Lifting systems of Galois representations associated to abelian varieties

Rutger Noot^{}

2nd October 2006

Abstract

This paper treats what we call ‘weak geometric liftings’ of Galois representations associated to abelian varieties. This notion can be seen as a generalization of the idea of lifting a Galois representation along an isogeny of algebraic groups. The weaker notion only takes into account an isogeny of the derived groups and disregards the centres of the groups in question. The weakly lifted representations are required to be geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question states that any irreducible geometric representation is a twist of a subquotient of an étale cohomology group of an algebraic variety over a number field.

It is shown that a Galois representation associated to an abelian variety admits a weak geometric lift to a group with simply connected derived group. In certain cases, such a weak geometric lift is itself associated to an abelian variety. This means that the conjecture of Fontaine and Mazur is confirmed for these representations. In other cases, one may find a lift which can not be found back in the étale cohomology of any abelian variety. The Fontaine–Mazur conjecture remains open for these representations. Nevertheless, certain consequences of the conjecture can be established.

2000 Mathematics Subject Classification 11G10, 11F80, 14K15

Introduction

This paper is motivated by a conjecture of Fontaine and Mazur, conjecture 1 of [FM95] and generalizes the observations made in [Noo01]. The conjecture in question aims to characterize the Tate twists of the irreducible subquotients of the Galois representations arising from the étale cohomology of algebraic varieties over number fields.

Partially supported by the European Marie Curie Research Training Network contract MRTN-CT2003-504917

To explain the conjecture, fix a number field and let 𝒢_{}be the absolute Galois group of . In [FM95, §1], a representation of 𝒢_{}on a finite dimensional Q_{}-vector space _{}is called geometric if it is unramified outside a finite set of non-archimedean places of and if, for each non-archimedean valuation ¯ of ¯ of residue characteristic , the restriction to the inertia group at ¯ is potentially semi-stable in the sense of Fontaine, see also 4.1. An irreducible representation of 𝒢_{}on _{}is said to come from algebraic geometry if it is isomorphic to a subquotient of the Galois representation on a cohomology group H_{ét}^{}( ¯ Q_{})( ) for some smooth and proper -scheme and integers and . The Tate twist ( ) has the effect of multiplying the action of 𝒢_{}on H_{ét}^{}( ¯ Q_{}) by the th power of the cyclotomic character. Fontaine and Mazur have expressed the conjecture that an irreducible Q_{}-linear representation _{}of 𝒢_{}is geometric if and only if it comes from algebraic geometry.

The ‘if’-part of the conjecture being resolved (see [Tsu99]), this paper is concerned with some reflexions on the implication ‘only if’ in the conjecture. We will actually investigate a particular type of geometric representations.

The construction of these representations starts with the choice of an abelian variety over . For such an abelian variety, the representation of 𝒢_{}on H_{ét}^{1}( ¯ Q_{}) factors through a map

: 𝒢_{}_{}(Q_{})

where _{}is the Mumford–Tate group of . After reading [Win95], one is tempted to look for an isogeny : _{}and a map _{} ˜ : 𝒢_{}(Q_{}) such that _{}= _{} ˜ and such that _{} ˜ defines a geometric representation of 𝒢_{}on { \atop _{}}for any Q_{}-linear representation { \atop _{}}of { \atop _{Q_{}}}.

This is not exactly the point of view adopted in this paper. As the conjecture of Fontaine–Mazur for representations with abelian image is quite well understood, cf. [FM95, §6], the centres of the groups occurring above can be considered to be less interesting than the derived groups. For this reason, given the representation _{}: 𝒢_{}_{}(Q_{}), we will search for a linear algebraic group , an isogeny ^{der} : { \atop ^{der}} _{}^{der} and a map _{} ˜ : 𝒢_{}(Q_{}) such that ad _{}= ^{ ˜ ad} _{} ˜ . Here ^{ad} : _{}_{}^{ad} and ^{ ˜ ad} : { \atop ^{ad}} are the natural projections and the groups _{}^{ad} and { \atop ^{ad}} have been identified using the map induced by ^{der}. If such a representation _{} ˜ defines a geometric representation of 𝒢_{}on any Q_{}-linear representation { \atop _{}}of { \atop _{Q_{}}}, then we will say that it is a weak geometric lift of _{}. An advantage of working with weak geometric liftings is that, by ‘correcting’ liftings using characters, it is easier to produce geometric representations than if one considers liftings along isogenies.

