Last revision 15 May 1997 8:1

Chapter 8

The Eisenstein symbol

8.0 Introduction

In this section we describe Beilinson’s construction of the Eisenstein symbol, and explain some consequences for the K-theory of elliptic curves and of cyclotomic fields.

Bloch [Blo] was the first to investigate those elements of K_{2}(E), for an elliptic curve E, which are linear combinations of symbols {f, g} where the divisors of f and g are supported in the torsion subgroup of E. He showed that for CM elliptic curves over ℚ, such elements always generated a nontrivial subgroup of K_{2}(E) (conjecturally all of it, up to torsion) and that the regulators of these elements were always rational multiples of L′(E, 0) (and for some elements, the regulator was non-zero). This was before Beilinson had formulated his conjectures (and Bloch’s work was one of the motivations for the formulation).

In the paper [Be2] Beilinson showed that Bloch’s elements came from a certain universal construction on the universal elliptic curve over a modular curve. More generally, he showed that, for every k ≥ 1, there are certain universal canonical elements of K_{k+1} of the Kuga-Sato variety X_{n}^{k} (??), and gave an entirely explicit construction of them. Later Deninger [De1,De2] used Beilinson’s construction to verify Beilinson’s conjecture (in the weak sense) for Hecke characters of imaginary quadratic fields, therebye generalising Bloch’s work.

At the same time, Beilinson’s construction made available families of elements of K-groups of non-CM elliptic curves and their self-products. For K_{2} these elements had been already investigated numerically by Bloch and Grayson [BG]. For K_{3}(E × E) the paper of Mestre-Schappacher [MS] contains many calculations as well as indications why, in general, the Eisenstein symbol will not provide many non-zero elements (except in the case of CM).

The picture is now changing significantly, with the discovery by Beilinson and Levin [BL] of the elliptic polylogarithm. For non-CM curves E, this should be an adequate source of non-trivial elements of K_{k+1}(E^{k}), for every k ≥ 1.

Unfortunately not very much has been proved in this direction at the time of writing — see [Wi,Go,?] for specific conjectures.

The main aim of this chapter is to give a complete construction of the Eisenstein symbol, consistent with the normalisations adopted in earlier chapters. In section 1 we define the symbol in the case of an elliptic curve over an arbitrary base. Here we follow Beilinson’s original treatment [Be2] fairly closely (with some improvements from [De1] and [SS2]).

The key property of the Eisenstein symbol is its regulator. In [Be1] Beilinson shows that the regulator of the Eisenstein symbol is the associated Eisenstein series in absolute Hodge cohomology, using an an explicit analytic computation. In [SS2] an alternative approach was outlined, which is purely algebraic, involving calculating the boundary of the Eisenstein symbol at a cusp. We shall not reproduce either of these proofs here, but simply give the statement (in the normalisations used here) of the formula and show how to obtain it from the boundary formula in [SS2] (whose normalisations are unfortunately inconsistent with those used here).

To associate Eisenstein series to cusps of the modular curve, one needs the horospherical isomorphism. We give a detailed account of this in 8.3.1 and 8.3.3.

In the last sections we shall sketch how the regulator formula implies Bloch’s theorem and the generalisations by Deninger. We shall also show, by degeneration to the boundary, how it can be used to give another proof of the Gross conjecture for cyclotomic fields (Theorem ??? of [Be1]; see also [Neu, E] and [HW]). Such a degeneration has been well known to experts for a long time (cf. [??]). Harder [HMM] has given another interpretation of cyclotomic higher regulators, in terms of some mixed motives, first introduced by Anderson (unpublished). Although to go into the theory of mixed motives here would take us too far afield, we give an interpretation of Harder’s calculations in terms of the Eisenstein symbol. Analogous extensions were also considered by Deninger [De3] for CM elliptic curves and Hecke characters — the reader is referred to his papers.

8.1 Definition of the Eisenstein symbol

8.1.1 General setting

In this section we shall work with an arbitrary elliptic curve E over a connected base scheme S. We assume that we are given a finite group of sections P ⊂ E(S). Recall that we denote the free abelian group on P as ℤ[P], and ℤ[P]^{0} it the subgroup of linear combinations of total degree 0. The spaces ℚ[P], ℚ[P]^{0} are obtained by tensor product with ℚ. We fix an integer k ≥ 1, and write E^{k} for the k-fold fibre product of E over S.

The basic construction in this context is the Eisenstein symbol map

Eis_{(E,P)}^{k}: ℚ[P]^{0} → H_{M}^{k+1}(Sym^{k} h^{1}(E/S), k + 1) = H_{M}^{k+1}(E^{k}, k + 1)(sgn_{k+1})

the second identity being ??. Before defining it we recall that in section 3.2.3 we have introduced the notation

pr_{i}: E^{k} → E, (1 ≤ i ≤ k)

for the projections onto the factors of the product, and have defined

pr_{0} = − ∑_{i=1}^{k} pr_{i}: E^{k} → E.

If (x_{1}, …, x_{k}) is a point of E^{k} (with values in some S-scheme) then we shall write x_{0} = − ∑_{i=1}^{k} x_{i}.

Define:

U = U_{P} = E – P;
U_{P}^{k} = ⋂_{i=0}^{k} pr_{i}^{−1}(U) ⊂ E^{k}

Thus

U_{P}^{k} (T) = {(x_{1}, …, x_{k}) ∈ E^{k}(T) | x_{0}, x_{1}, …, x_{k} ∉ P }

and U_{P}^{k} is stable under 𝔖_{k+1} (acting on the product as in §3.2.3). The group P^{k} also acts on U_{P}^{k} by translations. Define also:

θ: E → F =^{def} E/P
θ^{k}: E^{k} → F^{k} = F ×_{S} ⋯ ×_{S} F
V^{k} = θ^{k}(U_{P}^{k}) = ⋃_{i=0}^{k} pr_{i}^{−1}(F – {e})

The covering θ^{k}: U_{P}^{k} → V^{k} is finite; if P is a group of disjoint sections, then it is a Galois étale covering, with group P^{k}.

8.1.2 The construction

The contruction is in three stages. First recall that by ?? the divisor map gives an exact sequence

0 → O^{∗}(S)_{ℚ} → O^{∗}(U_{P})_{ℚ} →^{div} ℚ[P]^{0} → 0  (8.1.2.1)

where as usual A_{ℚ} is shorthand for A ⊗_{ℤ} ℚ. Choose h = h_{P} ∈ O^{∗}(U_{P})_{ℚ} with divisor η_{P} = P − #P · (e). We need an elementary lemma.

Lemma 8.1.2.1. For every k ≥ 1, y ∈ P

∑_{\begin{matrix}x_{1},...,x_{k}∈P \\ x_{1}+···+x_{k}=y \end{matrix}} η_{P} (x_{1}) … η_{P} (x_{k}) = #P^{k−1}η_{P} (y).

Proof: write η_{P} (x) = #P δ(x) − 1. Then multiplying out, we get

∑_{x_{1}+···+x_{k}=y}(#P δ(x_{1}) − 1) … (#P δ(x_{k}) − 1)

= ∑_{r=0}^{k}(−1)^{k−r}#P^{r} ∑_{i_{1}<···<i_{r}} ∑_{x_{1}+···+x_{k}=y} δ(x_{1}) … δ(x_{k}).

For r ≠ k the innermost sum is simply #P^{k−r−1}, and for r = k it is δ(y), giving the formula easily.

Now consider the (k + 1)-fold tensor product ⊗^{k+1}O^{∗}(U)_{ℚ}. It carries an action of P^{k} by

(x_{1}, …, x_{k}): f_{0} ⊗ ⋯ ⊗ f_{k} ↦ T_{x_{0}}^{∗}f_{0} ⊗ ⋯ ⊗ T_{x_{k}}^{∗}f_{k}

as does the symmetric power Sym^{k+1} O^{∗}(U)_{ℚ}.

Lemma 8.1.2.2. There is a unique isomorphism (the unlabelled arrow) making the following diagram commute. It does not depend on the choise of function h_{P}.

