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)].

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)].

Reviewer: W.W.J.Hulsbergen (Breda)

##### 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 |