Fitting ideals of class groups for CM abelian extensions

Let $K/k$ be a finite abelian CM-extension and put $G=\operatorname{Gal}(K/k)$. Let $S_{\infty}(k)$ be the set of archimedean places of $k$. Let $S_{\operatorname{ram}}(K/k)$ be the set of places of $k$ which are ramified in $K/k$, including $S_{\infty}(k)$. For each finite prime $v\in S_{\operatorname{ram}}(K/k)$, let $I_{v}\subset G$ denote the inertia group of $v$ in $G$ and $\varphi_{v}\in G/I_{v}$ the arithmetic Frobenius of $v$. We then define elements $g_{v}$ and $h_{v}$ by

where we put $\nu_{I_{v}}=\sum_{\tau\in I_{v}}\tau$. These elements are introduced in [3, Lemmas 6.1 and 8.3] and [9, §2.2, equations (7) and (10)] (up to the involution). Moreover, we define a $\mathbb{Z}[G]$-module $A_{v}$ by

We write $\mathbb{Z}[G]^{-}=\mathbb{Z}[1/2][G]/(1+j)$, where $j$ is the complex conjugation in $G$. For any $\mathbb{Z}[G]$-module $M$, we also define the minus component by $M^{-}=M\otimes_{\mathbb{Z}[G]}\mathbb{Z}[G]^{-}$. Note that we are implicitly inverting the action of $2$. For any $x\in M$, we write $x^{-}$ for the image of $x$ under the natural map $M\to M^{-}$.

In general, for a set $S$ of places of $k$, we write $S_{K}$ for the set of places of $K$ which lie above places in $S$. We take and fix a finite set $T$ of finite primes of $k$ satisfying the following.

• •

  $T\cap S_{\operatorname{ram}}(K/k)=\emptyset$.

• •

  $K_{T}^{\times}=\{x\in K^{\times}\mid\operatorname{ord}_{w}(x-1)>0\mbox{ for all primes }w\in T_{K}\}$ is torsion free. Here, $\operatorname{ord}_{w}$ denotes the normalized additive valuation.

Note that, if we fix an odd prime number $p$ and are concerned with the $p$-components, the last condition can be weakened to that $K_{T}^{\times}$ is $p$-torsion-free. We consider the $T$-ray ideal class group of $K$ defined by

where $w$ runs over the finite primes of $K$ which are not in $T_{K}$.

For a character $\psi$ of $G$, we write $L(s,\psi)$ for the primitive $L$-function for $\psi$; this function omits exactly the Euler factors of primes dividing the conductor of $\psi$. For any finite prime $v$ of $k$, we put $N(v)=\#\mathbb{F}_{v}$, where $\mathbb{F}_{v}$ is the residue field of $v$. We then define the $T$-modified $L$-function by

We define

where $\psi$ runs over the characters of $G$ and $e_{\psi}=\frac{1}{\#G}\sum_{\sigma\in G}\psi(\sigma)\sigma^{-1}$ is the idempotent of the $\psi$-component.

Now the first main theorem of this paper is the following, whose proof will be given in §3.

In the second main result below, we will obtain a concrete description of $h_{v}^{-}\operatorname{Fitt}^{[1]}_{\mathbb{Z}[G]^{-}}\left(A_{v}^{-}\right)$, which completes the description of the Fitting ideal of $\operatorname{Cl}_{K}^{T,-}$. We do not review the precise statement of eTNC (see e.g. [1, Conjecture 3.6]).

In order to compare with Theorem 1.1, we recall the result for the dualized version:

As already mentioned, Kurihara [9, Corollary 3.7] showed this formula under the validity of the eTNC, and recently Dasgupta-Kakde [2, Theorem 1.4] removed the assumption. Here, for a general $G$-module $M$, we equip the Pontryagin dual $M^{\vee}$ with the $G$-action by $(\sigma f)(x)=f(\sigma x)$ for $\sigma\in G$, $f\in M^{\vee}$, and $x\in M$. This convention is the opposite of [9] and [2], so the right hand side of the formula (1.2) differs from those by the involution.

