Capitulation in cyclotomic extensions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:09:19Z http://mathoverflow.net/feeds/question/40324 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40324/capitulation-in-cyclotomic-extensions Capitulation in cyclotomic extensions Franz Lemmermeyer 2010-09-28T14:36:30Z 2010-09-30T11:28:43Z <p>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?</p> http://mathoverflow.net/questions/40324/capitulation-in-cyclotomic-extensions/40605#40605 Answer by Chris Wuthrich for Capitulation in cyclotomic extensions Chris Wuthrich 2010-09-30T11:28:43Z 2010-09-30T11:28:43Z <p>Assume $p$ is an irregular prime for which <a href="http://en.wikipedia.org/wiki/Vandiver%20s_conjecture" rel="nofollow">Vandiver's conjecture</a> holds, e.g. $p&lt;12'000'000$. This conjecture asserts that $p$ does not divide the $+$-part of the class group.</p> <p>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 <a href="http://math.washington.edu/~greenber/book.pdf" rel="nofollow">Greenberg's book</a>, 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.</p> <p>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.</p>