Beilinson conjectures

Let be a smooth projective variety (cf. Projective scheme) defined over . For such one has, on the one hand, the algebraic -groups (cf. -theory) , , and on the other hand, various cohomology theories, such as Betti cohomology , de Rham cohomology and -adic cohomology . These cohomology theories can be considered as realizations of the (Chow) motive associated to . There are comparison isomorphisms between them. Decomposing the motive , , one may fix , , and define, via the Frobenius action on -adic cohomology (cf. [a6]), the -function , an infinite product which converges absolutely for . Here, is a pure motive of weight . Using the Hodge structure on the cohomology of the complex manifold , one defines the -factor "at infinity" , , essentially as a product of -factors. Finally, one defines . For one has a conjectural analytic continuation and functional equation , for a suitable function of the form , and with the dual motive of . Here, by Poincaré duality and hard Lefschetz, this means . In general, for an arbitrary motive of pure weight , one extends the above construction of , and . One should have .

On the -groups of one has the action of the Adams operators (cf. Cohomology operation). They all commute with each other. Write for the subspace on which acts as multiplication by , . A. Beilinson defines the absolute or motivic cohomology . As a matter of fact, this can be defined for any regular or affine (simplicial) scheme . It has many nice properties of a cohomology theory; in particular there is a motivic Chern character mapping (a sum of projections after tensoring with ) . A classical theorem of A. Grothendieck says that . Beilinson has extended motivic cohomology to the category of (Chow) motives with coefficients in a number field . Assuming that admits a regular model over , one defines

This is independent of the regular model, provided that it exists. The are conjectured to be finite-dimensional. Their construction may be applied to define groups for any Chow motive over with coefficients in .

Another main ingredient of Beilinson's conjectures is Deligne (or Deligne–Beilinson) cohomology. This is defined for any quasi-projective variety (cf. Quasi-projective scheme) defined over the complex numbers. For smooth projective it is easy to define. Let , or and write for the subgroup , where . Consider the following complex of sheaves on :

where (respectively, ) is placed in degree (respectively, ). (respectively, ) denotes the sheaf of holomorphic functions (respectively, holomorphic -forms) on . One defines the Deligne cohomology of as

the hypercohomology of the complex . For arbitrary one uses a smooth compactification of such that is a normal crossings divisor, and, using the associated logarithmic de Rham complex of along , it is possible to construct well-defined Deligne–Beilinson cohomology . Thus, one obtains a good cohomology theory, with supports, Poincaré duality, even a homological counterpart, satisfying the axioms of a Poincaré duality theory in the sense of S. Bloch and A. Ogus. In particular, there is again a Chern character mapping . For smooth projective defined over , one defines as the subspace of invariant under the induced action of complex conjugation acting on the pair , i.e., acting on differential forms by . Similarly for and . Then, for an integer , there is a short exact sequence

The -structures on the first two terms give rise to a natural -structure

on . In the general case of motives with coefficients in , one will have -structures, etc.

Taking things together, one sees that, for varieties over , there are natural transformations, called regulators, . Already the simplest explicit examples suggest one should restrict to "integral motivic cohomology" and one is led to Beilinson's regulator mappings . It should be remarked that one can extend the formalism to the category of Chow motives over with field of coefficients in the number field . The -functions will take their values in and the regulator mappings will be of the form . This is even expected to work for Grothendieck motives, i.e., motives modulo homological (which, conjecturally, coincides with numerical) equivalence.

Beilinson's first conjecture.

To state Beilinson's conjectures on special values of at integer arguments , one distinguishes between three cases:

i) , which by the functional equation corresponds to the region of absolute convergence;

ii) , even, which lies on the boundary of the critical strip ;

iii) , odd, the centre of the critical strip. It is easily shown that , for .

Beilinson's first conjecture reads as follows. Assume . Then: is an isomorphism; and

where is the first non-vanishing coefficient of the Taylor series expansion of at .

In [a1], Beilinson states this conjecture for general Chow motives with coefficients in .

Some special cases are as follows.

a) For , a number field, and , one recovers the situation studied by A. Borel [a4]. Beilinson showed that his regulator coincides with Borel's regulator (at least modulo ). Thus, by Borel's results, the first conjecture is true. Classically, for one obtains the Dirichlet regulator and Dedekind's class number formula.

b) Bloch and Beilinson were the first to construct a regulator mapping (or even ) for a Riemann surface , and make a conjecture about . For an elliptic curve without complex multiplication, and , Bloch and D. Grayson made computer calculations which actually gave rise to a formulation of the first conjecture in terms of the integral model . For elliptic curves with complex multiplication a weak form of the first conjecture was proved by Bloch and Beilinson.

c) Another conjecture which motivated Beilinson's first conjecture is due to P. Deligne [a6]. It is stated in terms of motives and predicts that the -function of such a motive (cf. also Motives, theory of) at a so-called critical value of the argument would be equal (modulo ) to a well-defined period.

d) J.-F. Mestre and N. Schappacher gave numerical evidence for the case of the symmetric square of an elliptic curve without complex multiplication.

e) For Dirichlet motives, Beilinson proved the conjecture. For general Artin motives one recovers Gross' conjecture.

f) C. Deninger has obtained results for motives of Hecke characters of imaginary quadratic number fields.

g) Beilinson has proved partial results for (products of) modular curves.

h) K.-I. Kimura has given numerical evidence for (a projective curve related to) the Fermat curve .

