On p-Adic Geometric Representations of G_{ℚ}

To John Coates

J.-P. Wintenberger

Received: August 30, 2005 Revised: September 15, 2006

Abstract. A conjecture of Fontaine and Mazur states that a geometric odd irreducible p-adic representation ρ of the Galois group of ℚ comes from a modular form ([10]). Dieulefait proved that, under certain hypotheses, ρ is a member of a compatible system of ℓ-adic representations, as predicted by the conjecture ([9]). Thanks to recent results of Kisin ([15]), we are able to apply the method of Dieulefait under weaker hypotheses. This is useful in the proof of Serre’s conjecture ([20]) given in [11], [14],[12],[13].

2000 Mathematics Subject Classification: : 11R32, 11R39

1 Introduction.

Let \overline{ℚ} be an algebraic closure of ℚ. For L a finite extension of ℚ contained in \overline{ℚ}, we write G_{L} for the Galois group of \overline{ℚ}/L. For ℓ a prime number, we write ℚ_{ℓ} for the field of ℓ-adic numbers and \overline{ℚ_{ℓ}} for an algebraic closure of ℚ_{ℓ}. An ℓ-adic representation ρ of G_{L} of dimension d is a continuous morphism ρ from G_{L} to GL_{d}(\overline{ℚ_{ℓ}}). In fact, ρ has values in GL_{d}(M), for M a finite extension of ℚ_{ℓ} contained in \overline{ℚ_{ℓ}} (lemma 2.2.1.1. of [6]). Such a representation ρ is said to be geometric if it satisfies the following two conditions ([10]):

- for L a prime of L above ℓ, the restriction of ρ to the decomposition subgroup D_{L} satisfies the potentially semi-stable condition of Fontaine’s theory (exp. 8 of [1]) ;

- there exists a finite set S of primes of L such that ρ is unramified outside S and the primes above ℓ. A geometric ℓ-adic Galois representation defines for each prime L of L an isomorphy class of representations of the Weil-Deligne group WD_{L} in GL_{d}(\overline{ℚ_{ℓ}})

([8], exp. 8 of [1], [10]). We call r_{L}(ρ) its F -semisimplification. It is attached to the restriction of ρ to the decomposition group D_{L}. When L is of characteristic ℓ, in order to define r_{L}, one needs to use the action of WD_{L} on the filtered Dieudonné module attached to the restriction of ρ to D_{L} via Fontaine’s theory (see remark 1 of section 4). Let E be a finite extension of ℚ contained in \overline{ℚ}. By a compatible system of geometric representations of G_{L} with coefficients in E of dimension d, we mean the following data :

- for each ℓ and for each embedding ι of E in \overline{ℚ_{ℓ}}, a geometric representation ρ_{ι} : G_{L} → GL_{d}(\overline{ℚ_{ℓ}}),

- a finite set S of primes of L, and for each prime L of L, an F -semisimple representation r_{L} of WD_{L} in GL_{d}(E), such that :

- r_{L} is unramified if L ∉ S ;

- for each ι as above, ι ∘ r_{L} is isomorphic to r_{L}(ρ_{ι}).

We fix a prime p. Let ρ be a p-adic geometric irreducible odd representation of dimension 2 of G_{ℚ}. By odd, we mean that ρ(c) has eigenvalues 1 et −1, for c a complex conjugation. We suppose that ρ has Hodge-Tate weights (0, k − 1), where k is an integer ≥ 2 : we shall say that ρ is of weight k. It is conjectured by Fontaine and Mazur that ρ comes from a modular form of weight k. More precisely, let k ≥ 2 and N ≥ 1 be integers. Let f = q + … + a_{n}q^{n} + … be a primitive modular form on Γ_{1}(N) of weight k. Let E(f) be its coefficient field, i.e. the field generated by the coefficients of f and the values of the character of f. The field E(f) is a finite extension of ℚ. It is classical that one can associate a p-adic representation ρ(f)_{ι} : G_{ℚ} → GL_{2}(\overline{ℚ_{p}}) to f and an embedding ι of E(f) in \overline{ℚ_{p}}. The representation ρ(f)_{ι} is unramified at ℓ if ℓ is ≠ p and does not divide N and is characterized by :