O^{∗}(U)_{ℚ} ⟶^{−⊗h_{P} ⊗···⊗h_{P}}  ⊗^{k+1}O^{∗}(U)_{ℚ}
div↓  ↓^{projection}
ℚ[P]^{0}  ⟶  Sym^{k+1} O^{∗}(U)_{ℚ}^{P^{k}}/ Sym^{k+1} O^{∗}(S)_{ℚ}

Proof: The group P acts trivially on the left hand of (8.1.2.1) and has no invariants on the right-hand term, so there is a unique P -invariant splitting, and we can write O^{∗}(U)_{ℚ} = ℚ[P]^{0} ⊕ Z say, where Z = O^{∗}(S)_{ℚ}. consider a P^{k}- submodule of ⊗^{k+1}O^{∗}(U)_{ℚ} of the form ℚ[P]^{0,⊗j} ⊗ Z^{⊗(k+1−j)} (0 ≤ j ≤ k + 1). Now if j ≤ k then there is a copy of P^{j} inside P^{k} which acts on ℚ[P]^{0,⊗j} in the obvious way (tensor product of j copies of the augmentation representation of P). In that case

(ℚ[P]^{0,⊗j})^{P^{j}} ⊗ Z^{⊗(k+1−j)} = (ℚ[P]^{0,P^{k}})^{⊗j} ⊗ Z^{⊗(k+1−j)}

which vanishes if j > 0. Therefore ⊗^{k+1}O^{∗}(U)_{ℚ}^{P^{k}} = Z^{⊗k+1} ⊕ (⊗^{k+1}ℚ[P]^{0})^{P^{k}} and therefore

Sym^{k+1} O^{∗}(U)_{ℚ}^{P^{k}}/ Sym^{k+1} O^{∗}(S)_{ℚ} = (Sym^{k+1} ℚ[P]^{0})^{⊗k+1}.

Now ⊗^{k+1}ℚ[P]^{0} can be identified as the space of functions ϕ: P^{k} → ℚ such that ∑_{x_{j}∈P} ϕ(\underline{x}) = 0 for every j. Under this isomorphism, (⊗^{k+1}ℚ[P]^{0})^{P^{k}} is mapped onto the subspace of ϕ invariant under P^{k} ≃ {(x_{0}, …, x_{k})| ∑ x_{j} = 0} ⊂ P^{k+1}. So we have an isomorphism

(⊗^{k+1}ℚ[P]^{0})^{P^{k}} ⟶^{∼} ℚ[P]^{0}

given by ϕ_{0} ⊗ ⋯ ⊗ ϕ_{k} ↦ ∏ ϕ_{j}.

To complete the proof it is enough to show that an inverse to this isomorphism is given by the map

ϕ ↦ P^{k}-invariant component of ϕ ⊗ η_{P} ⊗ ⋯ ⊗ η_{P}.

In other words, we have to compute

#P^{−k} ∑_{x_{0}+···+x_{k}=0}ϕ(z + x_{0})η_{P} (x_{1}) … η_{P} (x_{k})

= #P^{−1} ∑_{y∈P} ϕ(z − y)η_{P} (y) by lemma 8.1.2.1

= ∑_{y∈P} ϕ(z − y)(δ(y) − #P^{−1}) = ϕ(z).

Now consider the symbol map

⊗^{k+1}O^{∗}(U)_{ℚ} → H_{M}^{k+1}(U^{k}, k + 1)(sgn_{k+1})
f_{0} ⊗ ⋯ ⊗ f_{k} ↦ Π(sgn_{k+1})pr_{0}^{∗}f_{0} ∪ ⋯ ∪ pr_{k}^{∗}f_{k}.

If σ ∈ 𝔖_{k+1} then observe that (as in ??)

σ^{∗}(pr_{0}^{∗}f_{0} ∪ ⋯ ∪ pr_{k}^{∗}f_{k}) = pr_{σ(0)}^{∗}f_{0} ∪ ⋯ ∪ pr_{k}^{∗}f_{σ(k)}

= pr_{0}^{∗}f_{σ^{−1}(0)} ∪ ⋯ ∪ pr_{k}^{∗}f_{σ^{−1}(k)}

Therefore the symbol map factors through the symmetric product Sym^{k+1} O^{∗}(U)_{ℚ}. Moreover, if all the f_{i} belong to O^{∗}(S)_{ℚ}, then ∪_{i}pr_{i}^{∗}f_{i} is the pullback from S of the cup-product f_{0} ∪ ⋯ ∪ f_{k}, hence is invariant under 𝔖_{k+1}. Therefore the symbol map factors through Sym^{k+1} O^{∗}(U)_{ℚ}/ Sym^{k+1} O^{∗}(S)_{ℚ}.

Now the action of P^{k} on U_{P}^{k}, together with that the symmetric group 𝔖_{k+1} give as usual an action of the semi-direct product P^{k} ×𝔖_{k+1}. Write

H_{M}^{∗}(U_{P}^{k} , •)(sgn_{k+1})^{P^{k}}

for the subspace of the sgn_{k+1}-eigenspace fixed under P^{k}. It is a direct factor, with an associated projection

Π(P^{k}, sgn_{k+1}): H_{M}^{∗}(U_{P}^{k} , •)(sgn_{k+1}) → H_{M}^{∗}(U_{P}^{k} , •)(sgn_{k+1})^{P^{k}}.

We therefore obtain a composite homomorphism

′ Eis_{(E,P)} : ℚ[P]^{0} → H_{M}^{k+1}(U_{P}^{k} , k + 1)(sgn_{k+1})^{P^{k}}.

which is characterised by the identity

′ Eis_{(E,P)} : div g ↦ Π(P^{k}, sgn_{k+1})(pr_{0}^{∗}g ∪ pr_{1}^{∗}h_{P} ∪ ⋯ ∪ pr_{k}^{∗}h_{P})

At this stage is it useful to observe two simple compatibilities of the construction. let L ≥ 1 be an integer, and write P̃ = [×L]^{−1}(P) ⊂ E, and π : P̃ → P for the projection. We assume that P̃ is a union of sections of E/S. Multiplication by L induces a Galois covering [×L]: UP̃ → U_{P}, which by Galois descent gives an isomorphism

[×L]^{∗}: H_{M}^{k+1}(U_{P}^{k} , k + 1)^{P^{k}} ⟶^{∼} H_{M}^{k+1}(U k ; P̃, k + 1)P̃^{k}.

We then have a commutative diagram, in which the horizontal rows define the maps ′ Eis_{(E,P)}, ′ Eis_{(E,P̃)}:

\begin{matrix}ℚ[P̃]^{0} & ⟶^{−⊗hP̃···⊗hP̃} & ⊗^{k+1}ℚ[P̃]^{0} & ⟶^{(??)} & H_{M}^{k+1}(U k ; P̃, k + 1)(sgn_{k+1})P̃^{k} \\ _{π^{∗}}↑ & _{π^{∗⊗k+1}}↑ & & ↑^{[×L]^{∗}} \\ ℚ[P]^{0} & ⟶^{−⊗h_{P} ···⊗h_{P}} & ⊗^{k+1}ℚ[P]^{0} & ⟶^{(??)} & H_{M}^{k+1}(U_{P}^{k} , k + 1)(sgn_{k+1})^{P^{k}} \end{matrix} (8.1.2.2)

If we have two subgroups j: P ↪ P′ ⊂ E(S), let j_{!} : ℚ[P]^{0} → ℚ[P′]^{0} be the extension by zero map. Then we have a further commutative diagram

ℚ[P′]^{0} ⟶^{−⊗h_{P′} ···⊗h_{P′}} ⊗^{k+1}ℚ[P′]^{0} ⟶^{(??)} H_{M}^{k+1}(U_{P′}^{k}, k + 1)(sgn_{k+1})^{P′^{k}}
j_{!}↑  ^{j_{!}^{⊗k+1}}↑  ↑^{#(P′/P)^{k}×restriction}
ℚ[P]^{0} ⟶^{−⊗h_{P} ···⊗h_{P}} ⊗^{k+1}ℚ[P]^{0} ⟶^{(??)} H_{M}^{k+1}(U_{P}^{k} , k + 1)(sgn_{k+1})^{P^{k}}

(8.1.2.3)

These are easily checked from the proof of the lemma above.

Now observe that θ^{k} induces a diagram of 𝔖_{k+1}-equivariant maps

H_{M}^{k+1}(U_{P}^{k} , k + 1)^{P^{k}} ⟵_{θ^{k∗}}^{∼} H_{M}^{k+1}(V^{k}, k + 1)
↑^{j^{∗}}
H_{M}^{k+1}(E^{k}, k + 1) ⟵_{θ^{k∗}}^{∼} H_{M}^{k+1}(F^{k}, k + 1)

where j: V^{k} ↪ F^{k} is the inclusion. The horizontal arrows are isomorphisms by Galois descent ?? and by the invariance of H_{M}^{∗}(E^{k}, •) under P^{k} (??).