It turns out that for any abelian variety , there exist a finite extension , a group such that { \atop ^{der}} is the universal cover of _{}and, for every prime number , a weak geometric lift _{} ˜ : 𝒢_{^{}}(Q_{}) of the restriction of _{}to 𝒢_{^{}}. We refer to corollary 5.11 for a precise statement. Moreover, it follows from corollary 5.12 that every weak geometric lift of a _{}is dominated by a weak geometric lift to a group with { \atop ^{der}} simply connected. We can even conveniently ‘normalize’ the group .

It is natural to ask if the conjecture of Fontaine and Mazur is true for any weak geometric lift of a representation associated to an abelian variety. This question is not answered in this paper, but a number of partial results are obtained.

In section 4 we prove the following results. By combining proposition 3.4 and remark 4.10, it follows that for every abelian variety , there exists a group , a map { \atop ^{der}} _{}^{der} and, after replacing by a finite extension, a system ( _{} ˜ ) of weak geometric liftings of the _{}such that

– the group { \atop ^{der}} is ‘not far’ from the universal cover of _{}^{der} and

– for any representation of , the system of representations of 𝒢_{}on the Q_{}is isomorphic to the system of -adic representation associated to an abelian variety.

To be more precise where the first property is concerned, it means in the first place that the group { \atop _{C}^{der}} is the product of its simple factors { \atop _{}}. Secondly, it follows from well-known facts on the Mumford–Tate groups of abelian varieties that these factors are all of classical type ( , , or ). For the { \atop _{}}which are of type _{}, _{}or _{}, being ‘not far’ from the universal cover means that they are simply connected. Where the factors { \atop _{}}of type _{}are concerned, the condition is more difficult to state. We have to distinguish two subtypes, _{}^{R} and _{}^{H}, see 2.5 for the definitions. The { \atop _{}}which are of type _{}^{R} in this classification are also simply connected. The factors { \atop _{}}of type _{}^{H} are -maximal in the sense of 2.5, which means that every such { \atop _{}}is a quotient of its universal cover by a subgroup of order 2. Over R, these -maximal groups are orthogonal groups.

By theorem 4.9, the above system ( _{} ˜ ) is maximal in the following sense. For any group ^{}, any isogeny ^{der} _{}^{der} such that ^{der} is a quotient of { \atop ^{der}} and any weak geometric lift _{}^{}: 𝒢_{}^{}(Q_{}) of _{}(allowing a finite extension of ), the representation _{}^{}belongs to the tannakian category generated by the -adic Galois representations associated to abelian varieties and the representations with finite image.

The above statements about Galois representations follow from analog properties of Hodge structures associated to abelian varieties. If _{C} is an abelian variety over C, then the Hodge structure on the first Betti cohomology group H_{B}^{1}( _{C}(C) Q) is determined by a morphism : _{R} , where = C^{} as algebraic groups over R. It is show in section 2 that the group above is actually the Mumford–Tate group of an abelian variety _{C} and that the Hodge structure of corresponds to a morphism ˜ : { \atop _{R}} such that and ˜ have the same projection to _{R}^{ad} = { \atop _{R}^{ad}} . To underscore the analogy with the construction of weak geometric liftings of Galois representations, such an abelian variety will be called a weak Mumford–Tate lift of , cf. 2.1. This notion is very close to the notion of _{C} and _{C} being ‘adjoint-isogenous’ in the sense of definition 6.1 of Vasiu’s e-print [Vas03] and our variety _{C} should be equal to the variety obtained by Vasiu’s ‘shifting process’, loc. cit. 6.4.

In section 3 we prove some properties comparing the fields of definition of an abelian variety and a weak Mumford–Tate lift. Using the theory of absolute Hodge motives, the above statements concerning Galois representations are then derived from the corresponding statements about the Hodge structures associated to weak Mumford–Tate liftings. The arguments are based on those used in [Noo01] and [Pau04].

In the above statements, the abelian varieties with Mumford–Tate group of type _{}^{H} stand out as an exception to the general situation. These varieties deserve a separate treatment, and this is the subject of section 5. Let be a number field as above, an abelian variety over and assume that, for the Mumford–Tate group _{}, the derived group _{C}^{der} is isomorphic to a product of -maximal groups of type _{}^{H}. It can be shown that such abelian varieties do indeed exist. It turns out that the associated system ( _{}) of -adic Galois representations admits a system of non-trivial weak geometric liftings.