tr(ρ(f)_{ι}(Frob_{ℓ})) = ι(a_{ℓ}),

for these ℓ. Furthermore, ρ(f)_{ι} is absolutely irreductible, odd, geometric, of conductor N and of weight k (Hodge-Tate weights (0, k − 1)). The conjecture of Fontaine and Mazur states that ρ is isomorphic to ρ(f)_{ι} for an f and a ι. A consequence of the conjecture of Fontaine and Mazur is that ρ is a member of a compatible system of Galois representations. Dieulefait proved that it is the case under certain hypotheses ([9]). Using a recent result of Kisin ([15]), we give weaker hypotheses under which the result of Dieulefait is true. The main tool of the proof is a theorem of Taylor ([26] and [25]). There exists a totally real number field F which is Galois over ℚ and such that ρ_{|G_{F}} comes from an cuspidal automorphic representation π of GL_{2}(𝔸_{F}) of parallel weight k (or a Hilbert modular form for F). By Arthur-Clozel ([2]), for each F′ such that the Galois group of F/F′ is solvable, ρ_{|G_{F′}} comes from an automorphic representation π_{F′} for GL_{2}(𝔸_{F′}). Using Brauer’s theorem, we put together the compatible systems associated to the automorphic representations π_{F′}, and we obtain the compatible system of representations of G_{ℚ}.

On p-Adic Geometric Representations of G_{ℚ} 821

2 Taylor’s theorem.

Let ρ be an odd irreducible geometric p-adic representation of G_{ℚ} of dimension 2 of weight k, k an integer ≥ 2. We say that ρ is potentially modular if there exists a Galois totally real finite extension F of ℚ contained in \overline{ℚ} such that the restriction of ρ to G_{F} comes from a cuspidal automorphic representation π of GL_{2}(𝔸_{F}) of parallel weight k. The theorem of Taylor states in many cases that ρ is potentially modular. In fact, Taylor proves that the reduction \overline{ρ} of ρ is potentially modular, with F unramified (resp. split) at p if the restriction of \overline{ρ} to D_{p} is reducible (resp. irreducible). Then, the modularity of ρ_{|G_{F}} follows from modularity theorems. According to which modularity theorem one applies, one get different statements. We write the following statement which is needed for our work with Khare on Serre’s conjecture.

Théorème 1 Let ρ : G_{ℚ} → GL_{2}(\overline{ℚ_{p}}) be a p-adic representation, absolutely irreducible, odd, unramified outside a finite set of primes. One supposes that the reduction \overline{ρ} of ρ has non solvable image and, if p ≠ 2, that \overline{ρ} has Serre’s weight k(\overline{ρ}) in the range [2, p+1]. Then ρ is potentially modular in the following cases :

- a1) p ≠ 2 and ρ_{|D_{p}} is crystalline of weight k = k(\overline{ρ}) ;

- a2) p = 2, k(\overline{ρ}) = 2 and ρ_{|D_{2}} is Barsotti-Tate ;

- b) p ≠ 2 and k(\overline{ρ}) ≠ p + 1, ρ_{|D_{p}} is potentially Barsotti-Tate, Barsotti-Tate after restriction to ℚ_{p}(μ_{p}), and the restriction of the representation of the Weil-Deligne group WD_{p} to inertia is (ω_{p}^{k−2} ⊕ 1), where ω_{p} is the Teichmuller lift of the cyclotomic character modulo p ;

- c) p ≠ 2 and k(\overline{ρ}) = p + 1 or p = 2 and k(\overline{ρ}) = 4 and ρ_{|D_{p}} is semistable of weight 2.

The theorem follows from the potential modularity of \overline{ρ} ([26], [25]) and the modularity theorem stated in 8.3. of [13]. Remark. Using Skinner-Wiles modularity theorem ([22]), Taylor gives a variant of this statement in a lot of ordinary cases.

