Kummer's criterion for the special values of HeckeL-functions of imaginary quadratic fields and congruences among cusp forms

Abstract

Here, arrow (i) refers to our previous papers [9, 12] and [13], in which we have shown that a fixed primitive cusp form f of Sk(F I(N)) has congruences with other cusp forms of Sk(F~(N)) modulo the special value at s = k of a certain zeta function off . In this paper, we will construct the second arrow for the cusp form f associated with a Hecke character of an imaginary quadratic field K, and consequently obtain the third. Namely, in w167 6 we will show, as in the works of Coates-Wiles [3] and Robert [23], that the split primes of K which divide this special value are irregular in an appropriate sense. In this case, the zeta function o f f mentioned above is a Hecke L-function of K. Such a criterion of irregularity for Hurwitz numbers has already been obtained in [3] and [233. However, our result together with those of [3] and [23] does not cover all the L-values of Hecke L-functions of K whose algebraicity was given by Damerell [5] (for details, see below). Our method is analogous to that of Ribet [21] (see also Wiles [35]), who used congruences between cusp forms and Eisenstein series to prove the non-vanishing of certain eigenspaces (relative to the action of Gal(Q/Q)) of the p-primary part of the class group of the field of p-th roots of unity. By adopting this approach, we will obtain such an information of eigenspaces again in our case (see Theorem 0.1 below). To be more specific, let 2 be a Hecke character of K satisfying