Lifting Galois Representations,

and a Conjecture of Fontaine and Mazur

Rutger Noot^{1}

Received: May 11, 2001 Revised: November 16, 2001

Communicated by Don Blasius

Abstract. Mumford has constructed 4-dimensional abelian varieties with trivial endomorphism ring, but whose Mumford–Tate group is much smaller than the full symplectic group. We consider such an abelian variety, defined over a number field F, and study the associated p-adic Galois representation. For F sufficiently large, this representation can be lifted to G_{m}(Q_{p}) × SL_{2}(Q_{p})^{3}.

Such liftings can be used to construct Galois representations which are geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question predicts that these representations should come from algebraic geometry. We confirm the conjecture for the representations constructed here.

2000 Mathematics Subject Classification: 11G10, 11F80, 14K15 Keywords and Phrases: geometric Galois representation, Fontaine– Mazur conjecture, motives

Introduction

A construction due to Mumford shows that the Mumford–Tate group of a polarized abelian variety A/C is not, in general, determined by the polarization and the endomorphism ring of A. This construction, which can be found in [Mum69], proves the existence of families of polarized abelian fourfolds where the general fibre has trivial endomorphism algebra but Mumford–Tate group much smaller than the full symplectic group of H_{B}^{1}(A(C), Q).

1Partially supported by the Centre National de la Recherche Scientifique and by the European RTN network contract HPRN-CT-2000-00120

Documenta Mathematica 6 (2001) 419–445

Previous papers, [Noo95] and [Noo00], brought to light some properties of an abelian variety A arising from this construction. For example, if A can be defined over a number field F, then it can be shown that, after replacing F by a finite extension, A has good reduction at all non-archimedean places of F and ordinary reduction at “most” of those places. Many of these results are derived from the property that the p-adic Galois representations associated to A factor through maps ρ_{p}: 𝒢_{F} → G(Q_{p}), where G is the Mumford–Tate group of A. They are therefore to a large extent results on the properties of Galois representations of “Mumford’s type”. The present work continues the study of the Galois representations in question, in particular their lifting properties. In this respect, this paper is heavily inspired by Wintenberger’s wonderful paper [Win95], even though the results proved therein are never actually used. In fact, the conditions of the main theorem (1.1.3) of that paper are not verified in the case considered here and, indeed, our conclusions are weaker as well. Let G be the Mumford–Tate group of an abelian variety A over a number field F arising from Mumford’s construction. It admits a central isogeny G̃ → G, where the group G̃ is (geometrically) a product:

G̃\overline{Q} ≅ G_{m} × SL_{2}^{3}.

We fix a prime number p such that G̃ decomposes over Q_{p} and we attempt to lift the p-adic Galois representation ρ associated to A along the isogeny G̃(Q_{p}) → G(Q_{p}). This is possible after replacing F by a finite extension. The resulting lift ρ̃ naturally decomposes as a direct sum of a character and three 2-dimensional representations. These representations are studied in section 1. They have rather nice properties but for each one of them there exists a finite place of F where it is not potentially semi-stable. To be precise, the 2-dimensional representations are potentially unramified at all finite places of F of residue characteristic different from p, trivial at certain p-adic places of F but not potentially semi-stable (not even of Hodge–Tate type) at the other p-adic places. In section 1, the properties of the lifting ρ̃ and its direct factors are studied without using the fact that ρ comes from an abelian variety A with Mumford– Tate group G. If we do use that fact, we can say more about the representations in question. This is the subject of section 2, where it is shown in particular how to distinguish the p-adic places where a given 2-dimensional direct factor of ρ̃ is trivial, from the p-adic places where it is not. The results of section 2 are used in section 3 to show that considering two abelian varieties A and A′ of Mumford’s kind, and taking a tensor product of two representations of the type considered above, one obtains representations which are potentially crystalline at all p-adic places of F and are potentially unramified at all other finite places of F. These representations are unramified outside a finite set of non-archimedean places of F. For a suitable choice of the varieties A and A′, one obtains an irreducible representation.

According to a conjecture of Fontaine and Mazur ([FM95, conjecture 1], see section 0), such a representation should “come from algebraic geometry”, i. e. be a subquotient of a Tate twist of an étale cohomology group of an algebraic variety over F. Our construction is geometric in some large sense, but not in the sense of the conjecture. We show in the beginning of section 4 that, in general, our representations do not arise (in the sense of Fontaine and Mazur) from the geometric objects having served in their construction. It is therefore not clear if the conjecture holds in this case. Nevertheless, it is shown in section 4 that our representations do come from geometry, even from the geometry of abelian varieties, in the strong sense of Fontaine and Mazur. The construction of these abelian varieties is based on a construction going back to Shimura, cf. [Del72, §6]. In the final section, some generalizations of the construction of geometric representations are studied. It is shown in particular that any tensor product of the representations constructed in section 2 which is geometric comes from algebraic geometry.

Acknowledgements. It is a great pleasure to thank Johan de Jong for several extremely useful discussions on the subject of this paper. I thank Don Blasius for his questions, which led to the addition of section 5 to this paper.

0 Notations and terminology

0.1 Notations. For any field F, we denote by F̄ an algebraic closure of F and we write 𝒢_{F} = Gal(F̄/F) = Aut_{F} (F̄), the group of F -automorphisms of F̄. If v̄ is a valuation of F̄, then ℐ_{F,v̄} ⊂ 𝒟_{F,v̄} ⊂ 𝒢_{F} are the inertia group and decomposition group of v̄. If F is a local field, we write just ℐ_{F} ⊂ 𝒢_{F} for the inertia subgroup. In the case of the field Q of rational numbers, we let \overline{Q} be its algebraic closure in the field of complex numbers C. As usual, for any prime number p, Q_{p} is the p-adic completion of Q and C_{p} is the completion of \overline{Q}_{p}.

0.2 Representations of Mumford’s type. Let K be a field of characteristic 0, let G be an algebraic group over K and let V be a faithful K-linear representation of G. As in [Noo00, 1.2], we will say that the pair (G, V) is of Mumford’s type if

• Lie(G) has one dimensional centre 𝔠,

• Lie(G)K̄ ≅ 𝔠K̄ ⊕ 𝔰𝔩_{2,K̄}^{3} and

• Lie(G)K̄ acts on VK̄ by the tensor product of the standard representations.

We do not require G to be connected.

0.3 A conjecture of Fontaine and Mazur. Let F be a number field and let 𝒢_{F} = Aut_{F} (F̄). Fix a prime number p and suppose that ρ is a finite dimensional Q_{p}-linear representation of 𝒢_{F}. Following Fontaine and Mazur in [FM95, §1], we will say that ρ is geometric if

• it is unramified outside a finite set of non-archimedean places of F and

• for each non-archimedean valuation v̄ of F̄, the restriction of ρ to the decomposition group 𝒟_{F,v̄} is potentially semi-stable.

The meaning of the second condition depends on v̄. If v̄(p) > 0, then the notion of potential semi-stability is the one defined by Fontaine, see for example [Fon94]. If v̄(p) = 0 then the fact that ρ is potentially semi-stable means that the restriction of ρ to the inertia group ℐ_{F,v̄} is quasi-unipotent. By Grothendieck’s theorem on semi-stability of p-adic Galois representations, the condition of potential semi-stability at v̄ is automatically verified in the case where v̄(p) = 0. We can therefore restrict our attention to valuations v̄ with v̄|p. The only fact needed in what follows is that a crystalline representation of 𝒟_{F,v̄} is semi-stable and hence that a potentially crystalline representation is potentially semi-stable. We say that a finite dimensional, irreducible, Q_{p}-linear representation of 𝒢_{F} comes from algebraic geometry if it is isomorphic to a subquotient of a Tate twist of an étale cohomology group of an algebraic variety over F. We cite conjecture 1 from [FM95].

0.4. Conjecture (Fontaine–Mazur). Let F be a number field. An irreducible p-adic representation of 𝒢_{F} is geometric if and only if it comes from algebraic geometry.

0.5 The “if” part of the conjecture is true. A representation coming from a proper and smooth F -variety having potentially semi-stable reduction at all places of residue characteristic p is semi-stable by the C_{pst}-conjecture, [Fon94], proven by Tsuji in [Tsu99]. The general case of this part of the conjecture (for irreducible representations) follows by [dJ96]. The “only if” part is known for potentially abelian representations by [FM95, §6]. In [Tay00], Taylor proves the conjecture for representations of 𝒢_{Q} with values in GL_{2}(\overline{Q}_{p}) satisfying some supplementary conditions. These results can be generalized to certain two-dimensional \overline{Q}_{p}-linear representations of 𝒢_{F}, for totally real number fields F.

0.6 Remark. Let F′ be a finite extension of F. It follows from [FM95, §4], Remark (b) that the conjecture for representations of 𝒢_{F} is equivalent to the conjecture for representations of 𝒢_{F′}. For the implication which is still open, this can be seen as follows. Let ρ be a representation such that a Tate twist ρ_{|𝒢_{F′}} (n) of ρ_{|𝒢_{F′}} is a subquotient of an étale cohomology group of an algebraic variety X over F′. Then

Ind_{𝒢_{F′}}^{𝒢_{F}} (ρ_{|𝒢_{F′}} (n))

is a subquotient of an étale cohomology group of Res_{F′/F} (X). The representation ρ(n) is a subquotient of this induced representation.

1 Lifting Galois representations of Mumford’s type

1.1 Let p be a prime number, G_{p}/Q_{p} an algebraic group and V_{p} a faithful representation of G_{p} such that (G_{p}, V_{p}) is of Mumford’s type in the sense of 0.2. Throughout this section we will assume that F is a number field or a finite extension of Q_{ℓ} for some prime number ℓ and that ρ: 𝒢_{F} → G_{p}(Q_{p}) is a polarizable, continuous representation. The condition that ρ is polarizable means, by definition, that there exists a non-degenerate, alternating, bilinear, 𝒢_{F} -equivariant map V × V → Q_{p}(−1). Here Q_{p}(−1) is the 1-dimensional Q_{p}- linear representation where 𝒢_{F} acts through χ^{−1}, the inverse of the cyclotomic character. We will assume moreover that G_{p} is connected and that Lie(G_{p}) ≅ 𝔠 ⊕ 𝔰𝔩_{2}^{3}. In this case, there exists a central isogeny

N: G̃_{p} = G_{m} × SL_{2}^{3} ⟶ G_{p}

such that the induced representation of G̃_{p} on V_{p} is isomorphic to the tensor product V_{0} ⊗ V_{1} ⊗ V_{2} ⊗ V_{3} of the standard representations of the factors. We fix a 𝒢_{F} -stable Z_{p}-lattice V_{Z_{p}} ⊂ V_{p} such that there exists lattices V_{i,Z_{p}} ⊂ V_{i} with V_{Z_{p}} = ⊗_{i=1}^{4}V_{i,Z_{p}} and such that, on V_{Z_{p}}, the polarization form takes values in Z_{p}(−1).

1.2 Definition. We say that ρ(𝒢_{F}) is sufficiently small if it stabilizes V_{Z_{p}} and if all elements of ρ(𝒢_{F}) are congruent to 1 mod p if p > 2 and congruent to 1 mod 4 if p = 2.

1.3 Remarks.

1.3.1 For any polarizable, continuous representation ρ: 𝒢_{F} → G_{p}(Q_{p}) as in the beginning of 1.1, there exist a lattice V_{Z_{p}} as in 1.1 and a finite extension F′ ⊃ F such that ρ(𝒢_{F′}) is sufficiently small.

1.3.2 If ρ(𝒢_{F}) is sufficiently small, then it does not contain any non-trivial elements of finite order. This implies in particular that ρ is unramified at a finite place v of F of residue characteristic different from p if and only if it is potentially unramified at v.

1.4. Lemma. If ρ(𝒢_{F}) is sufficiently small, then ρ lifts uniquely to a continuous group homomorphism ρ̃: 𝒢_{F} → G̃_{p}(Q_{p}) with the property that all elements of ρ̃(𝒢_{F}) are congruent to 1 mod p if p > 2 and to 1 mod 4 if p = 2.

Proof. The map GL_{2}^{3} → G_{p} obtained by taking the tensor product of the standard representations is surjective on Q_{p}-valued points and even on Q_{p}- valued points of GL_{2}^{3} and G_{p} satisfying the above congruence condition. By Hensel’s lemma, the determinant of an element of GL_{2}(Q_{p}) which satisfies this congruence condition is the square of an element of G_{m}(Q_{p}) satisfying the same condition, which proves that if ρ(𝒢_{F}) is sufficiently small, then any element lifts to an element of G̃_{p}(Q_{p}), congruent to 1 mod 2p. One has ker(N) = {(ϵ_{0}, ϵ_{1}, ϵ_{2}, ϵ_{3}) | ϵ_{i} = ±1, ϵ_{0}ϵ_{1}ϵ_{2}ϵ_{3} = 1}. As the congruence condition implies that −1 ≢ 1, the lifting with this property is unique so we obtain the unique lift ρ̃: 𝒢_{F} → G̃_{p}(Q_{p}) from the lemma, for the moment just a continuous map. The uniqueness of the lifting from G_{p}(Q_{p}) to G̃_{p}(Q_{p}) implies that ρ̃ is a group homomorphism.

1.5 Definition. If ρ(𝒢_{F}) is sufficiently small, then we call the lifting ρ̃: 𝒢_{F} → G̃_{p}(Q_{p}) of the lemma the canonical lifting of ρ.

1.6 The representations ρ_{i}. For any ρ: 𝒢_{F} → G_{p}(Q_{p}) and any lifting ρ̃: 𝒢_{F} → G̃_{p}(Q_{p}), there exists a finite extension F′ of F such that ρ(𝒢_{F′}) is sufficiently small and such that the restriction of ρ̃ to 𝒢_{F′} is its canonical lifting. Replacing F by such a finite extension F′, we will assume in the sequel that this is already the case over F. By composing ρ̃ with the projections on the factors of G̃_{p}(Q_{p}), we get a character ρ_{0} : 𝒢_{F} → G_{m}(Q_{p}) and representations ρ_{i}: 𝒢_{F} → SL_{2}(Q_{p}) for i = 1, 2, 3. The facts that the image of ρ is sufficiently small and that ρ̃ is its canonical lift imply that the ρ_{i}(𝒢_{F}) do not contain any elements of finite order other than the identity. The following lemma is obvious.

1.7. Lemma. Let notations and hypotheses be as in 1.6. In particular, ρ(𝒢_{F}) is sufficiently small and ρ̃ is its canonical lifting. If ρ is unramified at some finite place of F of residue characteristic different from p, then so are ρ̃ and the ρ_{i}.

1.8. Lemma. Let ρ, ρ̃ and the ρ_{i} (for i = 0, 1, 2, 3) be as in 1.6 and lemma 1.7. Then the character ρ_{0} satisfies ρ_{0}^{2} = χ^{−1}, where χ: 𝒢_{F} → Z_{p}^{∗} is the cyclotomic character.

Proof. The condition that ρ is polarizable means that it respects a symplectic form up to a scalar and that the multiplier is the inverse of the cyclotomic character. This means that the determinant of ρ is χ^{−4}. The representation ρ_{1} ⊗ ρ_{2} ⊗ ρ_{3} has trivial determinant, so the determinant of its product by ρ_{0} is ρ_{0}^{8} and hence ρ_{0}^{8} = χ^{−4}. The lemma follows because ρ_{0}(𝒢_{F}) and ρ(𝒢_{F}) and hence χ(𝒢_{F}) are congruent to 1 mod 2p.

1.9. Proposition. Under the assumptions of the lemmas, let v be a p-adic valuation of F and let ℐ = ℐ_{F,v̄} be the inertia group of a valuation v̄ of F̄ lying over v. Assume that the restriction of ρ to ℐ is Hodge–Tate (resp. crystalline). For i ∈ {1, 2, 3}, either

• the restriction of ρ_{i} to ℐ is trivial, or

• the restriction of ρ_{0} · ρ_{i} to ℐ is Hodge–Tate (resp. crystalline).

Proof. One has an isomorphism G_{p}^{ad}(Q_{p}) ≅ PSL_{2}(Q_{p})^{3} and it can be shown as in [Noo00, 3.5] that the projection of ρ(ℐ) on exactly two of the factors is trivial. We sketch the argument. Let μ_{HT} : G_{m,C_{p}} → G_{p,C_{p}} be the cocharacter associated to the Hodge–Tate decomposition of ρ_{|ℐ}. As the kernel of the isogeny N: G̃_{p} → G_{p} is annihilated by 2, the square

μ_{HT}^{2} : G_{m,C_{p}} → G_{p,C_{p}}

lifts to a cocharacter μ̃: G_{m,C_{p}} → G̃_{p,C_{p}}. One has μ̃ = (μ̃_{0}, μ̃_{1}, μ̃_{2}, μ̃_{3}), where μ̃_{0} : G_{m,C_{p}} → G_{m,C_{p}} and μ̃_{i}: G_{m,C_{p}} → SL_{2,C_{p}} for i = 1, 2, 3. Since V_{p,C_{p}} is the direct sum of two eigenspaces for μ_{HT} and hence for μ̃, exactly one of the maps μ̃_{1}, μ̃_{2}, μ̃_{3} is non-trivial. The Zariski closure of the image of μ_{HT} therefore projects non-trivially on exactly one of the factors of G_{p,Q_{p}}^{ad} ≅ (PSL_{2,Q_{p}})^{3}. The theorem of Sen, [Ser78, Théorème 2], implies the statement concerning the projection of ρ(ℐ). To end the proof, we can assume that the projection of ρ(ℐ) on the first factor of G_{p}^{ad}(Q_{p}) is non-trivial. The hypotheses that ρ(𝒢_{F}) is sufficiently small and that ρ̃ is the canonical lifting of ρ imply that (ρ_{2})_{|ℐ} and (ρ_{3})_{|ℐ} are trivial. Since the tensor product ρ_{|ℐ} = (ρ_{0}ρ_{1})_{|ℐ}⊗(ρ_{2} ⊗ρ_{3})_{|ℐ} is Hodge–Tate (resp. crystalline), this implies that (ρ_{0}ρ_{1})_{|ℐ} is Hodge–Tate (resp. crystalline) as well.

1.10 Definition. Notations being as above, let 1 ≤ i ≤ 3 and let v̄ be a p-adic valuation of F̄. We say that (ρ_{i})_{|ℐ_{F,v̄}} is of the first kind if (ρ_{0}ρ_{i})_{|ℐ_{F,v̄}} is potentially crystalline. We say that (ρ_{i})_{|ℐ_{F,v̄}} is of the second kind if its image is trivial.

1.11 Remark. Let (G_{p}, V_{p}) be of Mumford’s type as in 1.1, let F be a number field and assume that A/F is a polarizable abelian fourfold such that, for some identification V_{p} = H_{ét}^{1}(AF̄, Q_{p}), the Galois representation of 𝒢_{F} on V_{p} factors through a map ρ: 𝒢_{F} → G_{p}(Q_{p}). Then ρ is a representation of the type considered in 1.1. This is the setting in which the results of this section will be applied. Abelian varieties with this property exist by [Mum69, §4] and [Noo95, 1.7]. By [Noo00, 2.2], such varieties have potentially good reduction at all nonarchimedean places of F. This implies that for each p-adic place v̄ of F̄, the restriction of ρ to the decomposition group 𝒟_{F,v̄} ⊂ 𝒢_{F} is potentially crystalline. At all other finite places, ρ is potentially unramified in any case and unramified if ρ(𝒢_{F}) is sufficiently small.

