Logarithmic functional and reciprocity laws

In this paper, we give a short survey of results related to the reciprocity laws over the field C. We announce a visual topological proof of Parshin’s multidimensional reciprocity laws over C. We introduce the logarithmic functional, whose argument is an n-dimensional cycle in the group (C∗)n+1. It generalizes the usual logarithm, which can be considered as the zero-dimensional logarithmic functional. It also generalizes the one-dimensional logarithmic functional that is a natural extension of the functional introduced by Beilinson for a topological proof of the Weil reciprocity law over C. 1. One-dimensional case 1.1. Weil reciprocity law. Let Γ be a complete connected complex onedimensional manifold (in other words, Γ is an irreducible complex algebraic curve). A local parameter u near a point a ∈ Γ is an arbitrary meromorphic function, whose order at a is equal to one. The local parameter u is a coordinate function in a small neighborhood of a. Let φ be a meromorphic function on Γ and let ∑ k≤m cmu m be its Laurent expansion at a. The leading monomial χ of φ is the first nonzero term in the expansion, i.e. χ(u) = cku. The leading monomial is defined for any meromorphic function φ not identically equal to zero. For each pair of meromorphic functions f , g on a curve Γ not identically equal to zero and each point a ∈ Γ, one defines the Weil symbol [f, g]a. It is a nonzero complex number given by the formula [f, g]a = (−1)amb n , where amu and bnu are the leading monomials of the functions f and g at a, with respect to the parameter u. The Weil symbol is defined with the help of the parameter u but it does not depend on the choice of u. By definition, the Weil symbol depends multiplicatively on functions f and g. The multiplicativity with respect to f means that if f = f1f2, then [f, g]a = [f1, g]a[f2, g]a. The multiplicativity with respect to g is defined similarly. The Weil symbol of functions f, g can differ from 1 only at points in the supports of the divisors of f and g. 1991 Mathematics Subject Classification. Primary 11S31, Secondary 14M25.