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A cohomological Tamagawa number formula. (English) Zbl 1230.11084
The Tamagawa number formula for a reductive algebraic group \(G\) over \(\mathbb{Q}\) relates the volume of \(G(\mathbb{A})/G(\mathbb{Q})\) (where \(\mathbb{A}\) denote the ring of adeles of \(\mathbb{Q}\)) to arithmetical invariants of \(G\). Bloch and Kato proposed a similar formula for all motives, and Bloch and Kato state that they see the intersection between the theory of algebraic groups and the theory of motives as consisting of abelian varieties and tori, and the authors’ philosophy is to view reductive groups in this intersection. Thus the purpose of the article is to propose a new definition of \(p\)-adic periods for reductive groups and obtain evidence that it is related to the Tamagawa number conjecture of Bloch and Kato.
Let \(G\) be a smooth linear group scheme over \(\mathbb{Z}\) whose generic fiber is connected reductive and assume that \(G(\mathbb{A})/G(\mathbb{Q})\) is compact. First the authors introduce a new definition of \(p\)-adic periods for such reductive groups. Let \(\mathfrak{g}\) be the \(\mathbb{Z}_p\)-Lie algebra associated to \(G/\mathbb{Z}_p\) and \([w]\) element in the Lie algebra cohomology of top degree which is represented by some invariant \(d\)-form \(\omega\) on \(G(\mathbb{Z}_p)\), then the authors construct a cohomological measure \(\mu_{[\omega]}^{\text{coh}}\) in the open subgroups of \(G(\mathbb{Z}_p)\) via the Lazard morphism [M. Lazard, Publ. Math., Inst. Hautes Étud. Sci. 26, 389–603 (1965; Zbl 0139.02302), ch. V, théor. 2.4.9] and via the compatibility of Lazard isomorphism with integral structures [A. Huber, G. Kings and N. Naumann, Compos. Math. 147, No. 1, 235–262 (2011; Zbl 1268.20051)] the authors prove \(\mu^{\text{coh}}_{[w]}\) coincides with the Tamagawa measure in the sense of Weil \(\mu^{\text{Tam}}_w\) [A. Weil, Adeles and algebraic groups. Progress in Mathematics, Vol. 23. Boston etc.: Birkhäuser (1982; Zbl 0493.14028), §2.2.1,]. As a corollary they obtain that for a basis element \([w]\in H^{n^2-1}(\text{Lie}(\text{SL}_n),\mathbb Z)\) one has \[ \prod_p\mu_{[w]}^{\text{coh}}(\text{SL}_n(\mathbb{Z}_p))=[\zeta(2)\zeta(3)\cdots\zeta(n)]^{-1}. \] The authors complete the construction of the measure for also the archimedean places: for \(\omega\neq 0\) an invariant \(d\)-form on \(G(\mathbb{R})\) define a cohomological measure with \(\mu_{w}^{Tam}=\mu_{[\omega]}^{\text{coh}}\).
Then they define a cohomological Tamagawa number with corrected local terms by the Euler factors with the above cohomological measure, similarly as the Tamagawa numbers are defined in the sense of Weil; the authors prove the equality of both Tamagawa numbers.
In the last section, the authors compare the cohomological measure with the Tamagawa measure in the sense of S. Bloch and K. Kato [L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)] and prove the Tamagawa number conjecture for some motives associated to tori. For this second part take \(T\) an algebraic torus of dimension \(d\) over \(\mathbb{Q}\) such that \(0=\text{rank} (\operatorname{Hom}_{\overline{\mathbb{Q}}}(T,\mathbb{G}_m))\) (then \(T(\mathbb{A})/T(\mathbb{Q})\) is compact) and consider the Artin-Tate motive \(h_1(T)\) with Betti realization \(\operatorname{Hom}_{\overline{\mathbb{Q}}}(\mathbb{G}_m,T)\otimes\mathbb{Q}(1)\), and the usual integral structures in the realizations. Then the authors compute the local and global Bloch-Kato points of the motive in terms of the torus and prove that the Bloch-Kato Tamagawa measure on \(h_1(T)\) coincides with the cohomological measure. To deduce the Tamagawa number conjecture for \(h_1(T)\) (in the sense of Bloch and Kato), the authors use the result of T. Ono on the Tamagawa measure of Weil [Ann. Math. (2) 78, 47–73 (1963; Zbl 0122.39101), §5] and the relation proved in the paper under review between the Tate-Shafarevich group \(\text Ш(h_1(T))\) for \(h_1(T)\) (in the sense of Bloch-Kato [op. cit.]) and the Tate-Shafarevich group for tori [J. S. Milne, Arithmetic duality theorems. Boston etc.: Academic Press (1986; Zbl 0613.14019)], in particular, in this paper A. Huber and G. Kings obtain the relation between the order of \(\text Ш(h_1(T))\) with the Ono constant defined in [op. cit.].

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R42 Zeta functions and \(L\)-functions of number fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
22E41 Continuous cohomology of Lie groups
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