2 Galois representations of abelian varieties of Mumford’s type

2.1 Fix an abelian fourfold A over a number field F and an embedding F ⊂ C. Let V = H_{B}^{1}(A(C), Q) and let G ⊂ GL(V) be an algebraic subgroup such that (G, V) is of Mumford’s type, with G connected. We assume that the morphism h: S → GL(V ⊗ R) determining the Hodge structure of A_{C} factors through G_{R}. This condition is equivalent to the condition that the Mumford–Tate group of A_{C} is contained in G. Such abelian varieties exist by [Mum69, §4], and [Noo95] implies that there also exist abelian varieties of this type with the additional property that the Mumford–Tate group is equal to G. It is explained in [Noo01, 1.5] that any abelian variety satisfying the above condition whose Mumford–Tate group is equal to G can be obtained as a fibre of one of the families constructed in [Mum69, §4]. As in [Noo01] one can draw the following conclusions. There is a central isogeny G̃ → G, where G̃ = G_{m} × G̃^{der} and G̃^{der} is an algebraic group over Q such that G̃der \overline{Q} ≅ SL_{2,\overline{Q}}^{3}. The group G̃^{der} can be described in the following way. There exist a totally real cubic number field K and a quaternion division algebra D with centre K, with Cor_{K/Q}(D) ≅ M_{8}(Q) and with D ⊗_{Q} R ≅ M_{2}(R) × H × H such that

G̃^{der} = {x ∈ D | Nrd(x) = 1},

considered as an algebraic group over Q. Here H is the algebra of real quaternions and Nrd: D → K is the reduced norm. The group G can be identified with the image of D^{×} in Cor_{K/Q}(D) ≅ M_{8}(Q) under the norm map N: D → Cor_{K/Q}(D) and therefore has a natural 8-dimensional representation. This representation is isomorphic to V. Note that for any field L containing a normal closure of K, one has

G̃_{L}^{der} ≅ ∏_{φ: K↪L} G_{φ},

where each G_{φ} is an L-form of SL_{2}. In what follows, we will refer to K and D as the number field and the division algebra associated to G.

2.2 The Hodge decomposition on V_{C} = H_{B}^{1}(A(C), C) ≅ H_{dR}^{1}(A_{C}/C) is determined by a cocharacter

μ_{HdR} : G_{m,C} ⟶ G_{C}

such that G_{m} acts by the character ·^{i} on the subspace H^{i,1−i} ⊂ H_{B}^{1}(A(C), C). Let C_{HdR} ⊂ G_{C} be the conjugacy class of μ_{HdR}. It follows from [Noo01, 1.2] (and is easy to check) that the field of definition in C of C_{HdR} is isomorphic to K. In what follows we will identify K with this subfield of C and thus assume that K ⊂ C. One deduces that C_{HdR} can be defined over F ⊂ C if and only if K ⊂ F, and hence that, up to replacing F by a finite extension, one can assume that C_{HdR} can be defined over F. From now on, we will assume that this is the case. The arguments used in the proof of proposition 1.9 also apply to μ_{HdR}. Thus μ_{HdR}^{2} lifts to a map μ̃: G_{m,C} → G̃_{C}. As in loc. cit., the fact that there are two eigenspaces in V_{C} for the action of G_{m} implies that the projection of μ̃ on one and only one factor SL_{2,C} of G̃_{C} is non trivial. This implies that C_{HdR} projects non-trivially on exactly one factor PSL_{2,C} of G_{C}^{ad}, namely the factor corresponding to the embedding K ⊂ C fixed above.

2.3 Let p be a prime number such that G̃_{Q_{p}} ≅ G_{m} × SL_{2}^{3}. If K and D are the number field and the division algebra associated to G, this condition is equivalent to p being completely split in K and D being split at all places of K above p. The factors SL_{2} of the above product correspond to the padic valuations of K. As K ⊂ F by assumption, all Hodge classes on A_{C} that are invariant for the G(Q)-action on the rational cohomology are defined over F, so the Galois representation associated to A factors through a map ρ: 𝒢_{F} → G(Q_{p}). This implies that we find ourselves in the situation of 1.1, with F a number field and with G_{Q_{p}} playing the role of the group G_{p}. After further enlarging F, the image of ρ is sufficiently small. The constructions of 1.6 provide a character ρ_{0} : 𝒢_{F} → G_{m}(Q_{p}) and representations ρ_{1}, ρ_{2}, ρ_{3} : 𝒢_{F} → SL_{2}(Q_{p}). The above remarks give rise to a bijective correspondence between the ρ_{i} (i = 1, 2, 3) and the p-adic valuations v_{1}, v_{2}, v_{3} of K, with ρ_{i} corresponding to v_{i}.

2.4 We summarize the notations and hypotheses in effect at this point.

• (G, V) is a pair of Mumford’s type (as defined in 0.2) with G connected.

• K is the number field associated to G.

• p is a prime number which splits completely in K and such that the group G is split at p.

• F ⊂ C is a number field A/F an abelian variety.

• V = H_{B}^{1}(A(C), Q) is an identification such that the Mumford–Tate group of A is contained in G.

• Inclusions K ⊂ F ⊂ C are fixed such that the field of definition in C of the conjugacy class C_{HdR} ⊂ G_{C} of μ_{HdR} is equal to K.

• The image of the Galois representation ρ: 𝒢_{F} → G(Q_{p}) is sufficiently small.

2.5. Proposition. Under the above hypotheses, let v be a p-adic valuation of F and ℐ = ℐ_{F,v̄} be the inertia group of a valuation v̄ of F̄ lying over v. Suppose that i ∈ {1, 2, 3}. Then (ρ_{i})_{|ℐ} is of the first kind if and only if v_{|K} = v_{i}.

Proof. We already saw (cf. proposition 1.9 and its proof) that the projection of ρ(ℐ) on one and only one factor of G^{ad}(Q_{p}) is non-trivial. This is the factor corresponding to v_{i} if and only if (ρ_{i})_{|ℐ} is of the first kind. After renumbering the v_{i} and the ρ_{i}, we can assume that v_{|K} = v_{1} and it suffices to show that the projection of ρ(ℐ) on the factor of G^{ad}(Q_{p}) corresponding to v_{1} is non-trivial. By 2.2, the conjugacy class C_{HdR} ⊂ G_{C} projects non-trivially to exactly one factor PSL_{2,C} of

G_{C}^{ad} = ∏_{K↪C} PSL_{2,C}.

Since the field of definition of C_{HdR} is equal to K, this must be the factor corresponding to the embedding K ⊂ F ⊂ C fixed in 2.2 (cf. 2.4). Let ι: \overline{Q} ↪ C_{p} be an embedding such that the composite

F ⊂ \overline{Q} ⟶^{ι} C_{p}

induces the valuation v on F. For each K-variety Y, we denote by Y ⊗_{K} C_{p} the base extension of Y to C_{p} via the embedding induced by ι and the inclusion K ⊂ \overline{Q}. The Hodge–Tate decomposition associated to ρ_{|ℐ} is determined by a cocharacter μ_{HT} : G_{m,C_{p}} → G_{C_{p}} and we let C_{HT} be its conjugacy class in G_{C_{p}}. Since [Bla94, theorem 0.3] implies conjecture 1 of [Win88], it follows from [Win88, proposition 7] that C_{HdR} ⊗_{K} C_{p} = C_{HT}, as subvarieties of (G_{K})⊗_{K} C_{p} = G_{C_{p}}. The conjugacy class C_{HT} thus projects non-trivially onto the factor of G_{C_{p}}^{ad} corresponding to the inclusion ι. As ι: \overline{Q} ↪ C_{p} restricts to an inclusion K ↪ Q_{p} inducing the valuation v_{1} on K, it follows from proposition 1.9 and its proof that ρ(ℐ) projects non-trivially to the factor of G_{Q_{p}}^{ad} corresponding to v_{1}.

2.6 The special case. We keep the above notations. In this section, it was not assumed that the Mumford–Tate group of A is equal to G. We investigate the case where these two groups are not equal, so let T ⊂ G the Mumford–Tate group of A and assume that T ≠ G. As T is a reductive Q-group containing the scalars, it has to be a torus of G. This means that A corresponds to a special point of an appropriate Shimura variety associated to (G, X), where X denotes the G(R)-conjugacy class of the morphism h: S → G_{R} determining the Hodge structure on H_{B}^{1}(A(C), R). It follows from [Noo01, §3] that there exists a totally imaginary extension L of K which splits D (with D the division algebra associated to G as in 2.1) and such that T is the image in G of L^{×} ⊂ D^{×}. If T̃ ⊂ G̃ denotes the connected component of the inverse image of T, one has T̃ = G_{m} × (G̃^{der} ∩ L^{×}). There is an identification

G̃^{der} ∩ L^{×} = {x ∈ L^{×} | xx̄ = 1},

where the right hand side is considered as an algebraic group over Q. Again by [Noo01, §3], the field L is the reflex field of the CM-type of A.

Assume that p is completely split in L. Then

T̃_{Q_{p}} ≅ G_{m} × (G_{m})^{3},

where the first factor lies in the centre of G̃ and the other factors identify with maximal tori of the factors SL_{2,Q_{p}} of G̃_{Q_{p}}. In this case, the representations ρ_{1}, ρ_{2} and ρ_{3} each decompose as a direct sum of two 1-dimensional representations given by opposite characters, so ρ̃ decomposes as a sum of 1-dimensional representations.

3 Construction of geometric Galois representations

3.1 Fix two representations (G, V) and (G′, V′) of Mumford’s type and assume that the number field associated (as in 2.1) to both groups G and G′ is the same totally real cubic field K. We will assume that both G and G′ are connected. Also fix a number field F ⊂ C, two 4-dimensional abelian varieties A/F and A′/F and identifications V = H_{B}^{1}(A(C), Q) and V′ = H_{B}^{1}(A′(C), Q) such that the Mumford–Tate groups of A and A′ are contained in G and G′ respectively. In analogy with 2.2, let μ_{HdR} : G_{m,C} → G_{C} (resp. μ′_{HdR} : G_{m,C} → G′_{C}) be the morphism determining the Hodge decomposition on V ⊗C (resp. V′⊗C) and let C_{HdR} (resp. C′_{HdR}) be the conjugacy class of μ_{HdR} (resp. μ′_{HdR}). We assume that the field of definition in C of C_{HdR} is equal to that of C′_{HdR} and we identify K with this field, just as in 2.2. Replacing F by a finite extension if necessary, we will assume that K ⊂ F. Let D and D′ be the division algebras associated to G and G′ respectively and assume that p is a prime number which splits completely in K and such that

D ⊗_{Q} Q_{p} ≅ D′ ⊗_{Q} Q_{p} ≅ M_{2}(Q_{p})^{3}.

