On formulae for the class number of real Abelian fields. (English. Russian original) Zbl 1007.11065

Kuz’min, L. V.

Izv. Math. 60, No. 4, 695-761 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 4, 43-110 (1996).

Summary: For a given real Abelian field \(k\) and a given prime natural number \(\ell\) we obtain an index formula for the order of the group \(C\ell(k)_{\ell,\varphi}\), where \(C\ell(k)_\ell\) is the \(\ell\)-component of the class group of \(k\). \(C\ell(k)_{\ell,\varphi}\) denotes the \(\varphi\)-component of \(C\ell(k)_\ell\) corresponding to a \(\mathbb{Q}_\ell\)-irreducible character \(\varphi\) of the Galois group \(G(k/\mathbb{Q})\) that is trivial on the Sylow \(\ell\)-subgroup of \(G(k/\mathbb{Q})\). This result generalizes a conjecture of Gras. The proofs rely on the ‘main conjecture’ of Iwasawa theory.

MSC:

11R29	Class numbers, class groups, discriminants
11R20	Other abelian and metabelian extensions
11R23	Iwasawa theory