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On Chinburg’s root number conjecture. (English) Zbl 0929.11054
This is a survey article, explaining the authors’ construction of a lifting \(\omega=\omega(K/k)\) of the (third) Chinburg class \(\Omega(3,K/k)\) attached to a \(G\)-Galois extension of number fields, where lifting means: specifying a preimage of \(\Omega(3)\) minus the root number class under the canonical surjection \(K_0T({\mathbb Z}G) \to Cl({\mathbb Z}G)\). For most details, the authors refer to forthcoming publications.
The invariant \(\omega\) is constructed as \(\Omega_\phi-A_\phi(\widehat\chi)W_{K/k}(\widehat\chi)\), where \(\Omega_\phi\) is constructed algebraically from an isogeny \(\Delta S\to E\), and \(A_\phi(\widehat\chi)\) is defined arithmetically, roughly speaking as a regulator divided by a leading term of an \(L\)-function. The whole construction is subject to the validity of the Stark conjecture, but the outcome does not depend on the choice of \(\phi\). The Lifted Root Number Conjecture in this context then simply says that \(\omega\) vanishes. The authors exhibit two situations where this conjecture can be proved: firstly, if \(K=\mathbb Q\) and \(L\) has odd prime degree, with exactly two primes ramifying; and secondly (here the argument is outlined in the paper) tame extensions \(L\) of prime degree over \(\mathbb Q\) with only one ramified prime. Concerning the first result, there is some recent progress due to R. Kučera and the reviewer. It should be pointed out that Burns and Flach have recently defined an invariant dubbed “equivariant Tamagawa number” \(T\Omega\) in the relative group \(K({\mathbb Z}[G], {\mathbb R})\) in general, by a rather different approach involving lifted Euler characteristics of perfect complexes; Burns has proved that \(T\Omega\) vanishes if and only \(\omega\) vanishes, provided of course that the latter is defined.

11R42 Zeta functions and \(L\)-functions of number fields
11R27 Units and factorization
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
19A31 \(K_0\) of group rings and orders
19B28 \(K_1\) of group rings and orders
11-02 Research exposition (monographs, survey articles) pertaining to number theory