The corollary 5.9 states that there exists a group such that { \atop ^{der}} _{}^{der} is the universal cover and, as always after replacing by a finite extension, a system ( _{} ˜ ) of weak geometric liftings of the _{}. These representations are called the -adic Galois representations of lifted abelian _{}^{H}-type. The construction makes use, again, of a lifting ˜ : { \atop _{R}} of : _{R} . This time, the existence of the system ( _{} ˜ ) follows from [Win95].

When considering weak geometric liftings of Galois representations associated to abelian varieties, these representations of lifted abelian _{}^{H}-type are the only ‘new’ representations that one may encounter. To be precise, let be an abelian variety with Mumford–Tate group _{}. For any group ^{}and isogeny ^{der} _{}^{der} and any weak geometric lift _{}^{}: 𝒢_{}^{}(Q_{}) of _{}, the representation _{}^{}belongs to the tannakian category generated by the -adic Galois representations associated to abelian varieties, the representations of lifted abelian _{}^{H}-type and those with finite image, see corollary 5.12.

In the final section 6 we study the representations _{} ˜ of lifted abelian _{}^{H}- type. It is proved in 6.2 that such a representation _{} ˜ generally does not belong to the tannakian category generated by the Galois representations associated to abelian varieties. The question if _{} ˜ belongs to the tannakian category generated by the Galois representations associated to algebraic varieties, as predicted by the Fontaine–Mazur conjecture, remains open.

Combined with ‘standard’ conjectures, the conjecture of Fontaine and Mazur implies that the _{} ˜ have certain ‘motivic’ properties. For example, any Frobenius element in 𝒢_{}should act semi-simply on any representation { \atop _{}}of { \atop _{Q_{}}}and the eigenvalues of the image should be Weil numbers, i. e. algebraic integers of which all complex absolute values coincide. Several of these motivic properties are proved in the final part of section 6.

In addition to their relevance to the Fontaine–Mazur conjecture, the construction of weak geometric liftings has applications to the study of Galois representations associated to abelian varieties. We have seen that for an abelian variety , and for large enough, the representations _{}: 𝒢_{}_{}(Q_{}) have weak geometric liftings _{}: 𝒢_{}_{}(Q_{}), where is an abelian variety and _{}^{der} _{}^{der} is not far from the universal cover. The Galois representation on H_{ét}^{1}( ¯ Q_{}) belongs to the tannakian category generated by H_{ét}^{1}( ¯ Q_{}) and the representations associated to abelian varieties of CM-type. This can be useful because the representation H_{ét}^{1}( ¯ Q_{}) is in general easier to study than H_{ét}^{1}( ¯ Q_{}). A first example of such an application can be found in the paper [Pau04] of F. Paugam. In [Vas03], A. Vasiu uses his technique of adjoint isogenous abelian varieties to study new instances of the Mumford–Tate conjecture. Other applications are to follow in forthcoming publications. Acknowledgements. This paper has benefited from discussions and correspondence with a number of people and I thank them heartily for their contribution. In particular, I would like to thank D. Blasius and P. Deligne for (independently) pointing out, after my previous work on the conjecture of Fontaine and Mazur, that the case of abelian varieties with Mumford–Tate group of type _{}^{H} deserved special attention.

1 Preliminaries

1.1 Basic notations. For any field , we denote by ¯ an algebraic closure of . The absolute Galois group of is the group 𝒢_{}= Aut_{}( ¯ ).

For any prime number , the field Q_{}is the -adic completion of Q and C_{} is the completion of an algebraic closure \overline{Q}_{}.

If is a group and a field, then Rep_{}( ) is the category of finite dimensional -linear representations of . If is a topological group (a Galois group for example) and a topological field, then the representations in Rep_{}( ) are assumed to be continuous. If is a linear algebraic group, then Rep_{}( ) is the category of algebraic representations of .

If is an algebraic group, a quasi-cocharacter of is an element of the direct limit

lim_{{ \atop N}} Hom (G_{}^{( )} )

where the transition map G_{}^{( )} G_{}^{( )} is the morphism ^{}. Giving a quasi-cocharacter of is equivalent to giving an integer and a cocharacter : G_{}. Intuitively the quasi-cocharacter given by ( ) is the th root of .

1.2 Absolute Hodge motives. We will freely use the language of tannakian categories. Everything we need here can be found in [DM82] which is to be considered the authoritative reference for all notions used but not explained in this paper. In particular, if 𝒞 is a tannakian category, then the subcategory -generated by a collection of objects of 𝒞 is the smallest subcategory of 𝒞 containing all objects which are isomorphic to a subquotient of a polynomial expression with coefficients in N in the objects contained in . In such a polynomial expression, + and are to be interpreted as and respectively.

For any field of characteristic 0, let Mot_{AH}( ) be the category of motives for absolute Hodge cycles as described in [DM82], especially section 6 of that paper. It is constructed as Grothendieck’s category of Chow motives except where it concerns the morphisms which are given by absolute Hodge classes, not by cycle classes as for Grothendieck motives. It must be pointed out that by definition, the morphisms between two motives _{1} and _{2} defined over are the appropriate absolute Hodge classes on the product which are defined over , i. e. Hodge classes on ( _{1} _{2}) ¯ invariant under the action of 𝒢_{}.

Motives are graded objects, each motive is a finite direct sum ^{}, where each ^{}is a pure motive of weight . The Tate motive Q(1) is the dual of ^{2}(P_{}^{1}) in the category Mot_{AH}( ).

1.3 Mumford–Tate groups. In what follows, we will assume that is contained in C and we write ¯ for the algebraic closure of in C. Assume that _{1} _{} are absolute Hodge motives over , and let 𝒞 = _{1} _{}Q(1) be the tannakian subcategory of Mot_{AH}( ) which is -generated by _{1} _{}and the Tate motive Q(1). Let H_{B}( Q) be the fibre functor of 𝒞 over Q defined by the Betti cohomology H_{B}^{}( (C) Q) of complex algebraic varieties. Then the Mumford–Tate group of 𝒞 is by definition the automorphism group of H_{B}( Q). The connected component of is reductive and acts on H_{B}( Q) for every object of 𝒞. The fibre functor H_{B}( Q) induces an equivalence between 𝒞 and the category Rep_{Q}( ). In particular, for ^{}𝒞, one has

Hom_{𝒞} ( ^{}) = Hom_{}(H_{B}( Q) H_{B}( ^{}Q))

i. e. the action of fixes all absolute Hodge classes defined over on all objects of 𝒞. Moreover, is the smallest Q-algebraic group with this property. Note that the connected component of is the Mumford–Tate group of the subcategory of Mot_{AH}( ¯ ) which is -generated by _{1 } _{ ¯}¯ and Q(1). When considering only one motive , the Mumford–Tate group _{}of is the Mumford–Tate group of the subcategory Q(1) of Mot_{AH}( ).

For a general subcategory 𝒞 of Mot_{AH}( ), we define the Mumford–Tate group in a similar way, obtaining a pro-algebraic group.

1.4 Subcategories of Mot_{AH}. An Artin motive is an object of Mot_{AH}( ) with finite Mumford–Tate group. The category of Artin motives is the tannakian subcategory of Mot_{AH}( ) generated by the finite -schemes. An abelian motive over is an AH-motive which belongs to the tannakian subcategory of Mot_{AH}( ) generated by the motives of abelian varieties, the Tate motive and the Artin motives. We will write (Artin)_{}for the category of Artin motives and (AV)_{}for the category of abelian motives over . An abelian variety is potentially of CM-type if the connected component of its Mumford–Tate group is commutative and (CM)_{}is the tannakian subcategory of Mot_{AH}( ) generated by the Artin motives, Q(1) and the motives of the abelian varieties which are potentially of CM-type.

If is an abelian variety over , then ( ) = _{}^{}^{1}( ), so in most questions concerning abelian motives, it suffices to consider the motives ^{1}( ) instead of the ( ).

1.5 Betti realization. For any absolute Hodge motive , the Betti realization H_{B}( Q) carries a Hodge structure. Giving this Hodge structure is equivalent to giving an action of the group = C^{}, viewed as an algebraic group over R, on the R-vector space H_{B}( Q) R. As the the Mumford–Tate group _{}of fixes all (absolute) Hodge classes, this action is given by a morphism of algebraic groups : _{R} . The couple ( _{}) is the Mumford–Tate datum associated to . More generally, if 𝒞 is a -subcategory of Mot_{AH}( ), with Mumford– Tate group , then there is a morphism : _{R} which functorially defines the Hodge structures on H_{B}( Q) for all objects of 𝒞.

We restrict our attention to the category of abelian motives (AV)_{}. As explained in [DM82, 6.25], it follows from [Del82a, Theorem 2.11] that the Betti realization induces an equivalence of (AV)_{}with its essential image in the category Hodge of Hodge structures.

Let 𝒞 be a -subcategory of Mot_{AH}( ) and let ( ) be the associated Mumford–Tate datum. For any object of 𝒞, the Betti realization H_{B}( Q) is a representation of and the Hodge structure on H_{B}( Q) is given by the action of on H_{B}( Q) R induced by : _{R}. Since every Q-linear representation of is the Betti realization of an object of 𝒞, this construction defines a -equivalence of Rep_{Q}( ) with the essential image of 𝒞 in Hodge .

If is an abelian variety, its Mumford–Tate group _{}is equal to the Mumford–Tate group of ^{1}( ). The connected component of _{}coincides in turn with the Mumford–Tate group of the Hodge structure H_{B}^{1}( (C) Q).

Let ( ) be a Mumford–Tate datum. Composition of with the cocharacter G_{C} _{C} dual to : _{C} G_{C} gives rise to Hodge cocharacter : G_{C} _{C} . Alternatively, is determined by the condition that G_{C} acts on the factor ^{}of the Hodge decomposition as multiplication by ^{}. Conversely, a pure Hodge structure is determined by giving its weight and the Hodge cocharacter.

1.6 -adic Galois representations. Let be a prime number. The étale cohomology with coefficients in Q_{}of algebraic varieties over ¯ defines the -adic étale realization on the category of absolute Hodge motives. The -adic étale realization of a motive over is a Q_{}-vector space H_{ét}( ¯ Q_{}) endowed with a continuous action of the group 𝒢_{}. It follows from standard tannakian theory that the image of 𝒢_{}in GL(H_{ét}( ¯ Q_{})) lies in _{}(Q_{}), where _{}is the Mumford–Tate group of . The representation thus gives rise to a morphism : 𝒢_{}_{}(Q_{}).

For any prime number , let (Artin)-Rep_{Q_{}}(𝒢_{}) be the tannakian subcategory of Rep_{Q_{}}(𝒢_{}) generated by the -adic étale realizations of the objects of (Artin)_{}. This category coincides with the category of Q_{}-linear representations of 𝒢_{}with finite image. Similarly, we let let (AV)-Rep_{Q_{}}(𝒢_{}) be the tannakian subcategory of Rep_{Q_{}}(𝒢_{}) generated by the -adic étale realizations of the objects of (AV)_{}. As before, in the case of an abelian variety , it is usually sufficient to consider just the representation on the first étale cohomology group H_{ét}^{1}( ¯ Q_{}).

Restricting to an inertia group at a -adic place of , these representations give rise to representations of Hodge–Tate type, cf. [Fon94, §3] or [Ser78]. If ℐ 𝒢_{}is such an inertia subgroup and _{}is a Q_{}-linear representation of ℐ of Hodge–Tate type, there is a canonical decomposition

Q_{}C_{}= _{}^{} the Hodge–Tate decomposition of _{}C_{}. The Hodge–Tate cocharacter is the cocharacter : G_{C_{}}GL( _{}C_{}) such that action of G_{C_{}}on ^{}is the multiplication by ^{}. It can be shown (see [Ser78, §1]) that the connected component of the Zariski closure of the image of ℐ in GL( _{}) coincides with the smallest subgroup GL( _{}) such that factors through _{C_{}}.

2 Mumford–Tate liftings of abelian varieties over C

2.1 Mumford–Tate liftings. Let C be an abelian motive, _{}its Mumford– Tate group and ( _{}_{}) the associated Mumford–Tate datum. This means that the Hodge structure on _{}= H_{B}( Q) is defined by _{}: _{R} .

We say that an abelian motive with Mumford–Tate datum ( _{}_{}) provides a Mumford–Tate lift of if there exists a central isogeny : _{}_{} such that _{R} _{}= _{}. We say that is Mumford–Tate liftable if there exists an abelian motive C giving a Mumford–Tate lift of and such the morphism : _{}_{}is not an isomorphism. We say that is Mumford–Tate unliftable if it is not Mumford–Tate liftable.

If there exists a central isogeny ^{der} : _{}^{der} _{}^{der} such that

_{}^{ad} _{}= _{}^{ad} _{}

then provides a weak Mumford–Tate lift of . Here the maps _{}^{ad} and _{}^{ad} are the projections _{}_{}^{ad} and _{}_{}^{ad}. As it is a central isogeny, it follows that ^{der} induces an isomorphism _{}^{ad} = _{}^{ad}, giving a sense to the equality _{}^{ad} _{}= _{}^{ad} _{}. Finally, is essentially Mumford–Tate unliftable if there does not exists any abelian motive C giving a weak Mumford–Tate lift of for which ^{der} is not an isomorphism.

We will often write M-T (un)liftable instead of Mumford–Tate (un)liftable.

2.2 Remark. If is a weak Mumford–Tate lift of , then and are adjointisogenous in the sense of [Vas03, 6.1] and conversely, if and are adjointisogenous and if there is an isogeny _{}^{der} _{}^{der}, then is a weak Mumford– Tate lift of .

2.3 The Mumford–Tate datum of an abelian variety. Let C be an abelian variety, ( _{}) the associated Mumford–Tate datum and : G_{C} _{C} be the Hodge cocharacter. It follows from [Del79], in particular from 1.3 and 2.3, that the simple factors of _{R}^{ad} are absolutely simple and of classical type ( , , or ). The group 𝒢_{Q} acts on the set of these simple factors and each factor is conjugate to a non-compact one. The compact factors are exactly the factors to which projects trivially.

Let be a Q-simple factor _{}^{ad} and decompose _{C} = _{_{}}_{C}for some finite set _{}with 𝒢_{Q}-action. Here we have identified \overline{Q} with the algebraic closure of Q in C. Note that the complex conjugation in 𝒢_{Q} acts trivially on because the simple factors of _{R}^{ad} are absolutely simple. Let _{}be the set of indices such that the Hodge cocharacter projects trivially to _{C}and let _{} be the complement of _{}in _{}. The _{}are exactly the indices for which the corresponding real factor of _{R} is compact. For each _{}, the Hodge cocharacter lifts to a quasi-cocharacter _{} ˜ of the universal cover { \atop _{C}} of _{C}. If the simple factors of _{C} are of type , or then it follows from [Del79, 1.3] that { \atop _{C}} admits a faithful representation _{C}such that _{} ˜ acts on _{C} with exactly two weights. Contemplating the tables [Del79, 1.3.9] or [Pin98, Table 4.2] one sees that the highest weight of _{C}is

– either _{1} or _{}if the simple factors of _{C} are of type _{},

– _{}if these simple factors are of type _{}and

– _{1} if the factors are of type _{}.

For each _{}, we define a representation _{C}of { \atop _{C}} as follows. Let _{C} be the direct sum of the representations with highest weights _{1} and _{}if is of type _{}with 2 and define _{C}to be the representation with highest weight _{1} (resp. _{}) if is of type _{1} or _{}(resp. _{}). For _{}, the cocharacter _{} ˜ still acts on each irreducible factor of _{C} with exactly two weights. The product over _{}of the { \atop _{C}} descends to an algebraic group over Q and a multiple of the direct sum of the _{C}descends to a faithful Q-linear representation of .

2.4 Remarks. If the simple factors of _{C} are of type _{}or _{}then the condition that _{} ˜ acts on _{C}with exactly two weights uniquely determines the highest weight. Similarly, if the simple factors of _{C} are of type _{}and if _{} ˜ is not dual to _{1}, only the representations with highest weight _{1} or _{}fulfill this condition. Obviously, all these representations are faithful.

In the case where the simple factors of _{C} are of type _{}and where _{} ˜ is dual to _{1}, the cocharacter _{} ˜ acts with exactly two weights on the representation with highest weight _{}for any 1 , see [Pin98, Table 4.2]. Only for = 1 and for = does one obtain a faithful representation.

Also note that _{C}is self dual in each case, see the aforementioned table in Pink’s paper.

