BEILINSON’S CONJECTURES

Jan Nekovář

University of California at Berkeley^{1}

Abstract. We give a survey of Beilinson’s conjectures about special values of Lfunctions, with emphasis on the underlying philosophy of mixed motives and motivic cohomology.

Introduction. In his seminal paper [1], A.A. Beilinson formulated far reaching conjectures about values of motivic L-functions at integers, and produced a compelling body of evidence in their favour by making ingenious calculations in several special cases. The main gist of [1] was a construction of “higher regulators”, expected to explain these L-values in the same spirit as the (slightly modified) classical Dirichlet regulator

r : O_{F}^{∗} ⊕ Z ⟶ R^{r_{1}+r_{2}}

does for the zeta function of a number field F at s = 0 (resp. s = 1). In this case, ζ_{F} (s) satisfies a functional equation relating its values at s and 1 − s. It has a simple pole at s = 1 and a zero of order r_{1} + r_{2} − 1 at s = 0. Its leading Taylor coefficient at s = 0 is equal to

lim_{s⟶0} ζ_{F} (s)s^{−(r_{1}+r_{2}−1)} = −\frac{♯Pic(O_{F}) · R}{♯(O_{F}^{∗})_{tors}},  (0.1)

where R denotes the covolume of the lattice Im(r) in R^{r_{1}+r_{2}}.

The quest for higher regulators, extending (0.1) to other values of ζ_{F} (s), has been initiated by Lichtenbaum [47]. He observed that for m > 1, the order d_{m} of vanishing of ζ_{F} (s) at s = 1 − m is equal to the dimension of the higher K-group K_{2m−1}(F) ⊗ Q. This led to a conjecture that the leading coefficient

lim_{s⟶1−m} ζ_{F} (s)(s + m − 1)^{−d_{m}}

should be equal, up to a rational factor, to the covolume of Im(r_{m}) for a certain map

r_{m} : K_{2m−1}(F) ⟶ R^{d_{m}}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 19F27, 11G40, 14A20; Secondary 14C35, 14G10.

1Miller Fellow

This conjecture has been proved by Borel [14],[15], for a slightly modified version of the regulator map r_{m}.

The next step was taken by Bloch [6],[7], who defined a regulator

r : K_{2}(E) ⟶ H^{1}(E(C), R)

for elliptic curves E_{/C} and verified that r computes the value L(E, 2) for curves with complex multiplication defined over Q. Later, Bloch [8] defined a regulator

K_{2}(X) ⟶ H^{1}(X(C), C^{∗})

for any curve X_{/C}.

Beilinson defines, for a given quasi-projective variety X_{/Q}, its motivic cohomology H_{M}^{.}(X, Q(∗)) as a suitable piece of K-theory of X. Let X be a smooth projective variety over Q. The L-function L(h^{i}(X), s), associated to the i^{th} cohomology of X, is expected to satisfy a functional equation relating its values at s and i + 1 − s. Suppose that n is an integer greater than 1 + i/2. Beilinson defines a regulator map

r : H_{M}^{i+1}(X, Q(n)) ⟶ H_{D}^{i+1}(X_{/R}, R(n))  (0.2)

from the motivic cohomology of X into Deligne cohomology. The significance of the latter lies in the fact that its dimension is equal to the order of vanishing of L(h^{i}(X), s) at i + 1 − n. Deligne cohomology also admits a natural Q-structure and Beilinson conjectures that det(r) ∈ R^{∗}/Q^{∗} with respect to this Q-structure is equal to the leading coefficient of the L-function at i + 1 − n. At the central point n = (i+1)/2 resp. near central point n = i/2+1 the conjecture has to be modified.

In a letter to Soulé [22], Deligne suggested a motivic formulation of the regulator 0.2, writing it as a ‘Hodge realization’

r : Ext^{1}(Q(0), M) ⟶ Ext^{1}(R(0), M_{Hodge}),

where the first Ext is the group of “motivic extensions” of the trivial motive by M = h^{i}(X)(n) and the second one is the group of extensions in a suitable category of Hodge structures. Deligne also introduced another Q-structure on the target of r, related to the value of L(h^{i}(X), s) at s = n.

In [3],[4], Beilinson followed this suggestion and put his conjectures about special values of L-functions into a general perspective of mixed motives and motivic sheaves. In this context, H_{M}^{i}(X, Q(j)) are expected to form a universal ‘absolute’, or ‘arithmetic’, cohomology theory of X, as opposed to Grothendieck’s h^{i}(X), which should provide only ‘geometric’ information about X ×_{Q} \overline{Q}.

Following this motivic thread, Scholl [63] proposed a unified formulation of Beilinson’s conjectures at all integers (including the central and near central points) as Deligne’s conjecture [21] for critical mixed motives.

In a separate development, Bloch and Kato [12] formulated a conjecture about the precise value of L(h^{i}(X), n), eliminating the undetermined rational factor in Beilinson’s approach. Recently, Fontaine and Perrin-Riou [32], [33] found a common generalization of the conjectures of Bloch-Kato and Scholl.

Apart from Beilinson’s original papers [1]–[4], there exist excellent surveys of his conjectures [29],[53],[56],[61], [67]. For geometric aspects of the conjectures, [43] is indispensable.

This survey attempts to explain not only the K-theoretic formulation of the conjectures, but also the underlying motivic intuition. The reader will have no trouble in distinguishing real mathematical statements from a mere wishful thinking (which prevails) simply by counting frequency of expressions “should”, “is expected” and the like.

1. Pure motives and realizations. (1.1) We first recall basic notions of (pure) motives. To a smooth projective variety X, defined over a number field K, and integers i ≥ 0, n ∈ Z, one hopes to associate a ‘motive’ M = h^{i}(X)(n) (pure of weight w = i − 2n), being a universal cohomology group of X. In Sec. 1–2, M will intervene only through its realizations, namely

• étale realizations (for each prime number ℓ)

M_{ℓ} = H^{i}((X ×_{K} \overline{K})_{et}, Q_{ℓ})(n)

— a finite dimensional ℓ-adic (continuous) representation of G(\overline{K}/K), pure of weight w. The last condition has the following meaning: there is a finite set S of places at which X has bad reduction. If v ∉ S is nonarchimedean and prime to ℓ, then M_{ℓ} is unramified at v and all eigenvalues of the geometric Frobenius F r_{v} on M_{ℓ} are algebraic numbers with absolute value (Nv)^{w/2} (by [20]).

• Betti realizations (for each embedding σ : K ↪ C)

M_{σ,B} = H^{i}((X ×_{K,σ} C)(C), Q(n))

— a pure Q-Hodge structure of weight w. If σ is a real embedding, then the action of the complex conjugation c ∈ G(C/R) on both (X ×_{K,σ} C)(C) and Q(n) = (2πi)^{n}Q induces an involution ϕ_{σ} on M_{σ,B} such that ϕ_{σ} ⊗ c preserves the Hodge decomposition

M_{σ,B} ⊗_{Q} C = ⨁_{p+q=w} H^{p,q}.

• de Rham realization

M_{dR} = H^{i}(X_{Zar}, Ω_{X/K}^{.})(n)

— a finite dimensional K-vector space with a decreasing filtration

F^{k}M_{dR} = H^{i}(X_{Zar}, Ω_{X/K}^{≥k+n}).

(1.2) There are standard comparison isomorphisms between different realizations:

• I_{σ} : M_{σ,B} ⊗_{Q} C ⟶^{∼} M_{dR} ⊗_{K,σ} C

(“de Rham theorem”), under which (F^{k}M_{dR}) ⊗ C corresponds to ⨁_{p≥k} H^{p,q}. If σ is a real embedding, then ϕ_{σ} ⊗ c corresponds to the action of 1 ⊗ c on the R.H.S.

• I_{ℓ,\overline{σ}} : M_{σ,B} ⊗_{Q} Q_{ℓ} ⟶^{∼} M_{ℓ}

(depending on the choice of an extension \overline{σ} : \overline{K} ↪ C of σ). For a real embedding σ, the action of ϕ_{σ} ⊗ 1 corresponds to the automorphism \overline{σ}^{∗}(c) ∈ G(\overline{K}/K) induced on \overline{K} by c (via \overline{σ}). The common dimension

rk(M) := dim_{Q}(M_{σ,B}) = dim_{K}(M_{dR}) = dim_{Q_{ℓ}}(M_{ℓ})

is called the rank of M (over K). (1.3) All constructions of linear algebra (duals, tensor products, …) apply to M, at least on the level of realizations. In particular,

M = h^{i}(X) ⊗ Q(1)^{⊗n},

where the Tate motive Q(1) = h^{2}(P_{K}^{1})^{∨} has realizations

Q(1)_{ℓ} = \underleftarrow{lim}_{k}(μ_{ℓ^{k}}(\overline{K})) ⊗ Q
Q(1)_{B} = (2πi)Q, Hodge type (−1, −1), ϕ_{σ} = −1
Q(1)_{dR}= K, F^{0} = 0, F^{−1} = K

By Poincaré duality and hard Lefschetz theorem (true in all realizations), we have

M^{∨}(1) = h^{i}(X)^{∨}(1 − n) = h^{2d−i}(X)(d + 1 − n)  (1.3.1)
= h^{i}(X)(i + 1 − n) = M(w + 1),

where d is the dimension of X and w = i − 2n the weight of M. There is also an operation of restriction of scalars for motives:

R_{K/Q}(M) = h^{i}(X_{/Q})(n),

where X_{/Q} is X viewed as a Spec(Q)-scheme. (1.4) The local L-factor of M at a non-archimedean place v of K is defined as

L_{v}(M, s) = det(1 − F r_{v}(Nv)^{−s} | M_{ℓ}^{I_{v}})^{−1} (ℓ |̸ Nv)

(where I_{v} is the inertia group of v), conjecturally independent of ℓ. For v ∉ S, L_{v}(M, s) is indeed independent of ℓ by [20] and all its poles have real part equal to Re(s) = w/2. For v ∈ S, independence of ℓ is not known; if true, purity conjecture for monodromy filtration then predicts that all poles of L_{v}(M, s) have real part Re(s) = w/2, (w − 1)/2, …, (w − i)/2 (see [40]). Local L-factors satisfy the following relations:

L_{v}(M(m), s) = L_{v}(M, s + m)
L_{p}(R_{K/Q}(M), s) = ∏_{v|p} L_{v}(M, s)  (1.4.1)
L_{v}(M_{1} ⊕ M_{2}) = L_{v}(M_{1}, s)L_{v}(M_{2}, s)

For an archimedean place v, corresponding to an embedding σ : K ↪ C, L_{v}(M, s) depends only on the real Hodge structure H = M_{σ,B} ⊗_{Q} R. The relations (1.4.1) are satisfied for archimedean places as well, and they determine L_{v}(M, s) for all M, once they are known for three basic Hodge structures (for a real place v): For M of rank 2 with H of Hodge type (k, 0)+(0, k) with k > 0, L_{v}(M, s) = Γ_{C}(s); for M of rank 1 with H of Hodge type (0, 0), L_{v}(M, s) = Γ_{R}(s) (resp. Γ_{R}(s + 1)), if ϕ_{σ} acts on H by +1 (resp. −1). Here we use the standard notation

Γ_{R}(s) = π^{−s/2}Γ(s/2), Γ_{C}(s) = Γ_{R}(s)Γ_{R}(s + 1) = 2(2π)^{−s}Γ(s).

The total L-factor at infinity will be denoted by

L_{∞}(M, s) = ∏_{v|∞} L_{v}(M, s).

See [26],[27],[28] for a unified definition of L-factors at all places.