Some further examples are known. They all deal with modular curves, Shimura curves, products of such curves, Hilbert modular surfaces, or products of elliptic modular surfaces. A general phenomenon occurs: in all these examples there exists a subspace of giving rise, via the regulator mapping , to a -structure on the corresponding Deligne–Beilinson cohomology with equal (up to a non-zero rational number) to the first non-vanishing coefficient of the -function at a suitable integer value of its argument.

Beilinson's second conjecture.

The second conjecture takes into account the possible pole of at the Tate point . One shows that

for .

Beilinson's second conjecture reads as follows. Let be even and write . Then:

i) , where is the group of -codimensional algebraic cycles on modulo homological equivalence, i.e., the image of the morphism , and is the inclusion;

ii) .

iii) .

This conjecture can also be stated in terms of motives.

a) For Artin motives it gives Stark's conjecture.

b) For a Hilbert modular surface , D. Ramakrishnan proved the existence of a subspace such that

gives a -structure on Deligne cohomology with equal (up to a non-zero rational number) to .

Beilinson's third conjecture.

The third conjecture deals with the centre , odd, of the critical strip. Let be a smooth projective variety (cf. Projective scheme) of dimension , and assume that admits a regular, proper model over . One has an isomorphism

giving a period matrix . Let . Beilinson [a3] showed that there exists a unique bilinear pairing of an arithmetic nature, i.e., closely related to the Gillet–Soulé arithmetic intersection pairing (on ), and generalizing Arakelov's intersection pairing on arithmetic surfaces.

Beilinson's third conjecture reads as follows. Let be a smooth projective variety defined over , and assume that has a regular, proper model over . Let . Then:

i) ;

ii) the pairing is non-degenerate.

iii) .

iv) modulo .

For an elliptic curve, , one recovers the Mordell–Weil theorem and the Birch–Swinnerton-Dyer conjectures for .

Generalizations.

Deligne observed that, for , one can interprete as a Yoneda extension , where is the category of -mixed Hodge structures with a real Frobenius. This made the search for a category of "mixed motives" over , (or, even better, over , ) very tempting. The regulator mapping in this setting would be just the Betti realization functor

The category should contain Grothendieck's category of pure motives and allow the treatment of arbitrary varieties over . Analogously, for other base fields , one should have categories , , etc. Also, the role of the Chow groups in the theory of Grothendieck motives might be enlarged to include all the algebraic -groups of the variety. In this respect one may mention a very geometric construction by Bloch of generalized Chow groups . For they coincide with . They are integrally defined and satisfy . A series of other conjectures, mainly about filtrations on Chow groups (Beilinson, J.P. Murre), emerges, and the ultimate formulation of Beilinson's conjectures appears in terms of derived categories, mixed motivic sheaves, mixed perverse sheaves, etc., cf. [a3], [a8].

In [a2], Beilinson introduced the notion of absolute Hodge cohomology . This generalizes Deligne–Beilinson cohomology by taking the weight filtration into account. It is a derived functor cohomology defined for any scheme over . For , or , let denote the category of -mixed Hodge structures. In this setting, for , one can define the Abel–Jacobi mappings of as . For smooth projective this gives the classical Abel–Jacobi mappings , where is Griffiths' intermediate Jacobian.

The following conjecture generalizes the classical Hodge conjecture. In this form it is due to Beilinson and U. Jannsen.

The Beilinson–Jannsen conjecture. Let be a smooth variety defined over . Then, for all , the regulator mapping has dense image.

In [a5] there is a formulation of Beilinson's conjectures in terms of (mixed) motives, without the modulo ambiguity. There is also a very precise conjecture in terms of Tamagawa numbers, cf. [a5], [a7] and the contribution by J.-M. Fontaine and B. Perrin-Riou in [a8].

References

[a1]	A. Beilinson, "Higher regulators and values of -functions" J. Soviet Math. , 30 (1985) pp. 2036–2070 (In Russian)
[a2]	A. Beilinson, "Notes on absolute Hodge cohomology" , Contemp. Math. , 55 , Amer. Math. Soc. (1985) pp. 35–68
[a3]	A. Beilinson, "Height pairings for algebraic cycles" , Lecture Notes in Mathematics , 1289 , Springer (1987) pp. 1–26
[a4]	A. Borel, "Cohomologie de et valeurs de fonctions zeta aux points entiers" Ann. Sci. Pisa (1976) pp. 613–636
[a5]	S. Bloch, K. Kato, "-functions and Tamagawa numbers of motives" , The Grothendieck Festschrift I , Progress in Mathematics , 86 , Birkhäuser (1990) pp. 333–400
[a6]	P. Deligne, "Valeurs de fonctions et périodes d'intégrales" , Proc. Symp. Pure Math. , 33 , Amer. Math. Soc. (1979) pp. 313–346
[a7]	J.-M. Fontaine, B. Perrin-Riou, "Autour des conjectures de Bloch et Kato, I--III" C.R. Acad. Sci. Paris , 313 (1991) pp. 189–196; 349–356; 421–428
[a8]	"Motives" U. Jannsen (ed.) etAAsal. (ed.) , Proc. Symp. Pure Math. , 55 , Amer. Math. Soc. (1994)