2.5 Factors of type _{}. In what follows, we will focus on the factors of _{}^{ad} of type _{}, first considering the case 5. Fix a Q-simple factor of _{}^{ad} such that the simple factors of _{C} are of type _{}with 5. Let _{C} = _{_{}}_{C} and let _{}and _{}be as before. The Dynkin diagram of _{C} is the disjoint union, indexed by _{}, of diagrams of type _{}, and it follows from [Del79, 1.3] that, for each _{}, the conjugacy class of the cocharacter _{}is dual to an endpoint ^{( )} of the -component of the Dynkin diagram.

2.6 Lemma. Assume that  5 and that ^{( )} = _{1} for some  _{}. Then

( ) = _{1} for all _{}.

Proof. Assume that ^{( )} = _{1} for _{}and let _{C}^{der} be the simple factor of _{C}^{der} projecting onto _{C}. Via the representation of _{}on = H_{B}^{1}( (C) Q), the Hodge cocharacter acts on _{C} with two weights. By the table in [Del79, 1.3], this implies that every non-trivial irreducible direct factor of the representation of _{C}^{der} on _{C} has highest weight _{1} or _{}. As is Q-simple, the same thing is true for the representation on _{C} of the simple factors of _{C}^{der} mapping to the factors _{C} for the other _{}. Using the tables [Del79, 1.3.9], the fact that acts with two weights implies that all non-trivial _{}are dual to 1.

If any factor of _{C} satisfies the conditions of the lemma, then we say that and the factors of _{R} and _{C} are of type _{}^{R}. As was assumed to be Qsimple, the vertices _{1}^{( )}form a single orbit for the action of 𝒢\overline{Q} in this case. In the opposite case we say that they are of type _{}^{H}. In the latter case, the conjugacy class of the projection of to any factor of _{C} is either trivial or dual to _{1} or _{}.

Let _{C}be a factor of _{C}^{ad} of type _{}^{R} and let { \atop _{C}} be its universal cover. For each _{}, let _{C}be the direct sum of the representations of _{C}with highest weights _{1} and _{}. As above, for _{}, the projection of the Hodge cocharacter to _{C}lifts to a quasi-cocharacter _{} ˜ of { \atop _{C}} and _{} ˜ acts on _{C} with two weights 1 2. The product over of the { \atop _{C}} and a multiple of the direct sum of the _{C}descend to an algebraic group over Q and a faithful Q-linear representation of .

Next assume that _{C}is a factor of _{C}^{ad} of type _{}^{H}. As before, for _{}, the projection of the Hodge cocharacter lifts to a quasi-cocharacter _{} ˜ of the universal cover { \atop _{C}} , but this time the group { \atop _{C}} does not have a faithful representation on which _{} ˜ acts with two weights. This property can only be achieved for a quotient of { \atop _{C}} which can be constructed as follows. For each , let _{C} be the representation of { \atop _{C}} with highest weight _{1}. We will refer to the quotient of { \atop _{C}} which acts faithfully on _{C}as the -maximal cover of _{C}. This -maximal cover descends over R to an inner form of the orthogonal group SO(2 ). For each _{}, the quasi-cocharacter _{} ˜ acts on C with weights 1 2. The product over of these -maximal covers descends to an algebraic group over Q and a multiple of the direct sum of the Cdescends to a faithful Q-linear representation of this group.

2.7 Factors of type _{4}. We finally consider the case where = 4, so let be a Q-simple factor of _{}^{ad} such that the simple factors of _{C} are of type _{4}. In this case, the automorphism group of the Dynkin diagram permutes the set of endpoints, so the types _{4}^{R} and _{4}^{H} can not be distinguished in the same manner as before. The fundamental difference between the two types is the existence of a Q-linear representation as above. We formalize this as follows.

Let _{C} = _{_{}}_{C} and _{}and _{}as in the general case. For each let _{}: G_{C} _{C}be the projection of the Hodge cocharacter. The conjugacy class of each _{}is dual to an endpoint of the corresponding component of the Dynkin diagram. Let Δ be the set of these vertices, it contains exactly one endpoint of the -component of the Dynkin diagram if _{}and no vertices in the other components. Choose _{}, let _{C}^{der} be the almost simple factor of _{C}^{der} projecting onto _{C} and let be the highest weight of some non trivial factor of the representation of _{C}^{der} on H_{B}^{1}( (C) Q). As the Hodge cocharacter acts on H_{B}^{1}( (C) C) with two weights, the arguments of [Del79, 1.3, 2.3] imply that corresponds to an endpoint of the Dynkin diagram of and that the 𝒢_{Q}-orbit Δ^{}of does not meet Δ. In particular, there exists a 𝒢_{Q}-stable set Δ^{}of endpoints such that Δ Δ^{}= . The set Δ^{}contains at least one endpoint in each connected component of the Dynkin diagram.