As in 2.3, the Galois representations associated to A and A′ factor through morphisms ρ: 𝒢_{F} ⟶ G(Q_{p}) and ρ′: 𝒢_{F} ⟶ G′(Q_{p}). After replacing F by a finite extension, we may assume that ρ and ρ′ have sufficiently small image in the sense of 1.2. Let v_{1}, v_{2} and v_{3} be the p-adic valuations of K and order the factors of

G̃_{Q_{p}} ≅ G_{m,Q_{p}} × ∏_{K↪Q_{p}} SL_{2,Q_{p}} ≅ G̃′_{Q_{p}}

in such a way that the ith factor SL_{2} in each product corresponds to the embedding K ↪ Q_{p} inducing v_{i}. Applying the construction of 1.6 to ρ and ρ′, we obtain a character ρ_{0} = ρ′_{0} and Galois representations

ρ_{i}, ρ′_{i}: 𝒢_{F} ⟶ SL_{2}(Q_{p})

for i = 1, 2, 3. We will consider the representation

τ = ρ_{1} ⊗ ρ′_{1} : 𝒢_{F} ⟶ SL_{2}(Q_{p})^{2} ⟶ GL_{4}(Q_{p}),  (3.1.∗)

where the last arrow is defined by the action of SL_{2}(Q_{p})^{2} on Q_{p}^{2} ⊗ Q_{p}^{2} ≅ Q_{p}^{4}.

3.2 Let v̄ be a p-adic place of F̄ and let ℐ = ℐ_{F̄,v̄} ⊂ 𝒢_{F} be the corresponding inertia group. If the restriction of v̄ to K is equal to v_{1}, then the representations (ρ_{1})_{|ℐ} and (ρ′_{1})_{|ℐ} are both of the first kind by proposition 2.5. It follows that the tensor product (ρ_{1} ⊗ ρ′_{1})_{|ℐ} is potentially crystalline, as it is the twist by χ of the tensor product of the two potentially crystalline representations ρ_{0}ρ_{1} and ρ_{0}ρ′_{1}. If the restriction of v̄ to K is different from v_{1}, then the representations (ρ_{1})_{|ℐ} and (ρ′_{1})_{|ℐ} are both of the second kind by proposition 2.5. In this case, the condition that ρ(𝒢_{F}) and ρ′(𝒢_{F}) are sufficiently small implies that (ρ_{1} ⊗ ρ′_{1})_{|ℐ} is trivial. It follows that (ρ_{1} ⊗ ρ′_{1})_{|ℐ} is potentially crystalline in this case as well. In both cases, it follows from [Fon94, 5.1.5] that (ρ_{1} ⊗ ρ′_{1})_{|𝒟_{F,v̄}} is potentially crystalline. This proves the following theorem.

3.3. Theorem. Assume that the hypotheses of 2.4 are verified for K ⊂ F ⊂ C, A/F, G, p and ρ: 𝒢_{F} → G(Q_{p}) and for K ⊂ F ⊂ C, A′/F, G′, p and ρ′: 𝒢_{F} → G′(Q_{p}). Let

τ = ρ_{1} ⊗ ρ′_{1} : 𝒢_{F} ⟶ SL_{2}(Q_{p})^{2} ⟶ GL_{4}(Q_{p})

be the representation of (3.1.∗). Then τ is a geometric Galois representation in the sense of 0.3. If F is sufficiently large then τ has good reduction everywhere.

3.4. Proposition. Let p, K, G and G′ be as in 3.1, subject to the condition that the associated division algebras D and D′ both satisfy

D ⊗ R ≅ D′ ⊗ R ≅ M_{2}(R) × H × H

and are both split at the same real place of K. Then one can choose the abelian varieties A and A′ used in the construction of τ in 3.1 such that Lie(τ(𝒢_{F})) = 𝔰𝔩_{2}^{2}. In that case τ is an irreducible representation of 𝒢_{F}.

Proof. From the surjections G_{m,Q} × G^{der} → G and G_{m,Q} × (G′)^{der} → G′ one deduces a surjection G_{m}^{2} × G^{der} × (G′)^{der} → G × G′. Let Δ ⊂ G_{m}^{2} be the diagonal and let G′′ ⊂ G × G′ be the image of Δ × G^{der} × (G′)^{der}. The representations V and V′ and the projections G′′ → G and G′′ → G′ induce a representation of G′′ on V′′ = V ⊕ V′. The representations V and V′ both carry a bilinear form which is G- resp. G′-invariant up to a scalar. These forms are also G′′-invariant up to a scalar and their multipliers are equal. It follows that V′′ can be endowed with a bilinear form, G′′-invariant up to scalars. Fix isomorphisms D ⊗ R ≅ M_{2}(R) × H × H ≅ D′ ⊗ R. Any pair of morphisms h_{D}: S → D_{R}^{×} and h_{D′}: S → (D′)_{R}^{×} both conjugate to

S ⟶ GL_{2,R} × H^{×} × H^{×}
z = a + bi ↦ (( \begin{matrix}a & b \\ −b & a \end{matrix} ) , 1, 1)

for the above identifications defines a morphism h′′: S → G′′_{R}. As in [Mum69, §4], one shows that these data define the Hodge structure of an abelian variety. One can choose h_{D} and h_{D′} in such a way that the image of h′′ is Zariski dense in G′′, i. e. is not contained in a proper subgroup of G′′ defined over Q. This implies that there exists an abelian variety A′′_{C} over C with Mumford– Tate group equal to G′′ and such that the representation of its Mumford–Tate group on H_{B}^{1}(A′′_{C}(C), Q) is isomorphic to the representation of G′′ on V′′. It follows from [Noo95, 1.7] and its proof that there exists an abelian variety A′′ over a number field F′′ ⊂ C with Mumford–Tate group G′′, such that the representation of its Mumford–Tate group on H_{B}^{1}(A′′(C), Q) is isomorphic to (G′′, V′′) and with the property that that the image of the Galois representation ρ′′: 𝒢_{F′′} → G′′(Q_{p}) is open. By construction of the representation (G′′, V′′) as a direct sum, A′′ is isogenous to a product A × A′, where the Mumford–Tate groups of A and A′ are G and G′ respectively. The image of

(ρ, ρ′): 𝒢_{F} ⟶ G(Q_{p}) × G′(Q_{p})

is open in G′′(Q_{p}). This implies that the image of τ: 𝒢_{F} → GL_{4}(Q_{p}) is open in H(Q_{p}), where H ⊂ GL_{4,Q_{p}} is the image of (SL_{2,Q_{p}})^{2} acting on Q_{p}^{2} ⊗ Q_{p}^{2} ≅ Q_{p}^{4} by the tensor product of the standard representations. Since the representation of H on Q_{p}^{4} is irreducible, the same thing is true for τ.

3.5 Remark. As noted in 2.1, the condition that

D ⊗ R ≅ D′ ⊗ R ≅ M_{2}(R) × H × H

is equivalent to the conditions that there are abelian varieties A and A′ such that D and D′ are the algebras associated to the Mumford–Tate groups of A and A′ respectively.

3.6 A special case. Let us return to the notations of theorem 3.3, i. e. we assume that the hypotheses of 2.4 are verified for K ⊂ F ⊂ C, A/F, G, p and ρ: 𝒢_{F} → G(Q_{p}) and for K ⊂ F ⊂ C, A′/F, G′, p and ρ′: 𝒢_{F} → G′(Q_{p}). From now on we moreover assume that the Mumford–Tate group of A is equal to G whereas that of A′ is strictly contained in G′. The condition on A implies that End(AF̄) = Z. Since the representation ρ is semisimple, as ρ(𝒢_{F}) ⊂ G(Q_{p}) and because

End_{𝒢_{F}} (H_{ét}^{1}(AF̄, Q_{p})) = Q_{p},

one easily deduces that ρ(𝒢_{F}) is open in G(Q_{p}). As was seen in 2.6, the condition on A′ implies that, after replacing F by a finite extension again if necessary, A′ is of CM-type and its Mumford–Tate group is a maximal torus T′ ⊂ G′. Let L′ be the reflex field of the CM-type of A′. The assumption that F is sufficiently large for the Mumford–Tate group of A′ to be a torus is equivalent to the condition that L′ ⊂ F. By 2.6, the map ρ̃′: 𝒢_{F} → G̃′(Q_{p}) factors through T̃′(Q_{p}), where T̃′ ⊂ G̃′ is a maximal torus which identifies with a subgroup of G_{m} × (L′)^{×}. Suppose that p splits completely in L′. Then the representation ρ′_{1} decomposes as a direct sum of two opposite characters ψ_{1} and ψ_{1}^{−1} and it follows from 3.3 that the product ψ_{1}ρ_{1} is geometric with potentially good reduction everywhere. By construction, the image of ψ_{1}ρ_{1} is Zariski dense in GL_{2}, so ψ_{1}ρ_{1} is an irreducible representation. Note that, as F contains the reflex field L′ of the CM-type of A′ (cf. 2.6), it is a totally imaginary field. We have thus constructed a 2-dimensional geometric representation of the absolute Galois group of a totally imaginary field F. Of course, the same statement holds for ψ_{1}^{−1}ρ_{1} and similar constructions can be carried out using the decompositions of ρ′_{2} as ψ_{2}⊕ψ_{2}^{−1} and of ρ′_{3} as ψ_{3}⊕ψ_{3}^{−1}. All the above statements are true for the 6 representations ψ_{i}^{±1}ρ_{i} (for i = 1, 2, 3). The representation that will be of interest in section 4 is the direct sum

σ = ψ_{1}ρ_{1} ⊕ ψ_{1}^{−1}ρ_{1} ⊕ ψ_{2}ρ_{2} ⊕ ψ_{2}^{−1}ρ_{2} ⊕ ψ_{3}ρ_{3} ⊕ ψ_{3}^{−1}ρ_{3}.  (3.6.∗)

Let T_{L′} be the kernel of the map (L′)^{×} → K^{×} induced by the field norm and define T_{K} = K^{×} ∩ T_{L′}, seen as group schemes over Q. This implies that T_{L′} = T̃′ ∩ (G̃′)^{der} and T_{K} = {x ∈ K^{×} | x^{2} = 1}. Let H′ be the algebraic Q-group defined by the short exact sequence

1 ⟶ T_{K} ⟶^{x↦(x,x^{−1})} G̃^{der} × T_{L′} ⟶ H_{′} ⟶ 1.  (3.6.†)

The Galois representation on H_{ét}^{1}(A\overline{Q}, Q_{p}) ⊗ H_{ét}^{1}(A′ ; \overline{Q}, Q_{p}) factors through (G × T′/G_{m})(Q_{p}), where

G_{m,Q} ⟶ G × T′
z ↦ (z · id, z^{−1} · id).

