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Equivariant Tamagawa numbers and Galois module theory. I. (English) Zbl 1014.11070

In this important paper the author defines an element \(T\Omega(L/K,0)\) attached to a Galois extension \(L/K\) and provides links with many other constructions and conjectures. The Galois group \(G\) of \(L/K\) is permitted to be non-Abelian, and \(T\Omega(L/K,0)\) is an element of the relative K-group K\(^0(\mathbb Z[G], \mathbb R)\). It is conjectured that \(T\Omega(L/K,0)\) is always zero. The author proves the following:
(1) The class \(T\Omega(L/K,0)\) comes from K\(^0(\mathbb Z[G], \mathbb Q)\) iff the Stark conjecture holds for \(L/K\), and if this case occurs, \(T\Omega(L/K,0)\) is zero iff the Strong Stark conjecture holds.
(2) The nullity of \(T\Omega(L/K,0)\) is equivalent to the Lifted Root Number Conjecture of K. W. Gruenberg, J. Ritter and A. Weiss [cf. Proc. Lond. Math. Soc. (3) 79, 47–80 (1999; Zbl 1041.11075); Jahresber. Dtsch. Math.-Ver. 100, 36–44 (1998; Zbl 0929.11054)]; more precisely, the element \(\omega_{\varphi_S}\) defined by the latter authors agrees with \(T\Omega(L/K,0)\) up to a canonical involution.
(3) The nullity of \(T\Omega(L/K,0)\) implies Chinburg’s \(\Omega(3)\)-conjecture for \(L/K\).
(4) The element \(T\Omega(L/K,0)\) (which was constructed globally, that is, without using any localization result for relative K-groups) agrees with an element, or rather, a family of elements indexed by all prime numbers \(p\) that were defined in an earlier preprint of the author and called equivariant Tamagawa numbers.
(Comment: One now speaks of the conjecture that \(T\Omega(L/K,0)\) vanishes as “the Equivariant Tamagawa number conjecture” for a certain motive.)
The construction of \(T\Omega(L/K,0)\) fits into a larger framework; actually \(T\Omega(L/K,0)=\break T\Omega(L/K,M)\) where the motive \(M\) is specialised to \(\mathbb Q(0)_L\). The main ingredients are reduced determinants and refined Euler characteristics of perfect complexes. To define the latter one needs also to fix a so-called “trivialisation fibre”, that is, a class of isomorphisms between the odd-degree cohomology and the even-degree cohomology of the given complex. The complexes that are fed into the machinery arise from earlier work of the author with M. Flach [Am. J. Math. 120, 1343–1397 (1998; Zbl 0929.11050)] and can be thought of as a kind of Tate four term sequence, and the trivialisation fibre comes from regulator maps.
Two small remarks: The objects of \({\mathcal M}_K(A)\) are \(K\)-motives endowed with \(A\)-action; of course there is a forgetful functor \({\mathcal M}_K(A) \to{\mathcal M}_K \) but perhaps \({\mathcal M}_K(A)\) is not a subcategory of \({\mathcal M}_K \) in the obvious way. On p. 217, if \(R\) is allowed to be any commutative ring, \(R[G]\) will not quite be Gorenstein, but the used properties of \(R\)-duals certainly hold true.
There is a lot of recent work on the equivariant Tamagawa number conjecture, much of it by the author and his coworkers. The article under review is fundamental for all this.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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