## 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

### Keywords:

Tamagawa numbers; Stark’s conjecture; Galois structures

### Citations:

Zbl 1041.11075; Zbl 0929.11054; Zbl 0929.11050
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