# zbMATH — the first resource for mathematics

Autour des conjectures de Bloch et Kato. II: Structures motiviques $$f$$- closes. (On the conjectures of Bloch et Kato. II: $$f$$-closed motivic structures). (French. Abridged English version) Zbl 0752.14001
[For part I see ibid. 313, No. 5, 189-196 (1991; Zbl 0749.11052).]
For a number field $$E$$ a ($$\mathbb{Q}$$-linear) neutral Tannakian category $$SPM_ \mathbb{Q}(E)$$ of premotivic structures (over $$E$$) is defined and by a Tannakian subcategory of $$SPM_ \mathbb{Q}(E)$$ is meant a full subcategory of $$SPM_ \mathbb{Q}(E)$$ which is stable under subobject, quotient, direct sum, tensor product and dual. It is hoped that, with the right definition of a (mixed) motive over $$E$$ with coefficients in $$\mathbb{Q}$$, such objects determine objects of $$SPM_ \mathbb{Q}(E)$$, their realizations, and thus lead to the smallest Tannakian subcategory $$SM_ \mathbb{Q}(E)$$ of $$SMP_ \mathbb{Q}(E)$$, the category of motivic structures (over $$E$$ with coefficients in $$\mathbb{Q})$$. Examples of motivic structures are the $$H^ i(X)$$, $$i\in\mathbb{N}$$, of a smooth proper algebraic variety $$X$$ over $$E$$, or the motivic structures associated with Deligne 1-motives, etc. In the sequel the existence of $$SM_ \mathbb{Q}(E)$$ is assumed. An object $$M$$ of $$SM_ \mathbb{Q}(E)$$ has Betti, de Rham and $$\ell$$-adic realizations with the usual filtrations, Galois actions and compatibilities. In particular, for a finite place $${\mathcal P}$$ of $$E$$ lying over $$p$$, the characteristic of the residue field $$k_{\mathcal P}$$ of $$E_{\mathcal P}$$, the $$\ell$$-adic realization $$M_ \ell$$, i.e. the $$\ell$$-adic representation of $$G_{E_{{\mathcal P}}}=\text{Gal}(\overline E_{\mathcal P}/E_{\mathcal P})$$, defines a finite dimensional $$\mathbb{Q}_ \ell$$- (resp. $$E_{0,{\mathcal P}}$$- )vector space, where $$E_{0,{\mathcal P}}$$ is the quotient field of the Witt vector ring of $$k_{\mathcal P}$$, $D_{\mathcal P}(M_ \ell)=\begin{cases} M_ \ell^{I_{E_{\mathcal P}}}, &\text{ if }\ell\neq p \\ (B_{cris}\otimes_{\mathbb{Q}_ p}M_ \ell)^{G_{E_{\mathcal P}}},&\text{ if }\ell=p \end{cases}.$ One defines the $$L$$-function of $$M$$ as $L(M,s)=\prod_{{\mathcal P} finite}L_{\mathcal P}(M,s),$ where $$L_{\mathcal P}(M,s)=P_{\mathcal P}(M,q^{-s})^{-1}$$, $$q=p^ h$$ the number of elements of $$k_{\mathcal P}$$, and $$P_{\mathcal P}(M,t)=\text{det}_{E_{0,{\mathcal P}}}(1-ft\mid D_{\mathcal P}(M_ p))$$, where $$f$$ is induced by the absolute Frobenius. A basic conjecture $$C_{pr.an}(M)$$ states that, for any finite place $${\mathcal P}$$ of $$E$$, $$P_{\mathcal P}(M,t)\in\mathbb{Q}[t]$$, and $$L(M,s)$$ converges absolutely for $${\mathfrak R}(s)>>0$$ and admits a meromorphic continuation on a connected open neighborhood of 0. Also, one expects that $$P_{\mathcal P}(M,t)$$ is equal to $$P_{{\mathcal P},\ell}(M,t)=\text{det}_{\mathbb{Q}_ \ell}(1-ft\mid D_{\mathcal P}(M_ \ell))$$ for any prime $$\ell$$, where $$f$$ is the geometric Frobenius at $${\mathcal P}$$. In what follows the truth of $$C_{pr.an}(M)$$ is assumed. Thus there exist $$r_ M\in\mathbb{Z}$$ and $$L^*(M,0)\in\mathbb{R}^ \times$$ such that $$\lim_{s\to 0}L(M,s)/s^{r_ M}=L^*(M,0).$$
