  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 ${\cal D}^{(l)}$ of $\mathbb{P}^1$, 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 ${\cal D}^{(l)}$ along the pro-$l$ longitudes of Frobenius
elements for $l=2,3$.
