Reciprocity laws through formal groups

A relation between formal groups and reciprocity laws is studied following the approach initiated by Honda. Let ξ denote an mth primitive root of unity. For a character χ of order m, we define two one-dimensional formal groups over Z[ξ] and prove the existence of an integral homomorphism between them with linear coefficient equal to the Gauss sum of χ. This allows us to deduce a reciprocity formula for the mth residue symbol which, in particular, implies the cubic reciprocity law.