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.

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.

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

##### 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 |