(1.5) Assuming that L-factors at places of bad reduction are defined and behave as expected (i.e. the location of their poles is that predicted by the purity conjecture), then the L-function

L(M, s) = ∏_{v |̸∞} L_{v}(M, s),

apriori only a formal Dirichlet series with rational coefficients, is absolutely convergent (and without zeroes) for Re(s) > 1 + w/2. The full L-function

Λ(M, s) = L_{∞}(M, s)L(M, s)

conjecturally satisfies a functional equation

Λ(M, s) = ε(M, s)Λ(M^{∨}(1), −s)

with an ε-factor of the form ε(M, s) = a · b^{s} (see [28]). We have, of course,

Λ(M^{∨}(1), −s) = Λ(M, w + 1 − s)

by (1.3.1) and (1.4.1), so the functional equation becomes

Λ(M, s) = ε(M, s)Λ(w + 1 − s)  (1.4.2)

Suppose that we are interested in the behaviour of L(h^{i}(X), s) at an integer s = n. Restricting the scalars to Q, applying a Tate twist by Q(n) and using (1.4.1), we may assume that K = Q and that the point of our interest is s = 0. Using the functional equation (1.4.2) (which we assume to hold), it is sufficient to treat only the case when s = 0 lies to the right of the central point (w + 1)/2 (or coincides with it), which happens iff w ≤ −1. In fact,

w = −1 ⟺ s = 0 is the central value (w + 1)/2
w = −2 ⟺ s = 0 is the near central value w/2 + 1
w ≤ −3 ⟺ s = 0 is in the convergence region

From now on, we shall assume that X is a smooth projective variety over Q and that M = h^{i}(X)(n) is of weight w = i − 2n ≤ −1. Since Q admits only one embedding ∞ : Q ↪ C, we shall often drop it from the notation.

(1.6) An easy calculation, based on basic properties of the Gamma function, shows that

− ord_{s=0} L_{∞}(M^{∨}(1), s) = ∑_{0>p>q} dim(H^{p,q})(+ dim(H^{w/2,w/2})^{−})
= dim_{K}(M_{dR}/F^{0}) − dim_{Q}(M_{B}^{+})  (1.6.1)
ord_{s=0} L_{∞}(M, s) = 0

(where the ± superscript denotes the (±1)-eigenspace for ϕ_{∞}) and that leading terms of the Taylor expansions of L_{∞} at s = 0 satisfy

\frac{L_{∞}^{∗}(M, 0)}{L_{∞}^{∗}(M^{∨}(1), 0)} ∈ (2π)^{w·rk(M)/2+dim(M_{B}^{−})}Q^{∗}  (1.6.2)

Here we define f^{∗}(a) = lim_{z⟶a}(z − a)^{−r}f(z) = b, where r = ord_{z=a} f(z) (for a function f meromorphic in a neighbourhood of a ∈ C).

2. Deligne’s period map. (2.1) Under the comparison map I_{∞} : M_{B} ⊗_{Q} C ⟶^{∼} M_{dR} ⊗_{Q} C, M_{dR} ⊗ R corresponds to (M_{B}^{+} ⊗ R) ⊕ (M_{B}^{−} ⊗ R(−1)). Deligne’s period map is the induced map

α_{M} : M_{B}^{+} ⊗ R ⟶ (M_{dR}/F^{0}) ⊗ R,

first introduced in [21]. As M has negative weight by assumption, we have

Ker(α_{M}) ⊆ (F^{0} ∩ \overline{F}^{0})(M_{B} ⊗ C) = 0.

We may, therefore, reformulate (1.6.1) as

ord_{s=0} L_{∞}(M, s) = dim_{R} Ker(α_{M}) = 0
− ord_{s=0} L_{∞}(M^{∨}(1), s) = dim_{R} Coker(α_{M})

(2.1.1)

The isomorphism M(−1)_{B}^{+} ⊗ R = M_{B}^{−} ⊗ R(−1) ⟶^{∼} M_{dR} ⊗ R/I_{∞}(M_{B}^{+} ⊗ R) also induces a map

β_{M} : Ker(α_{M(−1)}) ⟶ Coker(α_{M}).

For w ≠ −2, the domain of β_{M} vanishes; for w = −2, β_{M} is injective for the same reason as α_{M} is. (2.2) The Q-structures M_{B}^{+} and M_{dR}/F^{0}, on the domain and target of α_{M} respectively, define a natural Q-structure D(M) on the real vector space det(Coker(α_{M})) (where by det(V) we denote the highest exterior power of a vector space V). Deligne [21] calls the motive M critical, if the period map α_{M} is an isomorphism. If this is the case, he defines the period of M as

c^{+}(M) = det(α_{M}) ∈ R^{∗}/Q^{∗},

the determinant being taken with respect to the Q-structures M_{B}^{+}, M_{dR}/F^{0}. Of course, Coker(α_{M}) vanishes for such M, so its determinant is canonically isomorphic to R, and the Q-structure D(M) is equal to c^{+}(M)^{−1} · Q.

In general, there is a natural commutative diagram with exact rows:

0 ⟶ (F^{0}M_{dR} ⊕ M_{B}^{+}) ⊗ R ⟶ (M_{B}^{−}(−1) ⊕ M_{B}^{+}) ⊗ R ⟶ Ker(α_{M^{∨}(1)})^{∨} ⟶ 0
∥  ↓^{≀I_{∞}}  ↓^{≀}
0 ⟶ (F^{0}M_{dR} ⊕ M_{B}^{+}) ⊗ R ⟶  M_{dR} ⊗ R  ⟶ Coker(α_{M}) ⟶ 0,

which defines a canonical isomorphism

Coker(α_{M})^{∨} ⟶^{∼} Ker(α_{M^{∨}(1)})  (2.2.1)

Similarly, Coker(α_{M^{∨}(1)}) ⟶^{∼} Ker(α_{M})^{∨} = 0, and thus det(Ker(α_{M^{∨}(1)})) also has a natural Q-structure, inherited from the Q-structures M^{∨}(1)_{B}^{+} and M^{∨}(1)_{dR}/F^{0} on the domain and target of α_{M^{∨}(1)}. The above diagram shows that this Q-structure on det(Ker(α_{M^{∨}(1)})) corresponds, via (2.2.1), to the Q-structure

B(M) = (2πi)^{− dim(M_{B}^{−})}δ(M)D(M)

on det(Coker(α_{M})). Here δ(M) ∈ C^{∗}/Q^{∗} denotes the determinant of I_{∞} with respect to the Q-structures M_{B}, M_{dR}. The calculation in [21, 5.6], based on a conjectural description of motives of rank 1 over Q, shows that

(2πi)^{− dim(M_{B}^{−})}δ(M) ∈ (2π)^{− dim(M_{B}^{−})−w·rk(M)/2}ε(M, 0)Q^{∗},

hence, in view of (1.6.2),

L^{∗}(M^{∨}(1), 0)B(M) = L^{∗}(M, 0)D(M)  (2.2.2)

(2.3) We have seen that (by 1.4.2 and 2.1.1)

dim_{R}(Coker(α_{M})) = ord_{s=0} L(M^{∨}(1), s) − ord_{s=0} L(M, s)

(where the last term vanishes if w ≤ −3). This makes Coker(α_{M}) a natural candidate for the target of a regulator map, which should “explain” the values L^{∗}(M, 0) and L^{∗}(M^{∨}(1), 0). What we need is a Q-vector space A_{M} ‘of arithmetic nature’ (a generalization of Q ⊕ O_{F}^{∗} ⊗ Q for ζ_{F} (s) at s = 0) and regulator map r : A_{M} ⟶ Coker(α_{M}), inducing an isomorphism after tensoring with R. The value L^{∗}(M, 0) ∈ R^{∗}/Q^{∗} (resp. L^{∗}(M^{∨}(1), 0)) should be then equal to the determinant of r with respect to the Q-structure D(M) (resp. B(M)) on det(Coker(α_{M})). (2.4) As a first step towards the construction of a regulator map we give an interpretation of Coker(α_{M}) in terms of Hodge theory. For a subring A ⊂ R, denote by MH_{A} (resp. MH_{A}^{+}) the category of mixed A-Hodge structures (resp. mixed A-Hodge structures with infinite Frobenius, i.e. an involution ϕ_{∞} compatible with the weight filtration and such that ϕ_{∞} ⊗ c preserves the Hodge filtration). Both MH_{A} and MH_{A}^{+} are tensor categories with a unit object 1 = A(0). For H ∈ Ob MH_{R}, the complex (in degrees 0 and 1)

W_{0}H ⊕ F^{0}(W_{0}H)_{C} ⟶^{i_{W} −i_{F}} (W_{0}H)_{C}  (2.4.1)

(where i_{W} and i_{F} denote the obvious inclusions) represents R Hom(R(0), H), i.e. there are (natural) isomorphisms

Hom_{MH_{R}}(R(0), H) ⟶^{∼} W_{0}H ∩ F^{0}H_{C}  (2.4.2)
Ext_{MH_{R}}^{1}(R(0), H) ⟶^{∼} W_{0}H\(W_{0}H)_{C}/F^{0}(W_{0}H)_{C}  (2.4.3)

and higher Ext^{i} vanish for i > 1. For a proof of this basic fact of life in MH_{R} we refer the reader to [3],[16],[17]. Note that (2.4.2) is obvious: we associate to a morphism f : R(0) ⟶ H in MH_{R} the value f(1). Let us indicate how the morphism in (2.4.3) is defined: given an extension

0 ⟶ H ⟶ E ⟶ R(0) ⟶ 0

in MH_{R}, we can lift 1 ∈ R(0) to 1_{W} ∈ W_{0}E (defined modulo W_{0}H) and to 1_{F} ∈ F^{0}(W_{0}E)_{C} (defined modulo F^{0}(W_{0}H)_{C}). The class of 1_{F} − 1_{W} in

W_{0}H\(W_{0}H)_{C}/F^{0}(W_{0}H)_{C}

is then well defined, depends only on the extension class of E and is additive in E. (2.5) Similarly, for H ∈ Ob MH_{R}^{+}, we just take ϕ_{∞}⊗c-invariants: R Hom(R(0), H) is represented by

W_{0}H^{+} ⊕ F^{0}(W_{0}H_{dR})⟶W_{0}H_{dR},  (2.5.1)

hence

Hom_{MH}_{R}^{+}(R(0), H) ⟶^{∼} W_{0}H^{+} ∩ F^{0}H_{C}  (2.5.2)
Ext_{MH_{R}^{+}}^{1}(R(0), H) ⟶^{∼} W_{0}H^{+}\W_{0}H_{dR}/F^{0}(W_{0}H_{dR}),  (2.5.3)

where H_{dR} = H_{C}^{ϕ_{∞}⊗c=1}, H^{±} = H^{ϕ_{∞}=±1}.

In particular, for H = M_{B} ⊗ R ∈ Ob MH_{R}^{+}, we have H = W_{0}H and H_{dR} = M_{dR} ⊗ R, hence

Ext_{MH_{R}^{+}}^{1}(R(0), H) ⟶^{∼} M_{B}^{+} ⊗ R\M_{dR} ⊗ R/F^{0}M_{dR} ⊗ R = Coker(α_{M})

(2.6) Having interpreted Coker(α_{M}) as an Ext-group in the category of mixed Hodge structures, it is quite tempting to make a guess as to what the regulator map should be: simply the canonical map (‘Hodge realization’)

Ext_{MM_{Q}}^{1}(Q(0), M) ⟶ Ext_{MH_{R}^{+}}^{1}(R(0), M_{B} ⊗ R),  (2.6.1)

where the first Ext-group is computed in a suitable category of “mixed motives”, which extends Grothendieck’s category of pure motives over Q. In the following two sections we shall try to make this idea more precise.

