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Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B_k$ (for some even $k\in[2,p-3]$). This implies that the class number of the field $K$ of $p$-th roots of unity is divisible by $p$. Let $L$ be the field of $p^2$-th roots of unity. What, if anything, is known about the capitulation of ideal classes in $L/K$ ( we say that an ideal class from $K$ capitulates in $L$ if an ideal generating this class becomes principal there)? It is possible to write down criteria in terms of units that are or are not norms from $L$, but this does not seem to help a lot. I am mainly interested in the question whether there is a connection between the index $k$ and the capitulation of the subgroup of order $p$ corresponding to $k$ via Herbrand-Ribet. I am pretty sure that classical algebraic number theorists did not do an awful lot in this direction but I am not familiar with any advances in Iwasawa theory: whether an ideal class capitulates in $L/K$ is encoded in the Hilbert class field, so the structure of the maximal abelian unramified $p$-extension of the cyclotomic Iwasawa extension of $K$ might contain relevant information. Does it?

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I love the usage of «capitulation» :) – Mariano Suárez-Alvarez Sep 28 '10 at 14:39
@Mariano: the word capitulation was coined by Arnold Scholz; the German word for principal (as in principal ideal) is Haupt, which is caput in Latin; but caputilation would sound silly in both German and English -) – Franz Lemmermeyer Sep 28 '10 at 16:21
I prefer that they give up, as in capitulation, rather than if they would be beheaded, as in enthaupted (=decapitation). – Chris Wuthrich Sep 28 '10 at 17:20
Yeah: it is a nice image: those classes doing the best to survive, extension after extension until finally, well, they just have to give in and submit to principalization :P – Mariano Suárez-Alvarez Sep 28 '10 at 21:26
up vote 8 down vote accepted

Assume $p$ is an irregular prime for which Vandiver's conjecture holds, e.g. $p<12'000'000$. This conjecture asserts that $p$ does not divide the $+$-part of the class group.

Then there is no capitulation in the class group from the first layer of the cyclotomic $\mathbb{Z}_p^{\times}$-tower to any other in this tower. See Proposition 1.2.14 in Greenberg's book, which says that the capitulation kernel lies in the $+$-part. See also the discussion on page 102 where it is discussed what happens when Vandiver's conjecture does not hold.

Generally capitulations in Iwasawa theory are well studied. The capitulation is linked to the question of whether there are non-trivial finite sub-$\Lambda$-modules in the Iwasawa module $X$, here the projective limit of the $p$-primary parts of the class groups in the tower, or equivalently the Galois group mentioned in the question.

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But I guess the main conjecture of Iwasawa theory does not say anything about this because we can ignore finite submodules. The advances in Iwasawa theory have only been on formulating main conjectures in more general situations and not very much on getting finer information about the Iwasawa modules (Kurihara's work on computing all Fitting ideals in some situations is an exception to this that I know but not enough for these kinds of question I think). Does anyone if ETNC says anything for such conjectures? – Mahesh Kakde Sep 30 '10 at 13:10
Slightly more is true (as I should have known): the whole minus part of the class group cannot capitulate. This leaves us with the question which eigenspaces of the plus class group may or may not capitulate. Since this is entirely a problem involving the structure of the unit group of the real subfield, I guess it is not wise to expect an answer one way or another. – Franz Lemmermeyer Sep 30 '10 at 14:22

The Greenberg Conjecture for the p-cyclotomic field has not been proved, but is widely assumed to hold. Personally I trust the Vandiver conjecture too, but opinions about this conjecture are divided. Assuming that Vandiver is false and Greenberg true, then you have capitulation for every positive, non - trivial eigenspace. In fact, the same can be proved also in general (without assuming Greenberg), for the eigenspaces with non trivial torsion subgroups. However, I speak here of capitulation along the Iwasawa tower (K_n)_(n in N), in which the p^2 extension L = K_2. It is not necessary that capitulation begin from the first extension K_2/K_1. Since Vandiver is true for all p < 12 000 000, little numeric evidence can help. I nbrief, I believe that the best we can say by now, is that capitulation depends on the existence of torsion modules in a given eigenspace; but it needs not start from the first extension. And Vandiver denies its existence.

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