[For the entire collection see Zbl 0717.00008.]

*Deligne* (1979) and *Beilinson* (1985) stated conjectures on the values at integer points of $L$-functions associated with motives. These values were conjectured to be expressible in terms of periods and regulators, but only up to (non-zero) rational factors. In spite of several deep results (due to many people) on special cases of the conjectures, most of them have remained unproven until today. In the paper under review the ${\mathbb{Q}}^{\times}$-ambiguity is removed by the introduction of the notion of a Tamagawa number for motives. The definition of such a Tamagawa number is (as usual) by means of suitable measures on adelic and global points of suitable abelian groups associated with a motive. The local Haar measures on the groups of local points then give the inverse local factors of the $L$-function of the motive. Roughly speaking, these groups are determined by the (Galois) cohomology of some structure (called a motivic pair) believed to be determined by a motive. The conjecture says that this Tamagawa number can be expressed as the quotient of two integers: the order of a well defined (zeroth) Galois cohomology group, and the order of a Tate-Shafarevich group associated with the motive. This last group is conjectured to be finite. Partial results supporting the conjecture were obtained by the second author in the case of the Tate motive $\mathbb{Q}\left(r\right)$, related to the Riemann zeta function, and the case of the motive ${H}^{1}\left(E\right)\left(2\right)$ of an elliptic curve $E$ with complex multiplication.

After a motivational introduction where the notion of a Tamagawa number of motives is already advocated by the remark that the conjectures of Deligne and Beilinson are equivalent to the rationality of a kind of Tamagawa number, a short overview of the most important properties of the Fontaine-Messing rings ${B}_{DR}^{+}$, ${B}_{DR}$, ${B}_{crys}^{+}$ and ${B}_{crys}$ is given.

The next section is concerned with a formula relating the Coates-Wiles homomorphism in the local theory of cyclotomic fields and the Fontaine-Messing theory of $p$-adic periods. This homomorphism turns out to be closely related to the boundary map in an exact cohomology sequence for ${B}_{crys}^{+}$. This result is used later on in the proof of the main conjecture for the Tate motive $\mathbb{Q}\left(r\right)$.

In the following two sections the necessary local tools for the Tamagawa number and the Tate-Shafarevich group of a motive are introduced. Here a motive is to be thought as some universal cohomology ${H}^{i}(X,\mathbb{Z}\left(n\right))$ of a smooth, complete variety $X$. Let $K$ be a finite extension of ${\mathbb{Q}}_{p}$, with maximal unramified subfield ${K}_{0}$ (which is just the fraction field of the Witt vectors $W\left(k\right)$ of the residue field $k$ of $K)$, and write ${H}^{i}(K,V)$ for the Galois cohomology ${H}^{i}(\text{Gal}(\overline{K}/K),V)$, where $V$ is a finite dimensional ${\mathbb{Q}}_{p}$-vector space with a continuous action of ${G}_{K}=\text{Gal}(\overline{K}/K)$. For such $V$ define $\text{Crys}\left(V\right)={H}^{0}(K,{B}_{crys}\otimes V)$ and $DR\left(V\right)={H}^{0}(K,{B}_{DR}\otimes V)$. $\text{Crys}\left(V\right)$ is a ${K}_{0}$-vector space with a frobenius $f$, and there is an embedding $K{\otimes}_{{K}_{0}}\text{Crys}\left(V\right)\hookrightarrow DR\left(V\right)$. The filtration on ${B}_{DR}$ induces a decreasing filtration ${\left\{DR{\left(V\right)}^{i}\right\}}_{i\in \mathbb{Z}}$ on $DR\left(V\right)$. One has the following inequalities ${dim}_{{K}_{0}}\text{Crys}\left(V\right)\le {dim}_{K}DR\left(V\right)\le {dim}_{{\mathbb{Q}}_{p}}V$. When ${dim}_{{K}_{0}}\text{Crys}\left(V\right)={dim}_{{\mathbb{Q}}_{p}}V$ (resp. when ${dim}_{K}DR\left(V\right)={dim}_{{\mathbb{Q}}_{p}}V)$ $V$ is called a crystalline (resp. de Rham) representation of ${G}_{K}$. Then, for a prime number $\ell $ and a finite dimensional ${\mathbb{Q}}_{\ell}$-vector space $V$ with continuous ${G}_{K}$- action one defines the exponential, finite and geometric parts of ${H}^{1}(K,V)={H}^{1}({G}_{K},V)$, ${H}_{e}^{1}(K,V)\subset {H}_{f}^{1}(K,V)\subset {H}_{g}^{1}(K,V)\subset {H}^{1}(K,V)$, as follows. If $\ell \ne p$, let ${H}_{e}^{1}(K,V)=\left\{0\right\}$, ${H}_{f}^{1}(K,V)=\text{Ker}({H}^{1}(K,V)\to {H}^{1}({K}_{nr},V))$, ${H}_{g}^{1}(K,V)={H}^{1}(K,V)$, where ${K}_{nr}$ is the maximal unramified extension of $K$. If $\ell =p$, let

