Quaternionic exercises in \(K\)-theory Galois module structure. II
For Part I, see Algebraic \(K\)-theory, 2nd Great Lakes conf. 1996, Fields, Inst. Commun. 16, 4-29 (1997; Zbl 0886.11063).NEWLINENEWLINENEWLINEThe purpose of this paper is to investigate, for a certain infinite family of totally real quaternionic fields \(N/\mathbb{Q}\), the relationship between \(W_{N/\mathbb{Q}}\) and \(\Omega_1(N/\mathbb{Q},3)\) in \(Cl(\mathbb{Z}[Q_8])\). Recall that \(W_{N/\mathbb{Q}}\) is the Cassou-Noguès-Fröhlich root number class, and \(\Omega_1(N/\mathbb{Q},3)\) is the ``higher Chinburg invariant'' constructed by Chinburg-Kolster-Pappas-Snaith, namely the ``Euler characteristic'' arising from a canonical 2-extension NEWLINE\[NEWLINEK_3^{\text{ind}} (O_{N,S})\to A\to B\to K_2(O_{N/S})'.NEWLINE\]NEWLINE Here \(S\) is any equivariant finite set of primes containing the ramified primes, and the dash indicates a modified \(K_2\) (in order to deal with real places). It is of course conjectured that \(\Omega_1(N/\mathbb{Q},3)= W_{N/\mathbb{Q}}\) (\(=1\) in the present situation), and the main result of the authors is a partial computation of \(\Omega_1(N/\mathbb{Q},3)\) in terms of special values of Artin \(L\)-functions. Not surprisingly, the calculations are concentrated at the prime~2.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00018].
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