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Autour des conjectures de Bloch et Kato. III: Le cas général. (On the conjectures of Bloch and Kato. III: The general situation). (French. Abridged English version) Zbl 0758.14001
[For part I and II see ibid. No. 5, 189-196 and No. 6, 349-356 (1991; Zbl 0749.11052 and 752.14001).]
This paper is the third of a series on the Bloch-Kato conjectures about the values at $$s=0$$ of the $$L$$-function of a ‘motive’ (or a ‘motivic structure’) over a number field. We use concepts and notation defined in the two foregoing articles I and II cited above. Let $$E$$ be a number field, and define a premotivic category $${\mathcal M}$$ to be a full subcategory of the $$\mathbb{Q}$$-linear neutral Tannakian category $$SPM_ \mathbb{Q}(E)$$ of premotivic structures over $$E$$ such that $${\mathcal M}$$ is stable under subobject, quotient, direct sum, tensor product and dual. Such $${\mathcal M}$$ is an abelian category. For an object $$M$$ of $${\mathcal M}$$ one defines the $$\mathbb{Q}$$-vector spaces $$H^ 0_ M(E,M):=\text{Hom}(1,M)$$ and $$H^ 1_ M(E,M):=\text{Ext}^ 1(1,M)$$. One has maps $$u_ M:H^ 0(E,M)_ \mathbb{R}\to\text{Ker}(\alpha_ M)$$ and $$u_ M':\text{Coker}(\alpha_ M)\to H^ 0(E,M^*(1))^*_ \mathbb{R}$$.
The
‘$$\ell$$-adic realization functor’ induces a $$\mathbb{Q}$$-linear map $$H^ 1_ M(E,M)\to H^ 1(E,M_ \ell)$$, $$x\mapsto x_ \ell$$. Write $$H^ 1_{M,f}(E,M)$$ for the $$\mathbb{Q}$$-subspace of $$H^ 1_ M(E,M)$$ consisting of all of $$x$$ such that $$x_ \ell\in H^ 1_ f(E,M_ \ell)$$ for all $$\ell$$. Then $${\mathcal M}$$ is called $$f$$-admissible if
(i) for any object $$M$$ of $${\mathcal M}$$, $$\dim_ \mathbb{Q} H^ 1_{M,f}(E,M)<\infty$$;
(ii) $$H^ 1_{M,f}(E,1)=0$$;
(iii) for any short exact sequence $$0\to M\to N\to 1\to 0$$, whose class in $$H^ 1_ M(E,M)$$ is contained in $$H^ 1_{M,f}(E,M)$$, the map $$H^ 1_{M,f}(E,N^*(1))\to H^ 1_{M,f}(E,M^*(1))$$ is surjective;
(iv) the map $$u_{M^*(1)}:H^ 0(E,M^*(1))_ \mathbb{R}\to\text{Ker}(\alpha_{M^*})$$ is surjective for any object $$M$$ of $${\mathcal M}$$ such that $$H^ 1_{M,f}(E,M)=0$$ and the sequence $0\to H^ 0(E,M)_ \mathbb{R}\to\text{Ker}(\alpha_ M)\to H^ 1_{M,f}(E,M^*(1))^*_ \mathbb{R}\to 0$ is exact.
One also defines $$f$$-equivalence on $${\mathcal M}$$ as the strongest equivalence relation on $${\mathcal O}b({\mathcal M})$$ such that (i) if $$0\to M\to N\to 1\to 0$$ is a short exact sequence with class in $$H^ 1_{M,f}(E,M)$$, then $$M$$ and $$N$$ are $$f$$-equivalent; (ii) if $$M$$ and $$N$$ are $$f$$-equivalent on $${\mathcal M}$$, then $$M^*(1)$$ and $$N^*(1)$$ are also $$f$$-equivalent.
