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
N1 - Funding Information:
Acknowledgment We would like to thank Yasushi Mizusawa, Hiroaki Nakamura, Yuji Terashima, Hiroshi Tsunogai and Zdzisław Wojtkowiak for useful communications. Especially, we thank Terashima for discussions on the subsection 2.2 . We would like to thank the referee for useful comments which improved the earlier version. The second author is partly supported by JSPS KAKENHI Grant Number JP17H02837 , Grant-in-Aid for Scientific Research (B).
Funding Information:
Acknowledgment? We would like to thank Yasushi Mizusawa, Hiroaki Nakamura, Yuji Terashima, Hiroshi Tsunogai and Zdzis?aw Wojtkowiak for useful communications. Especially, we thank Terashima for discussions on the subsection 2.2. We would like to thank the referee for useful comments which improved the earlier version. The second author is partly supported by JSPS KAKENHI Grant Number JP17H02837, Grant-in-Aid for Scientific Research (B).
Publisher Copyright:
© 2018 Elsevier Inc.
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
SN - 0022-314X
VL - 198
SP - 211
EP - 238
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -