The Brauer-Kuroda formula for higher S-class numbers indihedral extensions of number fields

摘要

For a num-ber field E and a finite set of places S of E containing the archimedean places, let ζ_E~S be the Dedekind S-zeta function of E. For a complex number s, let ζ_E~S(s)~* denote the special value of ζ_E~S at s (i.e. the first nontrivial coefficient of the Laurent expansion of ζ_E~S around s). Dedekind, inspired by previous work of Dirichlet, proved the formula hs (1.1)ζ_E~S(0)~* =-(h_E~S)/(W_E)R_E~S where h_E~S is the class number of the ring O_E~S of S-integers of E, w_Eis the order of the group of roots of unity of E, and R_E~S is the regulator of (O_E~S)~× There are conjectural analogues of this formula when 0 is replaced by negative integers. More precisely, we recall the definition of motivic coho-mology groups in terms of Bloch's higher Chow groups: H~j (O_E~S, Z(m)) := CH~m(Spec(O_E~S, 2m - j).