Last revision 15 April 1997 7:1

Chapter 7

Motives for modular forms

7.0 Introduction

In this chapter we give the construction of a Chow motive associated to the space S_{k+2}(n) of cusp forms of weight k + 2 > 2 and level n ≥ 1. In so doing we follow closely [MF], although we fill in a number of details whose proofs were left to the reader there.

The 2-dimensional Galois representations attached to cusp forms were first constructed by Deligne [DF], who showed that they were constituents of the étale cohomology of a suitably compactified Kuga-Sato variety. To construct a motive means to isolate this constituent of the cohomology using algebraic cycles. This therefore involves explicitly analysing the compactification.

It may be useful to describe the key point. Consider for the moment classical cohomology groups H, H_{c} (cohomology with compact support), say with real coefficients. Recall that H_{c} is covariant for open immersions. It is easy to compute the cohomology of a Néron polygon Nér_{n} — the inclusion of 𝔾_{m} in the polygon induces an isomorphism of H^{1}(Nér_{n}) with H_{c}^{1}(𝔾_{m}) ≃ ℝ, and H^{2}(Nér_{n}), H^{0}(Nér_{n}) have dimension n, 1 respectively. Using the action of the symmetric group and multiplication by −1, it is easy then to cut out H^{1}(Nér_{n})^{⊗k} ⊂ H^{∗}(Nér_{n}^{k}) as the subspace H^{∗}(Nér_{n}^{k})(ϵ_{n}^{k}) on which the symmetric group Σ_{k} acts by the sign character, the group of translations (ℤ/n)^{k} acts trivially, and μ_{2}^{k} acts by the product map μ_{2}^{k} → μ_{2}.

The Kuga-Sato variety \overline{X}_{n}^{k} (??) has for its boundary a union of copies of Nér_{n}^{k}, indexed by cusps. But for k ≥ 2, \overline{X}_{n}^{k} is singular. When we resolve its singularities, the fibre over a cusp becomes much more complicated than Nér_{n}^{k}. The ‘miracle’ is that the ϵ_{n}^{k}-part of the cohomology is unchanged by the blowup. To prove this we have to write down explicitly the desingularisation of the k-fold self product of the Tate curve, and compute its cohomology. This computation can be done in essentially any cohomology theory in which one has a localisation sequence and can compute the cohomology of 𝔾_{m}.

The outline of the chapter is as follows: in the first section we introduce the cone singularities which arise as fibre product of ordinary double points, and their desingularisations. On the way we show that these are the same as the desingularisations originally given by Deligne, and also write them toroidally. The second section is devoted to the computation of the homology and cohomology. In the third section we apply these results to the canonical desingularisation of the k-fold product of a Tate curve, and in the fourth section we write down the exact sequences for the cohomology of modular form motives.

7.1 Cones and desingularisation of products of double points

7.1.1 Definitions

Let κ be an arbitrary ring (for the most part we can restrict to the case of a field). For any integer k ≥ 2, let I_{k} ⊂ κ[x_{1}, y_{1}, …, x_{k}, y_{k}] be the ideal generated by the differences x_{i}y_{i} − x_{j}y_{j}, 1 ≤ i < j ≤ k. We put I_{k} = {0} for k < 2. We are interested in the cones

C^{k} = Spec κ[x_{1}, y_{1}, …, y_{k}]/I_{k} ⊂ 𝔸_{κ}^{2k}

which arise as fibre products of ordinary curve singularities. Indeed, the morphism

C^{k} → 𝔸_{κ}^{1}
(\underline{x}, \underline{y}) ↦ t = x_{1}y_{1}

makes C^{k} the k-fold fibre product of an ordinary double point singularity over the affine line:

C^{k} = \underbrace{C^{1} ×_{𝔸^{1}} ⋯ × C^{1}}_{k copies}

C^{1} =𝔸^{2} → 𝔸^{1}
(x, y) ↦ xy

For k ≥ 2, C^{k} fails to be smooth over κ. Define:

• F^{k} = F_{k}^{k} = the closed subscheme {t = 0} ⊂ C^{k}

• For 0 ≤ j ≤ k, F_{j}^{k} = the closed subscheme of F^{k} on which at least (k − j) pairs (x_{i}, y_{i}) simultaneously vanish.

So F_{j}^{k} has relative dimension j over κ, and F_{0}^{k} ≃ Spec κ is the origin. The subset of C^{k} where it is not smooth over κ is then C^{k,sing} = F_{k−2}^{k}.

We shall at the same time consider the projectivisations of C^{k} and its subcones F_{j}^{k}. We denote them P^{k}, G_{j}^{k}. Thus

P^{k} = Proj κ[x_{1}, y_{1}, …, y_{k}]/I_{k} ⊂ ℙ_{κ}^{2k−1}
= (C^{k} – F_{0}^{k})/𝔾_{m}

and G_{j}^{k} is the image of F_{j}^{k} – F_{0}^{k} in P^{k}. In particular, G_{0}^{k} = ∅ and G_{k−2}^{k} = P^{k,sing}; so P^{k} is smooth if and only if k ≤ 2.

Let Γ^{k} be the semidirect product μ_{2}^{k} ×Σ_{k} of μ_{2}^{k} and the symmetric group Σ_{k}. Explicitly,

Γ^{k} = {(a_{1}, …, a_{k}, σ) | a_{i} ∈ {±1}, σ ∈ Σ_{k}}

with multiplication

(a_{1}, …, a_{k}, σ)(a′_{1}, …, a′_{k}, σ′) = (a_{1}a′_{σ^{−1}(1)}, …, a_{k}a′_{σ^{−1}(k)}, σσ′)

Then Γ^{k} acts on κ[x_{1}, …, y_{k}] on the right: elements of the symmetric group act by

σ: x_{i} ↦ x_{σ^{−1}(i)}
y_{i} ↦ y_{σ^{−1}(i)}

and \underline{a} ∈ μ_{2}^{k} acts by interchanging x_{i} and y_{i} if a_{i} = −1, and fixing them if a_{i} = +1. For 1 ≤ j ≤ k −1 there are obvious inclusions Γ^{j} ×Γ^{k−j} ↪ Γ^{k} coming from the partition {1, …, k} = {1, …, j} ∪ {j + 1, …, k}.

This gives an induced action (on the left) of Γ^{k} on 𝔸^{2k} and ℙ^{2k−1} preserving C^{k}, P^{k} and their subschemes F_{j}^{k}, G_{j}^{k}. For each j, the irreducible components of F_{j}^{k} are permuted transitively by Γ^{k}. The stabiliser of the component

{y_{1} = ⋯ = y_{j} = 0 = x_{j+1} = y_{j+1} = ⋯ = x_{k} = y_{k}} ⟶_{(x_{1},...,x_{j})}^{∼} 𝔸^{j} (7.1.1.1)

is Σ_{j} × Γ^{k−j}.

7.1.2 The desingularisation

Let

ϕ: Ĉ^{k} → C^{k}

be the blowup of the origin F_{0}^{k}. Since C^{k} is a cone, Ĉ^{k} is the tautological line bundle over P^{k}:

Ĉ^{k} = {(p, l) ∈ C^{k} × P^{k} | p ∈ l}
[figure: ϕ=pr_{1}; π=pr_{2}; s]
C^{k}  P^{k}

Here s is the zero section of π: Ĉ^{k} → P^{k}; its image is the exceptional divisor of ϕ. We then have:

• The singular locus Ĉ^{k}^{sing} = π^{−1}(P^{k,sing}) = π^{−1}(G_{k−2}^{k}); moreover

ϕ^{−1}(C^{k,sing}) = Ĉ^{k}^{sing} ∪ ϕ^{−1}(F_{0}^{k})  (7.1.2.1)
= π^{−1}(G_{k−2}^{k}) ∪ s(P^{k})

• The proper transform F̂_{j}^{k} ⊂ Ĉ^{k} of F_{j}^{k} is π^{−1}(G_{j}^{k}), for 1 ≤ j ≤ k.

• P^{k} (resp. Ĉ^{k}) has an open affine cover by 2k copies of C^{k−1} (resp. 𝔸^{1} × C^{k−1}), permuted transitively by Γ^{k}. The stabiliser of any one of them is isomorphic to Γ^{k−1}, acting on C^{k−1} as in 7.1.1 above (resp., and acting trivially on 𝔸^{1}).

• If ρ : C^{k−1} ↪ P^{k} is any of these charts, and 1 ≤ j ≤ k, then ρ^{−1}(G_{j}^{k}) = F_{j−1}^{k−1}. Likewise, the restriction to 𝔸^{1} × C^{k−1} of F̂_{j}^{k} ⊂ Ĉ^{k} is 𝔸^{1} × F_{j−1}^{k−1}.

Suppose that k = 2. The P^{k} = P^{2} is smooth (it is the quadric x_{1}y_{1} = x_{2}y_{2} in ℙ^{3}), so ϕ: Ĉ^{2} → C^{2} is a desingularisation of C^{2}, and we have

Ĉ^{2} = C^{2,reg} ∪ ϕ^{−1}(C^{2,sing})

with ϕ^{−1}(C^{2,sing}) ≃ P^{2}. In general, we have a similar, albeit more complicated, picture. It is convenient to state the result for P^{k} first.

Theorem 7.1.2.1. For each k ≥ 2 there is a proper birational morphism θ: P̃^{k} → P^{k} with the following properties:

(i) There is an action of Γ^{k} on P̃^{k} with respect to which θ is equivariant.

(ii) P̃^{k} is smooth; P̃^{2} = P^{2}.

(iii) There is a Γ^{k}-invariant filtration of P̃^{k} by divisors:

P̃^{k} ⊃ Y_{1}^{k} ⊃ ⋯ ⊃ Y_{k−2}^{k} ⊃ Y_{k−1}^{k} = ∅

such that:

(a) θ: P̃^{k} – Y_{1}^{k} ⟶^{∼} P^{k,reg}, so that in particular Y_{1}^{k} = θ^{−1}(P^{k,sing}); and

(b) For each j with 1 ≤ j ≤ k − 2 there is a smooth connected κ-scheme Z_{j} with an action of Σ_{j}, and a Γ^{k}-equivariant pair of isomorphisms:

Y_{j}^{k} – Y_{j+1}^{k}  ⟶^{∼} Γ^{k} ×^{Σ_{j}×Γ^{k−j}} (Z_{j} × P^{k−j,reg})
↓  ↓
Y_{j}^{k} ⊃ \overline{Y_{j}^{k} – Y_{j+1}^{k}} ⟶^{∼} Γ^{k} ×^{Σ_{j}×Γ^{k−j}} (Z_{j} × P̃^{k−j})

In the affine case, define C̃^{2} = Ĉ^{2}, and for k > 2, let C̃^{k} be the pullback of P̃^{k} by π, which sits in a commutative diagram

C̃^{k} ⟶^{π̃} P̃^{k}
[figure: ψ; ↓^{ψ_{1}}; ↓^{θ}; s̃]
C^{k} ⟵_{ϕ} Ĉ^{k} ⟶^{π} P^{k}

We have

Ĉ^{k,sing} = π^{−1}(P^{k,sing})
ϕ^{−1}(C^{k,sing}) = Ĉ^{k,sing} ∪ s(P^{k})

The filtration on C̃^{k}

C̃^{k} ⊃ W_{0}^{k} ⊃ ⋯ ⊃ W_{k−2}^{k} ⊃ W_{k−1}^{k} = ∅

is then defined as

W_{j}^{k} = π̃^{−1}(Y_{j}^{k})  if j ≥ 1
W_{0}^{k} = W_{1}^{k} ∪ s̃(P̃^{k})
= ψ_{1}^{−1}(Ĉ^{k,sing} ∪ s(P^{k}))
= ψ^{−1}(C^{k,sing})

Everything constructed is compatible with the action of Γ^{k}.