The maps G̃^{der} → G and T_{L′} → T′ induce a map H′ → (G × T′/G_{m}) and the representation σ defined in (3.6.∗) is a lifting to H′(Q_{p}) of the representation of 𝒢_{F} on the Tate twist

H_{ét}^{1}(A\overline{Q}, Q_{p}) ⊗ H_{ét}^{1}(A′ ; \overline{Q}, Q_{p})(1).

The facts that ρ(𝒢_{F}) is open in G(Q_{p}) and that ρ′(𝒢_{F}) is open in T′(Q_{p}) imply that σ(𝒢_{F}) is open in H′(Q_{p}).

4 The geometric origin

4.1. Proposition. Let τ: 𝒢_{F} → GL_{4}(Q_{p}) be a representation as in theorem 3.3 and assume that Lie(im(τ)) ≅ 𝔰𝔩_{2}^{2} (resp. let ψ_{i}^{±1}ρ_{i} be as in 3.6). Let A/F and A′/F be the abelian varieties serving in the construction of τ (resp. ψ_{i}^{±1}ρ_{i}). Then there do not exist n, m ∈ Z such that τ (resp. ψ_{i}^{±1}ρ_{i}) is isomorphic to a subquotient of a Tate twist of

H_{ét}^{1}(AF̄, Q_{p})^{⊗n} ⊗ H_{ét}^{1}(A′ ; F̄, Q_{p})^{⊗m}.

Proof. We first give the proof for ψ_{1}ρ_{1}, the other ψ_{i}^{±}ρ_{i} are handled by identical arguments. Assume that n, m ∈ Z such that ψ_{1}ρ_{1} is isomorphic to a subquotient of a Tate twist of H_{ét}^{1}(AF̄, Q_{p})^{⊗n}⊗H_{ét}^{1}(A′ ; F̄, Q_{p})^{⊗m}. Then ρ_{1} is a twist by a character of a subquotient of H_{ét}^{1}(AF̄, Q_{p})^{⊗n} and hence a twist by a character of a subquotient of

ρ_{1}^{⊗n} ⊗ ρ_{2}^{⊗n} ⊗ ρ_{3}^{⊗n}.

It follows from 3.6 that ρ̃(𝒢_{F}) is open in G_{m}(Q_{p}) × SL_{2}(Q_{p})^{3}. The representation theory of SL_{2}^{3} therefore implies that ρ_{1}^{⊗n} contains an irreducible factor isomorphic to ρ_{1} and that ρ_{2}^{⊗n} and ρ_{3}^{⊗n} both contain an invariant line. The first condition implies that n is even and the second one that n is odd, a contradiction which proves the proposition. To prove the result for τ, we use the notations of theorem 3.3. The condition on im(τ) implies that ρ(𝒢_{F}) and ρ′(𝒢_{F}) are not commutative, so G and G′ are the Mumford–Tate groups of A and A′ respectively. As noted in 3.6, this implies that ρ(𝒢_{F}) (resp. ρ′(𝒢_{F})) is open in G(Q_{p}) (resp. G′(Q_{p})). The Mumford–Tate group of A × A′ is the group G′′ ⊂ G × G′ introduced in the proof of proposition 3.4, so (ρ, ρ′)(𝒢_{F}) ⊂ G′′(Q_{p}). The Zariski closure H_{p} of the image of (ρ, ρ′) is a reductive algebraic subgroup of G′′_{Q_{p}} containing the centre. We will show that H_{p} = G′′ by proving that its rank is equal to 6. Considering the restrictions to the different inertia subgroups, it is clear that ρ′_{1} is isomorphic to neither ρ_{2} nor ρ_{3}. The condition on im(τ) implies that ρ′_{1} is not isomorphic to ρ_{1} either. Together with the fact that ρ(𝒢_{F}) is open in G(Q_{p}), this implies that the rank of H_{p} is at least 4. Let ℓ be a prime number which is inert in the cubic number field K. The Zariski closure H_{ℓ} of the ℓ-adic Galois representation associated to A × A′ is a reductive algebraic subgroup of G′′_{Q_{ℓ}}. It follows from [Ser81, §3] (cf. [LP92, 6.12, 6.13]) that the ranks of H_{p} and H_{ℓ} are equal, so H_{ℓ} is of rank at least 4. As the adjoint group G′′_{Q_{ℓ}} is a product of two Q_{ℓ}-simple groups of rank 3, this gives H_{ℓ} = G′′_{Q_{ℓ}}. We conclude that H_{ℓ} and hence H_{p} are of rank 6. The fact that H_{p} = G′′_{Q_{p}} implies that (ρ, ρ′)(𝒢_{F}) is open in G′′(Q_{p}). The statement about τ can now be proved using the representation theory of SL_{2}^{6} in an argument similar to the one used for ψ_{1}ρ_{1}.

4.2 Remark. In the proposition, the condition that Lie(im(τ)) ≅ 𝔰𝔩_{2}^{2} implies that τ is irreducible. This is essential for the conclusion of the proposition to hold. To see this, consider the representation ρ_{1} ⊗ ρ_{1}, which is reducible and decomposes as the sum of the trivial representation ∧^{2}ρ_{1} and Sym^{2}ρ_{1}. The only interesting representation of the two is Sym^{2}ρ_{1}. It is a quotient of H_{ét}^{1}(AF̄, Q_{p})^{⊗2}.

4.3 The special case considered in 3.6. From now on and up to and including theorem 4.12, we place ourselves in the situation of 3.6. In particular, the abelian varieties A and A′ have Mumford–Tate groups G and T′ ⊂ G′ respectively. Moreover, we let D be the division algebra associated to G, as in 2.1, and let L′ be the reflex field of the CM-type of A′. We saw in in 2.6 that the Mumford–Tate group of A′ is the image of (L′)^{×} in G′.

4.4 The construction of the abelian variety B. By [Noo01, proposition 1.5], the morphism h: S → G_{R} determining the Hodge structure on V = H_{B}^{1}(A(C), Q) lifts uniquely to a map

h_{D}: S ⟶ D_{R}^{×} ≅ GL_{2,R} × H^{×} × H^{×}

conjugate to

z = a + bi ↦ (( \begin{matrix}a & b \\ −b & a \end{matrix} ) , 1, 1) .

As D ⊗_{Q} R = ∏_{K↪R} D ⊗_{K} R, the factors in the above product correspond to the real embeddings φ_{1}, φ_{2} and φ_{3} of K (in that order) and the composite of φ_{1} with the inclusion R ⊂ C is the embedding K ↪ C fixed in 2.2. On the ′-side, there is an isomorphism

(L′)_{R}^{×} ≅ ∏_{K↪R} S,  (4.4.∗)

the factors still being indexed by the φ_{i}. Let

h′_{L′} = (1, id, id): S ⟶ ∏_{K↪R} S ≅ (L′)_{R}^{×},  (4.4.†)

where the trivial component is the one corresponding to φ_{1}. Let H denote the algebraic group defined by the short exact sequence

1 ⟶ K^{×} ⟶^{x↦(x,x^{−1})} D^{×} × (L′)^{×} ⟶ H ⟶ 1.  (4.4.‡)

The group H′ defined by (3.6.†) identifies with a subgroup of this group and H and H′ naturally act on W_{B} = D⊗_{K} L′. For any pair of maps h_{1} : S → D_{R}^{×} and h_{2} : S → (L′)_{R}^{×}, we will write h_{1} · h_{2} for the composite of the product (h_{1}, h_{2}) with the projection onto H_{R}. The morphisms h_{D} and h′_{L′} give rise to h_{H} = h_{D} ·h′_{L′}: S → H_{R} and via the action of H on W_{B}, this gives a Hodge structure on W_{B}, cf. [Del72, §6]. By [Del72, pp. 161–162], this is the Hodge structure on the Betti cohomology of some polarizable complex abelian variety B_{C}. By construction, W_{B} = H_{B}^{1}(B_{C}(C), Q) and dim(B_{C}) = 12. The totally real field K is contained in the centre of D, so L′ is contained in the centre of D ⊗_{K} L′. It follows that L′ acts on B_{C} by isogenies.

4.5. Lemma. Let B_{C} be as above. There exist a number field F′ ⊂ C and an abelian variety B over F′ such that B_{C} = B ⊗_{F′} C and L′ ⊂ End^{0}(B).

Proof. It suffices to show that B_{C} admits a model over \overline{Q}. Let X be the conjugacy class of the morphism h: S → G_{R} defining the Hodge structure on H_{B}^{1}(A(C), Q), let X_{D} be the conjugacy class of h_{D}: S → D^{×} and Y that of h_{H}. By [Del72, §6], this gives rise to Shimura data (G, X), (D^{×}, X_{D}), ((L′)^{×}, {h′_{L′}}) and (H, Y) and morphisms (D^{×}, X_{D}) → (G, X) and

(D^{×} × (L′)^{×}, X_{D} × {h′_{L′}}) → (H, Y).

For appropriate compact open subgroups C ⊂ G(A^{f}), C_{D} ⊂ D^{×}(A^{f}), C_{H} ⊂ H(A^{f}) and C_{D,L} ⊂ (D^{×} × (L′)^{×})(A^{f}) we obtain a diagram

C_{D,L}M\overline{Q}(D^{×} × (L′)^{×}, X_{D} × \{h′_{L′}\}) ⟶ _{C_{H}} M\overline{Q}(H, Y)
↓
C_{D}M\overline{Q}(D^{×}, X_{D})
↓
CM\overline{Q}(G, X)

of morphisms between the weakly canonical models of the associated Shimura varieties. We dispose of faithful representations of G and of H, and via these representations, the C-valued points of the Shimura varieties _{C}M\overline{Q}(G, X) and C_{H} M\overline{Q}(H, Y) correspond to isogeny classes of polarized abelian varieties endowed with extra structure. If the above compact subgroups are sufficiently small, all the Shimura varieties in question are smooth and both _{C}M\overline{Q}(G, X) and _{C_{H}} M\overline{Q}(H, Y) carry families of abelian varieties such that the fibre over a C-valued point of the Shimura variety lies in the isogeny class corresponding to that point. One can moreover choose the compact subgroups above such that the vertical maps are finite. After fixing a level structure on A\overline{Q}, it corresponds to a point

a ∈ _{C}M\overline{Q}(G, X)(\overline{Q}).