3. Arithmetic vs. geometric cohomology. (3.1) The group Coker(α_{M}) (where, as before, M = h^{i}(X)(n) for a smooth projective variety X_{/Q} and w = i − 2n ≤ −1) can be obtained as a composition of two cohomological functors: H_{B}^{i}(X(C), −) (applied to R(n)) and Ext_{MH_{R}^{+}}^{1}(R(0), −).

This suggests that Coker(α_{M}) is, in fact, equal to H_{?}^{i+1}(X, R(n)) in some fancy cohomology theory H_{?}^{.}, and that the isomorphism

H_{?}^{i+1}(X, R(n)) ⟶^{∼} Ext_{MH_{R}^{+}}^{1}(R(0), H_{B}^{i}(X(C), R(n)))

comes from the standard spectral sequence for composition of derived functors. (3.2) This is indeed the case and the corresponding cohomology theory fits into the following general framework:

• Let T be a tensor category with a unit object 1 and Tate twists − ↦ −(j). Set A = End_{T} (1),

Γ(T , −) = Hom_{T} (1, −) : T ⟶ (A − mod).

• Let V be a sufficiently large subcategory of the category of schemes of finite type over a given field F (e.g. containing all smooth quasi-projective varieties over F).

• Suppose that for each X ∈ Ob V and j ∈ Z, there is a complex (contravariant in X)

\underline{RΓ}(X, j) = \underline{RΓ}(X, 0)(j) ∈ Ob D^{b}(T ),

whose cohomology

\underline{H}^{p}(X, j) = \underline{H}^{p}(X, 0)(j) ∈ Ob T

are ‘geometric cohomology groups’ of X. Applying the functor RΓ(T , −) : D^{b}(T ) ⟶ D^{+}(A − mod), we get a complex

RΓ_{T} (X, j) = RΓ(T , \underline{RΓ}(X, j)) ∈ Ob D^{+}(A − mod)

with cohomology groups

H_{T}^{p} (X, j) ∈ Ob (A − mod)

— ‘arithmetic’, or ‘absolute’ cohomology groups of X. The spectral sequence, referred to in 3.1, is then

Ext_{T}^{p} (1, \underline{H}^{q}(X, j)) ⟹ H_{T}^{p+q}(X, j)  (3.2.1)

Reasonable geometric cohomology theories are usually equipped with additional structure: cohomology with supports, cup products, dual homology theory (in the sense of [13]). See 3.5 below for more details. (3.3) As a basic example, consider étale cohomology. For a fixed integer n prime to the characteristic of F, let T be the category of finite Z/nZ[G]-modules, where G = G(F^{sep}/F).Then 1 = Z/nZ(0), A = Z/nZ, R^{p}Γ_{T} (−) = H^{p}(G, −) and the geometric resp. arithmetic cohomology groups are étale cohomology of X over F^{sep} resp. F:

\begin{aligned}\underline{H}^{p}(X, j) & = H^{p}((X ×_{F} F^{sep})_{et}, Z/nZ)(j) \\ H_{T}^{p} (X, j) & = H^{p}(X_{et}, (Z/nZ)(j)) \end{aligned}

They are related by the Hochschild-Serre spectral sequence.

If F is a finitely generated extension of Q, then \underline{RΓ}(X, j) exist also for ℓ-adic cohomology. In this case, T is the category of Q_{ℓ}-vector spaces of finite dimension equipped with a continuous action of G, 1 = Q_{ℓ}(0), A = Q_{ℓ},

\begin{aligned}\underline{H}^{p}(X, j) & = H^{p}((X ×_{F} \overline{F})_{et}, Q_{ℓ})(j) \\ H_{T}^{p} (X, j) & = H^{p}(X_{et}, Q_{ℓ}(j)), \end{aligned}

where the ℓ-adic cohomology over F is the continuous étale cohomology in the sense of [41]. Similarly, (3.2.1) becomes the Hochschild-Serre spectral sequence

H^{p}(G, H^{q}((X ×_{F} \overline{F})_{et}, Q_{ℓ}(j)) ⟹ H^{p+q}(X_{et}, Q_{ℓ}(j))

for continuous Galois cohomology ([41]). (3.4) Another example is provided by what Beilinson calls ‘absolute Hodge cohomology’. For any separated scheme X of finite type over C, Beilinson constructs in [3] a complex \underline{RΓ}(X, 0) ∈ Ob D^{b}(MH_{R}), whose cohomology objects are H_{B}^{i}(X(C), R) with Deligne’s Hodge structures [18],[19]. The formalism of 3.2 (with T = MH_{R}) then produces ‘absolute Hodge cohomology’ of X, sitting in an exact sequence

0 ⟶ Ext_{MH_{R}}^{1}(R(0), H_{B}^{i}(X(C), R(n))) ⟶ H_{MH_{R}}^{i+1}(X, n) ⟶
⟶ Hom_{MH_{R}}(R(0), H_{B}^{i+1}(X(C), R(n))) ⟶ 0.

For X over R, we just replace MH_{R} by MH_{R}^{+}, and get a similar sequence for H_{MH_{R}^{+}}^{i+1}(X, n). If X_{/R} is proper and smooth, then (writing w = i − 2n, as usual)

we get from (2.5.2–3)

H_{MH_{R}^{+}}^{i+1}(X, n) = \{ \begin{matrix}Ext_{MH_{R}^{+}}^{1}(R(0), H_{B}^{i}(X(C), R(n))) & w ≤ −2 \\ H_{B}^{2n}(X(C), R(n))^{+} ∩ F^{0} & w = −1 \\ 0 & w ≥ 0 \end{matrix}  (3.4.1)

(3.5) Grothendieck’s philosophy of motives stipulates existence of functors X ↦ h^{i}(X), which are universal among cohomology groups for smooth projective varieties over a given field F. The point of view advocated by Beilinson and Deligne is that the category M_{F} of motives with respect to homological equivalence embeds into a larger category of ‘mixed motives’ MM_{F}, which should be a universal target for ‘geometric cohomology theories’ in the sense of 3.2. A fundamental object associated to a quasi-projective variety X_{/F} should then be the complex

\underline{RΓ}(X, 0) ∈ Ob D^{b}(MM_{F}),

rather then its cohomology groups

h^{i}(X) = H^{i}(\underline{RΓ}(X, 0)) ∈ Ob MM_{F}

(which would coincide with Grothendieck’s h^{i}(X) for smooth projective X).

There should be versions with support \underline{RΓ}_{Y} (X, j) (for Y ⊆ X closed), cup product \underline{RΓ}(X, i)⊗^{L} \underline{RΓ}(X, j) ⟶ \underline{RΓ}(X, i+j)) and homology complexes \underline{RΓ}′(X, j) satisfying several axioms (see [1, 2.3.2]), which ensure that

\begin{aligned}h_{Y}^{i} (X)(j) & = H^{i}(\underline{RΓ}_{Y} (X, j)) and \\ h_{i}(X, j) & = H^{−i}(\underline{RΓ}′(X, −j)) \end{aligned}

form a twisted Poincaré duality theory in the sense of Bloch and Ogus [13]. The most important axiom is the duality isomorphism

\underline{RΓ}′(Y, j) ⟶^{∼} \underline{RΓ}_{Y} (X, j + d)[2d],

valid for Y ⊆ X closed in a smooth X of dimension d. If Y has pure codimension i, then its cycle class is a map

cl(Y) : 1 ⟶ h_{2d−2i}(Y)(d − i) ⟶^{∼} h_{Y}^{2i}(X)(i)

Motivic cohomology and homology are then defined by

H_{MM_{F}}^{i} (X, j) = Ext_{MM_{F}}^{i} (1, \underline{RΓ}(X, j)) = Hom_{D^{b}(MM_{F})}(1, \underline{RΓ}(X, j)[i])
H_{i}^{MM_{F}} (X, j) = Ext_{MM_{F}}^{−i} (1, \underline{RΓ}′(X, −j)) = Hom_{D^{b}(MM_{F})}(1, \underline{RΓ}′(X, −j)[−i]).

Of course, 1 = Q(0) = h^{0}(Spec(F)). (3.6) Beilinson conjectures that the spectral sequence (3.2.1)

Ext_{MM_{F}}^{i} (1, h^{j}(X)(n)) ⟹ H_{MM_{F}}^{i+j} (X, n)  (3.6.1)

degenerates for smooth projective X and that Ext_{MM_{F}}^{i} vanishes for i greater than the Kronecker dimension of F (equal to the transcendence degree tr.deg(F/F_{p}) in characteristic p, resp. to 1 + tr.deg(F/Q) in characteristic zero). In particular, if X is a smooth projective variety over Q, then the spectral sequence (3.6.1) should degenerate into exact sequences

0 ⟶ Ext_{MM_{Q}}^{1}(1, h^{i}(X)(n)) ⟶ H_{MM_{Q}}^{i+1}(X, n) ⟶ Hom_{MM_{Q}}(1, h^{i+1}(X)(n)) ⟶ 0,

with the third group vanishing, unless w = i − 2n = −1 (for weight reasons). This suggests another description of the regulator map (2.6.1) for w < −1: the Hodge realization

H_{MM_{Q}}^{i+1}(X, n) ⟶ H_{MH_{R}^{+}}^{i+1}(X, n)  (3.6.2)

For i + 1 = 2n, the above sequence should be isomorphic to

0 ⟶ CH^{n}(X)_{0} ⊗ Q ⟶ CH^{n}(X) ⊗ Q⟶^{cl} Hom_{MM_{Q}}(Q(0), h^{2n}(X)(n)) ⟶ 0,

where CH^{n}(X) is the Chow group of codimension n cycles on X modulo rational equivalence, and CH^{n}(X)_{0} the subgroup of homologically trivial cycles (cf. 4.2). (3.7) Furthemore, there should be a relative version of motivic cohomology for morphisms f : X ⟶ Y; there should be a notion of “motivic sheaves” on every variety X_{/F}, together with the standard formalism of Grothendieck’s six functors (f^{∗}, Rf_{∗}, Rf_{!}, Rf^{!}, R Hom, ⊗^{L}) between corresponding derived categories. Denoting the category of motivic sheaves on X by M(X), then MM_{F} should be identified with M(Spec(F)). Writing a : X ⟶ Spec(F) for the structural morphism, we should have

\underline{RΓ}(X, j) = Ra_{∗}a^{∗}1(j)
H_{MM_{F}}^{i} (X, j) = Ext_{M(Spec(F))}^{i}(1, Ra_{∗}a^{∗}1(j)) = Ext_{M(X)}^{i}(a^{∗}1, a^{∗}1(j)).

Note that such a relative theory exists in both étale cohomology (3.3) and Hodge theory (3.4) (see [58]).

4. Mixed motives. (4.1) The category MM_{Q} of mixed motives over Q is expected to enjoy (at least) the following four properties:

• The category of semisimple objects of MM_{Q} is equivalent to the category M_{Q} of motives with respect to homological equivalence (this makes sense only if homological and numerical equivalences of cycles coincide, which is one of Grothendieck’s Standard Conjectures [39]; otherwise M_{Q} itself would not be semisimple, by [44]).

• Each mixed motive E ∈ Ob MM_{Q} admits a functorial weight filtration W_{.}E (increasing) with graded factors Gr_{i}^{W} (E) ∈ Ob M_{Q} pure of weight i.

• Each quasi-projective variety (more generally, simplicial variety) X has cohomology h^{i}(X) ∈ Ob MM_{Q}.

