The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 333-400 (1990).
[For the entire collection see Zbl 0717.00008.]
Deligne (1979) and Beilinson (1985) stated conjectures on the values at integer points of -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 -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 -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 , related to the Riemann zeta function, and the case of the motive of an elliptic curve 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 , , and 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 -adic periods. This homomorphism turns out to be closely related to the boundary map in an exact cohomology sequence for . This result is used later on in the proof of the main conjecture for the Tate motive .
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 of a smooth, complete variety . Let be a finite extension of , with maximal unramified subfield (which is just the fraction field of the Witt vectors of the residue field of , and write for the Galois cohomology , where is a finite dimensional -vector space with a continuous action of . For such define and . is a -vector space with a frobenius , and there is an embedding . The filtration on induces a decreasing filtration on . One has the following inequalities . When (resp. when is called a crystalline (resp. de Rham) representation of . Then, for a prime number and a finite dimensional -vector space with continuous - action one defines the exponential, finite and geometric parts of , , as follows. If , let , , , where is the maximal unramified extension of . If , let
The and represent classes of extensions of the form , such that if is unramified and , is unramified iff its class is in . If , and is crystalline (resp. de Rham), then so is iff its class lies in (resp. . For a prime and a free -module of finite rank and with a continuous -action, one defines with and . In particular, one sees that contains the torsion part of . For a free -module of finite rank and with continuous -action, one defines , where and . Using exact sequences relating the , , and one obtains interesting relations between the various cohomology groups. Also, one can define an exponential map , which is surjective and has kernel . To relate these notions to - functions, define local factors by
where, for , denotes the action of an element of which acts on by . If , denotes the - linear map . Assume , then the following results are proved:
(i) If , if is unramified and is a -stable -sublattice in , then , where denotes the normalized absolute value of ;
(ii) if , is unramified over and is crystalline, and some other condition is satisfied, then one may construct a Galois stable sublattice and one has , where is the absolute value on such that . is the Haar measure of induced from the Haar measure of having total measure 1 via the exponential map in this situation. Also, a formula for the measure is derived.
Now turn to the global situation (basically over . The notion of a motivic pair is introduced. This is a pair of finite dimensional -vector spaces with certain compatibilities and subject to a set of axioms. has a finite decreasing filtration by -vector spaces , and has a continuous -linear Galois action such that is stable under . Also, for any finite prime there is an isomorphism of -vector spaces preserving filtrations, and for , there is an isomorphism of -vector spaces , where denotes the -fixed part. The notion of weights for such a motivic pair is introduced. For a finite set of places of containing , and a motivic pair of weights , one defines the -function . This converges absolutely for . For a -lattice in such that is -stable in , one defines if ; if .
Also, one can define the ‘global points’ . All these matters are inspired by Beilinson’s conjecture on the regulator map from - theory to cohomology including the Bloch-Grayson suggestion of taking into account proper regular models of varieties over . For an isomorphism (which induces one locally for every place of , one obtains a measure for all , and for the -function one gets (assume the weights are : . (This must be modified for weights , .) It makes sense to define the total measure on and to define the Tamagawa number . If , one hopes to be able to associate with it a motivic pair , and to define the Tate-Shafarevich group as the kernel of the map
The final conjecture becomes: Assume comes from a motive, and let be a -lattice in such that is Galois stable in . Then is finite, and . The conjecture is shown to be ‘isogeny invariant’ in a suitable sense.