By construction, there is a C-valued point

ã ∈ _{C_{D,L}}M\overline{Q}(D^{×} × (L′)^{×}, X_{D} × \{h′_{L′}\})(C)

in the fibre over a of the vertical map such that ã maps to a point b ∈ _{C_{H}} M\overline{Q}(H, Y)(C) corresponding to the isogeny class of B_{C} endowed with an appropriate level structure. Because the vertical arrows are finite, it follows that ã and hence b are defined over \overline{Q} and this implies that B_{C} admits a model over \overline{Q}.

4.6 The construction of the abelian variety C. We keep the notations of 4.3 and 4.4. The morphism h: S → T′_{R} determining the Hodge structure of A′_{C} on V′ = H_{B}^{1}(A′(C), Q) lifts to a map h_{L′}: S → (L′)_{R}^{×} inducing an isomorphism of S with the factor of (L′)_{R}^{×} ≅ ∏_{K↪R} S corresponding to φ_{1} : K ↪ R and inducing the trivial map on the other factors. Let h′_{L′} be as in (4.4.†) and denote by \overline{h′_{L′}} its composite with the involution of S induced by complex conjugation. We define yet another map h′′_{L′}: S → (L′)_{R}^{×} as the product \overline{h′_{L′}}h_{L′}, where the product is taken for the commutative group structure on (L′)_{R}^{×}. Together with the natural action of L′ on W_{C} = L′, this defines a Hodge structure on W_{C}. This Hodge structure is the Hodge structure of C_{C} for an abelian variety C of CM-type over a number field F′′ ⊂ C. One has dim(C) = 3 and L′ ⊂ End^{0}(C). By construction, W_{C} = H_{B}^{1}(C(C), Q) and dim_{L′}(W_{C}) = 1.

4.7 Remark. Possibly after replacing the identification of equation (4.4.∗) on the first factor by its complex conjugate, one can assume that h_{L′} is given by

h_{L′} = (id, 1, 1): S ⟶ ∏_{K↪R} S ≅ (L′)_{R}^{×}.

Doing so, the map h′′_{L′} is given by z ↦ (z, z̄, z̄).

4.8 The construction of the motives m′ and m. In what follows, we will work in the category 𝒞 of motives for absolute Hodge cycles as described in [DM82], especially section 6 of that paper. Recall that this category is constructed as Grothendieck’s category of motives except where it concerns the morphisms. These are defined to be given by absolute Hodge classes, not by cycle classes as usual. We keep the assumptions of 4.3. Replacing F by a finite extension and extending the base field of the varieties in question, we will assume from now on that the abelian varieties A and A′ (of 3.6), B (of 4.5) and C (of 4.6) are F -varieties and that L′ acts on B and C (over F). The motives h^{1}(B) and h^{1}(C) belong to the category 𝒞_{(L′)} of objects of 𝒞 endowed with L′-action. One can thus form the tensor product

m′ = h^{1}(B) ⊗_{L′} h^{1}(C),

still belonging to 𝒞_{(L′)}, as in [DM82, pp. 155–156]. Let K̃ be the normal closure of K in C. As K is totally real, one has K̃ ⊂ R. The decomposition of the algebra K ⊗_{Q} K̃ as a product of fields gives rise to a system (u_{1}, u_{2}, u_{3}) of orthogonal idempotents in K ⊗_{Q} K̃, indexed be the real embeddings φ_{1}, φ_{2}, φ_{3} of K. For 1 ≤ i < j ≤ 3, let u_{i,j} = u_{i} + u_{j}. Let m′ ⊗_{Q} K̃ be the external tensor product as in [DM82, pp. 155–156]. As K ⊂ L′, it acts on m′ and one deduces a K̃-linear action of K⊗_{Q} K̃ on m′⊗_{Q} K̃, given by a Gal(K̃/K)-equivariant K̃-algebra morphism

K ⊗_{Q} K̃ ⟶ EndK̃(m′ ⊗_{Q} K̃) = End(m′) ⊗_{Q} K̃.

This action induces an action of K⊗_{Q}K̃ on ∧3 ; K̃(m′⊗_{Q}K̃) given by a Gal(K̃/K)- equivariant multiplicative map (not a morphism of algebras)

K ⊗_{Q} K̃ ⟶ EndK̃(∧3 ; K̃(m′ ⊗_{Q} K̃)) = End(∧^{3}m′) ⊗_{Q} K̃.

For 1 ≤ i < j ≤ 3, let u′_{i,j} be the image of u_{i,j} in End(∧^{3}m′) ⊗_{Q} K̃. As

{u_{2,3}, u_{1,3}, u_{1,2}} ⊂ K ⊗_{Q} K̃

is a Gal(K̃/K)-invariant subset, the sum u′_{2,3} + u′_{1,3} + u′_{1,2} ∈ End(∧^{3}m′) ⊗_{Q} K̃ is also Gal(K̃/K)-invariant. This element therefore determines an element u′ ∈ End(∧^{3}m′). The AH-motive m is defined to be the kernel of u′ on ∧^{3}m′.

4.9 Remark. Intuitively, the aim of this construction is to pass from an object m′ which can be expressed, after tensoring with K̃, as a direct sum, to the tensor product of the direct factors. This tensor product descends to an object m over Q.

4.10. Proposition. Let notations and assumptions be as in 4.3–4.8. In particular, the Mumford–Tate group of A is equal to G, that of A′ is T′, the varieties A, A′, B and C are defined over F and L′ acts on B and on C. Then there is an isomorphism

m ≅ (h^{1}(A) ⊗ h^{1}(A′)(−2))^{8}

of absolute Hodge motives.

Proof. The main theorem (2.11) of Deligne’s paper [Del82] states that the spaces of Hodge cycles and of absolute Hodge cycles on an abelian variety coincide. As noted in [DM82, 6.25], this implies that to prove the proposition, it suffices to show that the Hodge structures on the Betti realizations of the motives m and (h^{1}(A) ⊗ h^{1}(A′)(−2))^{8} are isomorphic. First consider h^{1}(A), the Betti realization of which is V = H_{B}^{1}(A(C), Q). The Mumford–Tate group G of A admits a morphism D^{×} → G, which makes V into a representation of D^{×}. The Hodge structure on V is determined by the action of S on V_{R} given by the morphism h_{D}: S → D_{R}^{×} of 4.4. Similarly, V′ = H_{B}^{1}(A′(C), Q), the Betti realization of h^{1}(A′), is made into a representation of (L′)^{×} by the morphism (L′)^{×} → T′, and the map h_{L′}: S → (L′)_{R}^{×} of 4.6 determines the Hodge structure on V′. In these representations, the subgroup K^{×} ⊂ D^{×}, resp. K^{×} ⊂ (L′)^{×} acts through the map K^{×} → G_{m,Q} induced by the field norm N_{K/Q}. It follows that the action of D^{×} × (L′)^{×} on V ⊗_{Q} V′, the Betti realization of h^{1}(A) ⊗ h^{1}(A′), factors through the group H defined by (4.4.‡). Passing to \overline{Q}, one has

H\overline{Q} ≅ ∏_{K↪\overline{Q}} (GL_{2} × G_{m}^{2}/ G_{m}) ,

where the map G_{m} → GL_{2}×G_{m}^{2} on each factor is given by z ↦ (z·id, z^{−1}, z^{−1}). The representation of H\overline{Q} on V ⊗ V′ ⊗ \overline{Q} induces a representation of

(GL_{2,\overline{Q}} × G_{m,\overline{Q}}^{2})^{3}

on this space which is isomorphic to the tensor product of the standard representations of its factors GL_{2} and G_{m}^{2} on \overline{Q}^{2}. The Hodge structure on V ⊗ V′, considered as the Betti realization of h^{1}(A) ⊗ h^{1}(A′), is given by h_{D} · h_{L′}: S → H_{R}. Multiplying the composite map

S ⟶^{h_{D}·h_{L′}} H_{R} ⟶ GL(V ⊗ V′ ⊗ R)

by the square of the norm map S → G_{m,R}, one gets the Hodge structure of the Betti realization of the Tate twist h^{1}(A) ⊗ h^{1}(A′)(−2). The Betti realization of m′ is H_{B}^{1}(B(C), Q) ⊗_{L′} H_{B}^{1}(C(C), Q) = W_{B} ⊗_{L′} W_{C} and the group H of (4.4.‡) naturally acts on this space. Over \overline{Q} we have a decomposition

(W_{B} ⊗_{L′} W_{C}) ⊗_{Q} \overline{Q} = (W_{1} ⊕ W_{2} ⊕ W_{3})^{2}

of the induced representation of H\overline{Q}. As before, the factors correspond to the embeddings K ↪ \overline{Q}. The representation of H\overline{Q} on W_{i} is isomorphic to the one induced by the representation of the ith factor GL_{2} × G_{m}^{2} on \overline{Q}^{2} ⊗\overline{Q} \overline{Q}^{2} by tensor product. The Hodge structure on the tensor product W_{B} ⊗_{L′} W_{C} is obtained by multiplying the map h_{H} defining the Hodge structure of B_{C} by h′′_{L′}, the map determining the Hodge structure of C. Since h_{H} is defined in 4.4 as the product h_{D} · h′_{L′} and since h′′_{L′} = h_{L′} \overline{h′_{L′}}, it follows that h_{H}h′′_{L′} = h_{D} · h_{L′}h′_{L′} \overline{h′_{L′}}. Writing N′ = h′_{L′} \overline{h′_{L′}}, this implies that the Hodge structure on W_{B} ⊗_{L′} W_{C} is determined by

h_{H}h′′_{L′} = h_{D} · h_{L′}N′: S ⟶ H_{R}.

The image of N′: S → (L′)_{R}^{×} lies in K_{R}^{×} and N′ is given by

N′: S ⟶ ∏_{K↪R} G_{m,R} ≅ K_{R}^{×}
z ↦ (1, zz̄, zz̄).