• Ext_{MM_{Q}}^{i} = 0 for i > 1. There is no Grothendieck style definition of MM_{Q} as yet. All definitions proposed so far ([23],[32],[43]) are based on the same principle: one constructs first a suitable Tannakian category of mixed realizations MR_{Q} and then defines MM_{Q} as a full subgategory of MR_{Q} consisting of objects of ‘geometric origin’, e.g. the smallest Tannakian subcategory of MR_{Q} containing cohomology realizations of all quasiprojective varieties. This is based on a tacit assumption that the realization functor MM_{Q} ⟶ MR_{Q} is fully faithful.

All realizations discussed in (1.1) in the context of pure motives have analogues for mixed motives. For arbitrary quasi-projective variety X_{/Q}, the mixed motive E = h^{i}(X) has realizations

\begin{aligned}E_{ℓ} & = H^{i}((X ×_{Q} \overline{Q})_{et}, Q_{ℓ}) \\ E_{B} & = H^{i}(X(C), Q) \\ E_{dR} & = H^{i}((Y_{.})_{Zar}, Ω_{Y_{.}/Q}^{.}), \end{aligned}

where Y_{.} ⟶ X is a smooth hypercovering of X (for Zariski topology). In this situation E_{B} is a mixed Hodge structure with infinite Frobenius ϕ_{∞} and the weight filtration on E_{B} corresponds, under comparison isomorphisms

I_{ℓ,\overline{σ}} : E_{B} ⊗_{Q} Q_{ℓ} ⟶^{∼} E_{ℓ}, I_{∞} : E_{B} ⊗_{Q} C ⟶^{∼} E_{dR} ⊗_{Q} C,

to natural filtrations W_{.}E_{ℓ} (by subrepresentations of G(\overline{Q}/Q)) and W_{.}E_{dR}. The graded objects Gr_{j}^{W} (E_{ℓ}) and Gr_{j}^{W} (E_{B}) are pure of weight j (as representations of G(\overline{Q}/Q) in the sense of (1.1) resp. as Q-Hodge structures).

(4.2) Let us give a few examples of mixed motives over arbitrary (say, finitely generated) field F. We shall confidently use cohomology with supports h_{Y}^{i} (X) resp. relative cohomology h^{i}(X, Y) for Y ⊆ X a closed subvariety of X (see [64] for a realization of the relative cohomology). The first example gives a motivic interpretation of Abel-Jacobi maps. Let X_{/F} be a smooth projective variety, Y ⊂ X a cycle of (pure) codimension i, homologically trivial. Consider the long exact cohomology sequence

0 ⟶ h^{2i−1}(X)(i) ⟶ h^{2i−1}(X − Y)(i) ⟶ h_{Y}^{2i}(X)(i)⟶^{β}h^{2i}(X)(i) ⟶ …

As Y is homologically trivial, the composition of β with the cycle class of Y

cl(Y) : Q(0) ⟶ h_{Y}^{2i}(X)(i)

vanishes. Taking pullback of the above exact sequence via β, we get an extension of motives

0 ⟶ h^{2i−1}(X)(i) ⟶ E ⟶ Q(0) ⟶ 0

Note that E is a motive with two weights, namely −1 and 0. The extension class of E depends, in fact, only on the rational equivalence class of Y and the map Y ↦ E induces a homomorphism

CH^{i}(X)_{0} ⟶ Ext_{MM_{F}}^{1} (Q(0), h^{2i−1}(X)(i))

This is a ‘motivic’ Abel-Jacobi map; if F ⊆ C, we may do the same with singular cohomology with integral coefficients, getting the usual Abel-Jacobi map with values in Griffiths’ Jacobian

Ext_{MH_{Z}}^{1}(Z(0), H_{Z}) = H_{Z}\H_{C}/F^{0},

where H_{Z} = H^{2i−1}(X(C), Z(i)) (cf. (2.4.3)).

(4.3) The second example is related to a motivic construction of height pairings (see [64]). Let X_{/F} be a smooth projective variety, equidimensional of dimension d, let Y, Z ⊂ X be homologically trivial cycles of pure codimensions i, j = d + 1 − i, with disjoint supports. The relative cohomology H = h^{2i−1}(X − Y, Z)(i) appears in exact sequences

0 ⟶ h^{2i−1}(X, Z)(i) ⟶ h^{2i}(X − Y, Z)(i) ⟶ h_{Y}^{2i}(X)(i) ⟶ h^{2i}(X, Z)(i)

h^{2i−2}(X −Y)(i) ⟶ h^{2i−2}(Z)(i) ⟶ h^{2i−1}(X −Y, Z)(i) ⟶ h^{2i−1}(X −Y)(i) ⟶ 0

Taking pullback of H via the cycle class cl(Y) : Q(0) ⟶ h_{Y}^{2i}(X)(i) and pushout via the trace map T r(Z) : h^{2i−2}(Z)(i) ⟶ Q(1), we get a mixed motive E sitting in a commutative diagram with exact rows and columns

0  0
↓ ↓
0 ⟶ Q(1) ⟶ E_{2} ⟶ M ⟶ 0
∥ ↓ ↓
0 ⟶ Q(1) ⟶ E ⟶ E_{1} ⟶ 0
↓ ↓
Q(0) = Q(0)
↓ ↓
0  0

where M = h^{2i−1}(X)(i) and E_{1} (resp. E_{2}^{∨}(1)) is associated to the cycle Y (resp. Z) as in the previous example. The weight filtration of E is given by

W_{−3}E = 0, W_{−2}E = Q(1), W_{−1}E = E_{2}, W_{0}E = E.

An important special case of this construction, when X = P^{1}, Y = (0) − (∞) , Z = (1) − (u) (u ∈ F^{∗} − {1}) gives a ‘Kummer motive’ E = h^{1}(G_{m}, {1, u})(1), which is an extension

0 ⟶ Q(1) ⟶ E ⟶ Q(0) ⟶ 0

It is believed that Kummer motives exhaust all motivic extensions of Q(0) by Q(1), in other words that

Ext_{MM_{F}}^{1} (Q(0), Q(1)) = F^{∗} ⊗ Q  (4.3.1)

Note that the extension class of E in various realizations is given by

[E_{ℓ}] = u ⊗ 1 ∈ H^{1}(F, Q_{ℓ}(1)) = F^{∗} ˆ⊗Q_{ℓ}
[E_{B}] = log(u) ∈ Ext_{MH_{R}^{+}}^{1}(R(0), R(1)) ⟶^{∼} R

(for ℓ different from the characteristic of F, resp. for an embedding F ↪ R), after a suitable normalization of signs [64]. In general, we get an extension of Q(0) by Q(1) whenever M = 0. It is by no means clear that extensions obtained in this way are Kummer (I am grateful to the referee for this remark). (4.4) The last example is borrowed from the appendix to [29]. Let

Δ_{n} = Spec(F [T_{0}, …, T_{n}]/(∑ T_{i} − 1))

be a simplex of dimension n, let ∂_{i} : Δ_{n} ↪ Δ_{n+1} (for 0 ≤ i ≤ n + 1) be the i^{th} face map (sending T_{i} to 0 and renumbering T_{j} ↦ T_{j−1} for j > i).

Let X_{/F} be an equidimensional smooth projective variety. Fix n > 0 and suppose that Y is a cycle of codimension i on X ×Δ_{n}, meeting all faces of X ×Δ_{n} properly. By a face we mean the image of X × Δ_{m} by any composition of face maps ∂_{i} (for m < n). We also assume that ∂_{j}^{∗}(Y) = 0 for all 0 ≤ j ≤ n.

We shall write Δ_{X}^{n} = X × Δ_{n}, ∂Δ_{X}^{n} for the union of all faces of codimension one ∂_{i}(X × Δ_{n−1}), |∂Y | = |Y | ∩ ∂Δ_{X}^{n}, U = Δ_{X}^{n} − Y, ∂U = U ∩ ∂Δ_{X}^{n}. There is a natural exact sequence ([29, (A.3)])

0 ⟶ h^{2i−n−1}(X)(i) ⟶ h^{2i−1}(U, ∂U)(i) ⟶ Ker(β) ⟶ h^{2i−n}(X)(i),

where

β : Ker[h_{Y}^{2i}(Δ_{X}^{n})_{0}(i) ⟶ h^{2i}(Δ_{X}^{n})_{0}(i)] ⟶ Ker[h_{∂Y}^{2i} (∂Δ_{X}^{n})_{0}(i) ⟶ h^{2i}(∂Δ_{X}^{n})_{0}(i)]

is induced by taking intersection with ∂Δ_{X}^{n} (the subscript 0 denotes kernel of the map ‘forget the supports’). The class of Y is a map cl(Y) : Q(0) ⟶ Ker(β) and its composite with Ker(β) ⟶ h^{2i−n}(X)(i) vanishes. Taking pullback by cl(Y), gives, finally, an extension

0 ⟶ h^{2i−n−1}(X)(i) ⟶ E ⟶ Q(0) ⟶ 0

(4.5) The example of Kummer motives in 4.3 shows that the motivic regulator (2.6.1) still needs a minor adjustment: for M = R_{F/Q}Q(1), we have

Ext_{MM_{Q}}^{1}(Q(0), M) = Ext_{MM_{F}}^{1} (Q(0), Q(1)) =^{?} F^{∗} ⊗ Q,

but the classical regulator is made up only of units O^{∗} (where O is the ring of integers of F). We need, therefore, a motivic interpretation of O^{∗} ⊗ Q.

We say, after A.J. Scholl, that a mixed motive E ∈ Ob MM_{F} is defined over O, if the weight filtration of E_{ℓ} splits as a representation of the inertia group I_{v}, for all ℓ and v |̸ ℓ. Mixed motives defined over O form a full subcategory MM_{O} of MM_{F}, containing M_{F}. For v |̸ ℓ, the valuation v induces isomorphisms

H^{1}(F_{v}, Q_{ℓ}(1)) = F_{v}^{∗} ˆ⊗Q_{ℓ} ⟶^{∼} H^{1}(I_{v}, Q_{ℓ}(1)) = F_{v,ur}^{∗} ˆ⊗Q_{ℓ} ⟶^{∼} Q_{ℓ},

which shows that the Kummer motive corresponding to u ∈ F^{∗} is defined over O iff u ∈ O^{∗}. Modulo (4.3.1), this gives the desired motivic formula for O^{∗} ⊗ Q:

O^{∗} ⊗ Q = Ext_{MM_{O}}^{1}(Q(0), Q(1)) = Ext_{MM_{Z}}^{1}(Q(0), R_{F/Q}Q(1)).

Similarly, we denote by H_{MM_{Z}}^{i+1}(X, n) (for smooth projective X_{/Q} and w = i−2n ≤ −2) the image of

Ext_{MM_{Z}}^{1}(Q(0), M) ↪ Ext_{MM_{Q}}^{1}(Q(0), M) ⟶^{∼} H_{MM_{Q}}^{i+1}(X, n).

The final version of the motivic regulator should then be given by restricting (3.6.1) to

H_{MM_{Z}}^{i+1}(X, n) ↪ H_{MM_{Q}}^{i+1}(X, n) ⟶ H_{MH_{R}^{+}}^{i+1}(X_{/R}, n)  (4.5.1)

5. Motivic cohomology.

(5.1) One of the most disturbing features of the motivic regulator map (4.5.1) is the way how the motivic cohomology H_{MM_{Z}}^{i+1} is ‘defined’, in terms of homological algebra in MM_{Q} and MM_{Z}, two Tannakian categories of rather dubious status. What we need is a direct description of motivic cohomology and the regulator map.

Beilinson [4] and Lichtenbaum [48] conjectured that motivic cohomology of X (even its version with integral coefficients) can be computed as hypercohomology of suitable complexes on Zariski resp. étale site of X. We shall not discuss this approach to motivic cohomology and instead refer the reader to the articles of S. Bloch, A.B. Gončarov and S. Lichtenbaum in these Proceedings.

