### 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 language | English |
---|---|

Pages (from-to) | 211-238 |

Number of pages | 28 |

Journal | Journal of Number Theory |

Volume | 198 |

DOIs | |

Publication status | Published - May 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

**Arithmetic topology in Ihara theory II : Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols.** / Hirano, Hikaru; Morishita, Masanori.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Arithmetic topology in Ihara theory II

T2 - Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols

AU - Hirano, Hikaru

AU - Morishita, Masanori

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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.

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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U2 - 10.1016/j.jnt.2018.10.010

DO - 10.1016/j.jnt.2018.10.010

M3 - Article

VL - 198

SP - 211

EP - 238

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -