On special values at $s=0$ of partial zeta-functions for real quadratic fields

Abstract

1.1 Let F be a totally real algebraic number field with finite degree, α a fractional ideal of F, and Fab the maximal abelian extension of F. We define a map ξa from the quotient space Fja to the group W(Fab) of roots of unity of Fab using the deep results of Coates-Sinnott [0—SI], [C—S2] and Deligne-Ribet [D— R] on special values of partial zeta functions of F. Under the action of the Galois group Gal(FabjF) of Fab over F this map behaves formally in a manner similar to Shimura's reciprocity law for elliptic curves with complex multiplication. This reciprocity law for the map ξa is also a direct consequence of those results of Coates-Sinnott and Deligne-Ribet. On the other hand we have studied in [Arl] a certain Dirichlet series and its relationship with parital zeta functions of real quadratic fields. In particular the special values at s=0 of partial zeta functions of real quadratic fields essentially coincide with the residues at the pole s=0 of our Dirichlet series. Using those residues, we give another expression for the map ξa in the case of F a real quadratic field. We also show that the expression works in a reasonable manner under the action of the Galois group Gal(Fab/F). 1.2 We summarize our results. For an integral ideal c of a totally real algebraic number field F> denote by HF(c) the narrow ray class group modulo c. For each integral ideal b prime to c, we define the partial zeta-function fc(b, *) to be the sum Σα(Λfa)~> α running over all integral ideals of the class of b in HF(t). Let α be a fractional ideal of F. For each class z of the quotient space jP/α, we take a totally positive representative element z^F of the class £, and write