We now briefly outline the proof of Theorem 1.1. An important ingredient is an exact sequence of $\mathbb{Z}[G]^{-}$-modules of the form

as in Proposition 3.2, where $\mathfrak{A}^{-}$ is free of finite rank $\#S^{\prime}$. Here, $S^{\prime}$ is an auxiliary finite set of places of $k$. This sequence was constructed by Kurihara [9], based on preceding work such as Ritter-Weiss [12] and Greither [3], and played a key role in proving (1.2) under eTNC. Our novel idea is to construct an explicit injective homomorphism from $W_{S_{\infty}}^{-}$ to $(\mathbb{Z}[G]^{-})^{\oplus\#S^{\prime}}$ whose cokernel is isomorphic to the direct sum of $A_{v}^{-}$ for $v\in S^{\prime}\setminus S_{\infty}(k)$. Moreover, assuming eTNC, we will compute the determinant of the composite map $\mathfrak{A}^{-}\hookrightarrow W_{S_{\infty}}^{-}\hookrightarrow(\mathbb{Z}[G]^{-})^{\oplus\#S^{\prime}}$. By using these observations, we obtain an exact sequence to which the theory of shifts of Fitting ideals can be applied, and then Theorem 1.1 follows.

In order to make the formula of Theorem 1.1 more explicit, in §4, we will compute $\operatorname{Fitt}^{[1]}_{\mathbb{Z}[G]}\left(A_{v}\right)$. This will be accomplished by using a similar method as Greither-Kurihara [5, §1.2], which was actually a motivation for introducing the shifts of Fitting ideals in [6].

As the problem is purely algebraic, we deal with a general situation as follows (it should be clear from the notation how to apply the results below to the arithmetic situation; simply add subscripts $v$ appropriately). Let $G$ be a finite abelian group. Let $I$ and $D$ be subgroups of $G$ such that $I\subset D\subset G$ and that the quotient $D/I$ is a cyclic group. We choose a generator $\varphi$ of $D/I$ and put

which are non-zero-divisors. We define a finite $\mathbb{Z}[G]$-module $A$ by

In order to state the result, we introduce some notations. We choose a decomposition

as an abelian group such that $I_{l}$ is a cyclic group for each $1\leq l\leq s$. Here, we do not assume any minimality on this decomposition, so we allow even the extreme case where $I_{l}$ is trivial for some $l$.

For each $1\leq l\leq s$, we put $\nu_{l}=\nu_{I_{l}}=\sum_{\sigma\in I_{l}}\sigma\in\mathbb{Z}[G]$. We also put $\mathcal{I}_{D}=\operatorname{Ker}(\mathbb{Z}[G]\to\mathbb{Z}[G/D])$.

Note that the definition of $Z_{i}$ does depend on the choice of the decomposition (1.3). On the other hand, it can be shown directly that the ideal $\mathcal{J}$ is independent from the choice. We omit the direct proof because, at any rate, the independency can be deduced from the discussion in §4.

In this setting, we can describe $\operatorname{Fitt}^{[1]}_{\mathbb{Z}[G]}(A)$ as follows. It is convenient to state the result after multiplying by $h$.

As an application of Theorems 1.1 and 1.4, we shall discuss the problem whether or not the Stickelberger element lies in the Fitting ideal of $\operatorname{Cl}_{K}^{T,-}$.

We return to the setup in §1.1. Let $p$ be a fixed odd prime number and we shall work over $\mathbb{Z}_{p}$. Let $G^{\prime}$ denote the maximal subgroup of $G$ of order prime to $p$. We put $k_{p}=K^{G^{\prime}}$, which is the maximal $p$-extension of $k$ contained in $K$. For each character $\chi$ of $G^{\prime}$, we regard $\mathcal{O}_{\chi}=\mathbb{Z}_{p}[\operatorname{Im}(\chi)]$ as a $\mathbb{Z}_{p}[G^{\prime}]$-module via $\chi$, and put $\mathbb{Z}_{p}[G]^{\chi}=\mathbb{Z}_{p}[G]\otimes_{\mathbb{Z}_{p}[G^{\prime}]}\mathcal{O}_{\chi}$. For a $\mathbb{Z}_{p}[G]$-module $M$, we put $M^{\chi}=M\otimes_{\mathbb{Z}_{p}[G]}\mathbb{Z}_{p}[G]^{\chi}$, which is an $\mathbb{Z}_{p}[G]^{\chi}$-module. For an element $x\in M$, we write $x^{\chi}$ for the image of $x$ by the natural map $M\to M^{\chi}$. We note that $\mathbb{Z}_{p}[G]$ is isomorphic to the direct product of $\mathbb{Z}_{p}[G]^{\chi}$ if $\chi$ runs over the equivalence classes of characters of $G^{\prime}$.

From now on, we fix an odd character $\chi$ of $G^{\prime}$. We define $K_{\chi}=K^{\operatorname{Ker}(\chi)}$. Then $K_{\chi}$ is a CM-field, $K_{\chi}\supset k_{p}$, and $K_{\chi}/k_{p}$ is a cyclic extension of order prime to $p$.

We put $S_{\chi}=S_{\operatorname{ram}}(K_{\chi}/k)$ and consider the $\chi$-component of the Stickelberger element defined by

where $\psi$ runs over characters of $G$ whose restriction to $G^{\prime}$ coincides with $\chi$ and we write

Note that, comparing (1.1) and (1.4), we have

Concerning the dualized version, by Dasgupta-Kakde [2, Theorem 1.3],

is always true. This is called the strong Brumer-Stark conjecture. More precisely, the displayed claim is a bit stronger than [2, Theorem 1.3] as we took $S_{\chi}$ instead of $S_{\operatorname{ram}}(K/k)$ in the definition of the Stickelberger element, but in any case it is an immediate consequence of the formula (1.2).

On the other hand, the corresponding claim without dual is known to be false in general (see [4]). However, we had only partial results and an exact condition was unknown. The following theorem is strong as it gives a necessary and sufficient condition.

This theorem will be proved in §5 as an application of Theorems 1.1 and 1.4. Note that there is an elementary equivalent condition for $\theta_{K/k,T}^{\chi}=0$ as in Lemma 5.3.

Theorem 1.5 indicates that the failure of the inertia groups to be cyclic is an obstruction for studying the Fitting ideal of the class group without dual. The same phenomenon will appear again in Theorem 1.6 below. We should say that this kind of phenomenon had been observed in preceding work, such as Greither-Kurihara [4]. It is also remarkable that the obstruction does not occur in the absolutely abelian case (i.e. when $k=\mathbb{Q}$), since in that case the inertia groups are automatically cyclic, apart from the $2$-parts. This seems to fit the fact that Kurihara and Miura [7], [10] succeeded in studying the class groups without dual in the absolutely abelian case.

Let us outline the proof of Theorem 1.5. We assume that $\chi$ is a faithful character of $G^{\prime}$ (i.e. $K_{\chi}=K$); actually we can deduce the general case from this case. Since $\omega_{T}^{\chi}$ is a non-zero-divisor of $\mathbb{Z}_{p}[G]^{\chi}$, by Theorem 1.1 and (1.5), we have $\theta_{K/k,T}^{\chi}\in\operatorname{Fitt}_{\mathbb{Z}_{p}[G]^{\chi}}((\operatorname{Cl}_{K}^{T}\otimes\mathbb{Z}_{p})^{\chi})$ if and only if

holds as fractional ideals of $\mathbb{Z}_{p}[G]^{\chi}$, where on both sides $v$ runs over the elements of $S_{\operatorname{ram}}(K/k)\setminus S_{\infty}(k)$.

Obviously we may assume that $\theta_{K/k,T}^{\chi}\neq 0$. The proof of (ii) $\Rightarrow$ (i) is the easier part. We will show that, under the assumption (ii), the inclusion of (1.6) holds even for every $v$ before taking the product. On the other hand, the opposite direction (i) $\Rightarrow$ (ii) is the harder part. That is because, roughly speaking, we have to work over the ring $\mathbb{Z}_{p}[G]^{\chi}$, whose ring theoretic properties are not very nice. A key idea to overcome this issue is to reduce to a computation in a discrete valuation ring. More concretely, we make use of a character $\psi$ of $G$ which satisfies $\psi|_{G^{\prime}}=\chi$ and a certain additional condition, whose existence is verified by Lemma 5.3, and we consider the $\mathbb{Z}_{p}[G]^{\chi}$-algebra $\mathcal{O}_{\psi}=\mathbb{Z}_{p}[\operatorname{Im}(\psi)]$. By investigating the ideals in (1.6) after base change from $\mathbb{Z}_{p}[G]^{\chi}$ to $\mathcal{O}_{\psi}$, we will show (i) $\Rightarrow$ (ii).

Even if we do not assume the validity of eTNC, our argument shows the following.

This theorem follows immediately from Corollaries 3.6 and 4.2. Furthermore, by similar arguments as the proof of Theorem 1.5, we can observe that the inclusion is often proper.

As already remarked, Dasgupta-Kakde [2] proved the formula (1.2) unconditionally. Therefore, if $I_{v}$ is cyclic for every $v\in S_{\operatorname{ram}}(K/k)\setminus S_{\infty}(k)$, we can also deduce from Theorem 1.6 that $\operatorname{Fitt}_{\mathbb{Z}[G]^{-}}\left(\operatorname{Cl}_{K}^{T,-}\right)$ also coincides with that ideal, and this removes the assumption on eTNC in Theorem 1.1. However, in Theorem 1.1 we still need to assume eTNC when $I_{v}$ is not cyclic for some $v$.

Let $R$ be a noetherian ring.

We will later make use of the following elementary lemma. We omit the proof (cf. Kurihara [8, Lemma 3.3]).

In this subsection, we review the definition of shifts of Fitting ideals introduced by the second author [6].

Although we can deal with a more general situation, for simplicity we consider the following. Let $\Lambda$ be a Dedekind domain (e.g. $\Lambda=\mathbb{Z}$, $\mathbb{Z}[1/2]$, or $\mathbb{Z}_{p}$). Let $\Delta$ be a finite abelian group and consider the ring $R=\Lambda[\Delta]$.

We define $\mathcal{C}$ as the category of $R$-modules of finite length. We also define a subcategory $\mathcal{P}$ of $\mathcal{C}$ by

where $\operatorname{pd}_{R}$ denotes the projective dimension over $R$. Note that any module $M$ in $\mathcal{C}$ satisfies $\operatorname{pd}_{\Lambda}(M)\leq 1$.

We also introduce a variant for the case where $d$ is negative.

We first review necessary ingredients from Kurihara [9], which in turn relies on preceding work, in particular Ritter-Weiss [12] and Greither [3].

For each place $w$ of $K$, let $D_{w}$ and $I_{w}$ denote the decomposition subgroup and the inertia subgroup of $w$ in $G$, respectively. These subgroups depend only on the place of $k$ which lies below $w$.

Let us introduce local modules $W_{v}$. For any finite group $H$, we define $\Delta H$ as the augmentation ideal in $\mathbb{Z}[H]$.

We take an auxiliary finite set $S^{\prime}$ of places of $k$ satisfying the following conditions.

• •

  $S^{\prime}\supset S_{\operatorname{ram}}(K/k)$.

• •

  $S^{\prime}\cap T=\emptyset$.

• •

  $\operatorname{Cl}_{K,S^{\prime}}^{T}=0$, where $\operatorname{Cl}_{K,S^{\prime}}^{T}=\operatorname{Cok}\left(K_{T}^{\times}\overset{\oplus\operatorname{ord}_{w}}{\longrightarrow}\bigoplus_{w\notin S^{\prime}_{K}\cup T_{K}}\mathbb{Z}\right)$.

• •

  $G$ is generated by the decomposition groups $D_{v}$ of $v$ for all $v\in S^{\prime}$.

We define a $\mathbb{Z}[G]$-module $W_{S_{\infty}}$ by

By using local and global class field theory, Kurihara constructed an exact sequence of the following form.

In [9], the author took the linear dual of this sequence, and the resulting sequence played an important role to study $\operatorname{Cl}_{K}^{T,\vee,-}$. In this paper, we do not take the linear dual but instead study the sequence itself for the proof of Theorem 1.1.

Our key ingredient for the proof of Theorem 1.1 is the following homomorphism $f_{v}$.

In §1.1 we introduced a finite $\mathbb{Z}[G]$-module $A_{v}=\mathbb{Z}[G/I_{v}]/(g_{v})$ with $g_{v}=1-\varphi_{v}^{-1}+\#I_{v}$. It is actually motivated by the following.

For any $v\in S^{\prime}\setminus S_{\infty}(k)$, we consider the homomorphism $f_{v}^{-}:W_{v}^{-}\longrightarrow\mathbb{Z}[G]^{-}$ which is induced by $f_{v}$. For any $v\in S_{\infty}(k)$, we have $(\oplus_{w\mid v}\Delta D_{w})^{-}\simeq\mathbb{Z}[G]^{-}$ by choosing $w$, so we fix this isomorphism and write $f_{v}^{-}$ for it. Using these $f_{v}^{-}$, we consider the following commutative diagram

where the upper sequence is that in Proposition 3.2 and the map $\psi$ is defined by the commutativity. By Lemma 3.4 and the snake lemma, we get the following proposition.

By Proposition 3.5, the Fitting ideal $\operatorname{Fitt}_{\mathbb{Z}[G]^{-}}(\operatorname{Cok}\psi)$ is a principal ideal of $\mathbb{Z}[G]^{-}$ generated by a non-zero-divisor. Then we can describe the Fitting ideals of $\operatorname{Cl}_{K}^{T,-}$ and of $\operatorname{Cl}_{K}^{T,\vee,-}$ as follows.

Recall the definitions of $\omega_{T}$ and of $h_{v}$ in §1.1.

We are now ready to prove Theorem 1.1.