The structure of some minus class groups, and Chinburg’s third conjecture for abelian fields.(English)Zbl 0919.11072

This paper, in some way, is a “multiplicative continuation” of the author’s work on Chinburg’s second invariant $$\Omega(L/\mathbb{Q},2)$$ for real abelian fields $$L$$ of odd prime power conductor $$l^m$$ [C. Greither, J. Reine Angew. Math. 479, 1-37 (1996; Zbl 0856.11051)]. In fact, for the same class of fields it is shown that also the third Chinburg class $$\Omega_3=\Omega(L/\mathbb{Q},3)$$ vanishes (actually, $$l=2$$ is not forbidden anymore). The restriction of the absolute ramification of $$L$$ clears away possible difficulties with the Tate class involved in the definition of the $$\Omega$$-invariants and arising from the global and local fundamental classes. There are some very nice and new ideas presented here in the multiplicative situation, which lead to even stronger results. Namely, the paper starts off from a splitting of $$\Omega_3$$ into $$\Omega^+Cl(\mathbb{Z} G_+)$$ and $$\Omega^-\in CL(\mathbb{Z} G_-)$$ with $$G$$ denoting the Galois group of a (non-necessarily real) abelian extension $$L/\mathbb{Q}$$ with conductor $$l^m$$. The splitting is induced by complex conjugation, $$c$$ say, and indeed, $$\Omega^+=\Omega(L^+/\mathbb{Q},3)$$, $$\mathbb{Z} G_+=\mathbb{Z} G/\langle c\rangle$$.
The first part of the paper concerns $$cl_L^-$$ and $$\Omega^-$$ under the restriction that $$L$$ is imaginary and $$l$$ is such that $$c\in\text{Gal}(\mathbb{Q} (\zeta_l)/\mathbb{Q})_p$$, the decomposition subgroups of the primes $$p$$ dividing $$l-1$$. The discussion is based on a result of Schoof, by which $$cl_L^-$$ has projective dimension $$\leq 1$$ over $$\mathbb{Z} G_-$$, and on Iwasawa theory by means of which the Fitting ideal of $$cl_L^-$$ is determined and (almost) realized as a Stickelberger ideal. On account of a Tate sequence $$E_S \rightarrowtail A\to B\twoheadrightarrow\Delta S$$ the two objects, $$\Omega^-$$ and $$cl_L^-$$, get related by $$\Omega^-=(E_S^-)-(\Delta S_-)$$, $$(E_S^-)+(cl_L^-)=(\mu_L)+\Delta S_-)$$ in $$K_0(\mathbb{Z} G_-)$$. Here, $$E_S$$ is the group of $$S$$-units in $$L$$ and $$\mu_L$$ its torsion subgroup. The first main theorem is now $$\Omega^-=0$$.
The second part of the paper regards real abelian field extensions $$L/\mathbb{Q}$$ with conductor $$l^m$$. Again, Iwasawa theory and impressive Fitting ideal computations with respect to the cohomologically trivial module $$E_S$$ lead to $$\Omega_3=0$$ in this case. – Related to this work is [D. Burns, Am. J. Math. 117, 875-903 (1995; Zbl 0863.11076); D. Burns and D. Holland, Proc. Lond. Math. Soc. 74, 29-51 (1997; Zbl 0885.11060); and J. Ritter and A. Weiss, J. Am. Math. Soc. 10, 513-552 (1997; Zbl 0885.11059)] and a preprint by Bley and Burns which concerns the Lifted Root Number Conjecture for real abelian field extension $$L/\mathbb{Q}$$ of prime power conductor.

MSC:

 11R29 Class numbers, class groups, discriminants 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R20 Other abelian and metabelian extensions 11R18 Cyclotomic extensions 11R23 Iwasawa theory
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