Computing the relative class group (with Galois action) of relatively large cyclotomic groups

Question

For a cyclotomic field $K = \mathbb Q(\zeta_n)$, let $K^+$ be its maximal totally real subfield. We know that $H^+ = Cl(K^+)$ injects into $H = Cl(K)$. I am interested in computing the group $H/H^+$ and the corresponding Galois action.

I found some references for $n$ prime (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.370.4613&rep=rep1&type=pdf) but I am really interested in specific composite n (n = 55,57,63,67, 69, 85...). The corresponding relative class numbers for these values are (10,9,7, 853513, 69, 6205...)

Does anyone know references or suggestions on how to compute the class group? I am specifically interested in figuring out if a specific character occurs in the class group.

Answer 1

Have you had a look at Schoof's paper The structure of the minus class group of abelian number fields, Séminaire de Théorie des Nombres de Paris, 1988--89? Schoof's idea is to conjecture that the Fitting ideal of the minus class group of any abelian complex number field $F$ (meaning the quotient $H_F/H_{F^+}$) should be equal to what he calls a "Stickelberger ideal" generated by Stickelberg elements. As he points out at the end of his introduction, this has somehow been superseeded by Kolyvagin's idea of Euler Systems, but the equality between Fitting and Stickelberger ideal remains open (there is a paper by Kurihara on this, entitled Iwasawa theory, higher Fitting ideals, and Kolyvagin systems of Gauss sum type).

Quite interesting is his Theorem 4.3 saying showing a criterion for freeness of the $\chi$-component of the minus class group.

Answer 2

This might help, although I am not sure if it is exactly what you need:

Aoki, Miho and Fukuda, Takashi An algorithm for computing p-class groups of abelian number fields, Algorithmic number theory, 56–71, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006. DOI:10.1007/11792086_5

From the abstract:

For an abelian number field $F$ and an odd prime number $p$ which does not divide the degree $[F:\mathbb{Q}]$, we propose a new algorithm for computing the $p$-primary part of the ideal class group of $F$ using Gauss sums and cyclotomic units.

I think they compute the $\chi$-components separately.