In the sequel it is assumed that one has at one’s disposal the category $$SM_ \mathbb{Q}(E)$$ of motivic structures, and we put $$H^ 1(E,M)=H^ 1_{SM_ \mathbb{Q}(E)}(E,M)$$ and $$H^ 1_ f(E,M)=H^ 1_{SM_ \mathbb{Q}(E),f}(E,M)$$. A motivic category is defined as a premotivic subcategory of $$SM_ \mathbb{Q}(E)$$. The importance of the notion of motivic category lies in the fact that, once one has a motivic structure $$M$$, one may hope to construct a motivic category containing $$M$$ without knowing the whole $$SM_ \mathbb{Q}(E)$$. Let $$M$$ be an object of the motivic category $${\mathcal M}$$. One defines $L_ f(M):=\text{det}_ \mathbb{Q} H^ 0(E,M)\otimes(\text{det}_ \mathbb{Q} H^ 1_ f(E,M))^{-1}$ and the fundamental line of $$M$$: $\Delta_ f(M):=L_ f(M)\otimes L_ f(M^*(1))\otimes(\text{det}_ \mathbb{Q} M^ +_ B)^{- 1}\otimes\text{det}_ \mathbb{Q}(t_ M(E)).$ Several conjectures can be stated, but one must be careful because they all involve the badly known category $${\mathcal M}$$, e.g. “$$C_{L,\text{weak}}(M,{\mathcal M})$$”:
(i) $$r_ M=\dim_ \mathbb{Q} H^ 1_ f(E,M^*(1))-\dim_ \mathbb{Q} H^ 0(E,M^*(1))$$;
(ii) if $$b$$ is a $$\mathbb{Q}$$-basis of $$\Delta_ f(M)$$, then one has $$L^*(M,0)\cdot| b|_{EP,\infty}\in\mathbb{Q}^ \times$$.
Also, “$$C_{\text{Gal}}(M,{\mathcal M})$$”:
(iii) One has isomorphisms $$\mathbb{Q}_ \ell\otimes_ \mathbb{Q} H^ 0(E,M)@>\sim>> H^ 0(E,M_ \ell)$$ and $$\mathbb{Q}_ \mathbb{Q}\otimes_ \mathbb{Q} H^ 1_ f(E,M)@>\sim>> H^ 1_ f(E,M_ \ell)$$ for all primes $$\ell$$ and similarly for $$M^*(1)$$.
Finally, under the assumption of “$$C_{\text{Gal}}(M,{\mathcal M})$$”, “$$C_ L(M,{\mathcal M})$$”:
If $$b$$ is a basis of $$\Delta_ f(M)$$, one has $$| b|_{EP,\ell}=1$$ for almost all $$\ell$$, and with $$\{\text{prime numbers}\}\cup\{\infty\}=\overline P$$, $$L^*(M,0)\cdot\prod_{\ell\in\overline P}| b|_{EP,\ell}\in\{1,-1\}$$.
This last “conjecture” can be translated into an equivalent conjecture in terms of Tamagawa measures. In case $$M$$ is $$f$$-closed, these “conjectures” amount to true conjectures, i.e. without knowledge of the whole category of motivic structures. One gets the following true conjecture $$C_{\text{ord}}(M,\ell)$$: For a prime $$\ell$$, one has $$r_ M=\dim_{\mathbb{Q}_ \ell}H^ 1_ f(E,M^*(1)_ \ell)-\dim_{\mathbb{Q}_ \ell}H^ 0(E,M^*(1)_ \ell)$$ . The “conjectures” are invariant under $$f$$-equivalence. The main result of the paper now says that one has a well defined Weil restriction functor $$\text{Res}_{E/\mathbb{Q}}$$ to reduce problems and conjectures over $$E$$ to corresponding ones over $$\mathbb{Q}$$, and, fortunately, over $$\mathbb{Q}$$ every object $$M$$ of a motivic category $${\mathcal M}$$ is $$f$$-equivalent to an $$f$$-closed object. This leads to an algorithm to reduce “conjectures” to true conjectures for $$f$$- closed objects.

##### MSC:
 14A20 Generalizations (algebraic spaces, stacks) 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 19E99 $$K$$-theory in geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry