Ideal Class Groups of CM-fields with Non-cyclic Galois Action

Abstract

Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group Cl L of L as a Gal(L/k)-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for L/k times the annihilator ideal of the roots of unity in L is in the Fitting ideal of Cl L , and also a problem whether it is in the Fitting ideal of the Pontrjagin dual (Cl L ) ∨ . We systematically construct extensions L/k for which these properties do not hold, and also give numerical examples.

Tate sequences and Fitting ideals of Iwasawa modules

“…The numerical example in [KM2] §2 satisfies all the conditions of Corollary 0.3 with s = 2, and Corollary 0.4 with s = 2, a = 1, q = 1. In §4 we will discuss this numerical example in detail.…”

“…We already hinted at this in the last remark. Now we study the numerical example in [KM2] §2. Take p = 3, k = Q( √ 1901) and L = k( √ −3, α, β) where α 3 − 84α − 191 = 0 and β 3 − 57β − 68 = 0.…”

“…We regard A L as a Z p [G] − = Z p [G (p) ]-module. We take generators σ 1 , σ 2 of G (p) = Gal(k(α)/k) × Gal(k(β)/k) (Z/pZ) ⊕2 such that σ 1 is σ in [KM2] and σ 2 is τ in [KM2] (in particular, σ 1 is a generator of Gal(k(α)/k) and σ 2 is a generator of Gal(k(β)/k)). Put τ i = σ i − 1 as before.…”

“…. On the other hand, the explicit long computation in [KM2] shows that Fit Zp[G (p) ] (A ∨ L ) = (p 4 , p 2 τ 1 , p 2 τ 2 , pτ 2 1 , pτ 2 2 , pτ 1 τ 2 ) (see the last line of page 425 of [KM2]). Let us check that these two descriptions agree.…”

“…In the following, we assume that L ∩ k ∞ = k. If L contains a primitive p-th root of unity, then we encounter a totally different phenomenon on the Fitting ideal of A ∨ L , namely the Fitting ideal of A ∨ L cannot equal the Stickelberger ideal, in general. In fact, if L contains µ p and G (p) = G ⊗ Z p is not cyclic, one knows, for example by Theorem 1.2 in [KM2], that…”