Any (pre)motivic structure $$M$$ admits a tangent space $$t_ M(E)$$, i.e. an $$E$$-vector space defined by its de Rham realization with its Hodge filtration. For an infinite place $${\mathcal P}$$ of $$E$$ and the Betti realization $$M_ B$$ of $$M$$, one defines $$M^ +_{B,{\mathcal P}}=(M_{B,{\mathcal P}})^{G_{\mathcal P}}$$. Taking the direct sum over all infinite places $${\mathcal P}$$ and tensoring with $$\mathbb{R}$$, gives rise to a map $$\alpha_ M:M^ +_{B,\mathbb{R}}\to t_ M(E)_ \mathbb{R}$$ and the so- called tautological exact sequence of $$\mathbb{R}$$-vector spaces $0\to\text{Ker}(\alpha_ M)\to M^ +_{B,\mathbb{R}}\to t_ M(E)_ \mathbb{R}\to\text{Coker}(\alpha_ M)\to 0.$ The motivic structure $$M$$ is called critical whenever one has isomorphisms $u_ M:H^ 0(E,M)_ \mathbb{R}:=\operatorname{Hom}_{SPM_ \mathbb{Q}(E)}(1,M)\otimes\mathbb{R}@> \sim >> \text{Ker} (\alpha_ M)$ and $u_{M^*(1)}:H^ 0(E,M^*(1))_ \mathbb{R}:=\operatorname{Hom}_{SPM_ \mathbb{Q}(E)}(1,M^*(1))\otimes\mathbb{R}@>\sim>> \text{Ker}(\alpha_{M^*(1)}).$ Such a critical $$M$$ defines a $$\mathbb{Q}$$- line $\Delta^ 0_ f(M)=\text{det}_ \mathbb{Q} H^ 0(E,M)\otimes\text{det}_ \mathbb{Q} H^ 0(E,M^*(1))\otimes (\text{det}_ \mathbb{Q} M^ +_ B)^{-1}\otimes\text{det}_ \mathbb{Q} t_ M(E)$ and an isomorphism $$i_{M,cr}:\Delta^ 0_ f(M)_ \mathbb{R}@>\sim>>\mathbb{R}$$. The usual absolute value on $$\mathbb{R}$$ then defines a norm $$|$$ $$|_{cr}$$ on $$\Delta^ 0_ f(M)$$. The following conjecture $$C_{cr,L,weak}(M)$$ is formulated: For critical $$M$$ one has $$r_ M=0$$ and $$L^*(M,0)\cdot| b|_{cr}\in\mathbb{Q}^ \times$$, where $$b$$ is a basis of $$\Delta^ 0_ f(M)\subset\Delta^ 0_ f(M)_ \mathbb{R}$$. This may be compared with Deligne’s conjecture for critical motives. The motivic structure $$M$$ is said to be $$f$$-closed if $$H^ 1_ f(E,M_ t)=H^ 1_ f(E,M^*(1)_ \ell)=0$$, where for an $$\ell$$-adic representation $$V$$, $$H^ 1_ f(E,V)$$ denotes the subspace of all $$x\in H^ 1(E,V)$$ with image $$x_{\mathcal P}\in H^ 1_ f(E_{\mathcal P},V)$$, the finite part of $$H^ 1(E_{\mathcal P},V)$$ in the terminology of Bloch and Kato, for all places $${\mathcal P}$$. A whole series of conjectures emerges:
(i) If $$M$$ is $$f$$-closed, it is critical.
(ii) For $$f$$-closed and critical $$M$$, one has $$r_ M=-\dim_ \mathbb{Q} H^ 0(E,M^*(1))$$ and $$L^*(M,0)\cdot| b|_{cr}\in\mathbb{Q}^ \times$$.
(iii) For $$f$$-closed $$M$$ one has an isomorphism $$\mathbb{Q}_ \ell\otimes_ \mathbb{Q} H^ 0(E,M)@>\sim>>H^ 0(E,M_ \ell)$$ and similarly for $$M^*(1)$$. Also, $$H^ 1_ f(E,M_ \ell)=H^ 1_ f(E,M^*(1)_ \ell)=0$$ for any prime $$\ell$$.
Finally, one has the conjecture $$C_{fc,L}(M)$$: For $$f$$-closed $$M$$, satisfying (i) and (iii), one has $$r_ M=-\dim_ \mathbb{Q} H^ 0(E,M^*(1))$$, and the Euler-Poincaré norm $$| b|_{\text{EP},\ell}=1$$, $$b$$ a basis of $$\Delta^ 0_ f(M)$$, for almost all $$\ell$$. — Furthermore, if $$\overline P$$ denotes the set of all finite prime numbers plus $$\infty$$, one has $$L^*(M,0)\cdot\prod_{\ell\in\overline P}| b|_{\text{EP},\ell}=\pm 1$$. In particular, if $$M$$ is the pure weight $$-1$$ motivic structure associated with an abelian variety $$A$$ over $$E$$ with finite Shafarevich-Tate group $$\text{ Ш}(A)$$, it follows that $$M$$ is critical.
$$M$$ if $$f$$-closed iff $$A(E)$$ is torsion, and then (i) and (iii) hold. Conjecture $$C_{cf,L}(M)$$ is equivalent with the Birch and Swinnerton- Dyer conjecture. Also, for $$M=1(-r)$$, $$r\text{ odd}$$, $$C_{fc,L}(M)$$ turns out to be Lichtenbaum’s conjecture. Finally, one can define a Haar measure $$\mu_{\text{EP}}$$ on $$t_ M(\mathbb{A}_ E)=\mathbb{A}_ E\otimes_ Et_ M(E)$$, where $$\mathbb{A}_ E$$ are the $$E$$-adèles, using the Euler- Poincaré norm on $$\Delta^ 0_ f(M)$$. On the other hand, one has a uniquely defined Tamagawa measure $$\mu_{\text{Tam}}$$ on $$t_ M(\mathbb{A}_ E)$$. Then, if (iii) holds, $$C_{fc,L}(M)$$ is equivalent with $$C'_{fc,L}:\mu_{\text{Tam}}=| L^*(M,0)|\cdot\mu_{\text{EP}}$$, or, in other words, $$\mu_{\text{EP}}(t_ M(\mathbb{A}_ E)/t_ M(E))=| L^*(M,0)|^{-1}$$.
[For part III see ibid. 313, No. 7, 421-428 (1991)].

##### MSC:
 14A20 Generalizations (algebraic spaces, stacks) 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 19E20 Relations of $$K$$-theory with cohomology theories