On the Galois module structure of minus class groups

It is our purpose here to show that our description of $\operatorname{Cl}_{L}^{T,-}$ up to $\sim$ can actually lead to concrete predictions concerning the structure of $\operatorname{Cl}_{L}^{T,-}$. Of course our predictions will fall short of determining the isomorphism class a priori – that would be way too ambitious.

Let $p\geq 3$ be a prime number. We consider the case where $L^{+}/K$ is a $p$-extension, where as usual $L^{+}$ denotes the maximal totally real subfield of $L$. In this case, $L/K$ has a unique intermediate field $F$ such that $F/K$ is a quadratic extension. Then $F$ must be a CM-field satisfying $F^{+}=K$.

The following is the main result here. Let $\operatorname{ord}_{p}(-)$ be the additive $p$-adic valuation normalized by $\operatorname{ord}_{p}(p)=1$. In Lemma 2.5, we will see that the $T$-modification is unnecessary because of the second assumption.

The proof will be given in §3. In §3.3, we will also check these predictions on some explicit examples. The results indicate that the theorem may be sharp.

The second topic in this paper is the question: Which finite $\mathbb{Z}[G]^{-}$-modules can be equivalent to a class group $\operatorname{Cl}_{L}^{T,-}$ a priori?

To be more precise, let us fix an abstract finite abelian group $\Gamma$. For simplicity, we assume that the order of $\Gamma$ is odd. Let $\mathcal{C}$ be the category of finite $\mathbb{Z}^{\prime}[\Gamma]$-modules, where we put $\mathbb{Z}^{\prime}=\mathbb{Z}[1/2]$. In this setting we have the notion of equivalence $\sim$ on $\mathcal{C}$. Let $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ be the set of equivalence classes. It is known that $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ can be regarded as a commutative monoid with respect to direct sums.

Let us consider various abelian CM-extensions $L/K$ such that $\operatorname{Gal}(L^{+}/K)\simeq\Gamma$. Since the order of $\Gamma$ is odd, as in §1.1, there is a unique intermediate CM-field $F$ satisfying $F^{+}=K$ and $\operatorname{Gal}(L/F)\simeq\Gamma$. Then we have an identification

induced by the inclusion $\Gamma\subset\operatorname{Gal}(L/K)$. Therefore, we may talk about the class of the minus class group $\operatorname{Cl}_{L}^{T,-}$ in $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$.

We call an element of $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ a realizable class if it is the class of $\operatorname{Cl}_{L}^{T,-}$ for some extension $L/K$ described above (both $L$ and $K$ vary). Let $\mathcal{Z}^{\operatorname{real}}\subset\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ be the subset of realizable classes. Now the question is to study the size of $\mathcal{Z}^{\operatorname{real}}$.

Our main results involve another subset $\mathcal{Z}^{\operatorname{adm}}$ of $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$, whose elements we call admissible classes. The definition of $\mathcal{Z}^{\operatorname{adm}}$ will be given in §2.6. Here we only mention that $\mathcal{Z}^{\operatorname{adm}}$ is defined in an algebraic way (independent from arithmetic) so that we have $\mathcal{Z}^{\operatorname{real}}\subset\mathcal{Z}^{\operatorname{adm}}$. Also, $\mathcal{Z}^{\operatorname{adm}}$ is by definition a submonoid, while it is not clear whether so is $\mathcal{Z}^{\operatorname{real}}$ a priori.

Now the problem splits naturally into two sub-problems:

• (a)

  Do we have $\mathcal{Z}^{\operatorname{real}}=\mathcal{Z}^{\operatorname{adm}}$?

• (b)

  What is the monoid structure of $\mathcal{Z}^{\operatorname{adm}}$?

Note that (a) is an arithmetic problem; to prove $\mathcal{Z}^{\operatorname{real}}=\mathcal{Z}^{\operatorname{adm}}$, we have to construct suitable extensions $L/K$. On the other hand, (b) is an algebraic problem.

As for (a), we will give the following affirmative answer, which will be proved in §4:

In particular, $\mathcal{Z}^{\operatorname{real}}$ is a submonoid of $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$. Note that we will moreover have a concrete condition on the base field $K$ to realize each admissible class. In particular, if $\Gamma$ is a cyclic group, every admissible class is realized by $K=\mathbb{Q}$ (see Remark 4.4).

As for (b), we need to introduce a finite set $\mathcal{T}$, which depends only on the group structure of $\Gamma$. When $\Gamma$ is a $p$-group, the set $\mathcal{T}$ is identified with the set of pairs $(I,D)$ such that

• •

  $I\subset D\subset\Gamma$ are subgroups,

• •

  $I$ is non-trivial, and

• •

  $D/I$ is cyclic.

The general definition will be given in Definition 5.1. The main result for (b) is the following:

Let us focus on $p$-groups. As an immediate corollary of Theorems 1.2 and 1.3, we obtain the following:

Here is a brief discussion on the relative size of $\mathcal{Z}^{\operatorname{real}}$ within $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ when $\Gamma$ is a non-trivial $p$-group for some prime number $p$. Note that the structure of $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ is already discussed in [5].

