Milnor invariants and Massey products for prime numbers

Abstract

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. Rédei's triple symbol.

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

“…In 2004 J. Labute, during his visits to the first author, pointed out the significance of Massey products for Galois theory, and he pointed out two remarkable works [34] and [45]. In 2011 the first author, while working with I. Efrat, observed the relevance of Massey products with work in [15].…”

Arithmetic Topology in Ihara Theory

“…In [Ms1] and [Ms2; Chapter 8], the arithmetic Milnor invariants for certain primes of a number field were introduced as multiple generalizations of power residue symbols and the Rédei triple symbol ( [R]). See also [Am].…”

“…This paper forms a (part of) elementary and group-theoretical foundation of arithmetic topology in Ihara theory. In the forthcoming papers, we shall study some connections of l-adic Milnor invariants and pro-l Johnson homomorphisms in this paper with arithmetic of multiple power residue symbols in [Am], [Ms1], [Ms2;Chapter 8] and the works of Wojtkowiak on l-adic iterated integrals and l-adic polylogarithms ( [NW], [W1] ∼ [W4] etc). See Remark 2.2.12.…”

On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field

“…We generalize this relation for our triple cubic residue symbol by describing them by the Massey product in Galois cohomology. It is also an extension of the earlier works [24,34] in the case of the rational number field to the Eisenstein number field.…”

“…In the late 1990s, Kapranov and the third author independently interpreted the Rédei symbol as a mod 2 arithmetic analogue of a triple linking number of a link, and further the third author introduced mod 2 arithmetic analogues for rational primes of the Milnor invariants (higher order linking numbers) in link theory [21], based on the analogies between primes and knots in arithmetic topology [16,[22][23][24][25]30]. For example, the mod 2 Milnor invariant μ 2 (12 .…”