3 Field of coefficients of ρ.

Let ρ : G_{ℚ} → GL_{2}(\overline{ℚ_{p}}) be as in the preceeding section. Furthermore, we suppose that ρ is potentially modular.

Proposition 1 There is a finite extension E of ℚ and an embedding ι_{p} : E ↪ \overline{ℚ_{p}} and for each prime ℓ, a F -semisimple representation r_{ℓ} of the Weil-Deligne group WD_{ℓ} with values in GL_{2}(E) such that for each ℓ, the F -semisimplification r_{ℓ}(ρ) of the representation of the Weil-Deligne group WD_{ℓ} associated to ρ is isomorphic to ι_{p} ∘ r_{ℓ}.

Proof. Let F and π as in the theorem of Taylor. Let F′ be a subfield of F such that F/F′ has solvable Galois group. By Arthur and Clozel, we know that the restriction of ρ to G_{F′} is also associated to a cuspidal representation π_{F′} of GL_{2}(𝔸_{F′}) ([2]). It follows that there exists a finite extension E_{F′} of ℚ such that the F -semisimplification of the representation of the Weil-Deligne group WD_{L} associated to the restriction of ρ to G_{F′} can be realized in E_{F′} for each prime L of F′. The rationality properties of π_{F′} follows from Shimura for the unramified primes and from Rogawski-Tunnell for the ramified primes ([21], see also [19] ; [18]). The compatiblity of global and local Langlands correspondances follows for L of characteristic ≠ p from Carayol completed by Taylor ([7],[23]) and for L of characteristic p from Saito and Kisin ([19],[15]). Take for E an extension of ℚ containing the images by all embeddings in \overline{ℚ} of the fields E_{F′}. Let L be a prime of F. Let F′_{L} be the subfield of F which is fixed by the decomposition subgroup of Gal(F/ℚ) for L. Let L′ be the restriction of L to F′_{L}. The representation of the Weil-Deligne group WD_{L′} defined by the restriction of ρ to F′_{L} can be realized in E_{F′_{L}}. As the Weil-Deligne groups WD_{ℓ} and WD_{L′} coincide, the proposition follows. Remark. Particular cases of the compatibility between global and local Langlands correspondences for the primes dividing the characteristic follows from Breuil, Berger and Taylor ([5],[3],[24]).

4 Construction of the compatible system.

Théorème 2 Let ρ be as in the preceeding section. Then, there exists a compatible system (ρ_{ι}) of geometric representations of G_{ℚ} with coefficients in a number field E such that there exists an embedding ι_{p} : E ↪ \overline{ℚ_{p}} with ρ_{ι_{p}} isomorphic to ρ. The ρ_{ι} are irreducible, odd and of weight k.

Proof. If ρ is induced from the p-adic representation associated to a Hecke’s character Ψ of an imaginary quadratic field, then one takes for (ρ_{ι}) the compatible system induced from the one defined by the Hecke character. Otherwise, ρ remains absolutely irreducible after restriction to any open subgroup of G_{ℚ}. We suppose this from now. Let F, π, E(π) and ι_{p} such that ρ_{|G_{F}} is isomorphic to the Galois representation ρ(π)_{ι_{p}} attached to π, and the embedding ι_{p} of the coefficient field E(π) of π in \overline{ℚ_{p}}. As in Taylor’s 5.3.3. of [27], one applies Brauer’s theorem to the trivial representation of Gal(F/ℚ). There exist fields F_{i} ⊂ F, such that each F/F_{i} has a solvable Galois group, integers m_{i} ∈ ℤ and characters Ψ_{i} of Gal(F/F_{i}) such that the trivial representation of Gal(F/ℚ) equals :

∑_{i} m_{i}Ind_{G_{F_{i}}}^{G_{ℚ}} Ψ_{i}.