With this information, the Betti realization W_{m} of m can be computed. It is the kernel of the endomorphism of ∧^{3}(W_{B} ⊗_{L′} W_{C}) induced by the map u′ of 4.8. For 1 ≤ i < j ≤ 3, the endomorphism u_{i,j} ∈ End\overline{Q}(W_{B} ⊗_{L′} W_{C} ⊗_{Q} \overline{Q}) induces the identity on W_{i} and W_{j} and zero on the remaining factor. Since W_{m} ⊗_{Q} \overline{Q} is the kernel of u′ ; \overline{Q} on ∧^{3}(W_{B} ⊗_{L′} W_{C}) ⊗_{Q} \overline{Q}, it follows that there is an isomorphism

W_{m} ⊗_{Q} \overline{Q} ≅ W_{1}^{2} ⊗ W_{2}^{2} ⊗ W_{3}^{2} = (W_{1} ⊗ W_{2} ⊗ W_{3})^{8}

of representations of H\overline{Q}. It follows that V ⊗ V′ ≅ W_{m} as representations of H. We still have to show that the Hodge structures are the same. In view of what we already know about these Hodge structures, it suffices to show that the action of S on W_{m}⊗R defined by N′: S → K_{R}^{×} ⊂ H_{R} and the representation of H on W_{m} is equal to scalar multiplication by the square of the norm S → G_{m,R}. This in turn follows immediately from the above description of N′ and the fact that K^{×} ⊂ H acts on W_{m} through the map K^{×} → G_{m,Q} induced by the field norm N_{K/Q}.

4.11. Corollary. Keep the hypotheses of the proposition. For any prime number p, there is an isomorphism of 𝒢_{F} -modules between

(H_{ét}^{1}(A\overline{Q}, Q_{p}) ⊗_{Q_{p}} H_{ét}^{1}(A′ ; \overline{Q}, Q_{p})(−2))^{8}

and the étale p-adic realization of m. This representation is a subquotient of H_{ét}^{6}(B\overline{Q} × C\overline{Q}, Q_{p}).

4.12. Theorem. Let notations and assumptions be as in 4.3. In particular, the Mumford–Tate group of A is equal to G and that of A′ is T′. Assume moreover that A, A′, B and C are defined over F and that ρ(𝒢_{F}) and ρ′(𝒢_{F}) are sufficiently small. Let the representation

σ: 𝒢_{F} → H′(Q_{p}) ⊂ H(Q_{p})

be as in (3.6.∗). Then there exists a finite extension F′ of F such that the restriction of σ ⊕ σ to 𝒢_{F′} is isomorphic to the representation of 𝒢_{F′} on

H_{ét}^{1}(B\overline{Q}, Q_{p}) ⊗_{L′⊗Q_{p}} H_{ét}^{1}(C\overline{Q}, Q_{p})(1).

In particular, the representations ψ_{i}^{±}ρ_{i} (for i = 1, 2, 3) defined in 3.6 come from algebraic geometry in the sense of 0.3.

Proof. The first statement of the theorem implies that σ: 𝒢_{F} → H′(Q_{p}) differs from a representation of 𝒢_{F} on a subquotient of an étale cohomology group of an algebraic variety by a finite character. As σ is the direct sum of the irreducible representations ψ_{i}^{±}ρ_{i}, the second statement of the theorem follows. This leaves the first statement to be proven. From the constructions of B and C and from the proof of proposition 4.10, it is clear that the Mumford–Tate group of the Hodge structure on H_{B}^{1}(B(C), Q) ⊗_{L′} H_{B}^{1}(C(C), Q) is contained in H and that the Mumford–Tate group of H_{B}^{1}(B(C), Q) ⊗_{L′} H_{B}^{1}(C(C), Q)(1) is contained in H′. By [Del82, 2.9, 2.11], this implies that there is a finite extension F′ of F such that L′ acts on B_{F′} and on C_{F′} and such that the representation of 𝒢_{F′} on

W^{(p)} = H_{ét}^{1}(B\overline{Q}, Q_{p}) ⊗_{L′⊗Q_{p}} H_{ét}^{1}(C\overline{Q}, Q_{p})(1) factors through a morphism σ′: 𝒢_{F′} → H′(Q_{p}). The hypotheses that D ⊗_{Q} Q_{p} ≅ M_{2}(Q_{p})^{3} and that p splits completely in L′ imply that W^{(p)} decomposes as the direct sum of two isomorphic representations of the group H′_{Q_{p}}. We contend that, possibly after replacing F′ by a finite extension, W^{(p)} is isomorphic to V^{(p)}, the 𝒢_{F′} module underlying σ ⊕ σ. The decomposition

σ = ⨁_{i=1,2,3} (ψ_{i}ρ_{i} ⊕ ψ_{i}^{−1}ρ_{i})

gives a decomposition V^{(p)} = (V_{1} ⊕ V_{2} ⊕ V_{3})^{2}. As σ lifts the Galois representation on H_{ét}^{1}(A\overline{Q}, Q_{p}) ⊗ H_{ét}^{1}(A′ ; \overline{Q}, Q_{p})(1) from (G × T′/G_{m})(Q_{p}) to H′(Q_{p}), one has

H_{ét}^{1}(A\overline{Q}, Q_{p}) ⊗ H_{ét}^{1}(A′ ; \overline{Q}, Q_{p})(1) = V_{1} ⊗ V_{2} ⊗ V_{3}.

If u′ ∈ End(∧^{3}V^{(p)}) is constructed as in 4.8, then ker(u′) = (V_{1} ⊗ V_{2} ⊗ V_{3})^{8}. In analogy with the proof of proposition 4.10, one has W^{(p)} = (W_{1} ⊕W_{2} ⊕W_{3})^{2} as representations of H_{Q_{p}} and (W_{1} ⊗ W_{2} ⊗ W_{3})^{8} is the étale p-adic realization of the motive m. It follows from the corollary 4.11 that

(V_{1} ⊗ V_{2} ⊗ V_{3})^{8} ≅ (W_{1} ⊗ W_{2} ⊗ W_{3})^{8}

as representations of 𝒢_{F′}. Hence σ and σ′ have the same projection to H′(Q_{p})/ ker(N)(Q_{p}). Here N: T_{K} → {±1} is the map induced by the norm N_{K/Q}, where T_{K} is, as in 3.6, the group scheme {x ∈ K^{×} | x^{2} = 1}. As T_{K}(Q_{p}) is finite, this implies that σ_{|𝒢_{F′}} and σ′_{|𝒢_{F′}} differ by a finite character and thus proves the theorem.

4.13. Corollary. We return to the hypotheses of theorem 3.3, i. e. we only assume that the Mumford–Tate groups of A and A′ are contained in G and G′ respectively. Then the representation τ: 𝒢_{F} → GL_{4}(Q_{p}) of (3.1.∗), which is geometric by theorem 3.3, comes from algebraic geometry.

Proof. By remark 0.6, it is sufficient to prove this after replacing F by a finite extension. Apart from the varieties A and A′ serving in the construction of τ in 3.1, we choose, after enlarging F if necessary, an auxiliary abelian variety A′′ with Mumford–Tate group contained in a group of Mumford’s type, which is of CM-type, and such that p splits completely in the reflex field L′′ of its CM-type. As in 3.6, let ψ_{1} be one of the characters in which the representation ρ′′_{1} decomposes. For F sufficiently large, the above theorem implies that ψ_{1}ρ_{1} and ψ_{1}^{−1}ρ′_{1} come from algebraic geometry. It follows that the same is true for τ = (ψ_{1}ρ_{1}) ⊗ (ψ_{1}^{−1}ρ′_{1}).

5 Other geometric representations

5.1 Let p be a prime number, F a number field and let Rep(𝒢_{F}) be the tannakian category of finite dimensional, continuous Q_{p}-linear representations of the absolute Galois group 𝒢_{F}. Consider all abelian fourfolds A/F such that the conditions of 2.4 are verified for some identification V_{A} = H_{B}^{1}(A(C), Q) and some algebraic group G_{A} ⊂ GL(V_{A}). We denote the p-adic Galois representation associated to A by ρ_{A}: 𝒢_{F} → G_{A}(Q_{p}) and define ℳ to be the full tannakian subcategory of Rep(𝒢_{F}) generated by the ρ_{A} for A running through the above set of abelian varieties and the Tate object Q_{p}(1). For any A as above, we denote by ρ̃_{A} the canonical lifting of ρ_{A} constructed in lemma 1.4. Let ℒ be the full tannakian subcategory of Rep(𝒢_{F}) generated by all liftings ρ̃_{A} and Q_{p}(1). Obviously, ℳ is a full subcategory of ℒ. Let ρ̃ be an object of ℒ, it is a representation of 𝒢_{F} on a finite dimensional Q_{p}-vector space V. We let G̃ be the Zariski closure of the image of ρ̃ in GL(V). The category generated by ρ̃ contains an object ρ of ℳ such that there is a central isogeny G̃ → G, where G denotes the Zariski closure of the image of ρ.

5.2. Theorem. Let τ be an irreducible representation of 𝒢_{F} contained in ℒ which is geometric. Then τ comes from algebraic geometry.

Proof. Each ρ̃_{A} is a direct sum of a (fixed) character ρ_{0} and three 2-dimensional representations ρ_{A,1}, ρ_{A,2} and ρ_{A,3}. It follows that ℒ is generated by ρ_{0} and the ρ_{A,i}, for i = 1, 2, 3 and A running through the abelian varieties considered in 5.1. This implies that each object of ℒ is a subquotient of a representation of the form

τ = ρ_{0}^{k} · ⨂_{j=1}^{r} ρ_{j},  (5.2.∗)

with k ∈ Z, and where ρ_{j} = ρ_{A_{j},i_{j}} for some an abelian variety A_{j} of the type considered above and some integer i_{j} ∈ {1, 2, 3}. It therefore suffices to consider representations of this form. Let τ be as in (5.2.∗). It follows from 3.6 and 4.12 that, for each j, there exist a finite extension F_{j} of F and a character ψ_{j} of 𝒢_{F_{j}} such that ψ_{j}τ_{|𝒢_{F_{j}}} comes from algebraic geometry. Let F′ be a finite extension of F containing all F_{j}, define the character ψ of 𝒢_{F′} by ψ = (ρ_{0}^{−k}) ∏ ψ_{j} and let

τ′ = ⨂_{j=1}^{r} (ψ_{i}ρ_{j})_{|𝒢_{F′}} = ψτ_{|𝒢_{F′}} .

This defines a geometric representation 𝒢_{F′}. After replacing F′ by a finite extension, τ′ is a Tate twist of a subquotient of an étale cohomology group of an algebraic (even abelian) variety over F′. Now assume that τ and hence τ_{|𝒢_{F′}} are geometric. This implies that ψ is a geometric representation of 𝒢_{F′} as well. If follows from [FM95, §6], that ψ comes from algebraic geometry and this in turn implies that τ_{|𝒢_{F′}} = ψ^{−1}τ′ is a Tate twist of a subquotient of an étale cohomology group of an algebraic variety. By remark 0.6, the same thing is true for τ.