There exist two candidates for motivic cohomology. Beilinson [1] defines, for a quasi-projective variety X_{/F},

H_{M}^{i}(X, Q(j)) = (K_{2j−i}(X) ⊗ Q)^{(j)}  (5.1.1)

as the subspace of weight j for Adams operations of a suitable K-group of X. Here K-theory enters the picture for two reasons. First, Beilinson was guided by the relationship between K-theory and singular cohomology in the topological situation (see the discussion of the Atiyah-Hirzebruch spectral sequence in [37]). The second hint came from arithmetic, through the works of Lichtenbaum, Borel and Bloch discussed in the introduction. In fact, in [1], Beilinson states that his work “owes its origin to an attempt to understand Bloch’s ideas and computations”.

The second construction is due to Bloch [9]. In the notation of 4.4, let Z^{q}(X)_{n} be the group of cycles of codimension q on X×Δ_{n}, meeting all faces transversally. Then \underline{\underline{n}} ↦ Z^{q}(X)_{n} is a simplicial abelian group Z^{q}(X). One defines the higher Chow group CH^{q}(X, p) as the homotopy group π_{p}(| Z^{q}(X) |) of the geometric realization of Z^{q}(X) (or as homology of the corresponding normalized chain complex; see [37]). Then

H^{i}(X, Z(j)) = CH^{j}(X, 2j − i)

is a candidate for motivic cohomology with integral coefficients.

At present, it is not even known if both recipes give the same result, i.e. if there exist canonical isomorphisms

(K_{p}(X) ⊗ Q)^{(j)} ⟶^{∼} CH^{j}(X, p) ⊗ Q  (5.1.2)

for all smooth varieties X_{/F}. This is certainly true for p = 0. In this case, CH^{j}(X, 0) = CH^{j}(X) is the standard Chow group of codimension j cycles on X modulo rational equivalence, and the isomorphism CH^{j}(X)⊗Q ⟶^{∼} (K_{0}(X)⊗Q)^{(j)} is a classical result of Grothendieck [66]. For p > 0, (5.1.2) still remains open (the argument in [9] runs into difficulties when applying various moving lemmas). An account of Beilinson’s conjectures in terms of higher Chow groups is presented in [29].

(5.2) It is possible to extend (5.1.1) and define K-theoretic cohomology with supports, homology theory (using K′(X)), cup products

∪ : H_{M}^{i}(X, Q(m)) ⊗ H_{M}^{j}(X, Q(n)) ⟶ H_{M}^{i+j}(X, Q(m + n))

and show that they satisfy Galois descent and almost all axioms of Bloch-Ogus [13] (see [68],[69]). Next we need a regulator map

r_{H} : H_{M}^{i}(X, Q(j)) ⟶ H_{MH_{R}^{+}}^{i}(X, j)  (5.2.1)

to replace (3.6.2) (say, for a quasi-projective variety X_{/Q}). Beilinson constructs r_{H} as a Chern class on higher K-theory, using the general machinery of characteristic classes due to Gillet [34]. We shall present a more direct construction of r_{H}, which works for X smooth and quasi-projective.

Fix an integer N and denote by B_{.}GL_{N/R} the classifying space of the algebraic group GL_{N/R}. It is a simplicial scheme (cf. [37]) and there is a universal simplicial bundle E of rank N over B_{.}GL_{N/R}. The Betti cohomology of B_{.}GL_{N/R} is well known:

\begin{aligned}H^{2n−1}(B_{.}GL_{N/R}(C), Q) & = 0 \\ ⨁_{n≥0} H^{2n}(B_{.}GL_{N/R}(C), Q(n)) & = Q[c_{1}, …, c_{N}], \end{aligned}

where

c_{i} = c_{i}(E) ∈ H^{2i}(B_{.}GL_{N/R}(C), Q(i))

are Chern classes of the universal bundle E. As E is defined over R, all c_{i} are fixed by ϕ_{∞}. According to [19, 9.1.1], all cohomology groups H^{2n}(B_{.}GL_{N/R}(C), Q(n)) have pure Hodge type (0, 0). From (3.4.1), we get isomorphisms

H_{MH_{R}^{+}}^{2n}(B_{.}GL_{N/R}, n) ⟶^{∼} H^{2n}(B_{.}GL_{N/R}(C), Q(n)) ⊗ R (n ≥ 0)

and we may view, therefore, c_{i} as elements of the first group.

Let A be an R-algebra of finite type. We have the evaluation map

ev : Spec(A) × B_{.}GL_{N}(A) ⟶ B_{.}GL_{N/R},

which is a morphism of (simplicial) R-schemes. Let us compute in general H_{MH_{R}^{+}}^{p}(X_{/R} × Y_{.}, q) for X_{/R} a separated scheme and Y_{.} a simplicial set. If we

denote by C^{.}(Y_{.}, R) the cochain complex of Y_{.} (with real coefficients), then, in the notation of 3.4,

\underline{RΓ}(X_{/R} × Y_{.}, q) = s(\underline{RΓ}(X_{/R}, q) ⊗_{R} C^{.}(Y_{.}, R)),

hence the Künneth formula and the cap product

H_{MH_{R}^{+}}^{p}(X_{/R} × Y_{.}, q) ⟶^{∼} ⨁_{j} H_{MH_{R}^{+}}^{p−j}(X_{/R}, q) ⊗_{R} H^{j}(Y_{.}, R)

∩ : H_{MH_{R}^{+}}^{2n}(X_{/R} × Y_{.}, n) ⊗ H_{i}(Y_{.}, R) ⟶ H_{MH_{R}^{+}}^{2n−i}(X_{/R}, n)

For i > 0, the Hurewicz map K_{i}(A) ⟶ H_{i}(GL(A), R) = \underrightarrow{lim}_{N} H_{i}(B_{.}GL_{N}(A), R) and cap product with ev^{∗}(c_{n}) induce a homomorphism

c_{i,n} : K_{i}(A) ⟶ H_{MH_{R}^{+}}^{2n−i}(Spec(A), n)

Suppose now that X_{/R} is a smooth quasi-projective variety. By Jouanolou’s trick [45], there is an affine variety Spec(A) and a morphism π : Spec(A) ⟶ X_{/R} which makes Spec(A) a vector bundle over X_{/R}. By the homotopy property, π induces isomorphisms both on K_{i} and H_{MH_{R}^{+}}^{i}, and we define

c_{i,n} : K_{i}(X_{/R}) ⟶ H_{MH_{R}^{+}}^{2n−i}(X_{/R}, n)

by transport of structure. Finally, one defines Chern character maps

ch_{i} = ∑_{n≥0} \frac{(−1)^{n−1}}{(n − 1)!} c_{i,n} : K_{i}(X_{/R}) ⊗ Q ⟶ ⨁_{n} H_{MH_{R}^{+}}^{2n−i}(X_{/R}, n)

for i > 0 and ch_{0} as the usual Chern character

K_{0}(X_{/R}) ⊗ Q ⟶ ⨁_{n≥0} H_{MH_{R}^{+}}^{2n}(X_{/R}, n).

The weight properties of Chern classes imply that (K_{i}(X_{/R}) ⊗ Q)^{(n)} is mapped by ch_{i} into H_{MH_{R}^{+}}^{2n−i}(X_{/R}, n). For a smooth quasi-projective variety X_{/Q}, Beilinson’s regulator r_{H} is defined as

r_{H} : H_{M}^{i}(X, Q(j)) = (K_{2j−i}(X)⊗Q)^{(j)} ⟶ (K_{2j−i}(X_{/R})⊗Q)^{(j)} ⟶^{ch_{i}} H_{MH_{R}^{+}}^{i}(X, j)

By multiplicativity of the Chern character, it satisfies

r_{H}(x ∪ y) = r_{H}(x) ∪ r_{H}(y)

The basic example of Beilinson’s regulator is the usual logarithm:

H_{M}^{1}(Spec(R), Q(1)) = R^{∗}⊗Q ⟶ H_{MH_{R}^{+}}^{1}(Spec(R), 1) = Ext_{MH_{R}^{+}}^{1}(R(0), R(1)) ⟶^{∼} R

maps x ⊗ 1 to log |x|. (5.3) Let X be, as before, a smooth projective variety over Q. Chose a proper flat model X_{Z} of X (it always exists) and put

H_{M}^{i}(X, Q(j))_{Z} = Im(K′_{2j−i}(X) ⊗ Q ⟶ (K_{2j−i}(X) ⊗ Q)^{(j)})

Beilinson conjectures that this subgroup is independent of the choice of X. This is true if we consider only proper and regular models of X; unfortunately, such models are rarely known to exist. The localization sequence

⋯ ⟶ K′_{j}(X) ⟶ K_{j}(X) ⟶ ⨁_{p} K′_{j−1}(X ×_{Z} F_{p}) ⟶ ⋯

and certain conjectures about K-theory of schemes over finite fields would imply that H_{M}^{i+1}(X, Q(n))/H_{M}^{i+1}(X, Q(n))_{Z} depends only on the bad fibres of X for w = i − 2n < −1 and vanishes for n > max(i, dim(X)) + 1 (cf. [1, 2.4.2.2]). (5.4) To sum up, Beilinson defines

H_{M}^{i+1}(X, Q(n))_{Z} ↪ H_{M}^{i+1}(X, Q(n))⟶^{r_{H}}H_{MH_{R}^{+}}^{i+1}(X_{/R}, n)

as a K-theoretic substitute for 4.5.1, which still remains only a wishful thinking.

6. Values of L-functions. We are now ready to formulate Beilinson’s conjectures concerning the special values of L-functions. Let X be a smooth projective variety over Q, i ≥ 0 and n ∈ Z integers satisfying w = i − 2n < 0.

(6.1) Conjecture. Assume w ≤ −3. Then:

(1) r_{H} ⊗ R : H_{M}^{i+1}(X, Q(n))_{Z} ⊗ R ⟶ H_{MH_{R}^{+}}^{i+1}(X_{/R}, n) is an isomorphism.

(2) r_{H}(det H_{M}^{i+1}(X, Q(n))_{Z}) = L(h^{i}(X), n)D(M) = L^{∗}(h^{i}(X), i + 1 − n)B(M) in det H_{MH_{R}^{+}}^{i+1}(X_{/R}, n), with D(M) and B(M) defined in 2.2 for M = h^{i}(n).

According to 2.3.1, the order of vanishing of L(h^{i}(X), s) at s = i + 1 − n is equal to dim_{R} H_{MH_{R}^{+}}^{i+1}(X_{/R}, n). If M = h^{i}(X)(n) is critical (see 2.2), then 6.1 reduces to

Deligne’s conjecture [21]

L(h^{i}(X), n) = L(M, 0) ∈ c^{+}(M)Q^{∗}

(6.2) For w = −2, L(h^{i}(X), s) can have a pole at s = n = 1 + i/2. The order of the pole is predicted by Tate’s conjecture:

− ord_{s=n} L(h^{2n−2}(X), s) = dim_{Q} N^{n−1}(X),

where

N^{n−1}(X) = (CH^{n−1}(X)/CH^{n−1}(X)_{0}) ⊗ Q =^{?} Hom_{MM_{Q}}(Q(0), h^{2n−2}(n − 1))

The cycle class in Betti cohomology cl_{B} : N^{n−1} ⟶ Ker(α_{M(−1)}) (for M = h^{2n−2}(X)(n)) and the map β_{M} : Ker(α_{M(−1)}) ⟶ Coker(α_{M}) of 2.1 define

r_{B} : N^{n−1}(X) ⟶ Coker(α_{M}) = H_{MH_{R}^{+}}^{2n−1}(X_{/R}, n).

