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Hecke modules, and suitable deformation problems at l.

“Table of contents”:

1. technicalities on Hecke modules;

2. a conjecture on acceptability;

3. deformation problems at l and group schemes;

4. some new things (if time).

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Hecke modules.

Input data:

• \overline{ρ}: G_{ℚ} → GL(V), V a k-vect. sp. of dim. 2, 𝔽_{l} → k finite, l ≠ 2, with \overline{ρ} abs. irr. over the quadratic subfield of ℚ(μ_{l});

• τ: I_{l} → GL_{2}(\overline{ℚ}_{l}) an l-type such that there exists a modular lift ρ of \overline{ρ} of type τ

• S: a finite set of primes not containing l.

Formula for the associated Hecke module:

L :=L_{(S,τ)} := Hom_{U}(M, lim_{V} H^{1}(Y_{V} , O))_{m}
=H^{1}(Y_{U}, F)_{m}

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Some explanation.

U, V open subgroups of GL_{2}(ℤ̂). U = ∏_{p} U_{p}, U_{p} ⊂ GL_{2}(ℤ_{p}) open.

Y_{V} = GL_{2}(ℚ)\GL_{2}(𝔸)/V ·ℂ^{∗}, the modular curve associated to V.

M = ⊗_{p}M_{p}, M_{p} free O-module finite rank, with U_{p}-action, irreducible.

F = GL_{2}(ℚ)\(GL_{2}(𝔸) × M^{∨})/U·ℂ^{∗}, a locally constant sheaf of free O-modules on Y_{V}.

H := lim_{V} H^{1}(Y_{V} , O) has GL_{2}(𝔸_{f})-action, hence Hom_{U}(M, H) has Hecke operators at all p with U_{p} = GL_{2}(ℤ_{p}) and M_{p} = O.

𝕋_{(S,τ)} the sub-O-algebra of End_{O}(Hom_{U}(M, H)) generated by the Hecke operators.

m is the maximal ideal of 𝕋_{(S,τ)} corresponding to \overline{ρ}.

𝕋 is the image of 𝕋_{(S,τ)} in End_{O}(L).

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Some properties.

L is free of finite rank as O-module, ℚ ⊗ L is free of rank 2 over ℚ ⊗ 𝕋.

L has a perfect pairing over O. This allows to control the congruence modules for S ⊂ S′.

L has a G_{ℚ}-action, and is a R_{(S,τ)}-module.

One can choose the U_{p} and M_{p} such that:

\overline{ℚ}_{l} ⊗_{O} L ∼ ⊕_{f∈N_{(S,τ)}}ρ_{f}

(the integral structure is very important).

Ingredients to choose the right U_{p}, M_{p}:

• Local Langlands (also at l);

• type theory: describes smooth rep’s of p-adic groups in terms of an irreducible part of their restriction to suitable compact open subgroups.

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Some examples.

1. l ≠ p ∉ S and \overline{ρ} unramified at p. Then U_{p} = GL_{2}(ℤ_{p}) and M_{p} = O.

2. l ≠ p ∈ S and \overline{ρ} unramified at p. Then

U_{p} = \{( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(ℤ_{p}) | p^{2}|c, \overline{d} ∈ (ℤ/p^{2}ℤ)_{l}^{∗}\}

and M_{p} = O.

3. Slightly more general when l ≠ p and p not vexing.

4. l ≠ p ∉ S, p vexing (\overline{ρ}|_{I_{p}} = Ind_{ℤ_{p^{2}}^{∗}}^{ℤ_{p}^{∗}} χ_{p}). Then U_{p} = GL_{2}(ℤ_{p}) and M_{p} = Θ(χ_{p}).

5. p ∈ S, p vexing. Then

U_{p} = \{( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(ℤ_{p}) | p|c, \overline{d} ∈ (ℤ/p^{2}ℤ)_{l}^{∗}\}

and M_{p} = Θ(χ_{p}).

6. l itself. Then U_{l} = GL_{2}(ℤ_{l}), and M_{l} is what τ Local Langlands and type theory suggest.

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A conjecture on acceptability.

Input data:

• \overline{ρ}: G_{l} → GL_{2}(k) continuous, with End_{G_{l}}(V) = k.

• τ: I_{l} → GL_{2}(\overline{ℚ}_{l}) a type.

We assume that there is a modular l-adic lift of \overline{ρ} that is of type τ.

Let R_{O}^{l} be the universal deformation ring of \overline{ρ}.

Then R_{O,τ}^{l} is the smallest quotient of R_{O}^{l} from which all l-adic lifts ρ of \overline{ρ} of type τ arise.

Definition. τ is weakly acceptable for \overline{ρ} if there exists a surjection from O[[X]] to R_{O,τ}^{l}.

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A question.

If f is a newform of l-type τ, then what can one say about \overline{ρ}_{f,l}: G_{l} → GL_{2}(\overline{𝔽}_{l}) ?

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An answer.

\overline{ρ} occurs in H^{1}(Y_{U}, k ⊗_{O} F_{τ}) with U = U_{l}U^{(l)}, F_{τ} coming from M_{l}, and U^{(l)} sufficiently small.

But \overline{F_{τ}} := k ⊗_{O} F_{τ} has a Jordan-Hölder filtration with successive quotients of the form F_{n,m} := Sym^{n}(\overline{F}) ⊗ det(\overline{F})^{⊗m}, with \overline{F} corresponding to the standard representation of GL_{2}(𝔽_{l}) over \overline{𝔽}_{l}, and 0 ≤ n < l and m in ℤ/(l − 1)ℤ.

Modular representation theory gives the multiplicities of the F_{n,m}.

The weight part of Serre’s conjectures does the rest.

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The conjecture.

Conjecture (BCDT). τ is weakly acceptable for \overline{ρ} if and only if the multiplicities of the F_{n,m} that lead to \overline{ρ}|_{I_{l}} add up to one.

Remark: the “only if” part is true, and follows from Diamond’s arguments that if the whole method works, then L is free of rank 2 over 𝕋.

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Deformation problems at l and group schemes.

Notation and assumptions as before. Remember: R_{O}^{l} → R_{O,τ}^{l} is the smallest quotient through which all lifts of \overline{ρ} of type τ factor. To prove, in enough cases: R_{O,τ}^{l} is a quotient of O[[X]]. Strategy: define a suitable deformation problem with universal ring R^{univ}, such that

1. R_{O}^{l} → R_{O,τ}^{l} factors through R_{O}^{l} → R^{univ}.

2. R^{univ} is a quotient of O[[X]].

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Shape of the deformation problem.

Let ℚ_{l} → F be a finite Galois extension (group Γ), such that τ is unramified over F.

A “descent datum” for a finite free group scheme G over O_{F} a right action of Γ on G, compatible with its action on Spec(O_{F}), i.e., for each γ in Γ, one has a commutative diagram:

G  ⟶_{∼}^{[γ]}  G
↓  ↓
Spec(O_{F}) ⟶_{Spec(γ)}^{∼} Spec(O_{F})

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Base change via O_{F} → k_{F} gives:

G_{k_{F}}  ⟶_{∼}^{[γ]} G_{k_{F}}
↓  ↓
Spec(k_{F}) ⟶_{Spec(γ)}^{∼} Spec(k_{F})

This gives a morphism W_{l}^{+} → End_{k_{F}} (G_{k_{F}}) by:

σ ↦ [σ^{−1}]·Frob_{abs}^{d(σ)}.

For G l-divisible, this morphism should give the type τ. This gives a set S relations in ℤ[F, V][Γ] that have to be satisfied.

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Shape of the deformation problem.

One considers: (R, ρ), with R an artinian O-algebra with residue field k, and ρ a deformation of \overline{ρ}, such that there exists (G, [·]) with ρ ∼ G(\overline{ℚ}_{l}) and G_{k_{F}} annihilated by S.

In the worst case (conductor 3^{5}), one also has to fix a model (G_{0}, [·]) of \overline{ρ}, and demand that (G, [·]) is a successive extension of copies of (G_{0}, [·]).

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The worst Γ.

{1} → I_{Γ} → Γ → ℤ/2ℤ → {0},

with I_{Γ} the semidirect product of μ_{4}(𝔽_{9}) by ℤ/3ℤ.

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Some new things.

Theorem (Taylor, Skinner-Wiles , …) Let l be an odd prime, and ρ: G_{ℚ} → GL_{2}(\overline{ℚ}_{l}) a continuous irreducible representation such that:

1. ρ is unramified at all but finitely many primes;

2. ρ is odd (i.e., det ρ(c) = −1);

3. ρ|_{G_{l}} is an extension of χ_{2} by ε^{n}χ_{1}, with χ_{1}|_{I_{l}} and χ_{2}|_{I_{l}} of finite order and n a non-zero positive integer, such that ε^{n}χ_{1}χ_{2}^{−1}(I_{l}) is not pro-l.

Then there is a totally real number field E, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{E}) and a place λ of the field of coefficients of π above l such that ρ_{π,λ} (the λ-adic representation associated to π) is equivalent to ρ|_{G_{E}}.