ON ℓ-ADIC GALOIS POLYLOGARITHMS AND TRIPLE ℓTH POWER RESIDUE SYMBOLS

抄録

The ℓ-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in ℓ-adic numbers, which arises from Galois actions on ℓ-adic étale paths on ℙ¹\{0, 1, ∞}. In the present paper, we discuss a relationship between ℓ-adic Galois polylogarithms and triple ℓth power residue symbols in some special cases studied in a work of Hirano and Morishita [J. Number Theory 198 (2019), 211-238]. We show that a functional equation of ℓ-adic Galois polylogarithms by Nakamura and Wojtkowiak [Non-abelian Fundamental Groups and Iwasawa Theory. Cambridge University Press, 2012, pp. 258-310] implies a reciprocity law of triple ℓth power residue symbols.