To obtain a map whose target is H_{M}^{k+1}(E^{k}, k + 1)(sgn_{k+1}), it is therefore enough to find a section of j^{∗} over the sgn_{k+1}-eigenspace. This is done in the next section.

8.1.3 The weight decomposition

Fix an integer N > 1, assumed invertible on S, and consider the morphism “multiplication by N”:

[×N]: F^{k} ⟶ F^{k}
∪  ∪
V^{k} ↩^{j̃} Ṽ^{k} ⟶^{ρ} V^{k}

where Ṽ^{k} = [×N]^{−1}(V^{k}). The induced covering ρ: Ṽ^{k} → V^{k} is finite étale, with group H = E[N]^{k} (a finite étale group scheme over S). Therefore by (??), ϕ induces an isomorphism

ρ^{∗}: H_{M}^{∗}(V^{k}, •) ⟶^{∼} H_{M}^{∗}(Ṽ^{k}, •)^{H}

and we can therefore define an endomorphism N of H_{M}^{∗}(V^{k}, •) by the commutativity of the diagram

H_{M}^{∗}(V^{k}, •)[figure] ⟶^{j̃^{∗}} H_{M}^{∗}(Ṽ^{k}, •) H-invariants projector ⟶ H_{M}^{∗}(Ṽ^{k}, •)^{H}

≀↑^{ρ^{∗}}
N  H_{M}^{∗}(V^{k}, •)

All the arrows in this diagram commute with the action of 𝔖_{k+1}. Clearly, on the image of H_{M}^{∗}(F^{k}, •), N is simply the map ([×N]^{∗})^{−1}.

For p ∈ ℤ define the weight p eigenspace to be

H_{M}^{∗}(V^{k}, •)^{(p)} =^{def} {a ∈ H_{M}^{∗}(V^{k}, •) | N (a) = N^{−p}a}

and similarly define H_{M}^{∗}(V^{k}, •)(sgn_{k+1})^{(p)} to be the sgn_{k+1}-part of H_{M}^{∗}(V^{k}, •)^{(p)}.

Theorem 8.1.3.1. (i) The weight p eigenspaces are independent of N, and there is a direct sum decomposition

H_{M}^{∗}(V^{k}, •)(sgn_{k+1}) = ⨁_{p∈ℤ} H_{M}^{∗}(V^{k}, •)^{(p)}(sgn_{k+1}).

(ii) If k ≤ p < 2k then

H_{M}^{∗}(V^{k}, •)^{(p)}(sgn_{k+1}) ≃ H_{M}^{∗−p+k}(F^{2k−p}, • − p + k)(sgn_{2k−p+1})

and if p < k or p ≥ 2k the left-hand group vanishes.

(iii) The restriction map

H_{M}^{∗}(F^{k}, •)(sgn_{k+1}) ⟶^{j^{∗}} H_{M}^{∗}(V^{k}, •)(sgn_{k+1})

is injective and identifies H_{M}^{∗}(F^{k}, •)(sgn_{k+1}) with H_{M}^{∗}(V^{k}, •)(sgn_{k+1})^{(k)}.

Proof: (following [SS2, 4.2]) We prove this in a manner analogous to the main theorem ?? for modular form motives by introducing a filtration on the complementary divisor F^{k} – V^{k}. Define closed subschemes Y_{j}^{k} ⊂ F^{k} by the condition that, for every S-scheme T,

Y_{j}^{k}(T) = {(x_{1}, … x_{k}) ∈ F^{k}(T) | at least j of x_{0}, … x_{k} equal e}.

This defines a filtration

F^{k} = Y_{0}^{k} ⊃ Y_{1}^{k} = F^{k} – V^{k} ⊃ Y_{2}^{k} ⊃ ⋯ ⊃ Y_{k}^{k} ⊃ Y_{k+1}^{k} = ∅

of F^{k}. A typical connected component of the stratum Y_{j}^{k} – Y_{j+1}^{k} is

{(e, …, e, x_{j+1}, …, x_{k}) ∈ E^{k}|x_{j+1}, …, x_{k} ≠ e}

on which the subgroup 𝔖_{j} ⊂ 𝔖_{k+1} acts trivially. The whole stratum is smooth of codimension j and can be written as

Y_{j}^{k} – Y_{j+1}^{k} ⟶^{∼} 𝔖_{k+1} ×^{𝔖_{j}×𝔖_{k−j+1}} V^{k−j}

where the action of 𝔖_{j} × 𝔖_{k−j+1} on V^{k−j} factors through the projection onto the second factor. Therefore

H_{M}^{∗}(Y_{j}^{k} – Y_{j+1}^{k}, •)(sgn_{k+1}) = 0 if j > 1,

since the restriction of sgn_{k+1} to 𝔖_{j} is non-trivial for j > 1. Now considering the long exact cohomology sequences

H_{M}^{∗}(F^{k} – Y_{j+1}^{k}, •)(sgn_{k+1}) → H_{M}^{∗}(F^{k} – Y_{j}^{k}, •)(sgn_{k+1}) →

H_{M}^{∗−2j+1}(Y_{j}^{k} – Y_{j+1}^{k}, • − j)(sgn_{k+1}) → …

we see that for 1 < j ≤ k

H_{M}^{∗}(F^{k} – Y_{j+1}^{k}, •)(sgn_{k+1}) ⟶^{∼} H_{M}^{∗}(F^{k} – Y_{j}^{k}, •)(sgn_{k+1}).

and therefore

H_{M}^{∗}(F^{k}, •)(sgn_{k+1}) ⟶^{∼} H_{M}^{∗}(F^{k} – Y_{2}^{k}, •)(sgn_{k+1}).

By Frobenius reciprocity,

H_{M}^{∗}(Y_{1}^{k} – Y_{2}^{k}, •)(sgn_{k+1}) ⟶^{∼} H_{M}^{∗}(V^{k−1}, •)(sgn_{k}).

We therefore have a long exact sequence:

H_{M}^{∗−2}(V^{k−1}, • − 1)(sgn_{k}) →^{δ} H_{M}^{∗}(F^{k}, •)(sgn_{k+1})
→ H_{M}^{∗}(V^{k}, •)(sgn_{k+1}) → H_{M}^{∗−1}(V^{k−1}, • − 1)(sgn_{k}) →^{δ} ….

and by (??) this sequence is compatible with the family of endomorphisms which are N on the middle two groups and N^{−2}N on the other ones. By induction we easily see that the following two statements hold:

• The boundary maps δ vanish.

• H_{M}^{∗}(V^{k}, •)(sgn_{k+1}) is the direct sum of its weight p eigenspaces for k ≤ p < 2k, and H_{M}^{∗}(V^{k}, •)^{(p)}(sgn_{k+1}) ≃ H_{M}^{∗−p+k}(F^{2k−p}, •−p+k)(sgn_{2k−p+1}).

It remains to observe that the weight eigenspaces are independent of N, but this is clear because the operators N for different N’s commute.

Now we can define the (abstract) Eisenstein symbol map:

Definition 8.1.3.2. The Eisenstein symbol of weight k for (E, P) is the composite homorphism Eis_{(E,P)}^{k}

ℚ[P]^{0}
↓′ ^{Eis}_{(E,P)}  [figure: ^{Eis{k \atop (E,P)}}]
H_{M}^{k+1}(U_{P}^{k} , k + 1)(sgn_{k+1})^{P^{k}}  H_{M}^{k+1}(E^{k}, k + 1)
↓_{projection∘θ^{k∗}}  ^{≀}↓_{θ^{k∗}}
H_{M}^{k+1}(V^{k}, k + 1)^{(k)}(sgn_{k+1}) ⟵_{restriction}^{∼} H_{M}^{k+1}(F^{k}, k + 1)(sgn_{k+1})

8.2 The boundary of Eis_{(E,P)}^{k}

8.2.1 The residue map

Now we assume that we are given a regular scheme S̄ of dimension 1, and a closed point s; let S = S̄ – {s} be its complement, and κ the residue field at s. Suppose an elliptic curve E/S as in the first section is given, and that its minimal model over S̄ has split multiplicatiove reduction at s. By this we mean that the connected component Ê of the Néron model of E has fibre at s isomorphic to 𝔾_{m}. In this case, the long exact sequence for the pair (Ê^{k}, E^{k}) becomes

H_{M}^{∗}(Ê^{k}•) → H_{M}^{∗}(E^{k}•) →^{δ} H_{M}^{∗−1}(𝔾_{m}^{k}, • − 1) → …

and taking sgn_{k+1}-eigenspaces, by (??), the boundary map can be written

H_{M}^{∗}(E^{k}, •)(sgn_{k+1}) →^{δ} H_{M}^{∗−1}(𝔾_{m}^{k}, • − 1)(sgn_{k+1}) →^{∼} H_{M}^{∗−k−1}(Spec κ, • − k − 1).

