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If a number field $K$ is a Galois extension of $\mathbb{Q}$, and $G = \operatorname{Gal}(K/\mathbb{Q})$, then the class group of $K$ is a $\mathbb{Z}[G]$-module, and since $N = \sum_{g \in G} g$ acts as the norm on ideals, sending every ideal to a principal ideal, the class group is in fact a $\mathbb{Z}[G]/\langle N \rangle$-module.

Roughly speaking, the Cohen-Lenstra heuristics predict that the class group of a totally real Galois number field $K$ should behave as a ``random" finite $\mathbb{Z}[G]/\langle N \rangle$-module modulo a random cyclic submodule.

My question is this: Can a similar kind of statement be made about totally complex number fields $K$ that are Galois over $\mathbb{Q}$? For example, would it simply be a random $\mathbb{Z}[G]/\langle N \rangle$-module, without quotienting out a random cyclic submodule, as in the case for imaginary quadratic fields? Or is that too naive?

Along the same lines, what can be said about the Cohen-Lenstra heuristics for the relative class group of a Galois extension of totally complex number fields?

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1 Answer 1

The general heuristic goes as follows (see the original paper by Cohen-Martinet, but I am being a bit more conservative, since some primes that Cohen-Lenstra-Martinet called "good" seem to not be all that "good", and I am leaving them out here):

Fix

  • a base field $K$,
  • a Galois group $G$,
  • any prime $p$ that is coprime to $\#G$ and to the order of the group of roots of unity $\mu(K)$,
  • any central idempotent $e$ of $\mathbb{Q}[G]$ (which automatically, by assumption on $p$, lives in $\mathbb{Z}_{(p)}[G]$) that is orthogonal to the trivial idempotent $\frac{1}{|G|}\sum_{g\in G} g$.
  • and finally a $\mathbb{Z}_{(p)}$-free $\mathbb{Z}_{(p)}[G]$-module $\Gamma$.

Let $F_i$ be the sequence of those $G$-extension of $K$ for which $\mathcal{O}_{F_i}^\times\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$ is isomorphic to $\Gamma$ as a $G$-module, ordered by absolute value of discriminant (and arbitrarily between fields of equal discriminant). Let $A_i$ be the $p$-primary part of the class group of $F_i$. Then the sequence $eA_i$ behaves like a random sequence of finite $e\mathbb{Z}_{(p)}[G]$-modules of $p$-power order, with the probability weight of such a module $A$ inverse proportional to $\#{\rm Hom}_G(\Gamma,A)\cdot\#{\rm Aut}_G(A)$.

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Alex - Thanks! This formulation of the heuristics raises for me a number of questions. To start with, is this formulation specialized to Galois extensions, or is it for general extensions? More specifically, by "$G$-extension" $F_i$ do you mean that $F_i$ is Galois over $K$ with Galois group $G$, or that the Galois closure of $F_i$ over $K$ has Galois group $G$? –  grad student Aug 19 at 1:30
    
Here, I was taking $F_i$ to be Galois over $K$. But in fact, you can extract from this general formulation the heuristic for non-Galois sub-extensions by taking suitable idempotents $e$. –  Alex B. Aug 19 at 5:31

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