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

Y1 - 2019/5

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 P1, 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 P1, 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

AN - SCOPUS:85057175239

VL - 198

SP - 211

EP - 238

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -