Arithmetic topology in Ihara theory II

Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols

Hikaru Hirano, Masanori Morishita

Research output: Contribution to journalArticle

Abstract

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 D (l) of 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 D (l) along the pro-l longitudes of Frobenius elements for l=2,3.

Original languageEnglish
Pages (from-to)211-238
Number of pages28
JournalJournal of Number Theory
Volume198
DOIs
Publication statusPublished - May 1 2019

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Covering
Frobenius
Topology
Invariant
Gerbe
Analogue
Galois Representations
Braid
Heisenberg Group
Monodromy
Galois
Prime number
Fundamental Group
Number field
Line

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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abstract = "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 D (l) of 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 D (l) along the pro-l longitudes of Frobenius elements for l=2,3.",
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AB - 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 D (l) of 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 D (l) along the pro-l longitudes of Frobenius elements for l=2,3.

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