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Let $K$ be an imaginary quadratic number field with $p$-Sylow-class group $A(K)$ and $L/K$ be an unramified cyclic extension of $K$ of degree $p$ ($p$ prime). Then I am looking for heuristics on

$ker(N_{L/K}:A(L)\rightarrow A(K))$,

where $N_{L/K}$ is the usual norm map on ideal classes. Numerical data suggest that:

$|KerN_{L/K}|\le p^2$ with high probability. But that is rather unprecise. Are there any more satisfactory known results ?

There is a result by Wittmann who gives some heuristics for cyclic $p$-extensions of $\mathbb{Q}$ and I wonder if one can possibly generalize his ideas to the above scenario.

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  • $\begingroup$ I would expect similar results to hold for cyclic extensions of number fields with trivial p-class group. Have you tried to generalize Wittmann's results to such fields? If yes, where does the proof break down? $\endgroup$
    – Franz Lemmermeyer
    Commented Aug 18, 2011 at 13:46
  • $\begingroup$ Wittmann has shown that his results also hold for cyclic p-extensions of imaginary quadratic fields with trivial p-class group. But I am interested in unramified cyclic p-extensions, which don't exist in that case. $\endgroup$
    – Tobias Bembom
    Commented Aug 19, 2011 at 11:52
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    $\begingroup$ If you take a complex quadratic field with large $p$-rank and unramified p-extensions, Golod-Shafarevich-type arguments will tell you that the relative class number tends to have large p-rank not only with high probability, but always. $\endgroup$
    – Franz Lemmermeyer
    Commented Aug 19, 2011 at 20:00
  • $\begingroup$ Thank you very much for your comment. I also read your paper on class field towers of 2010. It was very interesting and helpful. $\endgroup$
    – Tobias Bembom
    Commented Sep 2, 2011 at 14:21

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The forthcoming paper "Heuristics for p-class towers of imaginary quadratic fields," by Nigel Boston, Michael Bush, and Farshid Hajir, will answer your question. In fact, given any finite p-group G, it provides a heuristic for the question "What is the probability that the maximal everywhere-unramified p-extension of K has Galois group isomorphic to G?"

They don't seem to have posted a paper yet, so I'm hesitant to say more about the result without asking their permission, but I'm sure one of them will be happy to send you the preprint or at least tell you what the heuristic is.

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