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

Abstract

The theme of this paper may be summarized by the triangle: 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 off mentioned above is a Hecke L-function of K. Such a criterion of irregularity for Hurwitz numbers has already been obtained in , 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 (0.2) 2((c0) = ak-1 for aeK • and ~ = 1 mod • r with some ideals c of K,

A deformation problem for Galois representations over imaginary quadratic fields

“…In § 4 we show that there exists an ordinary one provided that val p (L int (0, φ)) > 0 for a certain Hecke character φ of F such that the reduction of φ p is χ 0 . Alternatively, one could invoke the generalizations of Kummer's criterion to imaginary quadratic fields (see, for example, [11,17,21,45]). Definition 3.1.…”

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

“…Indeed, if p ν, both κ and κ are unramified at p, and the same argument as in [14] shows thatκ λ =κ λ ρ provided p > k + 1.…”

“…Proof: We begin with a modification of the argument in the proof of Prop. 2.2 of [14]. Let κ be a Grossencharacter of K of type (k, 0) such that θ κ is congruent modulo λ to θ κ .…”

Constancy of adjoint L-invariant

“…This is a base of the proof by Mazur and Tilouine (e.g., [28]; see also [11] as a precursor of the result of Mazur and Tilouine) of the anticyclotomic main conjecture (see [19] and [21] for a version for CM fields).…”