An important point is that there is no distinguished choice of isomorphism Ê ×S̄ s ⟶^{∼} 𝔾_{m}. We shall have to keep track of the choice later on, so for the moment we introduce the 1-dimensional vector space

λ =^{def} Hom(Ê ×S̄ s, 𝔾_{m}) ⊗_{ℤ} ℚ.

By (??) there is a canonical isomorphism

H_{M}^{∗}(Ê^{k} ×S̄ s, •)(sgn_{k+1}) ⟶^{∼} λ^{⊗k} ⊗ H_{M}^{∗}(𝔾_{m}^{k}, •)(sgn_{k+1})

such that if λ: Ê ×S̄ s ⟶^{∼} 𝔾_{m} is an isomorphism, and λ^{k}: Ê^{k} ×S̄ s ⟶^{∼} 𝔾_{m}^{k} its k-fold product, then the map is given by

c ↦ λ^{⊗k} ⊗ λ^{k∗}(c).

The residue map we are interested in is the composite

Res_{M}^{k}: H_{M}^{k+1}(E^{k}, k + 1)(sgn_{k+1}) → λ^{⊗k} ⊗ H_{M}^{k+1}(𝔾_{m}^{k}, k + 1)(sgn_{k+1}) = λ^{⊗k} using the canonical isomorphism ℚ = H_{M}^{0}(Spec κ, 0) ≃ H_{M}^{k+1}(𝔾_{m}^{k}, k+1)(sgn_{k+1}).

In [SS2] a formula for the residue map is obtained. First we need a bit more notation. Consider the proper regular minimal model Ē of E over S̄; its fibre over s is a Néron n-gon for some n ≥ 1. The choice of an isomorphism λ: Ê_{s} ⟶^{∼} 𝔾_{m} determines an isomorphism of the group of irreducible components with ℤ/n, by stipulating that in the Néron polygon Ē_{s}, the intersection of the 0th and 1st components is λ^{−1}(∞). In terms of this isomorphism, if a = (a_{x}) ∈ ℚ[P] and i ∈ ℤ/n, let d_{i}(a) be the sum of all a_{x} where x runs over sections in P which meet the i-th component of the fibre at s.

Theorem 8.2.1.1.

Res_{M}^{k} ∘ Eis_{(E,P)}^{k}: a = (a_{x})_{x∈P} ↦ C_{P,n}^{k} (∑_{i=0}^{n−1} d_{a}(i)B_{k+2}(\frac{i}{n}))

where C_{P,n}^{k} is an explicit nonzero constant, and B_{k+2}(X) is the (k + 2)-th Bernoulli polynomial.

In fact C_{P,n}^{k} = n^{k} (k +1)/(k +2)! in the case when P consists of all n-torsion sections.

We shall not give the proof of this here; it is explained already in detail in [SS2]. We just want to point out that there are two parts to the computation:

• First, a computation of divisors on the the arithmetic surface Ē — essentially, we have to compute the divisor on Ē of the functions g_{x}, whose divisors are of the form (x) − (e) + (sum of components of Ē_{s}).

• Next, one has to compute the effect of projection onto the weight k eigenspace. It is here that the Bernoulli polynomials come in.

8.3 The modular case

8.3.1 Bernoulli polynomials

We recall for convenience the well-known properties of the Bernoulli polynomials and their connection with Dirichlet L-functions. See for example Lang, Cyclotomic Fields Ch.2 for proofs of these facts.

The Bernoulli polynomials are defined by the formula

\frac{te^{tX}}{e^{t} − 1} = ∑_{r=0}^{∞} B_{r}(X)\frac{t^{r}}{r!}  (8.3.1.1)

and enjoy the following properties:

• B_{r}(1 − X) = (−1)^{r}B_{r}(X), and B_{r}(0) is the r-th Bernoulli number (which in particular is 0 for odd r > 2); B_{0}(X) = 1.

• The Fourier expansion: if r ≥ 2 and 0 ≤ X ≤ 1 then

B_{r}(X) = ∑_{m∈ℤ−\{0\}} −\frac{r!}{(2πim)^{r}} e^{2πimx}.

• The distribution relation: for every n > 0,

B_{r}(X) = n^{r−1} ∑_{a=0}^{n−1} B_{k}(\frac{X + a}{n})  (8.3.1.2)

• The Bernoulli measure: for each r ≥ 0 there is a measure μ_{B,r} on 𝔸_{f} such that for all n ∈ ℕ, a ∈ ℚ, 0 ≤ a < n

∫_{a+nℤ̂} dμ_{B,r} = n^{r−1}B_{r}(\frac{a}{n}).

• Dirichlet L-series at integers: if ψ: ℤ/n → ℂ is any periodic function, with finite Fourier transform ψ̂(x) = ∑_{y∈ℤ/n} ψ(y)e^{2πixy/n} and r ≥ 2 then

∫ℤ̂ ψ dμ_{B,r} = −rL(ψ, 1 − r)

= −\frac{(r − 1)!n^{r−1}}{(2πi)^{r}} (L(ψ̂(x), r) + (−1)^{r}L(ψ̂(−x), r)) .

• For r = 0, μ_{B,0} = μ_{H}, Haar measure on 𝔸_{f} (normalised to give ℤ̂ measure 1), so if a ∈ 𝔸_{f}^{∗}, dμ_{B,0}(ax) = |a|_{f}dμ_{B,0}(x).

• For all r ≥ 0 and all a ∈ ℚ^{∗},

dμ_{B,r}(ax) = |a|_{∞}^{r−1} sgn(a)^{r} dμ_{B,r}(x) = a^{r−2}|a|_{f}^{−1} dμ_{B,r}(x). (8.3.1.3)

8.3.2 The horospherical isomorphism at finite level

Classically there are two different ways of defining Eisenstein series. Let k > 2 and n ≥ 1 be integers, and consider holomorphic Eisenstein series on the upperhalf plane for the principal congruence subgroup Γ(n) ⊂ SL_{2}(ℤ). Then one has for every \underline{a} = (a_{1}, a_{2}) ∈ (ℤ/n)^{2}

G_{k,(a_{1},a_{2})}(τ) =^{def} ∑_{\begin{matrix}\underline{m}∈ℤ^{2} – \underline{0} \\ \underline{m}≡\underline{a} (mod n) \end{matrix}} \frac{1}{(m_{1}τ + m_{2})^{k}}

and for every δ ∈ SL_{2}(ℤ)/Γ(n)

E_{k,δ}(τ) = ∑_{γ∈Γ_{∞}\Γ(n)} \frac{1}{(m_{1}τ + m_{2})^{k}} , where γδ^{−1} = (\begin{matrix}∗ & ∗ \\ m_{1} & m_{2} \end{matrix})

Each of these two sets of Eisenstein series spans the same complement to the space of cusp forms. They can be compared by computing their Fourier expansions: E_{k,δ} is normalised to that its constant term at the cusp δ(∞) is 1, and it vanishes at all cusps not equivalent to δ(∞). On the other hand, the constant term of G_{k,(a_{1},a_{2})} at ∞ is 0 if a_{1} ≠ 0, and if a_{1} = 0 the constant term is

∑_{\begin{matrix}m∈ℤ \\ m≡a_{2} (mod n) \end{matrix}} \frac{1}{m^{k}} = ζ_{m≡a_{2}(n)}(k) + (−1)^{k}ζ_{m≡−a_{2}(n)}(k).

The relation can be described by the horospherical isomorphism, which we now describe.

The left action of G(ℤ/n) on (ℤ/n)^{2} extends to ℚ[(ℤ/n)^{2}]. It is convenient to identify element of ℚ[(ℤ/n)^{2}] with functions on (ℤ/n)^{2} by identifying ∑ ϕ(\underline{x})\underline{x} with the function ϕ, and then the action of G(ℤ/n) becomes the representation (gϕ)(\underline{x}) = ϕ(g^{−1}\underline{x}). Write

F_{n}^{k} = \{f: GL_{2}(ℤ/nZ) → ℚ | f ((\begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix}) g) = f(g) = (−1)^{k}f(−g)\} .

The horospherical map at level n

ω_{n}^{k}: ℚ[(ℤ/n)^{2}]^{0} → F_{n}^{k}

is given by

(ω^{k}ϕ)(g) = n^{k} ∑_{\begin{matrix}\underline{x}∈ℤ^{2} \\ 0≤x_{i}<n \end{matrix}} ϕ(g^{−1}\underline{x})B_{k+2}(x_{2}/n).  (8.3.2.1)

This is a G(ℤ/n)-equivariant map which vanishes on the subspace ℚ[(ℤ/n)^{2}]^{0,(−1)^{k+1}}, where

ℚ[(ℤ/n)^{2}]^{0,±} = \{ϕ ∈ ℚ[(ℤ/n)^{2}]^{0} | ϕ(−\underline{x}) = ±ϕ(\underline{x})\}.

In terms of the Bernoulli measure this takes the form

(ω^{k}ϕ)(g) = ∫ℤ̂^{2} ϕ(g^{−1}\underline{x}) dμ_{B,0}(x_{1}) dμ_{B,k+2}(x_{2})

which shows that there is no dependence on n. If p is a prime dividing n, then define an operator T_{p}^{k} on ℚ[(ℤ/n)^{2}]^{0,(−)^{k}} by the formula

(T_{p}^{k}ϕ)(\underline{x}) = \begin{cases}ϕ(\underline{x}) & if \underline{x} ≢ 0 (mod p) \\ ϕ(\underline{x}) − p^{−k}ϕ(\underline{x}/p) & otherwise. \end{cases}

Proposition 8.3.2.1. For every k ≥ 1 and n > 1, the following sequence is exact:

⨁_{p|n} ℚ[(ℤ/(n/p))^{2}]^{0,(−)^{k}} ⟶^{(T_{p}^{k})} ℚ[(ℤ/n)^{2}]^{0,(−)^{k}} ⟶^{ω^{k}} F_{n}^{k} → 0.

Proof: We shall show first that ω^{k} ⊗ ℂ is surjective on functions at level n. Let χ: (ℤ/n)^{∗} → ℂ^{∗} be a Dirichlet character with χ(−1) = (−1)^{k}, and consider the function

f_{χ} : (\begin{matrix}a & b \\ c & d \end{matrix}) ↦ \begin{cases}0 & if c ≠ 0 \\ χ(d) & if c = 0 \end{cases}

These functions generate F_{n}^{k} ⊗ ℂ as a module under G_{ℤ/n}. We compute the image under the horospherical map of ϕ_{χ} ∈ ℂ[(ℤ/n)^{2}], defined by

ϕ_{χ}(\underline{x}) = ∑_{y∈(ℤ/nℤ)^{∗}} χ(y)e^{2πix_{2}y/n}.

We have, for g = (\begin{matrix}a & b \\ c & d \end{matrix})

∫ℤ̂ ϕ_{χ}(g^{−1}\underline{x}) dμ_{B,0}(x_{1}) = \begin{cases}0 if c ≠ 0; and otherwise \\ χ(d) ∑_{y∈(ℤ/n)^{∗}} χ(y)e^{2πix_{2}y/n} = χ(d)χ̂(x_{2}) \end{cases}

Here of course χ̂ is defined by extending χ by zero to a function on ℤ/n, and χ̂(x_{2}) is a Gauss sum. Therefore ω^{k}ϕ_{χ}(g) = 0 if c ≠ 0, and otherwise by ??

ω^{k}ϕ_{χ}(g) = −(k + 2)χ(d)L(χ̂, −k − 1)

= 2(−1)^{k+1}(k + 2)!(\frac{n}{2πi})^{k+2}χ(d)L(χ, k + 2)

≠ 0

This shows that ℚ[(ℤ/n)^{2}]^{(−)^{k}} surjects onto F_{n}^{k}. If k is odd then ℚ[(ℤ/n)^{2}]^{(−)^{k}} = ℚ[(ℤ/n)^{2}]^{0,(−)^{k}}, and we have proved that ω^{k} is surjective. If k is even, then ℚ[(ℤ/n)^{2}]^{(−)^{k}} is the sum of ℚ[(ℤ/n)^{2}]^{0,(−)^{k}} and constants. The horospherical map takes the constant function 1 to the constant function B_{k+2}(0), so it is enough to find ϕ ∈ ℚ[(ℤ/n)^{2}]^{0,(−)^{k}} which maps to a nonzero constant function. As n > 1, we can take ϕ = ∑_{\underline{x}∈(ℤ/n)^{2}}(\underline{x}) − (\underline{0}) which has ω^{k}(ϕ) = (1 − n^{k+2})B_{k+2}(0).

It remains to show identify the kernel. First we compute:

ω^{k}(T_{p}^{k}ϕ)(g) = ∫ℤ̂^{2} ϕ(g^{−1}\underline{x}) dμ_{B,0}(x_{1}) dμ_{B,k+2}(x_{2})
− p^{−k} ∫_{pℤ̂^{2}} ϕ(g^{−1}\underline{x}/p) dμ_{B,0}(x_{1}) dμ_{B,k+2}(x_{2})
= ω^{k}ϕ(g) − p^{−k}p^{k}ω^{k}ϕ(g) = 0.

Next, if d = p_{1} ⋯ p_{r} ≥ 1 is any squarefree positive divisors of n, define T_{d}^{k}: ℚ[(ℤ/(n/d))^{2}] → ℚ[(ℤ/n)^{2}] by

(T_{d}^{k}ϕ)(x) = ∑_{e|d}(−1)^{κ(e)}e^{k}ϕ(x/e)

where κ(e) is the number of prime divisors of e, and ϕ(x/e) is taken to be zero if x ≢ 0 (mod e). It is obvious that

T_{d}^{k} = T_{p_{1}}^{k} ∘ ⋯ ∘ T_{p_{r}}^{k}

where T_{p_{i}}^{k} is the operator defined earlier. We also have the elementary identity:

ϕ(x) = ∑_{e|d^{∞}} e^{k}(T_{d}^{k}ϕ)(x/e)  (8.3.2.2)

from which it follows that T_{d}^{k} is injective for every squarefree d ≥ 1. It is also easy to see that if (d, d′) = 1 then

im T_{d}^{k} ∩ im T_{d′}^{k} = im T_{dd′}^{k}.

In fact, suppose that T_{d}^{k}ϕ = T_{d′}^{k}ϕ′. Then setting

ψ = ∑_{e|d^{∞}} e^{k}ϕ′(x/e)

and using (8.3.2.2) one sees that T_{d}^{k}ψ = ϕ′ and also T_{d′}^{k}ψ = ϕ.

Therefore

dim im T_{d}^{k} = dim ℚ[(ℤ/(n/d))^{2}]^{0,(−)^{k}}

and in particular, for any k,

dim im T_{d}^{k} + dim im T_{d}^{k+1} = (n/d)^{2} − 1.

Now we can compute

dim ℚ[(ℤ/n)^{2}]^{0}− dim ∑_{p|n} im(T_{p}^{k}) − dim ∑_{p|n} im(T_{p}^{k+1})

= (n^{2} − 1) − ∑_{p|n}((\frac{n}{p})^{2} − 1) + ∑_{p,q|n}((\frac{n}{pq} )^{2} − 1) − …

= n^{2} ∏_{p|n}(1 − \frac{1}{p^{2}} )

= # (\begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix}) \G(ℤ/n) = dim F_{n}^{k} + dim F_{n}^{k+1}

which shows that (*) is exact at the middle also.

8.3.3 The horospherical isomorphism at infinite level

We now describe the adelic version of the map of the previous section. Write S(𝔸_{f}^{2}, U) for the Schwarz space of locally constant functions of compact support on 𝔸_{f}^{2} with values in a ℚ-vector space U. The group G_{f} acts on S(𝔸_{f}^{2}, U) on the left: (gf)(x) = f(g^{−1}x) if g ∈ G_{f} and f ∈ S(𝔸_{f}^{2}, U). The formula (8.3.2.1) can be extended to adelic functions, by writing, for any ϕ ∈ S(𝔸_{f}^{2}, U) and any g ∈ G_{f}

f_{ϕ}(g) = ∫_{𝔸_{f}^{2}} ϕ(g^{−1}\underline{x}) dμ_{B,0}(x_{1}) dμ_{B,k+2}(x_{2}).  (8.3.3.1)

This formula defines a G_{f}-equivariant map from S(𝔸_{f}^{2}) to the space of locally constant functions on G_{f}.

Lemma 8.3.3.1. Let ϕ ∈ S(𝔸_{f}^{2}, U). Then for every g ∈ G_{f} and every a ∈ 𝔸_{f}^{∗}, b ∈ 𝔸_{f} and d ∈ ℚ^{∗},

f_{ϕ} ((\begin{matrix}a & b \\ 0 & d \end{matrix}) g) = |\frac{a}{d}|_{f}d^{k}f_{ϕ}(g).

Proof: Since f_{ϕ}(g) = f_{ϕ∘g^{−1}}(1) it is enough to check this for g = 1, in which case, writing \underline{u} = g^{−1}\underline{v},

f_{ϕ} ((\begin{matrix}a & b \\ 0 & d \end{matrix})) = ∫_{𝔸_{f}^{2}} ϕ(g^{−1}\underline{v}) dμ_{B,0}(v_{1}) dμ_{B,k+2}(v_{2})

= ∫_{𝔸_{f}^{2}} ϕ(\underline{u}) dμ_{B,0}(au_{1}) dμ_{B,k+2}(du_{2})

= |\frac{a}{d}|_{f}d^{k}f_{ϕ}(1)

by (8.3.1.3).

This implies that f_{ϕ} belongs to the twist F^{k} ⊗ det^{⊗−k} of the induced representation F^{k} of G_{f} introduced earlier ??. Define

S(𝔸_{f}^{2}, U)^{0} = {ϕ ∈ S(𝔸_{f}^{2}, U)| ∫_{𝔸_{f}^{2}} ϕ = 0}

(the integral being with respect to Haar measure on 𝔸_{f}^{2}).

Theorem 8.3.3.2. The map ϕ ↦ f_{ϕ} given by (8.3.3.1) induces an isomorphism between F^{k}(U) and the maximal G_{f}-invariant quotient of S(𝔸_{f}^{2}, ℚ) on which

(\begin{matrix}a & 0 \\ 0 & a \end{matrix}) acts as multiplication by a^{k}, for any a ∈ ℚ^{∗}. The same remains true

when S(𝔸_{f}^{2}, ℚ) is replaced by S(𝔸_{f}^{2}, ℚ)^{0}.

Proof: By the Iwasawa decomposition G_{f} = B_{ℚ}Gℤ̂ restriction from G_{f} to Gℤ̂ defines an isomorphism between F^{k}(U) and the space

lim_{⟶}_{n} F_{n}^{k} = \{f: Gℤ̂ → ℚ | f ((\begin{matrix}∗ & ∗ \\ 0 & ±1 \end{matrix}) g) = (−1)^{k}f(g)\} .

Therefore the map is surjective, even when restricted to S(𝔸_{f}^{2}, ℚ)^{0}, by 8.3.2.1. Let ϕ belong to the kernel. Decompose ϕ = ϕ^{+} + ϕ^{−}, where ϕ^{±}(−x) = ±ϕ(x). Then ϕ^{(−)^{k+1}} vanishes in the claimed maximal quotient, so it is enough to consider the case when ϕ(−x) = (−1)^{k}ϕ(x). By acting on ϕ by a suitable rational scalar matrix we can assume that ϕ has support in ℤ̂^{2} and is invariant under nℤ̂^{2}, some n > 1. Then by 8.3.2.1 we can write

ϕ = ∑_{p|n} T_{p}^{k}ϕ_{p}

where ϕ_{p} is invariant under (n/p)ℤ̂^{2}. But for any \underline{x} ∈ 𝔸_{f}^{2}

t_{p}^{k}ϕ_{p}(\underline{x}) = ϕ_{p}(\underline{x}) − p^{−k}ϕ_{p}(p^{−1}\underline{x}) = (ϕ_{p} − p^{−k} (\begin{matrix}p & 0 \\ 0 & p \end{matrix}) ϕ_{p})(x)

i.e. T_{p}^{k}ϕ_{p} is in the kernel of the maximal quotient as required.

8.3.4 Application to the Eisenstein symbol

We now return to the definition of the Eisenstein symbol, specialising to the case where S in the modular curve M_{n} and E = X_{n} is the universal elliptic curve ??. We have the abstract Eisenstein symbol

Eis_{X_{n},n-torsion} : ℚ[(ℤ/n)^{2}]^{0} → H_{M}^{k+1}(X_{n}^{k}, k + 1)(ϵ_{n}^{k})

and the residue map at finite level ??

Res^{k}: H_{M}^{k+1}(X_{n}^{k}, k + 1)(ϵ_{n}^{k}) → F_{n}^{k}

Corollary 8.3.4.1. (i) The composite Res^{k} ∘ Eis_{X_{n},n-torsion}^{k}: ℚ[(ℤ/n)^{2}]^{0} → F_{n}^{k} is a non-zero constant multiple of the horospherical map ω_{n}^{k} (8.3.2.1).

(ii) The Eisenstein symbol map is surjective.

These follow at once from 8.2.1.1 and 8.3.2.1.

Now we consider what happens at infinite level. We can fit together the formal Eisenstein symbols Eis_{(X_{n},n-torsion)}^{k} into a G_{f}-equivariant mapping

Eis_{(X,torsion)}^{k}: S(𝔸_{f}^{2})^{0} ⊗ (det)^{⊗k} → H_{M}^{k+1}(X^{k}, k + 1)(ϵ_{n}^{k})

in the following way …

From the boundary formula and the properties of the horospherical isomorphism 8.3.3.2, we obtain:

Corollary 8.3.4.2. The composite Res^{k} ∘ Eis_{(X,torsion)}^{k} is a constant multiple of the horospherical isomorphism ω^{k}: ϕ → f_{ϕ}.

This gives the definition of the Eisenstein symbol E_{M}^{k} in the modular setting: it is the unique constant multiple of the composite map

F^{k} ⟶^{(ω^{k})^{−1}} ∼S(𝔸_{f}^{2})^{0} ⊗ (det)^{⊗k} ⟶^{Eis_{(X,torsion)}^{k}} H_{M}^{k+1}(X^{k}, k + 1)(ϵ_{n}^{k})

that satisfies Res^{k} ∘E_{M}^{k} = id.

8.3.5 The image in absolute Hodge cohomology

According to ??, together with the functoriality of the regulator, we have a commutative square, in which the lower horizontal arrow is an isomorphism:

H_{M}^{k+1}(X^{k}, k + 1)(ϵ_{n}^{k}) ⟶^{Res^{k}} F^{k}
r_{H}↓  ^{r_{H}}↓
H_{H}^{k+1}(X_{/ℝ}^{k}, k + 1)(ϵ_{n}^{k}) ⟶_{∼}^{Res^{k}} F_{ℂ}^{k,+}

By definition, the inverse of this isomorphism is the Eisenstein series map E_{H}^{k} in H-cohomology, given explicitly in terms of differential forms in ??. Since E_{M}^{k} is a section of the upper residue map, we obtain:

Corollary 8.3.5.1. r_{H} ∘ E_{M}^{k} = E_{H}^{k}.

This result was obtained by Beilinson by an entirely different, analytic argument. A more complicated argument, which proved the analogous result for a constant elliptic curve (i.e. for a fibre of X_{n}/M_{n}), was obtained by Deninger [De1], using some subtle analysis.

8.4 Applications of the Eisenstein symbol

8.4.1 Regulators of cyclotomic fields

Consider the long exact sequence in motivic cohomology:

0 → H_{M}^{k+1}(W_{n}^{k}, k + 1) → H_{M}^{k+1}(X_{n}^{k}, k + 1)(ϵ_{n}^{k}) ⟶^{Res} F_{n}^{k} ≃ H_{M}^{0}(M_{n}^{∞}, 0).

By the theory of the Eisenstein symbol, it is also exact at the right. We now

fix a cusp y ∈ M_{n}^{∞}, and let Y ⊂ \overline{\overline{X}}_{n}^{k} be its inverse image. If f ∈ F_{n}^{k} has the property that f(y) = 0, then the Eisenstein symbol E_{M}^{k}(f) belongs to the subspace

H_{M}^{k+1}(X_{n}^{k} ∪ Y, k + 1)(ϵ_{n}^{k}) ⊂ H_{M}^{k+1}(X_{n}^{k}, k + 1)(ϵ_{n}^{k})

(where X_{n}^{k} ∪ Y is regarded as an open subscheme of \overline{\overline{X}}_{n}^{k}).

Recall ?? that we have computed K_{∗}(Y_{•})_{ℚ}(ϵ_{n}^{k}) ≃ K_{2k+1}(F_{n})_{ℚ}. We therefore obtain a composite map:

λ_{y}: {f ∈ F_{n}^{k}|f(y) = 0} ⟶^{E_{M}^{k}} H_{M}^{k+1}(X_{n}^{k} ∪ Y, k + 1)(ϵ_{n}^{k}) ⟶^{restriction}

K_{k+1}(Y_{•})_{ℚ}(ϵ_{n}^{k}) ⟶^{∼} K_{2k+1}(F_{n})_{ℚ} = H_{M}^{1}(F_{n}, k + 1).

Now suppose that y is the standard cusp ∞.

Theorem 8.4.1.1. If ϕ ∈ S(𝔸_{f}^{2}, ℚ)^{0} such that f_{ϕ}(∞) = 0, then r_{H}(λ_{∞}(f_{ϕ})) ∈ H_{H}^{1}(F_{n}, k + 1) = [⊕_{σ: F_{n}↪ℂ}ℝ(k)]^{+} is given by

r_{H}(λ_{∞}(f_{ϕ})) = \frac{(k + 2)k!}{2} (L(ϕ̂_{2}(x), k + 1) + (−1)^{k}L(ϕ̂_{2}(−x), k + 1))

where ϕ̂_{2} is the Fourier transform of ϕ_{2}(u) = ϕ(u, 0).

Proof. To compute the regulator of λ_{y} we need to know the compatibility of the regulator with simplicial localisation. In the present situation this can be done directly: we have the following diagram:

H_{M}^{k+1}(X ∪ Y, k + 1)(ϵ_{n}^{k}) ⊂ K_{k+1}(X ∪ Y)_{ℚ}(ϵ_{n}^{k})
↓  ↙
K_{k+1}(Y_{•})_{ℚ}(ϵ_{n}^{k})  →^{∼} K_{k+1}^{c}(𝔾_{m}^{k}_{/F_{n}})_{ℚ}(ϵ^{k}) →^{∼}  K_{2k+1}(F_{n})_{ℚ}
↓  ↓  ↓
H_{H}^{k+1}(Y_{•/ℝ}, k + 1)(ϵ_{n}^{k}) →^{∼} H_{H,c}^{k+1}(𝔾_{m}^{k}_{/ℝ}, k + 1)(ϵ^{k}) →^{∼} H_{H}^{1}(F_{n/ℝ}, k + 1)
≀↓  ^{≀}↓  ^{≀}↓
H_{B}^{k}(Y, ℝ(k))^{+}(ϵ_{n}^{k}) →^{∼} H_{B,c}^{k}(𝔾_{m}^{k}, ℝ(k))^{+}(ϵ^{k}) →^{∼} H_{B}^{0}(Spec F_{n}, ℝ(k))^{+}

where the squares commute by the functoriality of the Chern class (chapter 2 ??).

We can assume that ϕ is supported on ℤ̂^{2} and invariant under nℤ̂^{2}, for some n. The class r_{H}(E_{M}^{k}(f_{ϕ})) = E_{H}^{k}(f_{ϕ}) is then represented by an explicit differential form on X_{n}^{k} which extends to X_{n}^{k} ∪Y. We therefore have to compute its restriction to Y. To do this we can take any k-cycle C_{y′} in a fibre of X_{n}^{k} over a point y′ near to ∞, whose specialisation at ∞ is non-zero, and evaluate

lim_{y′→∞} ∫_{C_{y′}} E_{H}^{k}(f_{ϕ}).

We take the cycle which is the k-fold self-product of the 1-cycle on ℂ/ℤ + τℤ which is the class of the straight line path from 0 to τ, and where y′ is the point (τ, 1) ∈ M_{n}(ℂ), Im(τ) → +∞. Then

∫_{C_{y′}} η^{j,k−j} = k!τ^{j}τ̄^{k−j}

and therefore by ?? we can compute (viewing ϕ as a function on (ℤ/n)^{2}):

∫_{C_{y′}} E_{H}^{k}(f_{ϕ}) = \frac{(−1)^{k}(k + 2)k!}{4πi}

× ∑_{j=0}^{k} ∑_{\begin{matrix}(c_{1},c_{2})≠\underline{0} \\ \underline{u}∈(ℤ/n)^{2} \end{matrix}} ϕ(\underline{u})e^{2πi\underline{c}·\underline{u}/n} \frac{(τ − τ̄)τ^{k}τ̄^{k−j}}{(c_{1}τ + c_{2})^{j+1}(c_{1}τ̄ + c_{2})^{k−j+1}} .

The constant term is the sum over c_{1} = 0, which vanishes under the assumption on ϕ. We can rearrange the remainder as

\frac{(−1)^{k}(k + 2)k!}{4πi} ∑_{j=0}^{k} ∑_{\begin{matrix}c_{1}≠0 \\ c_{2}∈ℤ/n \\ \underline{u}∈(ℤ/n)^{2} \end{matrix}}ϕ(\underline{u})e^{2πi(c_{1}u_{1}+sgn(c_{1})c_{2}u_{2})/n}

× ∑_{m≡c_{2} (n)} \frac{sgn(c_{1})^{k}(τ − τ̄)τ^{k}τ̄^{k−j}}{(|c_{1}|τ + c_{2})^{j+1}(|c_{1}|τ̄ + c_{2})^{k−j+1}} .

The inner sum is asymptotic to

\frac{1}{n}(−1)^{j}\binom{k}{j}\frac{2πi sgn(c_{1})^{k}(τ − τ̄)τ^{k}τ̄^{k−j}}{(|c_{1}|(τ − τ̄))^{k+1}}

by (??), and the c_{2}-summation gives nϕ(u_{1}, 0) if u_{2} = 0, and 0 otherwise. Therefore, summing over j we get

lim_{y′→∞} ∫_{C_{y′}} E_{H}^{k}(f_{ϕ}) = \frac{(k + 2)k!}{2} ∑_{c≠0, u∈ℤ/n} ϕ(u, 0)e^{2πicu/n} \frac{sgn(c)^{k}}{|c|^{k+1}}

= \frac{(k + 2)k!}{2} (L(ϕ̂_{2}(x), k + 1) + (−1)^{k}L(ϕ̂_{2}(−x), k + 1)) .

Corollary 8.4.1.2. (The Gross conjecture for F_{n}) For any Dirichlet character χ: (ℤ/n)^{∗} → \overline{ℚ}^{∗}, and any k > 1 with χ(−1) = (−1)^{k}, the image of the Gross regulator

(K_{2k+1}F_{n} ⊗ \overline{ℚ})(χ)⊂ K_{2k+1}F_{n} ⊗ \overline{ℚ}
↓^{r_{H}}
H_{H}^{1}(F_{n}, k + 1) ⊗ \overline{ℚ}
∩
⊕_{σ: F_{n}↪ℂ}ℝ(k) ⊗ \overline{ℚ}

is the \overline{ℚ}-subspace spanned by (2πi)^{k}L′(χ, 1 − k).

Proof. We know that f_{ϕ}(∞) = 0 as soon as ϕ_{1}(u_{2}) = ∑_{u_{1}} ϕ(\underline{u}) vanishes for all u_{2} ∈ ℤ/n. This means that in the conclusion of the theorem, ϕ_{2} can be any function on ℤ/n satisfying ∑_{u} ϕ_{2}(x) = 0. Extending by \overline{ℚ}-linearity, we can therefore arrange for ϕ_{2}(x) + ϕ_{2}(−x) to be any non-trivial Dirichlet character χ(x) mod n which has parity (−1)^{k}.

By Borel’s theorem ?? and the comparison of Borel and Beilinson regulators,

K_{2k+1}F_{n} ⊗ ℝ ⟶^{r_{H}} H_{H}^{1}(F_{n}, k + 1)

and the χ-eigenspaces are at most 1-dimensional, so it is enough to find one element of (K_{2k+1}F_{n} ⊗ \overline{ℚ})(χ) whose regulator is (2πi)^{k}L′(χ, 1 − k). For the trivial character Borel’s theorem already implies the Gross conjecture, and for non-trivial characters the previous theorem, together with the functional equation ?? shows that the regulator of χ the functional equation shows that if we choose ϕ as just described, E_{M}^{k}(f_{ϕ}) has regulator a non-zero rational multiple of (2πi)^{k}L′(χ, 1 − k).

8.4.2 The mixed motives of Anderson and Harder

Let z ≠ y = ∞ be another cusp of M_{n}^{∞}, and write Z for the inverse image of z

in \overline{\overline{X}}_{n}^{k}. Consider the relative cohomology

H^{k+1}(\overline{\overline{X}}_{n}^{k} − Y relZ)(ϵ_{n}^{k})

where H is either ℓ-adic cohomology (over \overline{ℚ}) or Betti cohomology with ℚ- or ℝ-coefficients. Then we have a diagram:

0

↓

H(W_{n}^{k})

↓

0 → H^{k}(Z)(ϵ_{n}^{k}) → H^{k+1}(\overline{\overline{X}}_{n}^{k} − Y relZ)(ϵ_{n}^{k}) → H^{k+1}(\overline{\overline{X}}_{n}^{k} − Y)(ϵ_{n}^{k}) → 0
∥  ↓
1  1(−k − 1) = H_{Y}^{k+2}(\overline{\overline{X}}_{n}^{k})(ϵ_{n}^{k})

↓

0

in the category of ℚ_{ℓ}-representations of Gal(\overline{ℚ}/F_{n}), or of mixed Hodge structures over Spec F_{n}. Here for once we are viewing X_{n}^{k} and related varieties as schemes over F_{n}.

We know (by the Manin-Drinfeld principle ??) that the row and column of this diagram each have a unique splitting which commutes with the Hecke algebra, and that this is a splitting in the category of Galois modules/Hodge struc-

tures. Therefore H(W_{n}^{k}) is canonically a direct factor of H^{k+1}(\overline{\overline{X}}_{n}^{k} − Y, Z)(ϵ_{k}^{n}). We call the quotient H(And^{k}(y, z)).

Remark. We have not formally introduced a category of mixed motives here. But using realisations, a suitable definition was given by Jannsen [J, chap 1] and Deligne [DFG]. Although he does not use relative cohomology, it is easy

to show that H^{k+1}(\overline{\overline{X}}_{n}^{k} − Y, Z)(ϵ_{k}^{n}) is the realisation of a motive over F_{n} in Jannsen’s category (the argument needed can be found in [??]), and that it splits as the direct sum of the image of W_{n}^{k} and a motive And^{k}(y, z), which is an extension of ℚ(−k − 1) by ℚ(0).

We shall determine the Hodge realisation of And^{k}(y, z) explicitly, which was first done by Harder. We have the following diagram, in which we have simply

written X in place of \overline{\overline{X}}_{n}^{k}, and where this time H_{H} denotes absolute Hodge cohomology with coefficients in A = ℝ or ℚ.

H_{M}^{k+1}(X − Z, k + 1)(ϵ_{n}^{k}) ⟶^{Res} K′_{k}(Z)_{ℚ}(ϵ_{n}^{k}) ≃ H_{M}^{0}(F_{n}, 0)
∩  ↓
K_{k+1}(X − Z)_{ℚ}  ⟶  K_{k}^{Z}(X)_{ℚ}
↓  ↓
K_{2k+1}(F_{n})_{ℚ} ≃ K_{k+1}(Y_{•})_{ℚ}(ϵ_{n}^{k})  ⟶  K_{k}(X, Y_{•})_{ℚ}(ϵ_{n}^{k})
↓  ↓  ↓
H_{H}^{1}(F_{n}, k + 1)≃H_{H}^{k+1}(Y_{•}, k + 1)(ϵ_{n}^{k}) ⟶ H_{H}^{k+2}(X, Y_{•}, k + 1)(ϵ_{n}^{k})
≀↓  ^{≀}↓
Ext_{H}^{1}(A(−k − 1), A(0)) ↪ Ext_{H}^{1}(A(−k − 1), H_{B}^{k+1}(X, Y_{•}))(ϵ_{n}^{k})

(8.4.2.1)

We have to explain this diagram. First, the top square. By definition, K_{k}^{Z}(X) is the π_{k+1} of the homotopy fibre of KX → K(X − Z), and as K′Z is also the homotopy fibre (by the localisation theorem), there is a unique isomorphism K_{k}^{Z}(X) ⟶^{∼} K′_{k}Z making the top square commute.

The second square is also formal. By definition,

K_{k}(X, Y_{•}) = π_{k+1}(homotopy fibre of KX → KY_{•})

and since Y ⊂ X − Z there is a natural map from K^{Z}X to this space. The commutativity of the square comes from the homotopy commutative square

K(X − Z)[−1] ⟶ hofib[KX → K(X − Z)]
↓  ↓
KY_{•}  ⟶ hofib[KX → KY_{•}]

The next square commutes by the functoriality of Chern classes ?? and the bottom one comes from the Manin-Drinfeld decomposition H^{k+1}(X, Y_{•})(ϵ_{n}^{k}) = ℝ(0) ⊕ H(W_{n}^{k}). Notice that the bottom arrow is in fact an isomorphism if we are working with ℝ-coefficients, since then Ext_{H}^{1}(ℝ(−k − 1), H(W_{n}^{k})) = H_{H}^{k+1}(W_{n}^{k}, ℝ(k + 1)) = 0 as we saw already in ??. (This would not be the case for A = ℚ, or in the ℓ-adic category.)

Now we take E_{M}^{k}(z) ∈ H_{M}^{k+1}(X − Z, k + 1)(ϵ_{n}^{k}) and follow it round the diagram. Going down the left-hand side of (8.4.2.1) we obtain r_{H}(λ_{y}(z)), which we computed in 8.4.1.1. Down the right-hand side we use the following general procedure. For the moment we allow X to be any smooth ℝ-scheme, Y, Z ⊂ X disjoint closed subschemes, with Y a divisor with strict normal crossings; write Y_{•} for the Čech nerve of its normalisation [??].

If α ∈ K_{q}^{Z}X = K_{q}^{Z}(X, Y_{•}) then it determines Betti cohomology classes with and without supports, by applying the Betti Chern class. We can view them as compatible morphisms:

A(−n)
c_{B}^{Z}(α)↙  ↘^{c_{B}(α)}
H_{B,Z}^{j}(X) = H_{B,Z}^{j}(X, Y) ⟶ H_{B}^{j}(X, Y)

for any n, j with j + q = 2n. We assume that c_{B}(α) = 0. In this case the regulator r_{H}(α) lies in the kernel of the map

H_{H}^{j}(X, Y; n) → Hom_{H}(ℝ(0), H_{B}^{j}(X, Y_{•}; n)).

On the other hand, we can by pullback from the long exact cohomology sequence form an extension E_{α}:

H_{B,Z}^{j−1}(X) →^{θ} H_{B}^{j−1}(X, Y) → H_{B}^{j−1}(X − Z, Y) → H_{B,Z}^{j}(X) → H_{B}^{j}(X, Y)
↓  ↑  ↑^{c_{B}^{Z}(α)}
0  → coker θ →  E_{α}  → ℝ(−n) →  0

Proposition 8.4.2.1. The class of E_{α} in Ext_{H}^{1}(ℝ(−n), coker θ) is given by the image of r_{H}(α) under the composite map

ker [H_{H}^{j}(X, Y; n) → Hom_{H}(ℝ(0), H_{B}^{j}(X, Y_{•}; n)).] → Ext_{H}^{1}(ℝ(0), H_{B}^{j−1}(X, Y)(n))

→ Ext_{H}^{1}(ℝ, coker θ(n)) = Ext_{H}^{1}(ℝ(−n), coker θ)

The proof of this is the same as [Ja], Lemma ??. We apply it in the present situation, with q = k, j = k + 2, and follow E_{M}^{k}(z) round the top and right hand edges of the diagram (8.4.2.1). It implies that the image of E_{M}^{k}(z) in Ext_{H}^{1}(ℝ(−k − 1), H^{k+1}(X, Y))(ϵ_{n}^{k}) is the extension class of the exact sequence

0 → H^{k+1}(X, Y)(ϵ_{n}^{k}) → H^{k+1}(X − Z, Y)(ϵ_{n}^{k}) → H_{Z}^{k+2}(X)(ϵ_{n}^{k}) → 0
∥
ℝ(−k − 1)_{/ Spec F_{n}}

But we know that this is the image of the class

[H(And^{k}(y, z))] ∈ Ext_{H}^{1}(ℝ(−k − 1), ℝ(0)).

Corollary 8.4.2.2. [HMM] The extension class of H(And^{k}(y, z)) in

Ext_{H}^{1}(ℝ(−k − 1), ℝ(0)) = H_{H}^{1}(F_{n}, k + 1)

is the regulator of λ_{y}(z) ∈ H_{M}^{1}(F_{n}, k + 1).

By carrying out the same construction in the ℓ-adic setting, one shows that the extension class of the ℓ-adic realisation of the Anderson motive is the ℓ-adic regulator of λ_{y}(z).