• •

  If $\Gamma$ is of order $p$, then both $\mathcal{Z}^{\operatorname{real}}$ and $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ are free of rank one. Indeed, we may even prove $\mathcal{Z}^{\operatorname{real}}=\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$, that is, all finite $\Gamma$-modules up to equivalence occur as minus class groups (see Theorem 4.1).

• •

  If $\Gamma$ is cyclic of order $p^{2}$, in [5] we have shown that $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ is not a free monoid and the rank of the abelian group associated to $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ is $4p-2$. This results from the Heller–Reiner classification on the $\Gamma$-lattices, given in [2]. On the other hand, it is easy to see that $\#\mathcal{T}=3$. Therefore, $\mathcal{Z}^{\operatorname{real}}$ is much smaller than $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ just as $3$ is smaller than $4p-2$.

• •

  For any other $\Gamma$, the classification of $\Gamma$-lattices is a deep result (see Heller and Reiner [7], [8]). In particular, it is known that the rank of the abelian group associated to $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ is infinite. On the other hand, $\#\mathcal{T}$ is of course always finite. Therefore, $\mathcal{Z}^{\operatorname{real}}$ is again much smaller than $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$.

So only a small portion of equivalence classes are realized as minus class groups. It is interesting to observe that the situation around plus components is totally in contrast (see Remark 2.14).

We begin by reviewing results from [5] in §2. We obtain a description of the equivalence classes of minus class groups, and then introduce the notion of realizable and admissible classes.

In §3, we prove Theorem 1.1 and check numerical examples. In §4, we prove Theorem 1.2. In §§5–6, we prove Theorem 1.3.

Let $R$ be a commutative ring that is Gorenstein of Krull dimension one. Typical examples include finite group rings such as $\mathbb{Z}[G]^{-}$ and $\mathbb{Z}^{\prime}[\Gamma]$, where

• •

  $G$ is a finite abelian group and $(-)^{-}$ is considered with respect to a fixed element $j\in G$ whose exact order is $2$.

• •

  $\Gamma$ is a finite abelian group.

Note that in [5] we established the general theory for Gorenstein rings of finite Krull dimension, but in this paper we only need dimension one cases.

Let $\mathcal{C}$ be the category of finitely generated torsion $R$-modules. Let us write $\mathcal{P}$ for the subcategory of $\mathcal{C}$ that consists of modules whose projective dimensions over $R$ are at most one. Note that when $R=\mathbb{Z}[G]^{-}$ or $R=\mathbb{Z}^{\prime}[\Gamma]$ as above, $\mathcal{C}$ consists of all finite $R$-modules and $\mathcal{P}$ consists of all finite $R$-modules that are $G$-c.t. or $\Gamma$-c.t., respectively (“c.t.” is an abbreviation of “cohomologically trivial”).

For basic properties of this notion and in particular the nontrivial fact that this is an equivalence relation we refer to [5, Remark 2.4, Propositions 2.5–2.6]. For now let us just remark that one easily shows: $M\sim 0$ is equivalent to $M\in\mathcal{P}$. Indeed, the definition of $\sim$ arose from the idea of forcing that property.

The set of equivalence classes $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ is equipped with a commutative monoid structure with respect to direct sums. In the following, $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$ will always be studied as a commutative monoid. For each module $M$ in $\mathcal{C}$, we write $[M]$ for the equivalence class of $M$ in $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$.

In [5, §4], we established an interpretation of the equivalence relation $\sim$ via lattices. Let us briefly review the results here.

Let $\operatorname{Lat}^{\operatorname{pe}}$ denote the commutative monoid of $R$-lattices up to projective equivalence. Here, an $R$-lattice (which we sometimes simply call a lattice) is by definition a finitely generated torsion-free $R$-module. Two $R$-lattices are projectively equivalent if they become isomorphic after adding finitely generated projective $R$-modules. We write

if $\mathcal{L}$ and $\mathcal{L}^{\prime}$ are projectively equivalent lattices. The monoid structure of $\operatorname{Lat}^{\operatorname{pe}}$ is defined by direct sums.

There are shift operators $\omega^{n}$ on the monoid $\mathcal{C}/\text{\lower 2.15277pt\hbox{$\sim$}}\,\,$, see [5, Definition 3.4] and following discussion. ($\omega^{n}(X)$ in this paper means $X[n]$ in [5].) Let us briefly define them.

On the lattice side, we have the Heller operator $\Omega$, which is an automorphism of $\operatorname{Lat}^{\operatorname{pe}}$. It works as follows: given a lattice $\mathcal{L}$, take an exact sequence $0\to\mathcal{M}\to F\to\mathcal{L}\to 0$, again with $F$ finitely generated projective over $R$. Then $\Omega$ sends the class of $\mathcal{L}$ to the class of $\mathcal{M}$.

We can link our shift operator $\omega^{1}$ and the Heller operator $\Omega$. The relation is as simple as possible.

We review the setting, which will be in force throughout the paper. Assume that $L$ is a CM-field which is an abelian extension of a totally real field $K$. Write $G$ for $\operatorname{Gal}(L/K)$. Note that $G$ must have even order; it contains a privileged element $j$ of order 2 given by complex conjugation.