Theorem 7.1.2.2. C̃^{k} is smooth, and the filtration satisfies:

(i) θ: C̃^{k} – W_{0}^{k} ⟶^{∼} C^{k,reg};

(b) For each j with 0 ≤ j ≤ k − 2 there is a Γ^{k}-equivariant isomorphism:

W_{j}^{k} – W_{j+1}^{k} ⟶^{∼} Γ^{k} ×^{Σ_{j}×Γ^{k−j}} (V_{j} × P^{k−j,reg})

for a smooth connected κ-scheme V_{j} with Σ_{j}-action.

Remark: We shall get a quite explicit description of the schemes V_{j} and Z_{j} in the course of the construction; they have open coverings by copies of 𝔸^{j}, 𝔸^{j−1} respectively, and can be constructed as iterated blowups of 𝔸^{j}, resp. ℙ^{j−1}, along linear subspaces. As the notation suggests, they do not depend on k.

7.1.3 The construction

We shall actually do the construction in reverse order, first explaining how to make C̃^{k}. We define a sequence of blowups and closed subschemes

C̃^{k} = C^{k}⟨k − 1⟩ ⟶^{ϕ_{k−2}} C^{k}⟨k − 2⟩ → … ⟶^{ϕ_{1}} C^{k}⟨1⟩ ⟶^{ϕ_{0}} C^{k}⟨0⟩ = C^{k}
∪  ∪  ∪
F^{k}⟨k − 2⟩  F^{k}⟨1⟩  F^{k}⟨0⟩

in the following iterative way:

• ϕ_{j}: C^{k}⟨j + 1⟩ → C^{k}⟨j⟩ is the blowup of C^{k}⟨j⟩ along F^{k}⟨j⟩;

• F^{k}⟨j + 1⟩ ⊂ C^{k}⟨j + 1⟩ is the strict transform of F_{j+1}^{k} ⊂ C^{k} under ϕ_{0} ∘ ⋯ ∘ ϕ_{j}: C^{k}⟨j + 1⟩ → C^{k}.

Observe that since no irreducible component of F_{j}^{k} lies in F_{j−1}^{k}, the morphism

ϕ_{0} ∘ ⋯ ∘ ϕ_{j−1} : F^{k}⟨j⟩ → F_{j}^{k}

is birational.

Because (Ĉ^{k}, {F̂_{j}^{k}}_{1≤j≤k}) is covered by 2k copies of (𝔸^{1} × C^{k−1}, {𝔸^{1} × F_{j−1}^{k−1}}_{1≤j≤k}), we get the following description of the pair (C^{k}⟨j⟩, F^{k}⟨j⟩):

Lemma 7.1.3.1. (i) There exists an open covering

⋃_{σ∈Γ^{k}/Γ^{k−j}} U^{k}⟨j⟩_{σ} = C^{k}⟨j⟩

whose elements are permuted transitively by Γ^{k}. Moreover

U^{k}⟨j⟩_{1} ⟶^{∼} 𝔸^{j} × C^{k−j}

and its stabiliser is Γ^{k−j} acting on 𝔸^{j} × C^{k−j} through the second factor. (ii) For 0 ≤ j ≤ k − 2 we have

F^{k}⟨j⟩_{σ} =^{def} F^{k}⟨j⟩ ∩ U^{k}⟨j⟩_{σ} ⟶^{∼} 𝔸^{j} × {0} ⊂ 𝔸^{j} × C^{k−j}.

In particular, F^{k}⟨j⟩ is smooth, and is contained in the singular locus C^{k}⟨j⟩^{sing}.

For later use we write down part of these open covers. At the first blowup, we take the open affine of C^{k}⟨1⟩ = Ĉ^{k}

U^{k}⟨1⟩_{1} = Spec κ [x_{1}, \frac{x_{2}}{x_{1}} , \frac{y_{2}}{x_{1}} , …, \frac{x_{k}}{x_{1}} , \frac{y_{k}}{x_{1}} ] ⟶^{∼} 𝔸^{1} × C^{k−1}

(notice that y_{1}/x_{1} = (x_{k}/x_{1})(y_{k}/x_{1}) from the relations in I_{k}). After j blowups:

U^{k}⟨j⟩_{1} = Spec κ [x_{1}, \frac{x_{2}}{x_{1}} , …, \frac{x_{j}}{x_{j−1}} , \frac{x_{j+1}}{x_{j}} , \frac{y_{j+1}}{x_{j}} , …, \frac{x_{k}}{x_{j}} , \frac{y_{k}}{x_{j}} ]
= Spec κ[x_{1}, x′_{2}, …, x′_{j}, x′_{j+1}, y′_{j+1}, …, x′_{k}, y′_{k}]

say, and

U^{k}⟨j⟩_{1} ⊃ F^{k}⟨j⟩_{1} = {x′_{j+1} = y′_{j+1} = ⋯ = x′_{k} = y′_{k} = 0}.

When j = k − 1 it is then clear that C̃^{k} = C^{k}⟨k − 1⟩ is smooth over κ. The desingularisation θ: P̃^{k} → P^{k} is defined in exactly the same way, as an iterate of (k − 2) blowups:

P̃^{k} = P^{k}⟨k − 1⟩ ⟶^{ξ_{k−2}} P^{k}⟨k − 2⟩ → … ⟶^{ξ_{1}} P^{k}⟨1⟩ = P^{k}

where ξ_{j}: P^{k}⟨j + 1⟩ → P^{k}⟨j⟩ is the blowup of P^{k}⟨j⟩ along G^{k}⟨j⟩, the strict transform of G_{j}^{k} under ξ_{1}∘⋯∘ξ_{j−1} : P^{k}⟨j⟩ → P^{k}. Clearly for each 1 ≤ j ≤ k−1, C^{k}⟨j⟩ is a line bundle over P^{k}⟨j⟩, and the inverse image of G^{k}⟨j⟩ is F^{k}⟨j⟩. We write

ψ_{j} =^{def} ϕ_{j} ∘ ϕ_{j−1} ∘ ⋯ ∘ ϕ_{k−1} : C̃^{k} → C^{k}⟨j⟩
θ_{j} =^{def} ξ_{j} ∘ ξ_{j−1} ∘ ⋯ ∘ ξ_{k−1} : P̃^{k} → P^{k}⟨j⟩

Now we compute the stratifications.

Definition 7.1.3.2. Define for j ≥ 0

W_{j}^{k} = ψ_{j}^{−1}(C^{k}⟨j⟩^{sing})

and for j ≥ 1

Y_{j}^{k} = θ_{j}^{−1}(P^{k}⟨j⟩^{sing}).

Clearly

ψ: C̃^{k} – W_{0}^{k} ⟶^{∼} C^{k,reg}
θ: P̃^{k} – Y_{1}^{k} ⟶^{∼} P^{k,reg}

Proposition 7.1.3.3. (i) For j ≥ 0, ψ_{j+1} induces an isomorphism

ψ_{j+1} : W_{j}^{k} – W_{j+1}^{k} ⟶^{∼} ϕ_{j}^{−1}(F^{k}⟨j⟩)^{reg} ⊂ C^{k}⟨j + 1⟩

between the j-th stratum of C̃^{k} and the smooth part of the exceptional divisor of ϕ_{j}.

(ii) For j ≥ 1, θ_{j+1} induces an isomorphism

θ_{j+1} : Y_{j}^{k} – Y_{j+1}^{k} ⟶^{∼} ξ_{j}^{−1}(G^{k}⟨j⟩)^{reg} ⊂ P^{k}⟨j + 1⟩

Proof:

W_{j}^{k} – W_{j+1}^{k} = ψ_{j+1}^{−1}(ϕ_{j}^{−1}(C^{k}⟨j⟩^{sing}) – C^{k}⟨j + 1⟩^{sing})
= ψ_{j+1}^{−1}(ϕ_{j}^{−1}(C^{k}⟨j⟩^{sing}) ∩ C^{k}⟨j + 1⟩^{reg}).

Since F^{k}⟨j⟩ ⊂ C^{k}⟨j⟩^{sing} and ϕ_{j} is an isomorphism over the complement of F^{k}⟨j⟩,

ϕ_{j}^{−1}(C^{k}⟨j⟩^{sing}) ∩ C^{k}⟨j + 1⟩^{reg} = ϕ_{j}^{−1}(C^{k}⟨j⟩) ∩ C^{k}⟨j + 1⟩^{reg}
= ϕ_{j}^{−1}(C^{k}⟨j⟩)^{reg}

by (7.1.2.1). The proof of (ii) is the same.

Let’s compute one of the connected components of W_{j}^{k} – W_{j+1}^{k}:

Definition 7.1.3.4. V_{j} denotes the component of F^{k}⟨j⟩ which is the proper transform of 7.1.1.1 under ψ_{j}.

Consider the j! open affines of C^{k}⟨j⟩:

U^{k}⟨j⟩_{σ} = Spec κ [x_{1}, \frac{x_{σ(2)}}{x_{σ(1)}} , …, \frac{x_{σ(j)}}{x_{σ(j−1)}} , \frac{x_{j+1}}{x_{σ(j)}} , \frac{y_{j+1}}{x_{σ(j)}} , …, \frac{x_{k}}{x_{σ(j)}} , \frac{y_{k}}{x_{σ(j)}} ]
= Spec κ[x_{1}^{σ}, x_{2}^{σ}, …, x_{j}^{σ}, x_{j+1}^{σ}, y_{j+1}^{σ}, …, x_{k}^{σ}, y_{k}^{σ}]  (7.1.3.1)

say, indexed by σ ∈ Σ_{j}. Then

F^{k}⟨j⟩_{σ} =^{def} F^{k}⟨j⟩ ∩ U^{k}⟨j⟩_{σ} = {x_{j+1}^{σ} = y_{j+1}^{σ} = ⋯ = x_{k}^{σ} = y_{k}^{σ} = 0} ≃ 𝔸^{j}

and

V_{j} = ⋃_{σ∈Σ_{j}} F^{k}⟨j⟩_{σ}.

We then have

ϕ_{j}^{−1}(F^{k}⟨j⟩_{σ}) = Spec κ[x_{1}^{σ}, …, x_{j}^{σ}] × Proj κ[x_{j+1}^{σ}, y_{j+1}^{σ}, …, y_{k}^{σ}]
⟶^{∼} F^{k}⟨j⟩_{σ} × P^{k−j}  (7.1.3.2)

and on the overlap U^{k}⟨j⟩_{σ} ∩ U^{k}⟨j⟩_{τ}, the last 2(k − j) coordinates are related by

(x_{j+1}^{σ}, y_{j+1}^{σ}, …, y_{k}^{σ}) = \frac{x_{τ(j)}}{x_{σ(j)}} (x_{j+1}^{τ}, y_{j+1}^{τ}, …, y_{k}^{τ}).

So the isomorphisms (7.1.3.2) glue to give a Σ_{j} × Γ^{k−j}-equivariant isomorphism

ϕ_{j}^{−1}(V_{j}) ⟶^{∼} V_{j} × P^{k−j}

where the action is the product of the actions of Σ_{j} on V_{j} and of Γ^{k−j} on P^{k−j}. Since Γ^{k} permutes the components of F^{k}⟨j⟩ transitively we get

ϕ_{j}^{−1}(F^{k}⟨j⟩) ⟶^{∼} Γ^{k} ×^{Σ_{j}×Γ^{k−j}} (V_{j} × P^{k−j})
↓^{ϕ_{j}}  ↓
F^{k}⟨j⟩ ⟶^{∼}  Γ^{k} ×^{Σ_{j}×Γ^{k−j}} V_{j}

From the definition of the resolution C̃^{k} it is clear that we can locally write ψ_{j} as

𝔸^{j} × C̃^{k−j}  ↪ C̃^{k}
↓  ↓^{ψ_{j}}
𝔸^{j} × C^{k−j} ≃ U^{k}⟨j⟩_{σ}↪C^{k}⟨j⟩

hence ψ_{j}^{−1}(F^{k}⟨j⟩) = \overline{W_{j}^{k} – W_{j+1}^{k}} is covered by copies of V_{j} × P̃^{k−j}.

7.1.4 C̃^{k} = Deligne’s desingularisation

Deligne in [DF] constructs in one step a desingularisation of C^{k}, by blowing up the ideal J_{k} generated by

{γ^{∗}(x_{1}^{k−1}x_{2}^{k−2} … x_{k−1}) | γ ∈ Γ^{k}}.

We can check easily that this gives C̃^{k}. Denote the monomial γ^{∗}(x_{1}^{k−1} … x_{k−1}) by m_{γ}. Then there are 2^{k−1}k! distinct such monomials, and the blowup of C^{k} along J_{k} is the union of the 2^{k−1}k! open affines

V_{γ} = Spec κ [x_{1}, y_{1}, …, x_{k}, y_{k}, \{\frac{m_{δ}}{m_{γ}} \}_{δ∈Γ^{k}}] .

But we can write

\frac{x_{i}}{x_{i−1}} = \frac{m_{τ}}{m_{id}} where τ = (i, i − 1), \frac{y_{1}}{x_{2}} = \frac{x_{1}^{k−1} … x_{k−2}^{2}y_{k}}{m_{id}} ,

and therefore

O(V_{1}) ⊃ κ [x_{1}, \frac{x_{2}}{x_{1}} , , \frac{x_{3}}{x_{2}} , …, \frac{x_{k−1}}{x_{k−1}} , \frac{x_{k}}{x_{k−1}} , \frac{y_{k}}{x_{k−1}} ] = O(U^{k}⟨k − 1⟩_{1})

(as in (7.1.3.1) above). On the other hand, the relation y_{i}/x_{j} = y_{j}/x_{i} shows that for every i, j with i ≠ k,

\frac{y_{j}}{x_{i}} = \frac{y_{i}}{x_{j}}

= \frac{x_{j+1}}{x_{j}} … \frac{x_{k}}{x_{k−1}} \frac{y_{i}}{x_{k}}

= \frac{x_{j+1}}{x_{j}} … \frac{x_{k}}{x_{k−1}} \frac{y_{k}}{x_{i}}

= (\frac{x_{j+1}}{x_{j}} … \frac{x_{k}}{x_{k−1}} ) (\frac{x_{i+1}}{x_{i}} … \frac{x_{k−1}}{x_{k−2}} ) \frac{y_{k}}{x_{k−1}}

∈ O(U^{k}⟨k − 1⟩_{1}).

Since m_{γ}/m_{id} can be written as a product of x_{j}/x_{i}, y_{j}/x_{i} with j ≥ i ≠ k, it follows that V_{1} = U^{k}⟨k − 1⟩_{1}, and more generally V_{σ} = U^{k}⟨k − 1⟩_{σ}. So Deligne’s desingularisation is the same as the one considered here.

7.1.5 C̃^{k} as a torus embedding

The complement

C^{k∗} = C^{k} − F^{k} = C^{k}[t^{−1}] = Spec κ[x_{1}^{±1}, … x_{k}^{±1}, t^{±1}]

is a torus over κ[t^{±1}]. There is a canonical isomorphism of its character group X with ℤ^{k}, given by attaching to \underline{m} ∈ ℤ^{k} the character

\underline{x} ↦ ∏_{i} x_{i}^{m_{1}}

Use this isomorphism to identify Hom(X, ℝ) with ℝ^{k}. Consider the cone over the unit cube [0, 1]^{k} ⊂ ℝ^{k}:

σ = {(μ_{1}, …, μ_{k}, ν) ∈ ℝ^{k+1} | 0 ≤ μ_{i} ≤ ν for all i}.

This is what in toroidal geometry [??] is called a rational polyhedral cone — it is the cone spanned by a finite subset of ℤ^{k+1}, containing no line. Associated to σ is the monoid

M_{σ} = {l = (m_{1}, …, m_{k}, n) ∈ ℤ^{k+1} | ⟨l, v⟩ ≥ 0 for all v ∈ σ}

= \{(\underline{m}, n) | n + ∑_{i∈I} m_{i} ≥ 0 for all I ⊂ \{1, …, k\}\}

= ⟨{e_{i}, e_{k+1} − e_{i}}_{1≤i≤k}⟩

and the torus embedding Z(σ) = Spec κ[M_{σ}], where

κ[M_{σ}] = \{linear cominations of monomials ∏ x_{i}^{m_{i}}t^{n}, (\underline{m}, n) ∈ M_{σ}\}

= κ[x_{1}, …, x_{k}, \frac{t}{x_{1}} , …, \frac{t}{x_{k}}]

= κ[x_{1}, y_{1}, …, x_{k}, y_{k}]/I_{k}

In toroidal geometry, to desingularise Z(σ) one finds a decomposition of σ into rational polyhedral cones which are smooth—this means that all their faces are simplices. In this case, to do this is equivalent to giving a rational triangularisation of the cube [0, 1]^{k} ⊂ ℝ^{k}. But this is easily done inductively: we first write the cube as the union of 2k cones, which are simply the cones on the codimension 1 faces with common vertex the point (1/2, …, 1/2). Triangulating each of these faces by induction gives a triangularisation of these cones, hence of the whole cube. Some pictures may help:

[figure: •]

[figure: •; •; •]

The groups Γ^{k} acts on the unit cube (it is its full group of symmetries) and permutes the 2^{k−1}k! k-simplices of this triangularisation transitively. A typical simplex is the one with vertices

(0, . . . . . . . . . . . . , 0)
(0, . . . . . . . . . , 0, 1)
(0, . . . . . . 0, \frac{1}{2}, \frac{1}{2})
(0, …, 0, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})
⋮
(\frac{1}{2}, … \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})

Let τ ⊂ σ be the cone on this simplex in ℝ^{k+1}. The associated monoid M_{τ} is the set of (\underline{m}, n) satisfying

\begin{cases}n ≥ 0 \\ m_{1} + n ≥ 0 \\ \frac{1}{2} ∑_{i=1}^{r} m_{i} + n ≥ 0 (2 ≤ r ≤ k) \end{cases}

which is freely generated by {e_{i} − e_{i+1}}_{1≤i≤k−1}, e_{k} and e_{k+1} − e_{1} − e_{2}. Hence Z(τ) = Spec κ[M_{τ}] and

κ[M_{τ}] = κ[\frac{x_{1}}{x_{2}} , …, \frac{x_{k−1}}{x_{k}} , xk, \frac{t}{x_{1}x_{2}} ]

= κ[\frac{x_{1}}{x_{2}} , …, \frac{x_{k−1}}{x_{k}} , xk, \frac{y_{1}}{x_{2}} ]

yand Z(τ) = U^{k}⟨k − 1⟩_{?}. The decomposition of σ into the Γ^{k}-translates of τ therefore gives the desingularisation C̃^{k}.

7.1.6 The absolute case [Agh]

Let R be any regular domain, and π ∈ R a non-zero element such that R/πR is also regular. Consider the construction above, with κ = ℤ. We have the desingularisation:

C̃^{k} ⟶ C^{k}=Spec ℤ[x_{1}, …, y_{k}]/I_{k}
↓
S = Spec ℤ[t]

Let C_{π}^{k}, C̃_{π}^{k} be the basechanges of C^{k}, C̃^{k} by the homomorphism ℤ[t] → R, t ↦ π. So C_{π}^{k} is the k-fold fibre product over R of C_{π}^{1} = Spec R[x, y]/(xy − π).

Proposition 7.1.6.1. C̃_{π}^{k} is regular.

Proof: It is enough to look at the basechange of U^{k}⟨k − 1⟩_{1} (7.1.3.1). We have

U^{k}⟨k − 1⟩_{1} = Spec ℤ[x_{1}, \frac{x_{2}}{x_{1}} , \frac{x_{3}}{x_{2}} , …, \frac{x_{k−1}}{x_{k−2}} , \frac{x_{k}}{x_{k−1}} , \frac{y_{k}}{x_{k−1}} ]
= Spec ℤ[x_{1}, x′_{2}, …, x′_{k−1}, x′_{k}, y′_{k}]

and t = x_{k}y_{k} = (x_{1}x′_{2} … x′_{k−1})^{2}x′_{k}y′_{k}. Hence

U^{k}⟨k − 1⟩ ⊗ R = Spec R[x_{1}, x′_{2}, …, x′_{k−1}, x′_{k}, y′_{k}]/(π − (x_{1}x′_{2} … x′_{k−1})^{2}x′_{k}y′_{k})

which we just have to show is regular. It is obviously regular at those points at which it is smooth over R. The nonsmooth locus consists of points P where either x_{1}x′_{2} … x′_{k−1} = 0 or x′_{k} = y′_{k} = 0, by the Jacobian criterion. So at such a point P either one of x_{1}, x′_{2}, …, x′_{k−1} is in the maximal ideal, or both of x′_{k}, y′_{k} do, hence in either case π = (x_{1}x′_{2} … x′_{k−1})^{2}x′_{k}y′_{k} ∈ m_{P}^{2} , from which it follows that we have a regular scheme.

7.2 Cohomology of the cones and their resolutions

7.2.1 Axioms for homology and cohomology

The calculations of homology and cohomology done in this and the next two sections require axioms rather weaker than those considered so far. We fix a coefficient ring A, and list what we need:

1. For every κ-variety X, there are A-modules H_{q}(X) for q ∈ ℤ, with the following properties:

(a) For closed (resp. open) immersions f: Y → X there are functorial module homomorphisms

f_{∗}: H_{q}(Y) → H_{q}(X) (resp., f^{∗}: H_{q}(X) → H_{q}(Y)).

(b) H_{∗}(−) is additive: if X = ⊔X_{ν} and i_{ν}: X_{ν} ↪ X are the inclusions,

then i_{ν}^{∗}i_{ν∗} = id and (i_{ν}^{∗}): H_{q}(X) ⟶^{∼} ⨁ H_{q}(X_{ν}).

(c) For i: Y ↪ X a closed immersion with complement j: X – Y = U ↪ X, there is a long exact sequence

⟶ H_{q}(Y) ⟶^{i_{∗}} H_{q}(X) ⟶^{j^{∗}} H_{q}(U) ⟶^{∂_{X,Y}} H_{q−1}(Y) ⟶

whose boundary map ∂_{X,Y} is A-linear and is functorial with respect to open immersion of pairs (X′ ↪ X, Y′ = Y ∪ X′ ↪ Y).

2. For every smooth κ-variety X there is a ℤ-graded anti-commutative Aalgebra H^{∗}(X), with the properties:

(a) X → H^{∗}(X) is a contravariant functor with respect to arbitrary morphisms, and is additive with respect to disjoint unions.

(b) There are functorial homomorphisms of abelian groups

λ = λ_{X}: O^{∗}(X) → H^{1}(X)

for smooth varieties X.

(c) If X = 𝔾_{m} × S = Spec κ[t, t^{−1}] × S then the map

H^{q}(S) ⊕ H^{q−1}(S) ⟶ H^{q}(𝔾_{m} × S)
(a, b) ↦ pr_{2}^{∗}a + λ(t) · pr_{2}^{∗}b

is an isomorphism.

3. For every smooth irreducible X there is an isomorphism

η_{X}: H^{q}(X) ⟶^{∼} H_{2 dim X−q}(X)

which is functorial with respect to open immersions, and satisfies the following compatibility: if S is irreducible and smooth of dimension d, and (X, Y) denotes the pair 𝔸^{1} × S, {0} × S), then the composite homomorphism

H^{q}(S) ⟶^{t·(−)} H^{q+1}(𝔾_{m} × S) ⟶_{η_{𝔾m×S}}^{∼} H_{2d+1−q}(𝔾_{m} × S)

⟶^{∂_{X,Y}} H_{2d−q}(S) ⟶^{η_{S}^{−1}} H^{q}(S)

is multiplication by ±1.

4. (As required) For every κ-variety X there is a graded A-module H_{c}^{∗}(X), covariant (resp. contravariant) with respect to open (resp. closed) immersions, with the properties:

(a) For every pair (X, Y) as in (1.ii) above there is a long exact sequence

⟶ H_{c}^{q}(U) ⟶^{j_{∗}} H_{c}^{q}(X) ⟶^{i^{∗}} H_{c}^{q}(Y) ⟶^{∂_{c}^{X,Y}} H_{c}^{q+1}(U) ⟶

functorial with respect to open immersions of pairs.

(b) There is an isomorphism for smooth and proper X

H_{c}^{∗}(X) ⟶^{∼} H^{∗}(X)

which is functorial with respect to closed immersions.

5. (For a weight-graded theory) The groups H_{q}(X), H^{q}(X), H_{c}^{q}(X) are ℤgraded: H_{q}(X) = ⊕_{n∈ℤ}H_{q}(X, n) etc., such that all morphisms in the above axioms are graded of degree 0, except:

(a) λ_{1} : O^{∗}(X) → H^{1}(X, 1) ⊂ H^{1}(X).

(b) η_{X}: H^{q}(X, n) ⟶^{∼} H_{2 dim X−q}(X, dim X − n).

In the case of X = 𝔾_{m}, then we simply write t for λ_{1}(t) ∈ H^{1}(𝔾_{m}, 1) = H_{1}(𝔾_{m}, 0). Note that by axiom 2(b), the automorphism x → x^{−1} maps t to −t, and so in 2(c) the decomposition is the decomposition of H^{∗}(𝔾_{m} × S) into its +1 and −1 eigenspaces under x ↦ x^{−1}. In homology the decomposition 2(c) becomes

H_{q−2}(S, n) ⊕ H_{q−1}(S, n) ⟶^{∼} H_{q}(𝔾_{m} × S, n).

7.2.2 Tori

We work throughout this section with a cohomology/homology theory (H^{∗}, H_{∗}) for varieties over κ which satisfies the axioms (1)–(2) of the previous section. We shall assume for the sake of the notation that we have a weight-graded theory (4).

We now consider k ≥ 1 copies of 𝔾_{m}. Let Γ^{k} act on 𝔾_{m}^{k} by:

• Σ_{k} permuting the coordinates; and

• (a_{i}) ∈ μ_{2}^{k} acting by the map (x_{i}) ↦ (x_{i}^{a_{i}}).

We get immediately:

Proposition 7.2.2.1. There is a canonical isomorphism, for every smooth S/κ

H^{∗}(𝔾_{m}^{k} × S, •) ≃ H^{∗}(S, •) ⊗ Λ^{∗}(t_{1}, …, t_{k})

where t_{i} = pr_{i}^{∗}(t) ∈ H^{1}(𝔾_{m}^{k}, 1) and Λ^{∗}(t_{1}, …, t_{k}) is the exterior algebra on t_{1}, …, t_{k}. This is functorial in S and Γ^{k}-equivariant, for the natural action of Γ^{k} on Λ^{∗}(t_{1}, …, t_{k}).

Corollary 7.2.2.2. (i) If q ∈ 𝔾_{m}^{k}(S) is any section of finite order, then translation by q acts trivially on H^{∗}(𝔾_{m}^{k} × S, •).

(ii) If f: 𝔾_{m}^{k} × S → 𝔾_{m}^{k} × S is an isogeny of tori, whose degree is invertible in A, it induces an isomorphism f^{∗}: on cohomology.

Proof: Let q = (q_{1}, …, q_{k}), and assume q^{N} = 1. Then λ_{1}(q_{i}) ∈ H^{1}(S, 1) is zero since λ_{1} is a homomorphism, and translation by q maps t_{i} ∈ H^{1}(𝔾_{m}^{k} ×S, 1) to t_{i} + pr_{S}^{∗}λ_{1}(q_{i}) = t_{i}. Therefore (i) follows from the proposition. For (ii), note that f is given by k × k integer matrix with determinant ± deg f, and under the hypotheses that induces an automorphism of Λ^{∗}(t_{1}, …, t_{k}).

Corollary 7.2.2.3. Cup-product with t_{1} ∪ ⋯ ∪ t_{k} defines isomorphisms

H^{q}(S, n) ⟶^{∼} H^{q+k}(𝔾_{m}^{k} × S, n + k)(ϵ^{k})
H_{q}(S, n) ⟶^{∼} H_{q+k}(𝔾_{m}^{k} × S, n)(ϵ^{k})  (7.2.2.1)

for every (q, n) and every smooth S/κ.

We also need to consider the exact sequence of tori

1 → 𝔾_{m}^{k} ⟶^{θ} 𝔾_{m}^{k+1} ⟶^{π} 𝔾_{m} → 1

where

π: (t_{0}, …, t_{k}) ↦ ∏_{i=0}^{k} t_{i}

θ: (t_{1}, …, t_{k}) ↦ (∏_{i=1}^{k} t_{i}^{−1}, t_{1}, …, t_{k})

The symmetric group Σ_{k+1} acts on 𝔾_{m}^{k+1} preserving the image of θ; this defines an action of Σ_{k+1} on H^{∗}(𝔾_{m}^{k}). The sign-eigenspace is then given as follows:

Proposition 7.2.2.4. Cup-product with t_{1} ∪ ⋯ ∪ t_{k} induces an isomorphism

H^{q}(S, n) ⟶^{∼} H^{q+k}(𝔾_{m}^{k} × S, n + k)(sgn^{k+1})

for every smooth S/κ and every (q, n).

Proof: Let r be an integer with 0 ≤ r ≤ k, and consider the action of the signprojector Π(sgn^{k+1}) on the subspace of H^{r}(𝔾_{m}^{k}, r) spanned by cup-products t_{α(1)} ∪ ⋯ ∪ t_{α(r)}. For any transposition τ ∈ Σ_{k+1} one has Π(sgn^{k+1}) ∘ τ = −Π(sgn^{k+1}), so if if r < k then

Π(sgn^{k+1})t_{α(1)} ∪ ⋯ ∪ t_{α(r)} = 0

(since we can find a transposition fixing α(1), …, α(r)). On the other hand, t_{1} ∪ ⋯ ∪ t_{k} clearly transforms under Σ_{k+1} by the sign character, so the result follows from the proposition.

Remark: One shows also ?? that for an elliptic curve E/S one has a similar identity

H^{∗}(E^{k}, •)(sgn^{k+1}) ≃ H^{∗}(E^{k}, •)(ϵ^{k});

the proof is not as trivial.

Finally we assume that our theory also admits groups with compact support H_{c} satisfying the axioms (3) of the last section. The reader will easily verify that one gets the following result:

Proposition 7.2.2.5. For any smooth S/κ,

H_{c}^{∗}(𝔾_{m} × S, •) ⟶^{∼} H_{c}^{∗−2}(S, • − 1) ⊕ H_{c}^{∗−1}(S, •)

the decomposition given by + and − eigenspaces for t ↦ t^{−1}, and for any k ≥ 1,

H_{c}^{∗}(𝔾_{m}^{k} × S, •)(ϵ^{k}) ⟶^{∼} H_{c}^{∗−k}(S, •).

7.2.3 Strata

The basic result we need is:

Proposition 7.2.3.1. For every k ≥ 2 and every smooth S/κ

H^{∗}(P^{k,reg} × S, •)(ϵ^{k}) = 0

(eigenspace for the action of Γ^{k}).

Proof: To simplify the notation we shall write

H(−) = H^{∗}(− × S, •).

Recall from 7.1.1 that

P^{k,reg} = \{(\underline{x}, \underline{y}) ∈ P^{k} |no two pairs of coordinates (x_{i}, y_{i}) simultaneously vanish\}

Define open subschemes P^{k∗} ⊂ P^{k∗∗} of P^{k,reg} by

P^{k∗} = \{(\underline{x}, \underline{y}) ∈ P^{k} | ∏ x_{i}y_{i} ≠ 0\}
P^{k∗∗} = \{(\underline{x}, \underline{y}) ∈ P^{k} | no pair (x_{i}, y_{i}) vanishes\}

The complement

P^{k,reg} – P^{k∗∗} = \{(\underline{x}, \underline{y}) ∈ P^{k} | exactly one pair (x_{i}, y_{i}) = (0, 0)\}

is the union of 2^{k−1}k components, permuted transitively by Γ^{k}. The component

{x_{1} = y_{1} = y_{2} = ⋯ = y_{k} = 0, x_{2}x_{3} … x_{k} ≠ 0}

is isomorphic to 𝔾_{m}^{k−2}, and its stabiliser is μ_{2} × Σ_{k−1} ⊂ Γ^{k}. Hence

H(P^{k,reg} – P^{k∗∗}) = Ind_{μ_{2}×Σ_{k−1}}^{Γ^{k}} H(𝔾_{m}^{k−2}).

But μ_{2} acts by the transposition x_{1} ↔ y_{1}, hence acts trivially this component and so on H(𝔾_{m}^{k−2}). As ϵ^{k}|_{μ_{2}} ≠ 1 we have

H(P^{k,reg} – P^{k∗∗})(ϵ^{k}) = 0  (7.2.3.1)

by Frobenius reciprocity. By the localisation sequence, the restriction map is an isomorphism

H(P^{k,reg})(ϵ^{k}) ⟶^{∼} H(P^{k∗∗})(ϵ^{k}).

We shall compute the cohomology of P^{k∗∗} by using the exact homology sequence for the pair (P^{k∗∗}, P^{k∗}). First we compute the cohomology of the open P^{k∗}. Observe that the map

ω: 𝔾_{m}^{k} → P^{k∗}
(z_{1}, …, z_{k}) ↦ (z_{1}, z_{1}^{−1}, …, z_{k}, z_{k}^{−1})

is an isogeny with kernel ⟨(−1, …, −1)⟩ ≃ μ_{2}, and that it is Γ^{k}-equivariant. Therefore by 7.2.2.2 ω induces a Γ^{k}-equivariant isomorphism

H(P^{k∗}) ⟶^{∼} H(𝔾_{m}^{k}).

Since ω(x_{i}/y_{i}) = z_{i}^{2}, we obtain by cup-product an isomorphism

H^{q}(S, n) ⟶^{∼} H^{q+k}(P^{k∗} × S, n + k)(ϵ^{k})
c ↦ c ∪ λ_{1}(\frac{x_{1}}{y_{1}}) ∪ ⋯ ∪ λ_{1}(\frac{x_{k}}{y_{k}}).

Now compute the boundary. We have

P^{k∗∗} – P^{k∗} = \{(\underline{x}, \underline{y}) ∈ ℙ^{2k−1} | for each i, exactly one of x_{i}, y_{i} vanishes\}

which therefore has 2^{k} components, permuted transitively by Γ^{k}. The stabiliser of the component

Q = \{x_{1} = ⋯ = x_{k} = 0, ∏ y_{i} ≠ 0\} ∼ ; ⟶ ; ( \frac{y_{2}}{y_{1}} ,..., \frac{y_{k}}{y_{1}} ) 𝔾_{m}^{k−1}  (7.2.3.2)

is Σ_{k} ⊂ G^{k}, and under the isomorphism 7.2.3.2, the action of Σ_{k} on 𝔾_{m}^{k−1} is that described in 7.2.2.4. So

H(P^{k∗∗} ⊂ P^{k∗}) = Ind_{Σ_{k}}^{Γ^{k}} H(Q)

and

H^{q}((P^{k∗∗} – P^{k∗}) × S, n)(ϵ^{k}) = H^{q}(Q × S, n)(sgn^{k}) ⟶^{∼} H^{q−k+1}(S, n − k + 1),

the last isomorphism given by cup-product with

λ_{1}(\frac{y_{2}}{y_{1}}) ∪ ⋯ ∪ λ_{1}(\frac{y_{k}}{y_{1}}).  (7.2.3.3)

We shall show that

∂: H^{q+1}(P^{k∗} × S, n + 1)(ϵ^{k}) → H^{q}((P^{k∗∗} – P^{k∗}) × S, n)(ϵ^{k})  (7.2.3.4)

is an isomorphism, and by what we have just said it is enough to prove that the boundary for the pair (P^{k∗} ∪ Q, Q)

∂: H^{k}(P^{k∗}, k) → H^{k−1}(Q, k − 1)

maps the ‘generator’ (axiom 2(c) above)

λ_{1}(\frac{x_{1}}{y_{1}}) ∪ ⋯ ∪ λ_{1}(\frac{x_{k}}{y_{k}})

to a non-zero multiple of 7.2.3.3. Explicitly,

P^{k∗} ∪ Q = Spec κ [\frac{x_{1}}{y_{1}} , (\frac{y_{2}}{y_{1}})^{±1}, …, (\frac{y_{k}}{y_{1}})^{±1}] ≃ 𝔸^{1} × Q

with Q identified with {0} × Q, given by the equation x_{1}/y_{1} = 0. Since x_{i}/y_{i} = (x_{1}/y_{1})(y_{1}/y_{1})^{2} on P^{k}, we can write

λ_{1}(\frac{x_{1}}{y_{1}}) ∪ ⋯ ∪ λ_{1}(\frac{x_{k}}{y_{k}}) = (−2)^{k−1}λ_{1}(\frac{x_{1}}{y_{1}}) ∪ λ_{1}(\frac{y_{2}}{y_{1}}) ∪ ⋯ ∪ λ_{1}(\frac{y_{k}}{y_{1}})

= λ_{1}(\frac{x_{1}}{y_{1}}) ∪ pr_{Q}^{∗}η

where

η = (−2)^{k−1}λ_{1}(\frac{y_{2}}{y_{1}}) ∪ λ_{1}(\frac{y_{k}}{y_{1}}) ∈ H^{k−1}(Q, k − 1).

Therefore (using axiom 3 to compute the boundary)

∂: λ_{1}(\frac{x_{1}}{y_{1}}) ∪ pr_{Q}^{∗}η ↦ η

as required. So 7.2.3.4 is an isomorphism, hence by the long exact cohomology sequence and 7.2.3.1 we obtain H(P^{k,reg})(ϵ^{k}) = 0.

Corollary 7.2.3.2. For every k ≥ 2 and any smooth S,

H^{∗}(P̃^{k} × S, •)(ϵ^{k}) = 0.

Proof: We compute in homology rather than cohomology (since P̃^{k} × S is smooth). Recall from Theorem 7.1.2.1 the filtration

P̃^{k} ⊃ Y_{1}^{k} ⊃ ⋯ ⊃ Y_{k−2}^{k} ⊃ Y_{k−1}^{k} = ∅

whose j-th successive stratum is

Y_{j}^{k} – Y_{j+1}^{k} ≃ Γ_{k} ×^{Σ_{j}×Γ^{k−j}} (Z_{j} × P^{k−j,reg})

(0 ≤ j ≤ k − 2). We therefore have

H_{∗}((Y_{j}^{k} – Y_{j+1}^{k}) × S, •)(ϵ^{k}) ≃ Ind_{Σ_{j}×Γ^{k−j}}^{Γ^{k}} H_{∗}(Z_{j} × P^{k−j,reg} × S, •)(ϵ^{k})
⊂ H_{∗}(Z_{j} × P^{k−j,reg} × S, •)(ϵ^{k−j}) = 0.

So by the long exact homology sequence we obtain the required result.

Suppose we have cohomology groups with compact support satisfying the axioms (4), and moreover that κ has characteristic zero. Then we get:

Proposition 7.2.3.3. For every k ≥ 2 and every smooth κ-scheme S

H_{c}^{∗}(P^{k,reg} × S)(ϵ^{k}) = 0.

Proof: Since κ has characteristic zero, we can find a smooth proper κ-scheme S̄ and a filtration by closed subschemes of strictly decreasing dimension

S̄ = S̄_{0} ⊃ S̄_{1} ⊃ ⋯ ⊃ S̄_{r} = ∅

such that S̄_{0} – S̄_{1} = S and each stratum S̄_{i} – S̄_{i+1} is smooth. By induction on dimension we may assume that

H_{c}^{∗}(P^{k,reg} × (S_{i} – S̄_{i+1}), •)(ϵ^{k}) = 0 for all k ≥ 2 and all i > 0;

hence by repeated application of the long exact sequence for H_{c} we are reduced to the case S = S̄; i.e., we can assume that S is proper.

For k = 2, P^{2,reg} = P^{2} is proper, hence

H_{c}^{∗}(P^{2,reg} × S)(ϵ^{2}) = H^{∗}(P^{2} × S)(ϵ^{2}) = 0

by (7.2.3.1). In general, we have

H_{c}^{∗}(P̃^{k} × S, •)(ϵ^{k}) = H^{∗}(P̃^{k} × S, •)(ϵ^{k}) = 0

and by induction on k, the cohomology of each stratum of strictly positive codimension vanishes:

H_{c}^{∗}((Y_{j}^{k} – Y_{j+1}^{k}) × S, •)(ϵ^{k}) ≃ Ind_{Σ_{j}×Γ^{k−j}}^{Γ^{k}} H_{c}^{∗}(Z_{j} × P^{k−j,reg} × S, •)(ϵ^{k})
⊂ H_{c}^{∗}(Z_{j} × P^{k−j,reg} × S, •)(ϵ^{k−j}) = 0 if j > 0.

So by the long exact sequence for H_{c}, we have also

H_{c}^{∗}(P^{k,reg} × S, •)(ϵ^{k}) = H_{c}^{∗}((P̃^{k} – Y_{1}^{k}) × S, •)(ϵ^{k}) = 0.

7.2.4 Gillet-Soulé theory

[eventually to move to Chap 2]

The simplest example of a theory with compact supports is obtained by taking κ = ℂ, H^{∗}, H_{∗} to be singular cohomology and Borel-Moore homology respectively, and H_{c}^{∗} to be singular cohomology with compact supports.

A much less trivial example has been worked out by Gillet and Soulé [GS]. The construction they give assumes characteristic zero (so as to be able to apply resolution of singularities) but can also be done in characteristic p, using de Jong’s method of alterations [dJ]. Let

H^{q}(X, n) = K_{2n−q}(X) ⊗ ℚ
H_{q}(X, n) = K′_{q−2n}(X) ⊗ ℚ
H_{c}^{q}(X, n) = K_{2n−q}^{c}(X) ⊗ ℚ

where K^{c} is K-theory with compact support, defined as follows: for a quasiprojective κ-variety X one takes a diagram of the form

Z_{•} ⟶_{j} Z̄_{•} ⊃ Y_{•}
↓
X

where:

• Z̄_{•} is a smooth, projective simplicial scheme over κ;

• each j_{ν}: Z_{n} ↪ Z̄_{ν} is an open immersion, and Z_{•} → X is a hyperenvelope [GS];

• Y_{ν} ⊂ Z̄_{ν} is a reduced divisor with normal crossings, whose compement is Z_{ν}.

Then K_{i}^{c}(X) is the “K-theory of Z̄_{•} relative to Y_{•}” — precisely,

K_{i}^{c}(X) = π_{i}(hofib(holimKZ̄_{•} → holimKY_{•})).

Gillet and Soulé show that K_{∗}^{c}(X) depends functorially only on X, and satisfies (amongst others) the properties of section 7.2.1.

7.2.5 Patching the cones

We need one further fact to use the results of the previous sections to compute the homology of the boundary of the Kuga-Sato variety. What follows is not strictly needed to obtain (7.4.1.3) below, but simplifies the original argument [MF] slightly, and in any case sheds more light on the structure of the boundary.

We define D^{1} to be the union of 2 copies of C^{1}:

D^{1} = Spec κ[x, y] ∪ Spec κ[x̄, ȳ]

glued by the identification x̄ = x^{−1}, ȳ = x^{2}y along the open affines

Spec κ[x, x^{−1}, y] ⟶^{∼} Spec κ[x̄, x̄^{−1}, ȳ].

The morphisms (x, y) ↦ t = xy, (x̄, ȳ) ↦ t = x̄ȳ glue to give a morphism D^{1} → 𝔸_{κ}^{1}. Notice that the open set D^{1}[1/t] = Spec[x^{±1}, y^{±1}] on which t is invertible is contained in both copeies of C^{1}. The special fibre {t = 0} ⊂ D^{1} is the union of 2 copies of 𝔸^{1} and one copy of ℙ^{1}:

[figure: M; y = 0; ȳ = 0; •; •; p_{0}; p̄_{0}; L; L̄; x = 0; x̄ = 0]

For later use we denote these lines by L, L̄ and M, and their intersections by p_{0}, p̄_{0} as depicted in the diagram. Let

D^{k} = \underbrace{D^{1} ×_{𝔸^{1}} × ⋯ × D^{1}}_{k copies}

be the k-fold fibre product. Clearly D^{k} has an open covering by 2^{k} copies of C^{k}. Let’s denote these \underline{a}C^{k}, \underline{a} ∈ \{±1\}^{k}.

We have an action of the group Γ^{k} on D^{k}, but it does not restrict to give the action on C^{k} used earlier — in fact, it does not even fix the \underline{a}C^{k}. Rather, we have an involution ι of D^{1} which interchanges the two copies of C^{1} by (x, y) ↔ x̄, ȳ. As xy = x̄ȳ, it is an automorphism over 𝔸^{1}. Therefore we get on the fibre product D^{k} a group of automorphisms isomorphic to μ_{1}^{k} generated by things of the form (1, …, ι, …, 1). This group action permutes simply transitively the elements \underline{a}C^{k} of the open affine cover of D^{k}.

Applying the same procedure as (7.1.2), we get a desingularisation

D̃^{k} = D^{k}⟨k − 1⟩ → D^{k}⟨k − 2⟩ → ⋯ → D^{k}⟨1⟩ → D^{k}⟨0⟩ = D^{k}

where D^{k}⟨j + 1⟩ is the blowup of D^{k}⟨j⟩ along the strict transform of ⋃\underline{a} \underline{a}F_{j}^{k}, for 0 ≤ j ≤ k − 2. We clearly have

D^{k}⟨j⟩ = ⋃_{\underline{a}} \underline{a}C^{k}⟨j⟩

where \underline{a}C^{k}⟨j⟩ is the inverse image of \underline{a}C^{k} in D^{k}⟨j⟩, and \underline{a}C^{k}⟨j⟩ is isomorphic to the C^{k}⟨j⟩ constructed earlier. We shall simply use the notation ϕ_{j}, ψ_{j} to denote the various birational morphisms for D^{k} which correspond to those denoted by the same symbol for C^{k}.

The filtration {W_{j}^{k}} of the fibre {t = 0} ⊂ C̃^{k} gives a filtration

D̃^{k} ⊃ {t = 0} ⊃ Q_{0}^{k} ⊃ Q_{1}^{k} ⊃ ⋯ ⊃ Q_{k−1}^{k} = ∅

and for 0 ≤ j ≤ k − 2

Q_{j}^{k} – Q_{j+1}^{k} → ⋃_{\underline{a}} \underline{a}F^{k}⟨j⟩  (7.2.5.1)

is a fibration for the Zariski toplogy, with fibre P^{k−j,reg}.