(6.3) Conjecture. Assume w = −2. Then

(1) (r_{H} ⊕ r_{B}) ⊗ R : H_{M}^{2n−1}(X, Q(n))_{Z} ⊗ R ⊕ N^{n−1}(X) ⊗ R ⟶ H_{MH_{R}^{+}}^{2n−1}(X_{/R}, n) is an isomorphism. (2)(r_{H} ⊕ r_{B})(det (H_{M}^{2n−1}(X, Q(n))_{Z} ⊕ N^{n−1}(X)) = L^{∗}(h^{2n−2}(X), n − 1)B(M) =

= L^{∗}(h^{2n−2}(X), n)D(M)

(6.4) In the remaining case of w = −1, the conjecture has to be modified. A new ingredient is the height pairing

h : CH^{n}(X)_{0} ⊗ Q ⊗ CH^{dim X+1−n}(X)_{0} ⊗ Q ⟶ R  (6.4.1)

We shall not discuss possible definitions of h (unfortunately, all definitions proposed so far are conditional, except for n = 1) and refer to [64] for more details.

(6.5) Conjecture. Assume w = −1. Then:

(1) The height pairing h in 6.4.1 is non-degenerate.

(2) ord_{s=n} L(h^{2n−1}(X), s) = dim_{Q} CH^{n}(X)_{0} ⊗ Q

(3) L^{∗}(h^{2n−1}(X), n) ∈ c^{+}(h^{2n−1}(X)(n)) det(h)Q^{∗}, where c^{+}(M) is Deligne’s period defined in 2.2.

(6.6) The only case when 6.1 is known is, essentially, if X = Spec(F) for a finite extension F/Q. In many cases, however, one can verify the following

Weak Conjecture. Assume w = i − 2n < 0. Then 6.1 (or 6.3), resp. 6.5 holds if H_{M}^{i+1}(X, Q(n))_{Z} (⊕N^{n−1}), resp. CH^{.}(X)_{0}⊗Q is replaced by a suitable Q-subspace.

(6.7) For quite a few interesting L-series one needs a refined version of the above conjectures, when L(h^{i}(X), s) is replaced by L(M, s) for a suitable ‘submotive’ M ⊂ h^{i}(X). This is indeed possible for Chow motives, which include, among others, motives of Dirichlet characters. Fix a number field E – the field of coefficients. Let V_{k} be the category of smooth projective varieties over a field k. The category MC(k, E) of Chow motives over k with coefficients in E has as objects triples

M = (X, p, m),

where X ∈ Ob V_{k}, p ∈ CH^{dim X}(X × X) ⊗ E a projector (p^{2} = p) and m ∈ Z. For N = (Y, q, n) one has

Hom(M, N) = q CH^{dim Y +m−n}(X × Y) ⊗ E p,

with composition of f : M_{1} ⟶ M_{2} and g : M_{2} ⟶ M_{3} defined by the intersection product: g ∘ f = p_{13∗}(p_{12}^{∗}f · p_{23}^{∗}g), where p_{ij} : X_{1} × X_{2} × X_{3} are the projections. There is a natural covariant functor V_{k} ⟶ MC(k, E), sending X to the triple (X, id, 0) and f : X ⟶ Y to the graph of f. This definition is due to U. Jannsen [42]; a more traditional construction works in two steps, first adjoining images of projectors and then adding Tate twists. The functors X ↦ H_{M}^{.}(X, Q(∗)) extend to MC(k, E): we set

H_{M}^{i}((X, p, m), Q(j)) = p^{∗}H_{M}^{i+2m}(X, Q(j + m)) ⊗ E

and, for f = q ∘ g ∘ p ∈ Hom((X, p, m), (Y, q, n)),

f^{∗} : H_{M}^{i}((Y, q, n), Q(j)) ⟶ H_{M}^{i}((X, p, m), Q(j))

by f^{∗}(α) = p^{∗}(π_{X})_{∗}(f ∪ π_{Y}^{∗} (q^{∗}α)), where

(π_{X})_{∗} : H_{M}^{i+2 dim Y} ((X × Y, id, m − n), Q(j + dim Y)) ⟶ H_{M}^{i}((X, id, m), Q(j))

is the Gysin map.

Similarly, for k = Q, one can extend to MC(Q, E) functors X ↦ H_{MH_{R}^{+}}^{.}(X_{/R}, ∗) and the regulator

r_{H} : H_{M}^{i}(M, Q(j)) ⟶ H_{MH_{R}^{+}}^{i}(M_{/R}, j).

The L-series L(M, s) has values in E ⊗ C and the above regulator is expected to determine its special value modulo E^{∗}. See [1], [24],[42] for more details. Beilinson conjectures [3, 8.5.1] that H_{M}^{i}(−, Q(j)) and the regulator in fact extend to the category of motives modulo homological equivalence.

7. Some computations. (7.1) In this section we present some explicit formulas for Beilinson’s regulator. First, we slightly modify the target H_{MH_{R}^{+}}^{i} and use instead a weaker cohomology theory, known as Deligne cohomology (see [31], [35]). Let us forget the weight filtration in the complex 2.4.1, which represents R Hom(R(0), H) in MH_{R}. The functor which associates to H ∈ Ob MH_{R} the complex

H ⊕ F^{0}H_{C} ⟶^{i_{W} −i_{F}} H_{C}  (7.1.1)

extends to a functor RΓ^{w}(MH_{R}, −) : D^{b}(MH_{R}) ⟶ D^{b}(R − mod). For a separated scheme X of finite type over C, its Deligne cohomology is defined as

H_{D}^{i}(X_{/C}, R(j)) = H^{i}(RΓ^{w}(MH_{R}, \underline{RΓ}(X, 0)(j))),

where \underline{RΓ}(X, 0) ∈ Ob D^{b}(MH_{R}) is Beilinson’s complex from 3.4. Almost by definition, Deligne cohomology sits in an exact sequence

⋯ ⟶ H^{i}(X(C), R(n)) ⟶ H_{dR}^{i}(X_{/C})/F^{n} ⟶ H_{D}^{i+1}(X_{/C}, R(n)) ⟶

⟶ H^{i+1}(X(C), R(n)) ⟶ H_{dR}^{i+1}(X_{/C})/F^{n} ⟶ ⋯ (7.1.2)

In particular, for n > i + 1, H_{D}^{i+1}(X_{/C}, R(n)) ⟶^{∼} H^{i}(X(C), C/R(n)). In a similar fashion, for H ∈ Ob MH_{R}^{+}, one replaces 2.5.1 by

H^{+} ⊕ F^{0}H_{dR}⟶H_{dR}

to get RΓ^{w}(MH_{R}^{+}, −) : D^{b}(MH_{R}^{+}) ⟶ D^{b}(R − mod) and Deligne cohomology

H_{D}^{i}(X_{/R}, R(j)) = H_{D}^{i}(X_{/C}, R(j))^{ϕ_{∞}⊗c=1}

for X_{/R} separated of finite type. There is a canonical map H_{MH_{R}^{+}}^{i}(X_{/R}, j) ⟶ H_{D}^{i}(X_{/R}, R(j)). It is an isomorphism for i ≤ j, or even for i ≤ 2j if X is proper. The composition with the regulator r_{H} defines a regulator with values in Deligne cohomology (for any quasi-projective variety X over Q)

r_{D} : H_{M}^{i}(X, Q(j)) ⟶ H_{D}^{i}(X_{/R}, R(j))

(7.2) For smooth varieties, Deligne cohomology can be computed as hypercohomology of quite explicit complexes of sheaves. Suppose first that X_{/C} is proper and smooth. The complex

R(n)_{D} = [R(n) ⟶ O_{X} ⟶ ⋯ ⟶ Ω_{X}^{n−1}]  (7.2.1)

in degrees 0 to n is part of an exact triangle

Ω_{X}^{<n}[−1] ⟶ R(n)_{D} ⟶ R(n)  (7.2.2)

By GAGA and the degeneration of the Hodge spectral sequence we have

H^{i}(X(C), Ω_{X}^{<n}) ⟶^{∼} H^{i}(X_{Zar}, Ω_{X/C}^{<n}) = H_{dR}^{i}(X_{/C})/F^{n}

Comparing 7.1.2 and the cohomology sequence of 7.2.2, we see that

H_{D}^{i}(X_{/C}, R(n)) ⟶^{∼} H^{i}(X(C), R(n)_{D})

If X_{/C} is smooth and separated, then there exists an open immersion j : X ↪ \overline{X} of X into a proper smooth variety \overline{X} such that the complement D = \overline{X} − X is a divisor with normal crossings. There are natural maps

Rj_{∗}R(n) ⟶ Rj_{∗}Ω_{X}^{.}, Ω≥n ; \overline{X}(log D) ⟶ Rj_{∗}Ω_{X}^{.}

and using their difference we define

R(n)_{D} = Cone(Ω≥n ; \overline{X}(log D) ⊕ Rj_{∗}R(n) ⟶ Rj_{∗}Ω_{X}^{.})[−1]

Again, the degeneration of the logarithmic Hodge spectral sequence

H^{i}(\overline{X}(C), Ω≥n ; \overline{X}(log D)) ⟶^{∼} F^{n}H_{dR}^{i}(X_{/C})

and 7.1.2 imply that

H_{D}^{i}(X_{/C}, R(n)) ⟶^{∼} H^{i}(\overline{X}(C), R(n)_{D})

(7.3) A version of the complex R(n)_{D} more suitable for calculations is given by

{ \atop R(n)}_{D} = Cone(Ω≥n ; \overline{X}(log D) ⟶ j_{∗}A_{X}^{.} ⊗ R(n − 1))[−1],

where A_{X}^{.} is the de Rham complex of smooth real valued differential forms on X(C). There is a canonical quasi-isomorphism between R(n)_{D} and { \atop R(n)}_{D} induced by quasi-isomorphisms

Rj_{∗}C ⟶ Rj_{∗}Ω_{X}^{.} ⟶ j_{∗}A_{X}^{.} ⊗ C

and the projection π_{n−1} : C ⟶ R(n − 1) along R(n). This gives an explicit description of H_{D}^{n}(X_{/C}, R(n)) ⟶^{∼} H^{n}(\overline{X}(C), { \atop R(n)}_{D}) as the R-vector space

\frac{\{φ ∈ H^{0}(X(C), A^{n−1} ⊗ R(n − 1)) | dφ = π_{n−1}(ω), ω ∈ H^{0}(\overline{X}(C), Ωn ; \overline{X}(log D))\}}{dH^{0}(X(C), A^{n−2} ⊗ R(n − 1))}

(7.3.1)

The cup product of the classes of φ_{m}, φ_{n} in H_{D}^{m}(X_{/C}, R(m)) resp. H_{D}^{n}(X_{/C}, R(n)) is represented by

φ_{m} ∪ φ_{n} = φ_{m} ∧ π_{n}ω_{n} + (−1)^{m}π_{m}ω_{m} ∧ φ_{n}  (7.3.2)

See [31] for general formulas for the cup product in Deligne cohomology. (7.4) Any successful attack on the Weak Conjecture 6.6 usually proceeds in three steps:

(1) Construction of elements in H_{M}^{i}(X, Q(j)).

(2) Calculation of their image under the regulator map.

