Modular motifs

William Stein, UCB

December 3, 1999

1 Introduction

Canonical disclaimer: I am not an expert!

Goal: I will describe Tony Scholl’s [Invent. 100, 419– 430 (1990)] theorem:

S_{k}(N) ↦ (Chow motif over Q)

Scholl also associates a Grothendieck motif to each newform f.

“To construct the motif means to isolate a constituent of the cohomology of a variety using algebraic cycles.”

2 Connectedness of SpecT

First, I’ll relay a question. Let

T ⊂ End(S_{k}(Γ_{0}(p)))

be the Hecke algebra. Question (Ash, Mazur): Is SpecT connected? When k = 2, yes, by Prop.II.10.2 of Mazur’s Modular curves and the Eisenstein ideal paper. Open problem when k > 2, but numerical evidence is encouraging: a related graph is connected for k ≤ 8 and p ≤ 53. Mazur’s proof involves Ribet’s correspondence between irreducible components of Spec(T) and isogeny factors of J_{0}(p); reducibility of the Hecke algebra quickly translates into a decomposition of the canonical principal polarization, a contradiction. Is there a higher weight motific analogue of Mazur’s proof?

“Well, the reason why the question is particularly intriguing is that we DON’T seem to have any hope of a proof (of connectedness for weight k > 2 Hecke algebras) that mimics the proof for weight 2. The higher weight motives just don’t (YET...) have the precise algebro-geometric structure that come from principally polarized abelian varieties. So one would need an utterly different kind of proof, (if the connectedness property is true... which, after your calculations looks possible).” Mazur, email, yesterday.

What sort of structure do we have??

3 Products of universal elliptic curves

Fix integers N, k ≥ 3, and set w = k − 2 (this is the opposite of the notation in Scholl, but I feel more confortable with it). The case k = 2 is J_{0}(N), which was already done by Eichler and Shimura.

[figure: M_{N}/Q; \{elliptic curves E, with (Z/N)^{2} ↪ E\}; smooth compactification; \overline{M}_{N} = M_{N} ∪ M_{N}^{∞}; (generalized elliptic curves + level structure)]

• X_{N} is the universal elliptic curve over M_{N} \overline{X}_{N} is the generalized elliptic curve; it is a smooth and proper Q-scheme \overline{X}_{N}^{∗} is the Néron model of X_{N}/\overline{M}_{N} (smooth locus of \overline{π}.)

\xymatrix{X_{N} \ar[dd]_{π} & \overline{X}_{N} \ar[dd]_{\overline{π}} \\  &  & _{\overline{X}{∗ \atop N}} \ar@{^{(}->}[ul] \ar@{-->}[dl] \\ M_{N} \ar@{^{(}->}[r]^{j} & _{\overline{M}_{N}}}

• The product \overline{X}_{N} ×_{\overline{M}_{N}} ⋯×_{\overline{M}_{N}} \overline{X}_{N} is singular over Q when there is more than one factor.

[figure: \overline{\overline{X}}_{N}^{w}; Deligne’s canonical desingularization; X_{N}^{w}; _{\overline{X}{w \atop N}}; _{\overline{π}_{w}}; _{\overline{X}_{N} ×_{\overline{M}_{N}} ⋯ ×_{\overline{M}_{N}} \overline{X}_{N} (w factors)}; \overline{π}_{w}^{−1}(M_{N}); _{\overline{M}_{N}}]

\overline{X}_{N}^{w,∞} := \overline{X}_{N}^{w} − X_{N}^{w}
\overline{\overline{X}}_{N}^{w,∞} := \overline{\overline{X}}_{N}^{w} − X_{N}^{w}

4 Automorphisms of Deligne’s desingularization

The level-N structure on the universal elliptic curve X_{N} over M_{N}:

(Z/N)^{2} ↪ X_{N} (over M_{N}).

By the Néron mapping property, this extends to a morphism of Néron models:

(Z/N)^{2} × \overline{M}_{N} ↪ \overline{X}_{N}^{∗} ↪ \overline{X}_{N},

so (Z/N)^{2} acts by translation on \overline{X}_{N}.

Also μ_{2} = \{±1\} acts on \overline{X}_{N}, so

(Z/N)^{2}⋊μ_{2} acts on \overline{X}_{N}.

(Both groups sit in Aut(\overline{X}_{N}) and (Z/N)^{2} has index 2.)

• Let Σ_{w} be the symmetric group on {1, 2, …, w}.

• The wreath product of (Z/N)^{2}⋊μ_{2} by Σ_{w}:

Γ_{w} = ((Z/N)^{2}⋊μ_{2}) ≀ Σ_{w}
= ((Z/N)^{2}⋊μ_{2})^{w}⋊Σ_{w}.

The wreath product is the following set of permutations θ of \overline{X}_{N} × \{1, 2, …, w\}:

{θ : θ(a, b) = (γ_{b}(a), η(b)) : γ_{b} ∈ (Z/N)^{2}⋊μ_{2}, η ∈ Σ_{w}},

where a ∈ \overline{X}_{N}, b ∈ \{1, …, w\}, and γ_{b} is chosen independently for each b. Automorphisms of \overline{X}_{N} × \{1, 2, …, w\} induce automorphisms of \overline{X}_{N}^{w}; and of \overline{\overline{X}}_{N}^{w} (because the desingularization is canonical).

5 The projector Π_{ε}

[figure: Γ_{w}; ε; {±1}; ((Z/N)^{2}⋊μ_{2})^{w}⋊Σ_{w}; ((a_{i}), (b_{i}), σ); (∏_{i=1}^{w} b_{i}) · sign(σ)]

Associate to ε a projector in the group ring:

Π_{ε} ∈ (Z [\frac{1}{2N · (w!)}]) [Γ_{w}].

We have

Π_{ε} = \frac{1}{#Γ_{w}} ∑_{g∈Γ_{w}} ε(g)g

because

(∑ ε(g)g)^{2} = #Γ_{w} ∑ ε(g)g,

which can be seen by considering how g ∈ ker(ε) and g ∉ ker(ε) act on ∑_{h∈ker(ε)} h and ∑_{h∉ker(ε)} h.

Define

V (ε) := Π_{ε}(V) ⊂ V.

6 Parabolic cohomology groups

Cohomological realization of cusp forms (Eichler-Shimura correspondence type stuff):

_{N}^{w}W_{ℓ} := H_{ét}^{1}(\overline{M}_{N} ⊗ \overline{Q}, j_{∗}Sym^{w}((R^{1}π_{∗})Q_{ℓ}))
= ker(H_{ét}^{1}(M_{N} ⊗ \overline{Q}, Sym^{w}R^{1}π_{∗}Q_{ℓ})
→ H_{ét}^{0}(M_{N}^{∞} ⊗ \overline{Q}, Q_{ℓ}(−k − 1)))

Theorem 6.1 (Scholl). For every prime ℓ,

_{N}^{w}W_{ℓ} = H_{ét}^{∗}(\overline{\overline{X}}_{N}^{w})(ε)

and a similar statement for the Betti cohomology.

Though desingularizing totally “messes up” the variety, miraculously a careful study reveals that the ε part of cohomology remains under control. Most of Scholl’s proof is devoted to understanding the desingularization of the compactification.

The pair

(\overline{\overline{X}}_{N}^{w}, Π_{ε})

has a right to be called the motif attached to the space of modular forms of weight k and level N. Let _{N}^{w}W be the motif defined by (\overline{\overline{X}}_{N}^{w}, Π_{ε}) in the category of Chow motifs over Q (rational equivalence).

7 Hecke operators on the motif _{N}^{w}W

It is possible to decompose _{N}^{w}W in the category of Grothendieck motives [homological equivalence, tensor with Q(… a_{n} …)]. “It should be possible to decompose _{N}^{w}W already in the category of Chow motives; but this seems very hard without assuming the standard conjectures.” (???)

• Fix p ∤ N a prime:

M_{N,p} := {(E, (Z/N)^{2} structure , C ⊂ E[p] order p)}

This is a curve over Q. •

\xymatrix{C \ar@{^{(}->}[r] & X_{N,p} \ar[d] \ar[r] & X_{N} \ar[d] \\  & M_{N,p} \ar[r] & M_{N}}

• Q := X_{N,p}/C + level N structure induced from X_{N,p}.

Q^{w} := Q ×_{M_{N,P}} × ⋯ ×_{M_{N,p}} Q (w-fold product)

• Consider

[figure: X_{N}^{w}; X_{N,p}^{w}; ϕ_{1}; ψ; Q^{w}; ϕ_{2}; X_{N}^{w}; M_{N}; M_{N,p}; M_{N,p}; M_{N}]

The first and third squares are cartesian.

• Define the Hecke correspondence T_{p} on X_{N}^{w} by

T_{p} = (ϕ_{1})_{∗}ψ^{∗}ϕ_{2}^{∗}.

The correspondence T_{p} on \overline{\overline{X}}_{N}^{w} is the closure of the graph of T_{p} in \overline{\overline{X}}_{N}^{w} × \overline{\overline{X}}_{N}^{w}.

Proposition 7.1 (Scholl).

• Γ_{w} on H_{B}^{∗}(\overline{\overline{X}}_{N}^{w}) commutes with T_{p}.

• Action of transpose of T_{p} on H^{0}(\overline{\overline{X}}_{N}^{w}, Ω^{k−1}) is the classical Hecke operator T_{p}. So we have an endomorphism of motives:

T_{p} : _{N}^{w}W → _{N}^{w}W