Proposition 7.2.5.1. The fibration (7.2.5.1) is trivial.

This is a matter of straightforward computation. First we should look at the base ∪\underline{a}\underline{a}F^{k}⟨j⟩ of this fibration. It is the proper transform of ∪\underline{a}\underline{a}F_{j}^{k}, and by definition

⋃_{\underline{a}} \underline{a}F_{j}^{k} = \left\{{all points (p_{1}, …, p_{k}) ∈ \{t = 0\} ⊂ D^{k} such \atop that at least (k − j) p_{i}’s equal p_{0} or p̄_{0}.}\right\}

Looking at the irreducible components of this subscheme, they are all obtained as Γ^{k}-translates of one of the following types:

\underbrace{M × ⋯ × M}_{r} × \underbrace{L × ⋯ × L}_{j−r} × \underbrace{\{p_{0}\} × ⋯ × \{p_{0}\}}_{k−j} .

We shall look at the component

\underbrace{M × ⋯ × M}_{j} × \underbrace{\{p_{0}\} × ⋯ × \{p_{0}\}}_{k−j} .

It lies in D^{j} ×_{𝔸^{1}} C^{k−j} ⊂ D^{k}, and its proper transform in D^{k}⟨j⟩ lies in the union of the open affines

\underline{a}U^{k}⟨j⟩_{σ} for σ ∈ Σ_{j}, a_{j+1} = ⋯ = a_{k} = 0.

For 1 ≤ i ≤ k − j we shall write

w_{i} = \begin{cases}x_{1} & if a_{i} = +1 \\ x̄_{1} & if a_{i} = −1. \end{cases}

Then if a_{j+1} = ⋯ = a_{k} = 0 and σ ∈ Σ_{j},

\underline{a}U^{k}⟨j⟩_{σ} = Spec κ [w_{σ(1)}, \frac{w_{σ(2)}}{w_{σ(1)}} , …, \frac{w_{σ(j)}}{w_{σ(j−1)}} , \frac{x_{j+1}}{w_{σ(j)}} , \frac{y_{j+1}}{w_{σ(j)}} , …, \frac{x_{k}}{w_{σ(j)}} , \frac{y_{k}}{w_{σ(j)}} ] . (7.2.5.2)

Let’s write

Ω_{j} = ⋃_{\underline{a}∈μ_{2}^{j},σ∈Σ_{j}} Spec κ [w_{σ(1)}, \frac{w_{σ(2)}}{w_{σ(1)}} , …, \frac{w_{σ(j)}}{w_{σ(j−1)}} ] ⊂ D^{k}⟨j⟩

which is a certain iterated blowup of (ℙ^{1})^{j}. The formula (7.2.5.2) shows that

ψ_{j}^{−1}(Ω_{j}) = \overline{Q_{j}^{k} – Q_{j+1}^{k}}

is isomorphic to the product Ω_{j} × P^{k−j}, and thus carries an action of Γ^{k−j}, which is compatible with the action on the various \underline{a}U^{k}⟨j⟩_{σ}’s. We also see that Ω_{j} itself carries an action of Γ^{j}.

7.3 Products of Tate curves

We can begin to reap the fruits of our study of cones. We assume that we are given a fixed complete discrete valuation ring R, with uniformiser π, residue field κ, and field of fractions F. Let n be a positive integer, and write q = π^{n} and S = Spec R. Recall (??) that in this case the n-sided Tate curve

Tate_{n}(q) → S

has the following properties:

• The fibre Tate_{n}(q) ⊗ κ is a Néron polygon with n sides.

• There is a closed immersion of S-schemes α: ℤ/n ↪ Tate_{n}(q), whose restriction to the generic fibre Tate_{n}(q) ⊗ F is a homomorphism. The automorphisms of Tate_{n}(q)⊗F which are translation by elements of im(α) extend to an action of ℤ/n on Tate_{n}(q). Together with inversion in the fibres, they generate a subgroup ℤ/n ×μ_{2} ⊂ Aut(Tate_{n}(q)/S).

• For n ≥ 2, the formal π-adic completion Tate_{n}(q)^{∧} of the Tate curve has an open covering by n copies of the formal completion of C_{π}^{1}. Each copy contains exactly one vertex of the special fibre.

• For n = 1, there is an étale covering C_{π}^{1} → Tate_{1}(q)^{∧}.

Remark. The action of μ_{2} ⊂ ℤ/n ×μ_{2} on Tate_{n}(q) does not necessarily restrict to the action of μ_{2} on C_{π}^{1}^{∧}. If n is even, the fixed points of the non-trivial element of this μ_{2} are the points 1 in the 0-th and n/2-th copies of ℙ^{1} ⊂ Tate_{n}(q) ⊗ κ. However in the whole dihedral subgroup ℤ/n ×μ_{2} acting on Tate_{n}(q), corresponding to each vertex ν of the n-gon Tate_{n}(q) ⊗ κ there is a reflection r_{ν} (i.e., an element of the form (b, −1) ∈ ℤ/n ×μ_{2}) which fixes ν and the copy of C_{π}^{1}^{∧} containing ν. On this copy, it acts by the automorphism x ↔ y (7.1.1).

We can therefore apply the results of the previous sections to the k-fold fibre product

T^{k} =^{def} Tate_{n}(q)^{k} = Tate_{n}(q) ×_{S} ⋯ ×_{S} Tate_{n}(q).

By successively blowing up singular strata of increasing dimension, we obtain a desingularisation (we use the same letters as for the desingularisation of C^{k}):

ϕ: T̃^{k} = T^{k}⟨k − 1⟩ ⟶^{ϕ_{k−2}} … ⟶^{ϕ_{1}} T^{k}⟨1⟩ ⟶^{ϕ_{0}} T^{k}⟨0⟩ = T^{k}
↓^{ψ_{j}}
T^{k}⟨j⟩

and a filtration by divisors

T̃^{k} ⊃ T̃^{k} ⊗ κ ⊃ Q_{0}^{k} ⊃ ⋯ ⊃ Q_{k−2}^{k} ⊃ Q_{k−1}^{k} = ∅

such that

Q_{j}^{k} = ψ_{j}^{−1} (T^{k}⟨j⟩^{sing}) .

We have for each j a closed subscheme

E^{k}⟨j⟩ ⊂ T^{k}⟨j⟩

which is smooth over κ, and which is the centre of the blowup ϕ_{j}. Acting on this picture we have the group

Δ_{n}^{k} = (ℤ/n ×μ_{2})^{k} ×Σ_{k}

where ℤ/n ×μ_{2} acts on each copy of Tate_{n}(q) as explained above, and Σ_{k} acts by permuting the factors of the product.

There are \binom{k}{j}n^{k} components of E^{k}⟨j⟩; in (k − j) of the factors one takes one of the n vertices of the polygon, and in the remaining j one of the sides. Each component is isomorphic to Ω_{j}, and they are permuted transitively by Δ_{n}^{k}, the stabiliser of one being Γ^{j} × Γ^{k−j}. Here Γ^{j} acts as described in (7.2.5) and Γ^{k−j} acts trivially. Putting together what we have obtained in the first and third sections of this chapter, we get the following description of the stratification on T̃^{k}:

Proposition 7.3.0.2. There are (for a suitable copy of Γ^{j} × Γ^{k−j} in Δ_{n}^{k}) Δ_{n}^{k}- equivariant isomorphisms

\overline{Q_{j}^{k} – Q_{j+1}^{k}} ⟶^{∼} Δ_{n}^{k} ×^{Γ^{j}×Γ^{k−j}} (Ω_{j} × P̃^{k−j})
∪  ∪
Q_{j}^{k} – Q_{j+1}^{k} ⟶^{∼} Δ_{n}^{k} ×^{Γ^{j}×Γ^{k−j}} (Ω_{j} × P^{k−j,reg})
↓^{ψ_{j}}  ↓
E^{k}⟨j⟩ ⟶^{∼}  Δ_{n}^{k} ×^{Γ^{j}×Γ^{k−j}} Ω_{j}

Now write ϵ_{n}^{k}: Δ_{n}^{k} → μ_{2} for the unique character which is trivial on (ℤ/n)^{k} and for which

ϵ_{n}^{k}|_{μ_{2}^{k}}: (a_{i}) ↦ ∏_{i} a_{i}
ϵ_{n}^{k}|_{Σ_{k}}: σ ↦ sgn(σ)

Notice that the restriction of ϵ_{n}^{k} to the subgroup Γ^{k} × Δ_{n}^{k} is the product ϵ^{j}ϵ^{k−j}. This is virtually obvious — one just has to remember that the embedding Γ^{j} ⊂ Δ_{n}^{k} is not the obvious one, but the one which maps μ_{2} to the subgroup generated by (b, −1) ∈ ℤ/n ×μ_{2}, for some b. But since ϵ_{n}^{k} is trivial on (ℤ/n)^{k} this makes no difference.

We consider a weight-graded theory (H^{∗}, H_{∗}) for varieties over κ, which satisfies the axioms (1)–(3) of 7.2.1.

Theorem 7.3.0.3. (i) The inclusion

𝔾_{m}^{k}_{/κ} ↪ T̃^{k} ⊗ κ

induced by the inclusion of the identity component

𝔾_{m}^{k} ↪ (Tate_{n}(q) ⊗ κ^{reg})^{k} ↪ T̃^{k} ⊗ κ

induces isomorphisms in homology:

H_{∗}(T̃^{k} ⊗ κ, •)(ϵ_{n}^{k}) ⟶^{∼} H_{∗}(𝔾_{m}^{k}_{/κ}, •)(ϵ_{n}^{k})

⟶^{∼} H_{∗−k}(Spec κ, •)

(ii) If there is an associated cohomology theory with compact supports (7.2.1), then

H_{c}^{∗}(T̃^{k} ⊗ κ, •)(ϵ_{n}^{k}) ⟶^{∼} H_{c}^{∗}(𝔾_{m}^{k}_{/κ}, •)(ϵ_{n}^{k})

⟶^{∼} H^{∗−k}(Spec κ, •).

Proof: first we consider homology, and show that

H_{∗}(Q_{j}^{k}, •)(ϵ_{n}^{k}) = 0 for all j ≥ 0

(in fact we only need the case j = 0). By descending induction on j and the homology exact sequence, it is enough to show that

H_{∗}(Q_{j}^{k} – Q_{j+1}^{k}, •)(ϵ_{n}^{k}) = 0 for all j ≥ 0.

But by the proposition and Frobenius reciprocity it is enough to show

H_{∗}(Ω_{j} × P^{k−j,reg}, •)(ϵ^{j} × ϵ^{k−j}) = 0 for all j ≥ 0

and this follows from 7.2.3.1. Therefore we get

H_{∗}(Q_{0}^{k}, •)(ϵ_{n}^{k}) = 0.

So to prove part (i) of the theorem we have to show that

H_{∗}(T̃^{k} ⊗ κ – Q_{0}^{k}, •)(ϵ_{n}^{k}) ⟶^{∼} H_{∗}(𝔾_{m}^{k}_{/κ}, •)(ϵ_{n}^{k})

since the second isomorphism was already proved in 7.2.2. But the blowups induce an isomorphism away from Q_{0}^{k}

ψ: T̃^{k} – Q_{0}^{k} ⟶^{∼} Tate_{n}(q)^{k,reg}

and we have

Tate_{n}(q)^{k,reg} ⊗ κ = {(p_{1}, …, p_{k}) ∈ Tate_{n}(q)^{k} ⊗ κ | at most one p_{i} is a vertex}

because the complement of 𝔾_{m}^{k}_{/κ} is the set of (p_{1}, …, p_{k}) such that exactly one p_{i} is a vertex. In other words,

Tate_{n}(q)^{k,reg} ⊗ κ – 𝔾_{m}^{k}_{/κ} ≃ ∐_{j=0}^{k−1}(𝔾_{m} × ℤ/n)^{j} × ℤ/n × (𝔾_{m} × ℤ/n)^{k−j−1}
≃ Δ_{n}^{k} ×^{μ_{2}×Γ^{k−1}} 𝔾_{m}^{k−1}

where μ_{2} acts trivially on 𝔾_{m}^{k−1} and Γ^{k−1} acts as in 7.1.1. Therefore

H_{∗}(Tate_{n}(q)^{k,reg} ⊗ κ – 𝔾_{m}^{k}_{/κ}, •)(ϵ_{n}^{k}) = H_{∗}(𝔾_{m}^{k−1}, •)(ϵ_{n}^{k}|_{μ_{2}×Γ^{k−1}}).

But ϵ_{n}^{k}|_{μ_{2}} ≠ 1 so this last group vanishes. By the long exact homology sequence we get (i).

The proof for cohomology with compact support is identical replacing the homology exact sequence with that for H_{c}^{∗} and using 7.2.3.3.

7.4 Modular form motives

7.4.1 Definitions of the motives

Let n ≥ 3, k ≥ 1 be integers. Recall from ??.that \overline{X}_{n} is the regular minimal model of the elliptic surface X_{n} → M_{n}. Define:

• \overline{X}_{n}^{k} ⟶^{f^{k}} \overline{M}_{n} the k-fold fibre product of \overline{X}_{n} over \overline{M}_{n}.

• \overline{X}_{n}^{k,∞} = \overline{X}_{n}^{k} ×_{\overline{M}_{n}} M_{n}^{∞}.

Recall that for k > 1, the variety \overline{X}_{n}^{k} is smooth over ℚ except above the cusps, and that the formal completion of \overline{X}_{n} along \overline{X}_{n}^{∞} is isomorphic to the union of copies the q-adic completion of the Tate curve

Tate_{n}(q) over Spf ℚ[[q^{1/n}]].

Therefore we can apply the desingularisation of the previous section to \overline{X}_{n}^{k}, and

obtain a birational projective morphism \overline{\overline{X}}_{n}^{k} → \overline{X}_{n}^{k}, which is an isomorphism over \overline{X}_{n}^{k}^{reg}.

Acting on \overline{X}_{n}^{k} is the finite group

Γ_{n}^{k} = ((ℤ/n)^{2} ×μ_{2})^{k} ×Σ_{k}

and we have the character ϵ_{n}^{k}: Γ_{n}^{k} → μ_{2} which is trivial on each factor ℤ/n, the identity on each factor μ_{2} and the sign character on Σ_{k}. The construction of

the previous section shows that the action of Γ_{n}^{k} extends to \overline{\overline{X}}_{n}^{k}.

Definition 7.4.1.1. The parabolic cohomology motive (of weight k+2 and level n) is the Chow motive

W_{n}^{k} =^{def} h(\overline{\overline{X}}_{n}^{k})(ϵ_{n}^{k}) = (\overline{\overline{X}}_{n}^{k}, Π(ϵ_{n}^{k}), 0).

Theorem 7.4.1.2. (i) For any homology/cohomology theory satisfying the axioms of 7.2.1, there are long exact sequences for m ∈ ℤ

H^{p−k−2}(M_{n}^{∞}, m − k − 1) → H^{p}(W_{n}^{k}, m) → H^{p}(X_{n}^{k}, m)(ϵ_{n}^{k})

→ H^{p−k−1}(M_{n}^{∞}, m − k − 1) …

(ii) For any associated theory with compact supports, there are long exact sequences

H^{p−k−1}(M_{n}^{∞}, m) → H_{c}^{p}(X_{n}^{k}, m)(ϵ_{n}^{k}) → H^{p}(W_{n}^{k}, m) → H^{p−k}(M_{n}^{∞}, m) …

Proof: we have long exact sequences

H_{p}(\overline{\overline{X}}_{n}^{k,∞}, m) → H_{p}(\overline{\overline{X}}_{n}^{k}, m) → H_{p}(X_{n}^{k}, m) → H_{p−1}(\overline{\overline{X}}_{n}^{k,∞}, m) …

H_{c}^{p−1}(\overline{\overline{X}}_{n}^{k,∞}, m) → H_{c}^{p}(X_{n}^{k}, m) → H^{p}(\overline{\overline{X}}_{n}^{k}, m) → H_{c}^{p}(\overline{\overline{X}}_{n}^{k,∞}, m) …

We take the ϵ_{n}^{k}-components and apply Theorem 7.3.0.3, choosing arbitrarily an isomorphism of the formal completion of \overline{X}_{n} along the fibre over a cusp and the n-sided Tate curve. We have

H_{p}(\overline{\overline{X}}_{n}^{k,∞}, m)(ϵ_{n}^{k}) ⟶^{∼} H_{p−k}(M_{n}^{∞}, m), H_{c}^{p}(\overline{\overline{X}}_{n}^{k,∞}, m)(ϵ_{n}^{k}) ⟶^{∼} H^{p−k}(M_{n}^{∞}, m)

and this gives the required sequences (for the cohomology sequence, first apply the Poincaré duality axiom 7.2.1(3)).

The parabolic cohomology groups were defined by Deligne as

BW_{n}^{k} = im(H_{1}^{c}(M_{n}(ℂ), Sym^{k} R^{1}f_{∗}ℚ) → H_{1}(M_{n}(ℂ), Sym^{k} R^{1}f_{∗}ℚ))
ℓW_{n}^{k} = im(H_{1}^{c}(M_{n} ⊗ \overline{ℚ}, Sym^{k} R^{1}f_{∗}ℚ_{ℓ}) → H_{1}(M_{n} ⊗ \overline{ℚ}, Sym^{k} R^{1}f_{∗}ℚ_{ℓ}))

We now show that they are the Betti and ℓ-adic realisations of the Chow motive W_{n}^{k}.

Theorem 7.4.1.3. There are canonical isomorphisms, for ? = B or ℓ;

H_{?}^{∗}(W_{n}^{k}) ⟶^{∼} _{?}W_{n}^{k}.

Proof: First we directly compute the ϵ_{n}^{k}-part of the cohomology of X_{n}^{k}. This is easy because in the first place we have the Leray spectral sequence (in, say, ℓ-adic coholomogy)

E_{2}^{ab} = H^{a}(M_{n} ⊗ \overline{ℚ}, R^{b}f_{∗}^{k}ℚ_{ℓ}) ⇒ H_{ℓ}^{a+b}(X_{n}^{k})

and moreover we can compute the higher direct images using the Künneth formula:

R^{b}F_{∗}^{k}ℚ_{ℓ} ⟶^{∼} ⨁_{b_{1}+···+b_{k}=b} R^{b_{1}}f_{∗}ℚ_{ℓ} ⊗ ⋯ ⊗ R^{b_{k}}f_{∗}ℚ_{ℓ}.

Now μ_{2} acts as (−1)^{j} on R^{j}f_{∗}ℚ_{ℓ}, so in the ϵ_{n}^{k}-component the only terms which occur have b_{1} = ⋯ = b_{k} = 1, b = k and

R^{k}f_{∗}^{k}ℚ_{ℓ}(ϵ_{n}^{k}) = ⊗^{k}R^{1}f_{∗}ℚ_{ℓ}(sgn_{k}) = Sym^{k} R^{1}f_{∗}ℚ_{ℓ}.

So the ϵ_{n}^{k}-component of E_{2}^{ab} vanishes unless b = k, so we get an isomorphism

H_{ℓ}^{a+k}(X_{n}^{k})(ϵ_{n}^{k}) ⟶^{∼} H^{a}(M_{n} ⊗ \overline{ℚ}, Sym^{k} R^{1}f_{∗}ℚ_{ℓ}).

For k > 0 the sheaves Sym^{k} R^{1}f_{∗}ℚ_{ℓ} have no global sections, and since M_{n} is an affine curve they have no cohomology in degree > 1, so we get

H_{ℓ}^{∗}(X_{n}^{k})(ϵ_{n}^{k}) = H_{ℓ}^{k+1}(X_{n}^{k})(ϵ_{n}^{k}) ⟶^{∼} H^{1}(M_{n} ⊗ \overline{ℚ}, Sym^{k} R^{1}f_{∗}ℚ_{ℓ}).

The same argument applies to cohomology with compact support, and we therefore have

im(H_{c,ℓ}^{∗}(X_{n}^{k})(ϵ_{n}^{k}) → H_{ℓ}^{∗}(X_{n}^{k})(ϵ_{n}^{k})) ⟶^{∼} _{ℓ}W_{n}^{k}.

Exactly the same arguments work in Betti cohomology. On the other hand, in either Betti or ℓ-adic cohomology, we have H_{?}^{p}(M_{n}^{∞}) = 0 unless p = 0. Putting this into the exact sequences of 7.4.1.2 gives

H_{?}^{p}(\overline{\overline{X}}_{n}^{k})(ϵ_{n}^{k}) = 0 if p ≠ k + 1

and exact sequences

0 → H_{?}^{k+1}(W_{n}^{k}) → H_{?}^{k+1}(X_{n}^{k})(ϵ_{n}^{k}) → H_{?}^{0}(M_{n}^{∞})(−k − 1) → 0
0 → H_{?}^{0}(M_{n}^{∞}) → H_{c,?}^{k+1}(X_{n}^{k})(ϵ_{n}^{k}) → H_{?}^{k+1}(W_{n}^{k}) → 0

Now as representations of Gal(\overline{ℚ}/ℚ), H_{ℓ}^{k+1}(W_{n}^{k}) ⊂ H_{ℓ}^{k+1}(\overline{\overline{X}}_{n}^{k}) and H_{ℓ}^{0}(M_{n}^{∞}) are pure of weights k + 1 and 0 respectively. So the exact sequences show that

H_{ℓ}^{k+1}(W_{n}^{k}) = im(H_{c,ℓ}^{k+1}(X_{n}^{k}) → H_{ℓ}^{k+1}(X_{n}^{k}))(ϵ_{n}^{k})

which proves the result for ℓ-adic cohomology. By the comparison isomorphism it holds for Betti cohomology as well.

7.4.2 Passage to infinite level

Suppose we have positive integer n, m ≥ 3 with n|m. Then the change of level morphism σ_{n,m}: X_{n}^{k} → X_{m}^{k} extend by closure to correspondences of degree zero

between \overline{\overline{X}}_{n}^{k} and \overline{\overline{X}}_{m}^{k}, hence induces a morphism W_{m}^{k} → W_{n}^{k}, which is a direct factor (since σ_{n,m} is finite). We can form the direct limit W^{k}; formally this would be an ind-motive, but we have no use for such things here, and are only concerned with its cohomology groups H^{∗}(W^{k}), which we define as the direct limit lim_{⟶}_{n}H^{∗}(W_{n}^{k}).

If g ∈ G_{f} then for every m we can find an n such that the composite X ⟶^{g} X → X_{m} factors as X → X_{n} ⟶^{g} X_{m}, by the definition of the inverse limit. This shows that G_{f} acts on W^{k} (in other words, on its cohomology groups).

We want to take the direct limit of the exact sequences of 7.4.1.2 and write them G − f-equivariantly. There are two problems which arise:

1. The boundary map depends on the choice, at each cusp, of a local uniformisation of the universal elliptic curve by the Tate curve. This cannot be done in a canonical way (as was already seen in ??).

2. The boundary maps are not compatible with change of level in a naive sense, since the coverings \overline{M}_{n′} → \overline{M}_{n} are ramified at the cusps.

Recall ?? that X = lim_{⟵}_{n}X_{n} is a pro-elliptic curve over M, and that over

the cusps, the “smooth” part \overline{X}^{reg} = lim_{⟵}_{n} \overline{X}_{n}^{reg} satisfies

\overline{X}^{reg} ×\overline{M} M^{∞} ⟶^{∼} 𝔾̃_{m} × ℤ̂ × M^{∞}

(the isomorphism being non-canonical). Recall that X̂ ⊂ \overline{X}^{reg} is the connected component of the identity section. Let A be one of ℚ, \overline{ℚ}, ℝ or ℂ. We define an enhanced cusp over A to be a triple (x, v, λ), where x ∈ M^{∞} ⊗_{ℚ} A is a cusp, v is a ℚ-valued valuation of the function field of M ⊗ A centred on x, and λ: 𝔾̃_{m} ⟶^{∼} X̂_{x} is an isomorphism of group schemes over the residue field x (which is F_{∞} = ℚ(μ_{∞}) in the case A = ℚ, and ℂ in the other two cases). We let M̃_{A}^{∞} be the set of enhanced cusps. If A = ℚ we write M̃^{∞} for M̃_{ℚ}^{∞}.

Now suppose we have a cohomology theory V ↦ H^{∗}(V) as in Theorem 7.4.1.2. Let u ∈ H^{p}(X_{n}^{k}, m)(ϵ^{k}). Define a function Res^{k}(u): M̃^{∞} → H^{p−k−1}(F_{∞}, m− k−1) as follows: if (x, v, λ) ∈ M̃^{∞}, then Res^{k}(u)(x, v, λ) is the image of u under the composite:

H^{p}(X_{n}^{k}, m)(ϵ^{k}) [figure]
↓_{v(q^{1/n})×∂_{x}}
H^{p−1}(X̂_{n}^{k} × x, m − 1)(ϵ^{k})  H^{p−k−1}(F_{∞}, m − k − 1)
↓  ^{≀}↓^{(7.2.2.1)}
H^{p−1}(X̂^{k} × x, m − 1)(ϵ^{k}) ⟶^{λ^{∗}} H^{p−1}(𝔾̃_{m}_{F_{∞}}^{k}, m − 1)(ϵ^{k})

We need a further compatibility of the cohomology, namely that if f: V′ → V is a finite flat morphism ramified along a smooth irreducible divisor D ⊂ V, and D′ = f^{−1}(D) ⟶^{∼} D, then the diagram below commutes, where e is the ramification degree:

H^{∗}(V – D) ⟶ H^{∗−1}(D)
f^{∗}↓  ^{≀}↓^{×e}
H^{∗}(V′ – D′) ⟶ H^{∗−1}(D′)

Assuming this (which holds in all the theories to be used in what follows), we see that the map Res^{k}(u)(x, v, λ) is compatible with change of levels, and therefore if u ∈ H^{p}(X^{k}, m)(ϵ^{k}) =^{def} lim_{⟶}_{n}H^{p}(X_{n}^{k}, m)(ϵ^{k}), then the formula above

defines unambiguously a map Res^{k}(u): M̃^{∞} → H^{p−k−1}(F_{∞}, m − k − 1). The construction is evidently G_{f}-equivariant, and we know that G_{f} acts on F_{∞} (the field of constants of M) by determinant followed by the Artin map (??). Therefore

Res^{k}(gu)(x, v, λ) = Res^{k}(u)((x, v, λ)g))^{[F_{∞}/ℚ,det g]}

for any g ∈ G_{f}.

Above the standard cusp at infinity we have the canonical uniformisation of \overline{X} by the Tate curve over F_{∞}[[\{q^{1/n}\}]]. Therefore we have a canonical valuation v^{can}, which assigns to q the value 1, and a canonical isomorphism λ^{can} between 𝔾̃_{m} and the identity component of the fibre X̂×∞. Let’s denote this “standard” enhanced cusp ˜∞ = (∞, v^{can}, λ^{can}). Then we an describe the structure of M̃^{∞} using the action of G_{f} as described in Chapter 5. The stabiliser of ∞ itself is the

subgroup of (\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ G_{f} with d ∈ ℚ^{∗}. The stabiliser of (∞, v^{can}) consists of

those elements with a/d ∈ ℤ̂^{∗}, and the stabiliser of ˜∞ consists of those matrices with a ∈ ℤ̂^{∗} and d = 1.

Now suppose a, d ∈ ℚ^{∗} with ad > 0. Then we get from ?? that

˜∞ (\begin{matrix}a & 0 \\ 0 & d \end{matrix}) = (∞, \frac{d}{a}v^{can}, a^{−1}λ^{can})

and therefore using the definition of Res^{k} we obtain, for any u,

Res^{k} ((\begin{matrix}a & 0 \\ 0 & d \end{matrix}) u) ( ˜∞) = Res^{k}(u)(∞, \frac{a}{d}v^{can}, a^{−1}λ^{can})^{[F_{∞}/ℚ,det g]}

= \frac{d}{a^{k+1}} Res^{k}(u)( ˜∞)^{[F_{∞}/ℚ,det g]}

We therefore see that the function given by g ↦ Res^{k}(u)( ˜∞g) belongs to the space of locally constant functions f: G_{f} → H^{∗}(F_{∞}, •) satisfying the identity

f ((\begin{matrix}a & 0 \\ 0 & d \end{matrix}) g) = |a|_{f}^{k+1}d sgn d^{k+1}f(g)^{[F_{∞}/ℚ,det g]}  (7.4.2.1)

for all a ∈ 𝔸_{f}^{∗}, d ∈ ℚ^{∗}. We will call this space of functions F^{k}(H^{∗}(F_{∞}, •)). We can work this out in detail in four cases:

1. Betti or ℓ-adic realisation H = H_{?}, with ? = B or ℓ: in this case the inverse limit of the long exact sequence of 7.4.1.2 can be written

0 → H_{?}^{k+1}(W^{k}) → H_{?}^{k+1}(X^{k})(ϵ^{k}) → F k ; \overline{ℚ} ⊗ A(−k − 1) → 0

where A = ℚ or ℚ_{ℓ}, and F k ; \overline{ℚ} = F^{k}(H_{?}^{0}(F_{∞})) is the space of locally con-

stant function f: G_{f} → ℚ such that f ((\begin{matrix}a & b \\ 0 & d \end{matrix}) g) = a^{−k−1}d f(g) for

all b ∈ 𝔸_{f} and a, d ∈ ℚ^{∗} with ad > 0. Notice that we can decompose F k ; \overline{ℚ} ⊗ \overline{ℚ} by induction by stages ??; it is the sum, over pairs (χ_{1}, χ_{2}) of Dirichlet characters whose product has parity k, of induced representations Ind(|−|_{f}^{k}χ_{1}, χ_{2}).

2. Algebraic de Rham cohomology: here we have an exact sequence

0 → H_{dR}^{k+1}(W^{k}) → H_{dR}^{k+1}(X^{k})(ϵ^{k}) → F^{k}(F_{∞}) → 0.

3. Absolute Hodge cohomology over ℝ: then by the vanishing statements ?? we get an isomorphism

H_{H}^{k+1}(X_{/ℝ}^{k}, k + 1)(ϵ^{k}) ⟶^{∼} F^{k}(H_{H}^{0}(F_{∞/ℝ}, 0))

and we can rewrite the second space as F_{ℂ}^{k,+}, where

F_{ℂ}^{k,±} = \{f ∈ F_{ℂ}^{k} |f ((\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) g) = ±f(g)\} .

We also have exact sequences (of interest for m > k + 1)

0 → H_{H}^{k+2}(W^{k}, m) → H_{H}^{k+2}(X_{/ℝ}^{k}, m)(ϵ^{k}) → F^{k}(H_{H}^{1}(F_{∞/ℝ}, m − k − 1)) → 0 (7.4.2.2)

and the right hand group is simply F_{ℂ}^{k,(−)^{m−k}} since H_{H}^{1}(F_{∞/ℝ}, m−k−1) = H_{B}^{0}(F_{∞}, m − k − 2).

4. Motivic cohomology: here we know by Borel’s thm ?? that H_{M}^{i}(F_{∞}, j) = 0 unless i = j = 0 or j ≥ i = 1. Consider first the case p = m = k + 1. Then we have an exact sequence

0 → H_{M}^{k+1}(W^{k}, k + 1) → H_{M}^{k+1}(X^{k}, k + 1)(ϵ^{k}) ⟶^{Res^{k}} F_{ℚ}^{k} =^{def} F^{k}(ℚ).

In the next chapter we will prove Beilinson’s fundamental result that this sequence is exact at the right as well, by constructing an explicit inverse or Res^{k}, the Eisenstein symbol map. In degree k + 2 we get for every m > k + 1 the analogous sequence to (7.4.2.2)

0 → H_{M}^{k+2}(W^{k}, m) → H_{M}^{k+2}(X^{k}, m)(ϵ^{k}) ⟶^{Res^{k}} F^{k}(H_{M}^{1}(F_{∞}, m − k − 1)) (7.4.2.3)

and in Chapter 9 we show (using the theorems of Borel and Beilinson) that this sequence is exact on the right as well. This implies that for every (p, m) other than (k + 1, k + 1) and (k + 2, m) (m > k + 1) we have an isomorphism

H_{M}^{p}(W^{k}, m) ⟶^{∼} H_{M}^{p}(X^{k}, m)(ϵ^{k})

Moreover, Beilinson’s conjecture tells us that since W^{k} has cohomology only in degree k + 1, we should have H_{M}^{p}(W^{k}, m) = 0 if p ≠ k + 2 or if p = k + 2 and m ≤ k + 1. (So Res^{k} in the first of the above sequences should be an isomorphism, just as it is in absolute Hodge cohomology.) However results of this strength are unknown at the time of writing.

7.4.3 Back to finite level

We can now pass to invariants under an open compact subgroup of G_{f} to recover a (canonical) exact sequence at finite level. Suppose that n ≥ 3 as usual, and let’s pass to invariants under K_{n}. Then if our cohomology theory has Galois descent, we get H(X_{n}^{k}) ⟶^{∼} H(X^{k})^{K_{n}}, and by the Iwasawa decomposition, restriction from G_{f} to Gℤ̂ gives an isomorphism

(F^{k})^{K_{n}} ⟶^{∼} F_{n}^{k} =^{def} \{f: G(ℤ/n) → ℚ | {f ((\begin{matrix}a & b \\ 0 & ±1 \end{matrix}) g) = (±1)^{k}f(g) \atop for all a ∈ (ℤ/n)^{∗}, b ∈ ℤ/n.}\}

Therefore we get a residue map Res^{k}: H_{M}^{k+1}(X_{n}^{k}, k +1) → F_{n}^{k}, which is G(ℤ/n)- equivariant. What is much more important is that (since it comes by restriction from a G_{f}-morphism) it is equivariant for the Hecke algebra H(G_{f}, K_{n}). Of course it is not possible to see the action of the Hecke algebra by looking only at the definition of F_{n}^{k}, since that “knows” only the G(ℤ/n)-structure of the cusps, and the twisting by automorphisms of 𝔾_{m}. To see the action of the complete Hecke algebra there is little alternative to writing the residue map at infinite level, and then passing to invariants.

Last revision 15 April 1997 7:33

Likewise, in the realisations, we get short exact sequences (with ? = B or ℓ)

0 → H_{?}^{k+1}(W_{n}^{k}) → H_{?}^{k+1}(X_{n}^{k})(ϵ^{k}) → F_{n,\overline{ℚ}}^{k} ⊗ A(−k − 1) → 0

where F_{n,\overline{ℚ}}^{k} is the space of functions f: N(ℤ/n)\G(ℤ/n) → ℚ such that f(−g) = (−1)^{k}f(g), and A = ℚ or ℚ_{ℓ}.