We also have to discuss $T$-modification, in order to make the results from [1] applicable. Let $T$ be any finite set of prime ideals of $K$ not containing any ramified prime. Let $T_{L}$ denote the set of primes of $L$ that lie above primes in $T$.

An ideal $J$ of $L$ is called $T$-principal, if it admits a generator $x\in L^{\times}_{T}$, where the latter group is defined as

The $T$-modified class group $\operatorname{Cl}_{L}^{T}$ is then defined as the group of all fractional ideals of $L$ having support disjoint from $T$, modulo the subgroup of $T$-principal ideals. This is a slight enlargement of the usual class group $\operatorname{Cl}_{L}$. More precisely, there is a canonical surjection $\operatorname{Cl}^{T}_{L}\to\operatorname{Cl}_{L}$, and its kernel is an epimorphic image of $\bigoplus_{\mathfrak{P}\in T_{L}}\kappa(\mathfrak{P})^{\times}$, where $\kappa(\mathfrak{P})$ denotes the residue field at $\mathfrak{P}$.

The requirement for $T$ in [1] and many other papers is the following: In addition to the conditions already stated, $T$ must be such that $L^{\times}_{T}$ is $\mathbb{Z}$-torsion-free. In other words, we must have $\mu(L)\cap L^{\times}_{T}=\{1\}$, where $\mu(L)$ denotes the group of roots of unity in $L$. Trivially this implies that $T$ cannot be empty. However, the following lemma implies that in certain cases this modification does not matter. Let ${}_{p}\operatorname{Cl}_{L}$ and ${}_{p}\operatorname{Cl}_{L}^{T}$ be the $p$-Sylow subgroups of $\operatorname{Cl}_{L}$ and $\operatorname{Cl}_{L}^{T}$, respectively.

When one takes [1] and [5] together, one sees that using the notions of equivalence and of shifting one can say a lot on $\operatorname{Cl}_{L}^{T,-}$.

We need a little more notation. Let $v$ run through the finite primes of $K$ ramifying in $L$. For each such $v$, let $I_{v}\subset G$ be the inertia group and $\varphi_{v}\in G/I_{v}$ the Frobenius at $v$. Define

where $\#I_{v}$ denotes the order of $I_{v}$.

We will work over the ring $\mathbb{Z}[G]^{-}$ and define $\mathcal{C}$ and $\mathcal{P}$ accordingly. Then [1, Proposition 3.6] shows the existence of a short exact sequence of $\mathbb{Z}[G]^{-}$-modules

where $P$ is a $G$-c.t. module (i.e., in $\mathcal{P}$). Given the good behaviour of shift under equivalence, this gives the following basic result:

Note that the right hand side does not depend on the set $T$, and that the variance under $T$ is hidden in the equivariant $L$-value, which is not a part of the statement here. Nevertheless, to keep things technically correct, one has to leave $T$ in at least formally.

Now Lemma 2.7 implies that Theorem 2.6 can be rephrased as follows:

Let us fix a finite abelian group $\Gamma$ whose order is odd. We study the category $\mathcal{C}$ over $\mathbb{Z}^{\prime}[\Gamma]$. As explained in the introduction, we define the set of realizable classes as follows:

Next we define the submonoid of admissible classes. The motivation for the definition will be clear in Corollary 2.12.

Let us employ a useful term from group theory. Given a prime number $p$, a finite group is called $p$-elementary if it is the product of a $p$-group and a cyclic group. A finite group is called elementary if it is $p$-elementary for some prime number $p$.

Now Theorems 1.2 and 1.3 are formulated, except for the general definition of $\mathcal{T}$.

First we compute the lattice $\Phi(\omega^{1}(A_{I,\varphi}))$.

By Corollary 2.12 and Theorem 3.1, we obtain the following:

Now we begin the proof of Theorem 1.1. As in §1.1, we consider the case where $L^{+}/K$ is a cyclic $p$-extension, where $p$ is an odd prime number. Let $F$ be the unique quadratic extension of $K$ in $L$. Let us write $R=\mathbb{Z}_{p}[G]^{-}\simeq\mathbb{Z}_{p}[\Gamma]$ and ${}_{p}\operatorname{Cl}_{L}^{-}=\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\operatorname{Cl}_{L}^{-}$.

We begin with the following, which implies that ${}_{p}\operatorname{Cl}_{L}^{-}$ is a cyclic $R$-module in the situation of Theorem 1.1.

From now on, let us assume the hypotheses of Theorem 1.1. By Proposition 3.3, we can write ${}_{p}\operatorname{Cl}_{L}^{-}=R/J$ for a suitable ideal $J$. By Theorem 3.2, taking Lemma 2.5 into account, $J$ is projectively equivalent, and even isomorphic, to $(N_{\Gamma},p^{r})$. Note that this lattice is non-free, so the case $r=1$ follows at once. By Proposition 3.3, $J$ must contain $N_{\Gamma}$.

Therefore, Theorem 1.1 follows from the following algebraic proposition: