚ of degree l (l being a prime number) in terms of the arithmetic higher linking matrices. In particular, our formula generalizes the classical formula of Redei on the 4 and 8 ranks of the 2-class group of a quadratic field.