(3) Comparison of the result with the value of the L-function. Let us give a simple example. If X is a smooth quasi-projective variety over Q, then the regulator

r_{D} : O(X)^{∗} ⊗ Q = H_{M}^{1}(X, Q(1)) ⟶ H_{D}^{1}(X_{/C}, R(1))  (7.4.1)

maps f ∈ O(X)^{∗} into φ = log |f| (with corresponding ω = d log(f)). For f_{1}, …, f_{n} ∈ H_{M}^{1}(X, Q(1)), denote by {f_{1}, …, f_{n}} their cup product in H_{M}^{n}(X, Q(n)). Both regulators r_{H}, r_{D} are multiplicative, hence

r_{D}({f_{1}, …, f_{n}}) = r_{D}(f_{1}) ∪ ⋯ ∪ r_{D}(f_{n}) ∈ H_{D}^{n}(X_{/C}, R(n))

In particular, if f_{1}, …, f_{n} are any rational functions on X, write U ⊆ X for the complement of their divisors. The symbol {f_{1}, …, f_{n}} lies in H_{M}^{n}(U, Q(n)) and the problem is how to extend it to X. It is possible, sometimes, to exploit an inherent symmetry of the situation in question and construct a natural projector π_{M} from H_{M}^{n}(U, Q(n)) to H_{M}^{n}(X, Q(n)), completing thus Step 1. As a next step, one copies the construction of π_{M} to get a similar projector π_{D} in Deligne cohomology. The regulator is then equal to

r_{D}(π_{M}({f_{1}, …, f_{n}})) = π_{D}(r_{D}(f_{1}) ∪ ⋯ ∪ r_{D}(f_{n}))

Consider the simplest case of H_{M}^{2}(X, Q(2)). Suppose that X is a smooth scheme over a field k. According to [68, Th.4], there is a spectral sequence

E_{1}^{pq} = ∐_{x∈X^{(p)}}(K_{−p−q}(k(x)) ⊗ Q)^{(j−p)} ⟹ (K_{−p−q}(X) ⊗ Q)^{(j)},

where X^{(p)} denotes the set of points of codimension p on X. For a field F and i = 1, 2, (K_{i}(F) ⊗ Q)^{(j)} vanishes unless j = i, so the spectral sequence reduces to

H_{M}^{2}(X, Q(2)) = Ker(K_{2}(k(X)) ⊗ Q⟶^{T} ∐_{x∈X^{(1)}} k(x)^{∗} ⊗ Q)

Here the value of T ({f, g}) at x ∈ X^{(1)} is equal to the tame symbol

(−1)^{ord_{x}(f)·ord_{x}(g)} · (\frac{f^{ord_{x}(g)}}{g^{ord_{x}(f)}} ) (x).

The following construction is due to Bloch. Suppose that X is a smooth projective curve over a number field k and f, g ∈ k(X)^{∗} two rational functions on X. Write D for the union of supports of the divisors of f and g and k_{D} for the splitting field of D. We assume that, for a fixed integer N, a difference of any two geometric points P, Q ∈ D(k_{D}) in the Jacobian of X is torsion of order dividing N. Fix O ∈ D(k_{D}). For any P ∈ D(k_{D}) − {O} there is a rational function f_{P} ∈ k_{D}(X)^{∗} with divisor N(P) − N(O). Write c_{P} ∈ k_{D}^{∗} for the value of the tame symbol T ({f, g}) at P ∈ D(k_{D}). Then

π(\{f, g\}) = \{f, g\} + ∑_{O≠P ∈D(k_{D})} \{f_{P} , c_{P} \} ⊗ \frac{1}{N}

lies in K_{2}(k_{D}(X)) ⊗ Q, is G(k_{D}/k)-invariant, does not depend on the choice of f_{P} (as K_{2} is torsion for number fields) and lies in the kernel of the tame symbol T (this is clear outside of O; vanishing of T at O then follows from Weil’s reciprocity law), hence represents an element of

H_{M}^{2}(X ×_{k} k_{D}, Q(2))^{G(k_{D}/k)} ⟶^{∼} H_{M}^{2}(X, Q(2))

Interestingly, there exist examples of families of elliptic curves E_{/Q} with elements in H_{M}^{2}(E, Q(2)) coming from functions with divisors supported at non-torsion points. (7.5) Fix an embedding k ↪ C. Let us compute the regulator r_{D} on π({f, g}). Set U = X − D. According to 7.3.1, we have

H_{D}^{2}(U_{/C}, R(2)) ⟶^{∼} H^{1}(U(C), R(1)), H_{D}^{2}(X_{/C}, R(2)) ⟶^{∼} H^{1}(X(C), R(1)).

By 7.4.1, r_{D}(f) ∈ H_{D}^{1}(U_{/C}, R(1)) is represented by φ_{f} = log |f| with df = π_{0}(ω_{f}) for ω_{f} = ∂f/f. According to 7.3.2, the cup product r_{D}({f, g}) ∈ H^{1}(U(C), R(2)) is represented by

ψ(f, g) = φ_{f}(π_{1}ω_{g}) − (π_{1}ω_{f})φ_{g} = log |f|d arg(g) − log |g|d arg(f)

The pairing

⟨ , ⟩ : ψ, ω ↦ \frac{1}{2πi} ∫_{U(C)} ψ ∧ ω

defines a map H^{1}(U(C), R(1)) ⟶ H^{0}(X(C), Ω_{X}^{1})^{∗}, whose restriction to H^{1}(X(C), R(1)) is an isomorphism (of R-vector spaces). An elementary calculation shows that, modulo an exact form,

ψ(f, g) ∧ ω ∼ log |g|d log(\overline{g}) ∧ ω,

which vanishes if g is a constant function. Consequently, for ω ∈ H^{0}(X(C), Ω_{X}^{1}),

⟨r_{D}(π(\{f, g\})), ω⟩ = \frac{1}{2πi} ∫_{X(C)} log |g|d log(\overline{g}) ∧ ω,  (7.5.1)

completing Step 2 of the program formulated in 7.4. For an elliptic curve, this integral can be computed explicitly in terms of certain Kronecker (– Eisenstein – Lerch) series. More precisely, suppose that X is an elliptic curve defined over R with complex points X(C) = C/Γ for Γ = Z + Zτ (Im(τ) > 0) and f, g two rational functions on X with divisors supported at torsion points. Writing dz for the canonical differential on C/Γ and

(z, γ) = exp (\frac{2πi}{τ − \overline{τ}} (z\overline{γ} − \overline{z}γ)) , z ∈ C/Γ, γ ∈ Γ = Z + Zτ

for the duality between C/Γ and Γ, the regulator is given by the formula

⟨r_{D}(π(\{f, g\})), dz⟩ = −\frac{1}{2}\sqrt{\frac{τ − \overline{τ}}{2πi}} ∑_{x,y∈X(C)} ord_{x}(f) ord_{y}(g) ∑_{0≠γ∈Γ} \frac{\overline{γ}}{|γ|^{4}} (y − x, γ)

(7.5.2)

The reader may wish to consult [1],[30],[57] for the details of the computation. If the curve has complex multiplication, then its L-function at s = 2 is a series of the same type and it is relatively easy to compare the regulator in 7.5.2 with L(X, 2). Historically, this computation was performed first by Bloch [6],[7], using another definition of the regulator.

If X_{/Q} is an elliptic curve without complex multiplication but with a nontrivial torsion over Q, one can repeat the construction of 7.4 and get an element of H_{M}^{2}(X, Q(2)), but not necessarily of H^{2}(X, Q(2))_{Z}. Amusingly, the obstruction to integrality is related to the third Bernoulli polynomial in [11],[60]. In fact, it was only after the calculations of [11] that the cohomology of the integral model was incorporated into the conjectures. (7.6) Note that the above construction works in a family: if S is the (open) modular curve over C classifying elliptic curves with a full level N structure (for a fixed N ≥ 3), consider the universal elliptic curve p : X ⟶ S and choose two sections u, v which generate the subgroup of N-torsion of X. There exist rational functions f, g on X with divisors equal to N(u) − N(0) resp. N(v) − N(0) and they can be normalized in such a way that f_{|v} = 1, g_{|u} = 1. The symbol {f, g} then represents an element of H_{M}^{2}(X, Q(2)). Its restriction to each fiber X_{s} has the same field of definition as s ∈ S(C).

In general, if p : X ⟶ S is a proper smooth map between two varieties over C and a ∈ H_{M}^{i}(X, Q(n)) a global element with i < n, then the regulator of its restriction to the fibre X_{s}

r_{D}(a_{s}) ∈ H_{D}^{i}(X_{s}, R(n))

is “locally constant” as a function of s. The intuition is quite clear: for i < n, the groups H_{D}^{i}(X_{s}, R(n)) ⟶^{∼} H^{i−1}(X_{s}(C), C/R(n)) form a locally constant sheaf on S(C). The formal argument goes as as follows: r_{D}(a) ∈ H^{i}(\overline{X}(C), R(n)_{D}) defines a global section of the sheaf R^{i}p_{∗}j^{∗}R(n)_{D}, where R(n)_{D} is the complex defined in 7.3, living on a suitable compactification j : X ↪ \overline{X}. The complex j^{∗}R(n)_{D} is quasi isomorphic to

F_{X} = [R(n) ⟶ O_{X} ⟶ ⋯ Ω_{X}^{n−1}]

Denote by F_{X/S} the analogous complex in which we replace differentials on X by relative differentials. We may assume that S is a curve. Then the exact triangle

p^{∗}Ω_{S}^{1} ⊗ Ω_{X/S}^{.}/F^{n−1}[−2] ⟶ F_{X} ⟶ F_{X/S}

induces an exact sequence

R^{i}p_{∗}F_{X} ⟶ R^{i}p_{∗}F_{X/S}⟶^{D}H_{dR}^{i−1}(X/S)/F^{n−1} ⊗_{O_{S}} Ω_{S}^{1}.

Here the map D comes from an exact sequence

H_{B}^{i−1}(X/S, R(n)) ⟶ H_{dR}^{i−1}(X/S)/F^{n} ⟶ R^{i}p_{∗}F_{X/S} ⟶ 0

and the Gauss-Manin connection

H_{dR}^{i−1}(X/S)/F^{n}⟶^{∇}H_{dR}^{i−1}(X/S)/F^{n−1},

with respect to which the Betti cohomology of the fibres is horizontal. Finally, r_{D}(a_{s}) ∈ H^{0}(S, R^{i}p_{∗}F_{X/S}) must be a section of Ker(D), which is (assuming i < n), nothing else than the locally constant sheaf of H^{i−1}(X_{s}(C), C/R(n)).

This digression explains why, for example, there is no “uniform” construction of elements in H_{M}^{2}(E, Q(n)), for elliptic curves E and n > 2 (corresponding to the values L(E, n) = L(h^{1}(E), n), conjecturally). Such a uniform construction works, however, for H_{M}^{n+1}(Sym^{n}(h^{1}(E)), n+1) (Eisenstein symbols, constructed by Beilinson [2]; see also [24],[29]). For curves with complex multiplication, Deninger [24] defines a map

H_{M}^{2m}(Sym^{2m−1}(h^{1}(E)), 2m) ⟶ H_{M}^{2}(h^{1}(E), m + 1)

into cohomology responsible for the value L(E, m + 1).

8. Evidence for the conjectures. In this section we make a survey of the progress towards the conjectures in various special cases; results on Deligne’s conjecture in the critical case will not be mentioned.

(1) Zeta functions of number fields. For ζ_{F} (s), the strong conjecture has been proved by Borel [14],[15] at all integers (with s = 0, 1 being classical), using his own definition of a regulator

K_{2n−1}(C) ⟶ R(n − 1)

It is proved in [1] (cf. [55]) that Borel’s and Beilinson’s regulator coincide.

In recent years, there has been a resurgence of activity centered around classical polylogarithm functions, values of ζ_{F} (s) and their relation to Borel’s regulator. See the articles [5],[10],[36] in these proceedings.

(2) Dirichlet L-series. For every Dirichlet character χ there is a Chow motive M_{χ} over Q such that L(s, χ) = L(M_{χ}, s). Beilinson [1] proved the weak conjecture at all integers; the results of Borel (1) then imply the strong conjecture.