The ${H}_{f}^{1}(K,V)$ and ${H}_{g}^{1}(K,V)$ represent classes of extensions of the form $0\to V\to E\to {\mathbb{Q}}_{\ell}\to 0$, such that if $V$ is unramified and $\ell \ne p$, $E$ is unramified iff its class is in ${H}_{f}^{1}(K,V)$. If $\ell =p$, and $V$ is crystalline (resp. de Rham), then so is $E$ iff its class lies in ${H}_{f}^{1}(K,V)$ (resp. ${H}_{g}^{1}(K,V))$. For a prime $\ell $ and a free ${\mathbb{Z}}_{\ell}$-module $T$ of finite rank and with a continuous ${G}_{K}$-action, one defines ${H}_{*}^{1}(K,T)\stackrel{\text{def}}{\phantom{\rule{0.0pt}{0ex}}=}{\iota}^{-1}\left({H}_{*}^{1}(K,T\otimes \mathbb{Q})\right)$ with $*=e,f,g$ and $\iota :{H}^{1}(K,T)\to {H}^{1}(K,T\otimes \mathbb{Q})$. In particular, one sees that ${H}_{*}^{1}(K,T)$ contains the torsion part of ${H}^{1}(K,T)$. For a free $\widehat{\mathbb{Z}}$-module $T$ of finite rank and with continuous ${G}_{K}$-action, one defines ${H}_{*}^{1}(K,T)=\prod {H}_{*}^{1}(K,{T}_{\ell})$, where $*=e,f,g$ and ${T}_{\ell}=T{\otimes}_{\widehat{\mathbb{Z}}}{\mathbb{Z}}_{\ell}$. Using exact sequences relating the ${B}_{crys}^{+}$, ${B}_{DR}^{+}$, ${B}_{crys}$ and ${B}_{DR}$ one obtains interesting relations between the various cohomology groups. Also, one can define an exponential map $exp:DR\left(V\right)/DR{\left(V\right)}^{0}\to {H}_{e}^{1}(K,V)$, which is surjective and has kernel $\text{Crys}{\left(V\right)}^{f=1}/{H}^{0}(K,V)$. To relate these notions to $L$- functions, define local factors $P(V,u)={P}_{\ell}(V,u)$ by

where, for $\ell \ne p$, ${f}_{K}$ denotes the action of an element of ${G}_{K}$ which acts on ${\mathbb{Z}}_{\ell}(-1)$ by ${p}^{[{K}_{0}:{\mathbb{Q}}_{p}]}$. If $\ell =p$, ${f}_{K}$ denotes the ${K}_{0}$- linear map ${f}^{[{K}_{0}:{\mathbb{Q}}_{p}]}$. Assume $P(V,1)\ne 0$, then the following results are proved:

(i) If $\ell \ne p$, if $V$ is unramified and $T$ is a ${G}_{K}$-stable ${\mathbb{Z}}_{\ell}$-sublattice in $V$, then $\#{H}_{f}^{1}(K,T)={\left|P(V,1)\right|}_{\ell}^{-1}$, where ${\left|\phantom{\rule{4pt}{0ex}}\right|}_{\ell}$ denotes the normalized absolute value of ${\mathbb{Q}}_{\ell}$;

(ii) if $\ell =p$, $K$ is unramified over ${\mathbb{Q}}_{p}$ and $V$ is crystalline, and some other condition is satisfied, then one may construct a Galois stable sublattice $T\subset V$ and one has $\mu \left({H}_{f}^{1}(K,T)\right)={\left|P(V,1)\right|}_{p}^{-1}$, where ${\left|\phantom{\rule{4pt}{0ex}}\right|}_{p}$ is the absolute value on $K={K}_{0}$ such that $\left|p\right|={p}^{-1}$. $\mu $ is the Haar measure of ${H}_{f}^{1}(K,V)$ induced from the Haar measure of $D/{D}^{0}$ having total measure 1 via the exponential map $exp:D\left(V\right)/D{\left(V\right)}^{0}@>\sim >>{H}_{e}^{1}(K,V)={H}_{f}^{1}(K,V)$ in this situation. Also, a formula for the measure $\mu \left({H}^{1}(K,\widehat{\mathbb{Z}}\left(r\right))\right)$ is derived.

