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We introduce mod $l$ Milnor invariants of a Galois element associated toIhara's Galois representation on the pro-$l$ fundamental group of a puncturedprojective line ($l$ being a prime number), as arithmetic analogues of Milnorinvariants of a pure braid. We then show that triple quadratic (resp. cubic)residue symbols of primes in the rational (resp. Eisenstein) number field areexpressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobeniuselements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramifiedcovering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded as a higheranalog of the dilogarithmic function, for the gerbe associated to the mod $l$Heisenberg group, and we study the monodromy transformations of certainfunctions on ${\cal D}^{(l)}$ along the pro-$l$ longitudes of Frobeniuselements for $l=2,3$.

Arithmetic topology in Ihara theory II: Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols