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

Abstract

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.