Now turn to the global situation (basically over $\mathbb{Q})$. The notion of a motivic pair $(V,D)$ is introduced. This is a pair of finite dimensional $\mathbb{Q}$-vector spaces with certain compatibilities and subject to a set of axioms. $D$ has a finite decreasing filtration by $\mathbb{Q}$-vector spaces ${D}^{i}$, and $V\otimes {\mathbb{A}}_{f}$ has a continuous ${\mathbb{A}}_{f}$-linear Galois action such that $V\subset V\otimes {\mathbb{A}}_{f}$ is stable under $\text{Gal}(\u2102/\mathbb{R})\subset \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Also, for any finite prime $p$ there is an isomorphism of ${\mathbb{Q}}_{p}$-vector spaces ${\theta}_{p}:{D}_{p}=D\otimes {\mathbb{Q}}_{p}@>\sim >>DR\left({V}_{p}\right)=DR(V\otimes {\mathbb{Q}}_{p})$ preserving filtrations, and for $p=\infty $, there is an isomorphism of $\mathbb{R}$-vector spaces ${\theta}_{\infty}:{D}_{\infty}@>\sim >>{\left({V}_{\infty}{\otimes}_{\mathbb{R}}\u2102\right)}^{+}$, where ${}^{+}$ denotes the $\text{Gal}(\u2102/\mathbb{R})$-fixed part. The notion of weights for such a motivic pair is introduced. For a finite set of places of $\mathbb{Q}$ containing $\infty $, and a motivic pair $(V,D)$ of weights $\le w$, one defines the $L$-function ${L}_{S}(V,s)={\prod}_{p\notin S}{P}_{p}{(V,{p}^{-s})}^{-1}$. This converges absolutely for $\Re \left(s\right)>w/2+1$. For a $\mathbb{Z}$-lattice $M$ in $V$ such that $M\otimes \widehat{\mathbb{Z}}$ is $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-stable in $V\otimes {\mathbb{A}}_{f}$, one defines $A\left({\mathbb{Q}}_{p}\right)={H}_{f}^{1}({\mathbb{Q}}_{p},M\otimes \widehat{\mathbb{Z}})$ if $p<\infty $; $A\left({\mathbb{Q}}_{p}\right)=(\left({D}_{\infty}{\otimes}_{\mathbb{R}}\u2102\right)/\left({D}_{\infty}^{0}{\otimes}_{\mathbb{R}}\u2102\right)+M){)}^{+}$ if $p=\infty $.

Also, one can define the ‘global points’ $A\left(\mathbb{Q}\right)$. All these matters are inspired by Beilinson’s conjecture on the regulator map from $K$- theory to cohomology including the Bloch-Grayson suggestion of taking into account proper regular models of varieties over $\text{Spec}\left(\mathbb{Z}\right)$. For an isomorphism $\omega :{\text{det}}_{\mathbb{Q}}(D/{D}^{0})@>\sim >>\mathbb{Q}$ (which induces one locally for every place $p$ of $\mathbb{Q})$, one obtains a measure for all $p\notin S:{\mu}_{p,\omega}\left(A\left({\mathbb{Q}}_{p}\right)\right)={P}_{p}(V,1)$, and for the $L$-function one gets (assume the weights are $\le -3)$: ${L}_{S}(V,0)={\prod}_{p\notin S}{\mu}_{p,\omega}\left(A\left({\mathbb{Q}}_{p}\right)\right)$. (This must be modified for weights $-1$, $-2$.) It makes sense to define the total measure $\mu ={\prod}_{p<\infty}{\mu}_{p,\omega}$ on ${\prod}_{p\le \infty}A\left({\mathbb{Q}}_{p}\right)$ and to define the Tamagawa number $\text{Tam}\left(M\right)=\mu ((\prod A\left({\mathbb{Q}}_{p}\right))/A\left(\mathbb{Q}\right))$. If $M={H}^{\overline{m}}\left(X\right)\left(r\right)$, one hopes to be able to associate with it a motivic pair $(V,D)$, and to define the Tate-Shafarevich group $\u0428\left(M\right)$ as the kernel of the map

The final conjecture becomes: Assume $(V,D)$ comes from a motive, and let $M$ be a $\mathbb{Z}$-lattice in $V$ such that $M\otimes \widehat{\mathbb{Z}}$ is Galois stable in $V\otimes {\mathbb{A}}_{f}$. Then $\text{\u0428}\left(M\right)$ is finite, and $\text{Tam}\left(M\right)=\#\left({H}^{0}(\mathbb{Q},{M}^{*}\otimes \mathbb{Q}/\mathbb{Z}\left(1\right))\right)/\#\left(\text{\u0428}\left(M\right)\right)$. The conjecture is shown to be ‘isogeny invariant’ in a suitable sense.