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arXiv · 1412.6894

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

Fumiya Amano·Yasushi Mizusawa·Masanori Morishita
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Abstract

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 Rédei'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 Rédei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.

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Fumiya Amano, Yasushi Mizusawa, Masanori Morishita. 2018-01-19. On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field. https://arxiv.org/abs/1412.6894

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