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Local Tamagawa numbers and the Bloch-Kato conjecture for the motives \(\mathbb Q(m)\) over an abelian field. (Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs \(\mathbb Q(m)\) sur un corps abélien.) (French) Zbl 1125.11351

In this instructive and nicely written article, the authors present a proof of the Bloch-Katō conjecture for the motive \(Z(m)\) over an arbitrary absolutely abelian number field \(F\), where \(m\) is an arbitrary integer. The cases \(m=0,1\) are known, and are not discussed any further. The main ingredients are a result on algebraic \(K\)-groups (cohomological Lichtenbaum conjectures) and explicit calculations. For \(F\) the rationals, the result is already in the ground-breaking paper of S. Bloch and K. Kato [in: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)]. A somewhat more general result (validity of the Bloch-Kato conjecture for Dirichlet motives) was recently proved by Huber and Kings by entirely different methods, and a yet stronger result (to wit: a result on equivariant Tamagawa numbers) was given by D. Burns and the reviewer [Invent. Math. 153, No. 2, 303–359 (2003; Zbl 1142.11076)], again by a different approach and in a rather different mathematical language.
Let us briefly explain the setting and the approach used by the authors of the present paper. Given a motive \(M\) defined over \(F\) (for example \(M=Z(m)\)), Bloch and Kato associated with it, under certain assumptions, a Tamagawa number \(\text{Tam}(M)\) which roughly speaking encompasses Euler factors on the one hand and regulators on the other hand. In fact, \(\text{Tam}(M)\) is the measure of a certain compact abelian group attached to \(M\), whose definition resembles the construction of the idéle class group; calculating this measure involves calculations at finite places and something like a regulator calculation at the infinite places.
A little more precisely: In the present article, one deals with a global number \(\text{Tam}^0(M)\) which is the product of local contributions \(\text{Tam}^0_v(M)\), \(v\) running over the finite places of \(F\) and \(\infty\). In the Bloch-Katō article [loc. cit.], only the global Tamagawa number \(\text{Tam}(M)\) is explicitly defined, but one implicitly has local contributions \(\text{Tam}_v(M)\), denoted \(\mu_p(A(Q_p))\) in [S. J. Bloch and K. Katō, loc. cit.] for \(v\) finite (they assume \(F=Q\), so \(v\) corresponds to a prime \(p\)); these differ from \(\text{Tam}^0_v(M)\) by an Euler factor; cf. p. 240 in J.-M. Fontaine’s article [Astérisque No. 206, Exp. No. 751, 205–249 (1992; Zbl 0799.14006)]. Hence the conjecture C\(_{\text{BK}}\) involving \(\text{Tam}^0(M)\) as formulated in the article under review shows a zeta function on its left hand side, whereas the corresponding statement involving \(\text{Tam}(M)\) due to Bloch and Katō [loc. cit. (Section 5.15)] contains no zeta or \(L\)-function.
The authors prepare everything that is needed for the proof of C\(_{\text{BK}}\) for \(Z(m)\) in Sections 2 to 4: they calculate the local Tamagawa numbers outside infinity (this generalizes the corresponding result for the unramified case in the Bloch-Kato paper [loc. cit.], they determine the Tamagawa number at infinity, using the Beilinson regulator, and also the Ш groups, which are closely related to higher \(K\)-groups. When all these calculations are finally assembled, to obtain the desired result it suffices to invoke the validity of the (cohomological) Lichtenbaum conjecture which relates the leading term of the zeta function at \(s=1-m\) (for \(m\geq2\)), the Beilinson regulator, and the size of the \((2m-2)\)nd cohomological \(K\)-group. This validity was proved, for the absolutely abelian case, by M. Kolster, T. Nguyen Quang Do and V. Fleckinger [Duke Math. J. 84, 679–717 (1996; Zbl 0863.19003)]. (Remark: In an appendix, the authors of the present article take the opportunity of carefully cleaning up a few technical inaccuracies in that work; this mainly concerns superfluous Euler factors and one inappropriate reference.) In the final part of the proof, there is a case distinction \(m\leq-1\) and \(m\geq 2\). In the proof of the latter case, the functional equation for the zeta function comes in. The authors remark at an early stage that their results imply the compatibility of the Bloch-Katō conjecture with the functional equation.
The calculation of the Tamagawa numbers at finite places for the motives \(Z(m)\) involves finding the order of the cokernel of the Bloch-Kato exponential. This is done using a result of Perrin-Riou generalizing Coleman theory, and the calculation of the relative index of two explicit lattices in \(F\).
In Section 2.4 the authors give some formulas for equivariant Tamagawa numbers, mentioning that these should suffice to get compatibility with the functional equation in the equivariant setting as well. It appears that Benois and Burns (work in progress) have results in this direction.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11G55 Polylogarithms and relations with \(K\)-theory
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