One has :

ρ = ∑_{i} m_{i}Ind_{G_{F_{i}}}^{G_{ℚ}} (ρ_{|G_{F_{i}}} ⊗ Ψ_{i}).

As in the proof of proposition 1, it follows from the theorems of Taylor and Arthur-Clozel that ρ_{|G_{F_{i}}} is the Galois representation ρ(π_{i}) attached to an automorphic representation π_{i} of GL_{2}(𝔸_{F_{i}}) whose coefficient field is embedded in E. Let ι be an embedding of E in \overline{ℚ_{q}} for a prime q. We enlarge E such that it contains the values of the characters Ψ_{i}. One defines the virtual representation R_{ι} in the Grothendieck group of irreducible representations of G_{ℚ} with coefficients in \overline{ℚ_{q}} by :

R_{ι} = ∑_{i} m_{i}Ind_{G_{F_{i}}}^{G_{ℚ}} (ρ(π_{i})_{ι} ⊗ Ψ_{i}).

Let us prove that R_{ι} is a true representation. For i and j, let {τ_{k}}, τ_{k} ∈ G_{ℚ} be a set of representatives of the double classes G_{F_{i}}\G_{ℚ}/G_{F_{j}}. Let us call F_{ijk} the compositum of F_{i} and τ_{k}(F_{j}). One has :

Ind_{G_{F_{j}}}^{G_{ℚ}} (ρ(π_{j})_{ι} ⊗ Ψ_{j})_{|G_{F_{i}}} = ∑_{k} Ind_{G_{F_{ijk}}}^{G_{F_{i}}} (((ρ(π_{j})_{ι} ⊗ Ψ_{j}) ∘ int(τ_{k}^{−1}))_{|G_{F_{ijk}}} ) .

It follows that the scalar product < R_{ι}, R_{ι} > in the Grothendieck group is equal to the sum over i, j, k of :

m_{i}m_{j} < ((ρ(π_{j})_{ι} ⊗ Ψ_{j}) ∘ int(τ_{k}^{−1}))_{|G_{F_{ijk}}} , (ρ(π_{i})_{ι} ⊗ Ψ_{i})_{|G_{F_{ijk}}} > .

We see that the scalar product of R_{ι} with itself is ∑_{i,j,k} m_{i}m_{j}t_{ijk} with t_{ijk} = 1 or 0 depending whether

((ρ(π_{j})_{ι} ⊗ Ψ_{j}) ∘ int(τ_{k}^{−1}))_{|G_{F_{ijk}}} ≃ (ρ(π_{i})_{ι} ⊗ Ψ_{i})_{|G_{F_{ijk}}}

or not. One has a similar calculation for the scalar product of ρ with itself in the Grothendieck group of irreducible representations of G_{ℚ} with coefficients in \overline{ℚ_{p}}. The calculation gives ∑_{ijk} m_{i}m_{j}t′_{ijk}, with t′_{ijk} = 1 or 0 depending whether

((ρ ⊗ Ψ_{j}) ∘ int(τ_{k}^{−1}))_{|G_{F_{ijk}}} ≃ (ρ ⊗ Ψ_{i})_{|G_{F_{ijk}}}

or not. As ρ(π_{i})_{ι} and ρ_{|G_{F_{i}}} are irreducible and have the same characteristic polynomial of Frobenius outside a finite set of primes, one has t_{ijk} = t′_{ijk}. As < ρ, ρ >= 1, it follows that the scalar product of R_{ι} with itself is 1. As the dimensions of R_{ι} and ρ are both ∑ 2m_{i}[G_{ℚ} : G_{F_{i}}], we have dim(R_{ι}) = 2. We see that R_{ι} is a true representation of dimension 2. We call it ρ_{ι}. It follows from the formula defining R_{ι} that the restriction of ρ_{ι} to G_{F} is associated to π. By Blasius-Rogawski ([4]), (ρ_{ι})_{|G_{F}} comes from a motive, except perhaps if k = 2. It then follows by Tsuji that the restriction of ρ_{ι} to the decomposition group for the characteristic q of ι is potentially semi-stable of weight k ([28]). The case k = 2 and ρ_{ι} is constructed as a limit of q-adic representations attached to automorphic forms with one local component discrete series is taken care by Kisin ([23],[15]).

The F -semisimple representation of the Weil-Deligne group WD_{ℓ} on ρ_{ι} is isomorphic to :

∑_{i} m_{i} (∑_{L} Ind_{D_{L}}^{D_{ℓ}}(r_{L}(π_{i}) ⊗ Ψ_{i})) ,

where L describes the set of primes of F_{i} over ℓ. The compatibility follows from the fact that π_{i} ↦ ρ(π_{i}) is compatibility with local Langlands correspondance (see the references quoted in the proof of proposition 1). By an argument of Ribet, it follows from compatibility that ρ_{ι} is absolutely irreducible ([17]). As the restriction of ρ_{ι} to G_{F} is associated to π, it is odd and ρ_{ι} is odd. This finishes the proof of the theorem. Remarks.

1) Let M be a finite extension of ℚ_{p} contained in \overline{ℚ_{p}} and let γ : G_{M} → GL_{d}(E) be a potentially semistable representation of the Galois group G_{M} with coefficients in a finite extension E of ℚ_{p}. Let WD_{M} be the Weil-Deligne group. Let M_{0} be the maximal unramified extension of ℚ_{p} contained in M. Fontaine has defined a representation of WD_{M} on the filtered Dieudonné D module attached to γ (exp. 8 of [1]). Let us recall how it defines, up to conjugacy, a representation r of WD_{M} in GL_{d}(\overline{ℚ_{p}}). The filtered Dieudonné module D is a L⊗_{ℚ_{p}} E-module D, L a finite unramified extension of M_{0} in \overline{ℚ_{p}}, with an action of WD_{M} commuting with the action of L ⊗_{ℚ_{p}} E. One knows that the E ⊗_{ℚ_{p}} L-module D is free. Let us briefly recall why. Let us choose such an embedding of E in \overline{ℚ_{p}}, and let us call E_{1} = E ∩ L. For each element τ of the Galois group of E_{1}/ℚ_{p}, let D_{τ} be the sub-module of the elements x of D such that (e ⊗ 1)x = (1 ⊗ τ(e))x for every e ∈ E_{1}. As the Frobenius ϕ of D acts semi-linearly relatively to the action of L and commutes with the action of E, ϕ transitively permutes the D_{τ}, and the D_{τ} have the same dimension. This implies the freeness. As the action of the Weil-Deligne group WD_{M} on D commutes with the action of E ⊗_{ℚ_{p}} L, it follows that WD_{M} acts on each D_{τ}. One defines r as the F-simplification of the action of WD_{M} on D_{id}.

2) One can describe the projective representation associated to ρ_{ι} as in [29]. Let F and π as in Taylor’s theorem. Let ρ_{ι} the Galois-representation associated to π and ι. The multiplicity one theorem ([16]) implies that for σ ∈ G_{ℚ}, the automorphic representations π and ^{σ}π are isomorphic. It follows that the Galois representations ρ_{ι} and ρ_{ι} ∘int(σ) are isomorphic. That means that there exists \overline{g_{σ}} ∈ PGL_{2}(\overline{ℚ}_{q}) such that :

ρ_{ι} ∘ int(σ) ≃ int(\overline{g_{σ}}) ∘ ρ_{ι}.

This characterizes \overline{g_{σ}} as ρ_{F,q} is absolutely irreducible. Then, σ ↦ g_{σ} defines a projective representation which is the projective representation associated to ρ_{ι}. As in [29], one can show directly that this projective representation lifts to a representation in GL_{2}(\overline{ℚ}_{q}).

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Jean-Pierre Wintenberger Université Louis Pasteur Département de Mathématiques IRMA 7, rue René Descartes 67084 Strasbourg Cedex France wintenb@math.u-strasbg.fr 828

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