5.3 Using the proof of theorem 5.2, one can describe the geometric representations contained in ℒ. We keep the notations used above. Let A/F be an abelian variety as in 5.1, G_{A} its Mumford–Tate group and v a p-adic valuation of F. Let

μ_{HT,A,v}: G_{m,C_{p}} → G_{A,C_{p}}

be the cocharacter defined by the Hodge–Tate decomposition of ρ_{|ℐ_{F,v̄}} for some valuation v̄ of F̄ with v = v̄_{|F}. We saw in the proof of proposition 1.9 that μ_{HT,A,v}^{2} lifts to μ̃_{A,v}: G_{m,C_{p}} → G̃_{A,C_{p}}. This lifting is conjugate to the map

G_{m,C_{p}} ⟶ G_{m,C_{p}} × SL_{2,C_{p}} × SL_{2,C_{p}} × SL_{2,C_{p}} ≅ G̃_{A,C_{p}}
z ↦ (z, ( \begin{matrix}z & 0 \\ 0 & z^{−1} \end{matrix} ) , id, id) .

The factor SL_{2} on which μ̃_{A,v} is non trivial is the factor to which the image of ℐ_{F,v̄} projects non-trivially. The representations

ψ_{j}ρ_{j}: 𝒢_{F} → GL_{2}(Q_{p})

occurring in the expression for τ′ are geometric and the maps μ̃_{A,v} define cocharacters μ̃_{j,v}: G_{m,C_{p}} → GL_{2,C_{p}}. The above description of the μ̃_{A,v} and the choice of the ψ_{j} imply that each μ̃_{j,v} is a square of a cocharacter of GL_{2,C_{p}}. Similarly, the liftings μ̃_{A,v} and the expression of the character

ψ: 𝒢_{F} → G_{m}(Q_{p})

as a product of ρ_{0}^{−k} and the ψ_{j} define a map μ̃_{v}: G_{m,C_{p}} → G_{m,C_{p}}. It follows from proposition 1.9, its proof, and the description of the μ̃_{A,v}, that ψ_{|ℐ_{F,v̄}} is the nth power of (ρ_{0})_{|ℐ_{F,v̄}} for some integer n and that μ̃_{v}(z) = z^{n}. This means that ψ is potentially semi-stable at v if and only if n is even, which is the case if and only if μ̃_{v} is a square. The condition that ψ is potentially semi-stable at v is equivalent to τ being potentially semi-stable at v. This proves the following proposition, characterizing the geometric representations in ℒ.

5.4. Proposition. For each object ρ̃: 𝒢_{F} → GL(V) of ℒ, there exists a central isogeny N: G̃ → G of algebraic groups over Q_{p} such that G̃ is the Zariski closure of the image of ρ̃ and such that N ∘ ρ̃ belongs to ℳ. The representation ρ is semi-stable at v if and only if the cocharacter

μ_{HT,v}: G_{m,C_{p}} → G_{C_{p}},

defined by the Hodge–Tate decomposition of ρ_{|ℐ_{F,v̄}} (for a place v̄ of F̄ lying over v), lifts to G̃_{C_{p}}. The representation ρ is geometric if and only if μ_{HT,v} lifts for each valuation v of F.

5.5. Corollary. Let F be a number field, p a prime number and m be a motive belonging to the category of absolute Hodge motives generated by the motives of the abelian varieties considered in 5.1. Assume that G is the Mumford– Tate group of m and let ρ: 𝒢_{F} → G(Q_{p}) the associated p-adic Galois representation. Suppose that G̃ → G is a central isogeny such that ρ lifts to a representation ρ̃: 𝒢_{F} → G̃(Q_{p}) belonging to ℒ. Then the following statements are equivalent.

1. ρ̃ is geometric.

2. For some p-adic valuation v̄ of F̄, the restriction ρ̃_{|ℐ_{F,v̄}} is potentially semi-stable.

3. The cocharacter μ_{HdR} : G_{m,C} → G_{C} defining the Hodge filtration on the Betti realization of m lifts to G̃_{C}.

4. ρ̃ comes from algebraic geometry.

Proof. By hypothesis, the conjugacy class C_{HdR} of μ_{HdR} can be defined over F. It was noted in the proof of proposition 2.5 that, for any embedding F̄ ↪ C_{p} inducing the valuation v̄, the conjugacy class of μ_{HT,v} is equal to C_{HdR} ⊗_{F} C_{p}. Using proposition 5.4, this proves that 2 implies 3 and that 3 implies 1. The theorem 5.2 shows that 1 implies 4. The remaining implications are left to the reader.

5.6 Remarks.

5.6.1 If condition 3 of the corollary is verified, then it follows from [Win95, Théorème 2.1.7] that there exists a finite extension F′ of F such that the representations 𝒢_{F′} → G(Q_{ℓ}), for ℓ ranging through the prime numbers, simultaneously lift to G̃(Q_{ℓ}).

5.6.2 The geometric representations considered in this paper are obtained by lifting representations coming from algebraic geometry along isogenies of algebraic groups H̃ → H over Q. Conjecturally, a representation 𝒢_{F} → H(Q_{p}) which comes from algebraic geometry corresponds to a morphism f: G_{M}^{0} → H, where G_{M}^{0} is the connected component of the motivic Galois group, see for example [Ser94]. The geometric representations considered in this paper are obtained by lifting along an isogeny H̃ → H with the property that the morphism G_{m} × G_{m} → H_{C} determining the (mixed) Hodge structure on the corresponding Betti cohomology group lifts to H̃. In [Ser94, 8.1], Serre asks whether the derived group of G_{M}^{0} is simply connected. He notes that an argument due to Deligne shows that if this is the case, then the fact that the morphism determining the Hodge structure lifts to H̃ implies that f: G_{M}^{0} → H lifts to a morphism G_{M}^{0} → H̃. This in turn should imply that the lifted Galois representation comes from algebraic geometry. An affirmative answer to Serre’s question, in conjunction with the standard conjectures involved in the theory of motives, therefore proves the Fontaine–Mazur conjecture for the representations considered in this paper. The above discussion also indicates that, conversely, the conjecture of Fontaine and Mazur is unlikely to be true if the answer to Serre’s question is negative.

References

[Bla94] D. Blasius. A p-adic property of Hodge classes on abelian varieties. In U. Jansen, S. Kleiman, and J.-P. Serre, editors, Motives, Proc. Sympos. Pure Math. 55, Part 2, pages 293–308. Amer. Math. Soc., 1994.

[Del72] P. Deligne. Travaux de Shimura. In Séminaire Bourbaki, Lecture Notes in Math. 244, pages 123–165. Springer-Verlag, 1972. Exposé 389, février 1971.

[Del82] P. Deligne. Hodge cycles on abelian varieties. In P. Deligne, J. S. Milne, A. Ogus, and K.-y. Shih, editors, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, chapter I, pages 9– 100. Springer-Verlag, 1982.

[dJ96] A. J. de Jong. Smoothness, semi-stability and alterations. Publ. Math. de l’I. H. E. S. 83 (1996), 51–93.

[DM82] P. Deligne and J. S. Milne. Tannakian categories. In P. Deligne, J. S. Milne, A. Ogus, and K.-y. Shih, editors, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, chapter II, pages 101–228. Springer-Verlag, 1982.

[FM95] J.-M. Fontaine and B. Mazur. Geometric Galois representations. In J. Coates and S. T. Yau, editors, Elliptic curves, modular forms, & Fermat’s last theorem, Ser. Number Theory I, pages 41–78. International Press Inc., 1995.

[Fon94] J.-M. Fontaine. Représentations p-adiques semi-stables. In Périodes p-adiques, séminaire de Bures 1988, Astérisque 223, pages 113–184. Soc. Math. France, 1994.

[LP92] M. Larsen and R. Pink. On ℓ-independence of algebraic monodromy groups in compatible systems of representations. Invent. Math. 107 (1992), 603–636.

[Mum69] D. Mumford. A note of Shimura’s paper “Discontinuous groups and abelian varieties”. Math. Ann. 181 (1969), 345–351.

[Noo95] R. Noot. Abelian varieties—Galois representations and properties of ordinary reduction. Compos. Math. 97 (1995), 161–171.

[Noo00] R. Noot. Abelian varieties with ℓ-adic Galois representation of Mumford’s type. J. Reine Angew. Math. 519 (2000), 155–169.

[Noo01] R. Noot. On Mumford’s families of abelian varieties. J. Pure Appl. Algebra 157, 1 (2001), 87–106.

[Ser78] J-P. Serre. Groupes algébriques associés aux modules de Hodge–Tate. In Journées de géométrie algébrique de Rennes (III), Astérisque 65, pages 155–188. Soc. Math. France, 1978. Also published in [Ser86, 119].

[Ser81] J-P. Serre. Letter to Ribet, 1–1–1981. Published in [Ser00, 133].

[Ser86] J-P. Serre. Œuvres — Collected papers. Springer-Verlag, 1986.

[Ser94] J-P. Serre. Propriétés conjecturales des groupes de Galois motiviques et des représentations ℓ-adiques. In U. Jansen, S. Kleiman, and J.-P. Serre, editors, Motives, Proc. Sympos. Pure Math. 55, Part 1, pages 377–400. Amer. Math. Soc., 1994. Also published in [Ser00, 161].

[Ser00] J-P. Serre. Œuvres — Collected papers, Volume IV. Springer-Verlag, 2000.

[Tay00] R. Taylor. Remarks on a conjecture of Fontaine and Mazur. e-print, Harvard university, to appear in Journal de l’Institut de Mathematiques de Jussieu., 2000.

[Tsu99] T. Tsuji. p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137, 2 (1999), 233–411.

[Win88] J.-P. Wintenberger. Motifs et ramification pour les points d’ordre fini des variétés abéliennes. In Séminaire de théorie des nombres de Paris 1986-87, Progr. Math. 75, pages 453–471. Birkhäuser, 1988.

[Win95] J.-P. Wintenberger. Relèvement selon une isogénie de systèmes ℓadiques de représentations galoisiennes associés aux motifs. Invent. Math. 120 (1995), 215–240.

Rutger Noot I. R. M. A. R. Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France noot@univ-rennes1.fr 446

Documenta Mathematica 6 (2001)