Remark. These are the only cases when the strong conjecture has been proved, except for (9) below, dealing with the central point.

(3) Eliptic curves with complex multiplication. For E_{/Q} an elliptic curve with complex multiplication, the weak conjecture for L(h^{1}(E), s) has been proved at s = 2 by Bloch [6], [7], Beilinson [1]; for elliptic curves of Shimura type, proved at all integers s ≥ 2 by Deninger [24].

(4) Motives of Hecke characters of an imaginary quadratic field. The weak conjecture is proved by Deninger [25], where he also reproves (2).

(5) Modular forms. If f is a newform of weight k + 2 on some congruence subgroup of SL_{2}(Z), then there is a Grothendieck motive M(f) associated to f ([62]) such that L(f, s) = L(M(f), s). The motive M(f) is constructed from a Chow motive M corresponding to all cusp forms of given type using a projector Π_{f} in Hecke algebra. For every integer n ≥ k + 2, Beilinson [2] for k = 0 and Scholl [65] in general construct a subspace P_{n} ⊆ H_{M}^{k+2}(M, Q(n)) such that det(r_{D}(Π_{f}(P_{n}))) gives the value of L(f, n). However, for k > 0, it is not known if P_{n} lies in H_{M}^{k+2}(M, Q(n))_{Z}. See also [29],[59] for more details.

(6) Shimura curves over Q. For any Shimura curve X coming from an automorphic form on an indefinite quaternion algebra B over Q, Ramakrishnan [52] proves the weak conjecture for L(h^{1}(X), s) at all integers s ≥ 2. Ramakrishnan uses Jacquet-Langlands correspondence between automorphic forms on B and GL_{2}, together with Faltings isogeny theorem to deduce this result from the corresponding statement for modular forms on GL_{2}, proved by Beilinson (see (5) above).

(7) Product of two modular curves. For two modular curves C_{1}, C_{2} defined over Q, Beilinson [1] proves the weak conjecture for L(h^{2}(C_{1} × C_{2}), s) at s = 2, but he makes an incorrect argument for integrality of elements in H_{M}^{3}(C_{1} × C_{2}, Q(2)) he constructs. A revised version of [54] is supposed to fill this gap.

(8) Hilbert-Blumenthal surfaces. Let X be a Hilbert-Blumenthal surface over a real quadratic field F for some congruence subgroup of GL_{2}(O_{F}). There is a smooth toroidal compactification \overline{X} of X. Ramakrishnan [54] proves the weak conjecture for an incomplete L-series L_{S}(h^{2}(\overline{X}), s) (with bad Euler factors removed) at s = 2. The integrality of relevant elements in the motivic cohomology is not known, however.

(9) Elliptic curves at the central point. If E is an elliptic curve over Q which is modular (i.e. admits a nontrivial map X_{0}(N) ⟶ E) and the order of vanishing of L(E, s) at s = 1 is equal to 0 or 1, then the conjecture of Birch and Swinnerton-Dyer is true for E, up to a controlled rational factor. This follows from the work of Kolyvagin [46], combined with [38] and nonvanishing theorems about L-functions of modular forms (see [51]).

(10) Numerical evidence. In [11], Bloch and Grayson give results of computations of the regulator on H_{M}^{2}(E, Q(2))_{Z} for certain elliptic curves without complex multiplication. The result compares favorably with the value L(E, 2), as expected. Mestre and Schappacher [50] report on similar computations for the symmetric square of an elliptic curve without complex multiplication.

9. Mixed motives revisited. (9.1) This section contains a reformulation of Beilinson’s conjectures in terms of mixed motives, due to Scholl [63]. We have to assume that the formalism of motivic cohomology over Q, described in Sec. 3–4, makes sense and that the categories MM_{Z}, MM_{Q} exist and behave as expected. For the reader’s convenience, we summarize a few relevant formulas. If X is a smooth projective variety over Q and M = h^{i}(X)(n), then we should have

Ext_{MM_{Z}}^{0}(Q(0), M) = \{ \begin{matrix}0 & if i ≠ 2n \\ CH^{n}(X)/CH^{n}(X)_{0} ⊗ Q & if i = 2n \end{matrix}
Ext_{MM_{Z}}^{1}(Q(0), M) = \{ \begin{matrix}H_{M}^{i+1}(X, Q(n))_{Z} & if i + 1 ≠ 2n \\ CH^{n}(X)_{0} ⊗ Q & if i + 1 = 2n \end{matrix}

The Ext group in MM_{Q} should be given by the same formula with H_{M}^{i+1}(X, Q(n))_{Z} replaced by H_{M}^{i+1}(X, Q(n)). A special case of the above formulas (for M = h^{2}(P^{1})(2)) is

Ext_{MM_{Q}}^{1}(Q(0), Q(1)) = Q^{∗} ⊗ Q
Ext_{MM_{Z}}^{1}(Q(0), Q(1)) = Z^{∗} ⊗ Q = 0

Finally, for w = i−2n < −1, the regulator map should be none else than the Hodge realization

Ext_{MM_{Z}}^{1}(Q(0), h^{i}(X)(n)) ⟶ Ext_{MH_{R}^{+}}^{1}(R(0), H^{i}(X(C), R(n)))

(9.2) Deligne’s period map

α_{E} : E_{B}^{+} ⊗ R ⟶ (E_{dR}/F^{0}) ⊗ R

makes sense for any mixed motive E ∈ Ob MM_{Q}. Scholl [63] calls E critical if α_{E} is an isomorphism. For a critical mixed motive E, Deligne’s period c^{+}(E) ∈ R^{∗}/Q^{∗} is defined as the determinant of α_{E} with respect to the Q-structures E_{B}^{+} and E_{dR}/F^{0}. One defines the L-function L(E, s) of a mixed motive in the same way as L(M, s) is defined in 1.4 (assuming the independence of ℓ of the local L-factors). If E is a mixed motive over Z, then

L(E, s) = ∏_{j} L(Gr_{j}^{W} (E), s)

(9.3) Scholl [63] makes the following three conjectures: (A) If E ∈ Ob MM_{Q} is critical, then L(E, 0) · c^{+}(E)^{−1} ∈ Q. (B) For any mixed motive E over Z,

ord_{s=0} L(E, s) = dim Ext_{MM_{Z}}^{1}(Q(0), E^{∨}(1)) − dim Ext_{MM_{Z}}^{0}(Q(0), E^{∨}(1)).

(C) If E is a mixed motive over Z satisfying

Ext_{MM_{Z}}^{i}(Q(0), E) = Ext_{MM_{Z}}^{1}(Q(0), E^{∨}(1)) for i = 0, 1,  (9.3.1)

then E is critical. (9.4) There is a natural construction which transforms any mixed motive M over Z into a new mixed motive E over Z satisfying 9.3.1, by taking successive universal extensions and killing inconvenient subgroups. It proceeds in four steps

0 ⟶ Hom(Q(0), M) ⊗ Q(0) ⟶ M ⟶ M_{1} ⟶ 0
0 ⟶ M_{1} ⟶ M_{2} ⟶ Ext_{MM_{Z}}^{1}(Q(0), M_{1}) ⊗ Q(0) ⟶ 0
0 ⟶ M_{3} ⟶ M_{2} ⟶ Hom(Q(0), M_{2}^{∨}(1))^{∨} ⊗ Q(1) ⟶ 0
0 ⟶ Ext_{MM_{Z}}^{1}(Q(0), M_{3}^{∨}(1))^{∨} ⊗ Q(1) ⟶ E ⟶ M_{3} ⟶ 0

which do not require special comment. In verifying that E indeed satisfies 9.3.1 one has to remember that Ext^{1}(Q(0), Q(1)) is supposed to vanish in MM_{Z}. The L-function of E is equal to

L(E, s) = L(M, s)ζ(s)^{a}ζ(s + 1)^{b},

where

a = dim Ext_{MM_{Z}}^{1}(Q(0), M) − dim Ext_{MM_{Z}}^{0}(Q(0), M)
b = dim Ext_{MM_{Z}}^{1}(Q(0), M^{∨}(1)) − dim Ext_{MM_{Z}}^{0}(Q(0), M^{∨}(1))

The conjectures (A)–(C) predict that this new motive E is critical, L(E, 0) ≠ 0 and

L^{∗}(M, 0) ∈ L(E, 0) · Q^{∗} = c^{+}(E) · Q^{∗}.  (9.4.1)

(9.5) We shall now consider the case of a pure motive M = h^{i}(X)(n) of weight w = i − 2n. Suppose first that w ≤ −2 and that Hom(Q(0), M^{∨}(1)) = 0. For weight reasons and by semisimplicity of M_{Q},

Ext^{0}(Q(0), M) = Ext_{MM_{Z}}^{1}(Q(0), M^{∨}(1)) = 0

This means that E is simply the universal extension

0 ⟶ M ⟶ E ⟶ Ext_{MM_{Z}}^{1}(Q(0), M) ⊗ Q(0) ⟶ 0

as α_{Q(0)} = 0 and Ker(α_{M}) = 0, snake lemma implies that E is critical if and only if the connecting homomorphism

Ext_{MM_{Z}}^{1}(Q(0), M)⟶^{∂_{M}} Coker(α_{M})

becomes an isomorphism after tensoring with R. This map, however, is nothing else than the regulator and the canonical Q-structure on its target is D(M), so 9.4.1 is equivalent to the first half of the conjecture 6.1.2

det(∂_{M}) = L(h^{i}(X), n)Q^{∗} ∈ R^{∗}/Q^{∗}

Continuing with the same notation, set N = M^{∨}(1). This is a pure motive of weight −2 − w ≥ 0 with Hom(Q(0), N) = 0 and its ‘universal extension’ is

0 ⟶ Ext_{MM_{Z}}^{1}(Q(0), M)^{∨} ⊗ Q(1) ⟶ E^{∨}(1) ⟶ N ⟶ 0

The period map for N is given by the connecting homomorphism

∂_{N} : Ker(α_{N}) ⟶ Ext_{MM_{Z}}^{1}(Q(0), M)^{∨} ⊗ R.

Using 2.2.1 and the fact that ∂_{M}, ∂_{N} are adjoint maps, we see that 9.4.1 for N becomes

det(∂_{N}^{∨}) = L^{∗}(N, 0)Q^{∗} = L^{∗}(h^{i}(X), i + 1 − n)Q^{∗} ∈ R^{∗}/Q^{∗},

which is the second half of the conjecture 6.1.2. If M is pure of weight −2 with no other restrictions, then E has to be constructed in two steps. Its period map accounts for Beilinson’s ‘thickened regulator’ r_{H} ⊕ r_{B} in 6.2, but is more canonical; the reader is invited to verify that (A)–(C) predict that the following sequence

0 ⟶ Ext_{MM_{Z}}^{1}(Q(0), M) ⊗ R ⟶ Coker(α_{M}) ⟶ Hom(Q(0), M^{∨}(1))^{∨} ⊗ R ⟶ 0

is exact. The natural Q-structures, say, E(M), D(M), H(M) on determinants of all three terms define the period

c^{+}(E) = E(M) · D(M)^{−1} · H(M) ∈ R^{∗}/Q^{∗},

equal to L^{∗}(M, 0) by 9.4.1. This is equivalent to the conjecture 6.3. The case of w = −1, related to height pairings, is discussed in detail in [64]. Scholl [63] shows that, modulo some other (reasonable) hypotheses about the structure of MM_{Q}, his conjectures (A)–(C) are in fact equivalent to Beilinson’s conjectures 6.1, 6.3, 6.5. Finally, let us mention that Fontaine and Perrin-Riou [32],[33] introduced a beautiful six-term exact sequence, which is effectively a translation of (A)–(C), but works for general mixed motives over Q.

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