We provide a generalization of Scholz’s reciprocity law using the subfields and of , of degrees and over , respectively. The proof requires a particular choice of primitive element for over and is based upon the splitting of the cyclotomic polynomial over the subfields.
Numerous important lattices (, the Coxeter-Todd lattice , the Barnes-Wall lattice , the Leech lattice , as well as the -modular -dimensional lattices found by Quebbemann and Bachoc) possess algebraic structures over various Euclidean rings, e.g. Eisenstein integers or Hurwitz quaternions. One obtains efficient algorithms by performing within this frame the usual reduction procedures, including the well known LLL-algorithm.
Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad.In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws.