Polylogarithms and the Bloch-Kato conjecture

on special values of L-functions

by Guido Kings Arbeitstagung 14.6.2001

In this talk we give an overview of the state of the Bloch-Kato conjecture on special values of L-functions and explain the role of the polylogarithm sheaf in the proofs.

1 What is the Bloch-Kato conjecture?

Let F be a number field, O_{F} its ring of integers. The Dedekind zeta function is defined by the Euler product

ζ_{F} (s) := ∏_{𝔭} \frac{1}{1 − \frac{1}{N𝔭^{s}}}

for Re(s) > 1. This function has a meromorphic continuation to ℂ and satisfies a functional equation. In the case, where F = ℚ we get the Riemann zeta function. An theorem due to Euler gives now:

ζ_{ℚ}(1 − n) = −\frac{B_{n}}{n}

for n ≥ 1, where B_{n} is the n-th Bernoulli number. We ask:

What is the arithmetic meaning of these numbers?.

A prototype answer is provided by the class number formula:

ζ_{F} (0)^{∗} = − vol (H/r_{∞}(O_{F}^{∗})) \frac{#Cl(O_{F})}{#μ(F)}.

Here ζ_{F} (0)^{∗} denotes the leading term of the Taylor series at 0, Cl(O_{F}) is the ideal class group and μ(F) is the group of roots of unity in F. Moreover r_{∞} is the Dirichlet regulator map

r_{∞} : O_{F}^{∗} ⟶ ( ⨁_{σ:F →ℂ} ℝ)^{+}
u ⟼ (log |σ(u)|)_{σ}.

Here + means invariants under complex conjugation.

The Bloch-Kato conjecture generalizes this formula to the L-values at all integers of the L-functions of all varieties over number fields.

Consider the motivic cohomology of F, which we take for definiteness to be the 2n−1 Quillen K-group and its n-th Adams eigenspace, thus H_{M}^{1}(F, ℚ(n)) := (K_{2n−1}(O_{F}) ⊗ ℚ)^{n}. One has the Beilinson regulator

r_{∞} : H_{M}^{1}(F, ℚ(n)) → ( ⨁_{σ:F →ℂ}(2πi)^{n−1}ℝ)^{+} =: H_{B}^{+}

and the Soulé regulator

r_{p} : H_{M}^{1}(F, ℚ(n)) → H^{1}(G_{F,p}, ℤ_{p}(n)) ⊗ ℚ_{p}

into Galois cohomology. The group G_{F,p} is the Galois group of the maximal unramified outside of p extension of F. One denotes ℤ_{p}(1) := \underleftarrow{lim}_{r}μ_{p^{r}} and ℤ_{p}(n) := ℤ_{p}(1)^{⊗n}. Conjecture 1.0.1. (Bloch-Kato conjecture for F) Let n > 1 and choose Ω ⊂ H_{M}^{1}(F, ℚ(n)) a ℤ-lattice, then

BK(F,1-n) ζ_{F} (1 − n)^{∗} = ± vol (H_{B}^{+}/r_{∞}(Ω)) ∏_{p} \frac{#H^{2}(G_{F,p}, ℤ_{p}(n))}{# (H^{1}(G_{F,p}, ℤ_{p}(n))/r_{p}(Ω ⊗ ℤ_{p}))}.

Here ζ_{F} (1 − n)^{∗} denotes again the leading term in the Taylor series expansion. Remark:

1. The conjecture is independent of Ω

2. There exists a version for all n ∈ ℤ and for Artin motives

3. In the case described above the conjecture is already due to Lichtenbaum 4. A similar formula is conjectured for the L-functions of all varieties

2 What is known?

The following is a nearly comprehensive list of known results: Theorem 2.0.2. (Bloch-Kato ’89) BK(ℚ, n) is true for all n ∈ ℤ. Theorem 2.0.3. (Wiles ’90) BK(F, 1−n) is true for totally real fields F and n ≥ 0 even. Theorem 2.0.4. (Huber-Kings ’00, Burns-Greither ’00 for n ≤ 0) BK(F, n) is true for all abelian field extensions F/ℚ (more generally for abelian Artin motives) and all n ∈ ℤ.

For higher dimensional varieties one has:

Theorem 2.0.5. (Kings ’99) Let E/K be a CM elliptic curve of conductor 𝔣 over an imaginary quadratic field. Fix k ≥ 0 and a prime p ∤ 6𝔣\overline{𝔣} such that H^{2}(G_{K,p𝔣}, T_{p}E(k + 1) is finite. Then there is a sublattice Ω ⊂ H_{M}^{2}(E, ℚ(k + 2)) such that the p valuation of

\frac{L(E/K, −k)^{∗}}{vol (H^{1}(E(ℂ), (2πi)^{k+1}ℝ)^{+}/r_{∞}(Ω))}

and of

\frac{#H^{2}(G_{K,p𝔣}, T_{p}E(k + 1))}{# (H^{1}(G_{K,p𝔣}, T_{p}E(k + 1))/r_{p}(Ω ⊗ ℤ_{p}))}

coincide.

The proofs have two essential steps:

1. Construct Ω ⊂ H_{M}^{1}(F, ℚ(n)) and compute r_{∞}(Ω) and r_{p}(Ω). Moreover, relate r_{∞}(Ω) to L-values. If this is possible, choose an Ω such that the value of the zeta function is exactly ζ_{F} (1 − n)^{∗} = ± vol (H_{B}^{+}/r_{∞}(Ω)). Then it remains to show:

#H^{2}(G_{F,p}, ℤ_{p}(n)) = # (H^{1}(G_{F,p}, ℤ_{p}(n))/r_{p}(Ω ⊗ ℤ_{p})) .

One should see this as an Euler characteristic type formula.

2. use Iwasawa theory to reduce to the case n = 1. More specificly consider

\underleftarrow{lim}_{r}H^{1}(G_{F (μ_{p^{r}}),p}, ℤ_{p}(n)).

This is up to isomorphism independent of n and H^{1}(G_{F,p}, ℤ_{p}(n)) is essentially a quotient of it. Thus, one can reduce to the case n = 1 provided one knows:

r_{p}(Ω ⊗ ℤ_{p}) is compatible with “twist”.

To show these two points one needs polylogarithms:

3 What is the polylogarithm sheaf?

We sketch this beautiful idea of Beilinson very roughly: Let X = 𝔾_{m}(ℂ) or = E(ℂ). Fix a point x ∈ X ∖ {1}. Denote by ℚ[π_{1}(X ∖ x, 1)]^{∧} the completion of the group algebra of π_{1}(X ∖ x, 1) with respect to the augmentation ideal. Similarly we consider ℚ[π_{1}(X, 1)]^{∧}. These are Hopf algebras and we have a map

ℚ[π_{1}(X ∖ x, 1)]^{∧} → ℚ[π_{1}(X, 1)]^{∧}.

This induces a map of the Lie algebras L_{x} → H of primitive elements L_{x} resp. H in ℚ[π_{1}(X ∖ x, 1)]^{∧} resp. ℚ[π_{1}(X, 1)]^{∧}. Let K_{x} := ker(L_{x} → H) and define:

Definition 3.0.6. The polylog Lie algebra at x is

Pol_{x} := \frac{L_{x}}{[K_{x}, K_{x}]}.

Note that when x varies, these form a sheaf of Lie algebras on X ∖ x. There is also a Hodge and an etale version of Pol_{x}.

The Lie algebra Pol_{x} sits in an exact sequence

0 → K_{x}/[K_{x}, K_{x}] → Pol_{x} → H → 0.

It turns out that H = H_{1}(X, ℚ) and we define the logarithm Lie algebra to be Log_{x}(1) := K_{x}/[K_{x}, K_{x}].

Theorem 3.0.7. If x ∈ X is a torsion point, then

Log_{x}(1) ≅ ∏_{k≥0} Sym^{k} H(1).

In particular one gets for X = 𝔾_{m}(ℂ) two extension classes (note H = ℚ(1)):

Pol_{x,∞} ∈ Ext_{ℝ−Hodge−str}^{1}(ℝ(1), ℝ(k + 1))
Pol_{x,l} ∈ Ext_{Gal(\overline{K}/K)}^{1}(ℚ_{l}(1), ℚ_{l}(k + 1)).

The properties of these classes are summarized in the next theorem:

Theorem 3.0.8. 1. X = 𝔾_{m}:

(a) (Huber-Wildeshaus): Pol_{x,∞} and Pol_{x,l} are in the image of the Ktheory regulator.

(b) (Beilinson-Deligne): Pol_{x,∞} is given by Li_{k}(x) = ∑_{n≥1} \frac{x^{n}}{n^{k}} for k > 1.

(c) (Deligne): Pol_{x,l} is given by “twists” of cyclotomic units 1 − x.

2. X = E:

(a) (Beilinson-Levin): Pol_{x,∞} and Pol_{x,l} are in the image of the Ktheory regulator.

(b) (Beilinson-Levin): Pol_{x,∞} is given by Eisenstein-Kronecker series of weight k + 2.

(c) (Kings): Pol_{x,l} is given by “twists” of elliptic units.

This theorem gives the desired construction of K-theory elements, the relation to L-values and the relation to Iwasawa theory.

References

[Hu-Ki] A. Huber, G. Kings: Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, preprint 2001.

[Ki] G. Kings: The Tamagawa number conjecture for CM elliptic curves, Invent. math. 143, 571–627 (2001)