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Equivariant Tamagawa numbers, Fitting ideals and Iwasawa theory. (English) Zbl 0987.11069
Specializing the much more general theory of “Equivariant Tamagawa numbers” for motives down to number fields, one obtains for any finite Galois extension \(L/K\) of number fields with Galois group \(G\) the lifted omega invariant \(T \Omega (L/K)\) as an element of the relative Grothendieck group \(K_0 (\mathbb Z [G], \mathbb R)\). The second author shows in [Compos. Math. 129, No. 2, 203-237 (2001; Zbl 1014.11070)] that the Stark (Strong Stark, resp.) conjecture holds for \(L/K\) if and only if \(T \Omega (L/K)\) already belongs to \(K_0 (\mathbb Z [G], \mathbb Q)\) (and is a torsion element, resp.).
In the present paper the authors describe \(T \Omega (L/K)\) in terms of finite \(G\)-modules (Proposition 2.5) and, in case that \(L/K\) is abelian, still more explicitly by using Fitting ideals (Theorem 2.7). The “Equivariant Tamagawa Number Conjecture” would imply that \(T \Omega (L/K)\) vanishes, which is up to now only proven for some special abelian extensions \(L\) over \(K=\mathbb Q\).
In the second part of this paper the authors use this explicit description to investigate \(T \Omega (L/\mathbb Q)\), where \(L\) is any subfield of \(L_1 L_2\). Here \(L_i\) denotes the maximal real subfield of the cyclotomic field generated by the roots of unity of order \(l_i^{a_i}\), with different odd primes \(l_1, l_2\), and it is further supposed that for \(i \neq j\) \(l_i\) does not split in \(L_j\). Using methods of Iwasawa theory, class field theory and cyclotomic units, it is shown that \(T \Omega (L/\mathbb Q)\) vanishes up to its \(2\)-component, which might be non trivial only for \([L:\mathbb Q]\) even (Theorem 1.1).
Reviewer: G.Lettl (Graz)

MSC:
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R18 Cyclotomic extensions
11R23 Iwasawa theory
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