Equivariant Tamagawa numbers, Fitting ideals and Iwasawa theory (Q2721351)

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.). NEWLINENEWLINENEWLINEIn 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 \(K=\mathbb Q\). NEWLINENEWLINENEWLINEIn 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).