If there exists a 𝒢_{Q}-stable set Δ_{max}^{}of endpoints of the Dynkin diagram with Δ Δ_{max}^{}= containing two endpoints in each connected component, then we say that and the factors of _{R} and _{C} are of type _{4}^{R}. If there does not exist such a Δ_{max}^{}, they are of type _{4}^{H}.

Assume that is a factor of _{C}^{ad} of type _{4}^{R}, and let _{}. Then the - component of the Dynkin diagram contains two endpoints _{}and _{}Δ_{max}^{}. Let _{C} be the direct sum of the representations of the universal cover { \atop _{C}} with highest weights corresponding to _{}and _{}respectively. It is a faithful representation of { \atop _{C}} and as _{}_{}Δ, it follows from [Del79, 1.3.9] that, for _{}, the lifting _{} ˜ of _{}to { \atop _{C}} acts on _{C}with weights 1 2. The product of the { \atop _{C}} and a multiple of the direct sum of the _{C}descend to a Q-algebraic group with a faithful Q-linear representation .

If _{C}is a factor of _{C}^{ad} of type _{4}^{H}, then the above set Δ^{}meets the Dynkin diagram of _{C}in one endpoint, corresponding to a fundamental weight . The -maximal cover of _{C} is the quotient of its universal cover { \atop _{C}} acting faithfully on the representation of { \atop _{C}} with highest weight . As in the case 5, the product of these data over all _{}descends to an algebraic group over Q together with a faithful Q-linear representation. This cover of a Qsimple factor of _{}^{ad} of type _{4}^{H} is the -maximal cover of _{}^{ad}.

2.8 Remark. For all factors _{C}of type , the representation _{C}is self dual. As in the case of the factors of types , and this can be read off from [Pin98, Table 4.2].

2.9 Theorem. Let C be an abelian variety and let ( _{}) be the associated Mumford–Tate datum. Then the following conditions are equivalent.

1. is essentially Mumford–Tate unliftable.

2. The group _{C}^{der} is the product of its simple factors. The simple factors of types , _{}, _{}and _{}^{R} are simply connected and the factors of type _{}^{H} are - maximal in the sense defined above.

2.10 Definition. Let C be an abelian variety, _{}its Mumford–Tate group and _{}= H_{B}^{1}( (C) Q). If _{}^{ad} is Q-simple, then we say that is Mumford–Tate decomposed if the following conditions hold.

– There are a totally real field _{0} and an absolutely simple algebraic group over _{0} such that _{}^{der} = Res_{_{0} Q} ^{}.

– There is a faithful representation ^{}of ^{}such that the representation of _{}^{der} on is isomorphic to Res_{_{0} Q} ^{}.

– There is no proper non-trivial abelian subvariety of with Mumford– Tate group _{}verifying _{}^{der} = _{}^{der}.

We say that an abelian variety C is Mumford–Tate decomposed if it is isogenous to a product _{}such that each _{}is Mumford–Tate decomposed with _{_{}}^{ad} simple and if _{}^{ad} is the product of the _{_{}}^{ad}. An abelian variety over a number field C is Mumford–Tate decomposed if _{C} is Mumford–Tate decomposed.

2.11 Remark. The upshot of this definition is that the abelian varieties arising from the construction of [Del79, 2.3] are Mumford–Tate decomposed. The representation of _{}^{der} on H_{B}^{1}( (C) Q) is a direct sum of the representations of the _{_{}}^{der} constructed in 2.3 and 2.5. See also the proofs given below.

The notion of being (essentially) Mumford–Tate unliftable is a condition on the Mumford–Tate datum of an abelian variety whereas the notion of being Mumford–Tate decomposed pertains to the action of the Mumford–Tate group on the first Betti cohomology group.

2.12 Theorem. For every abelian variety C there exists a weak Mumford–Tate lift C of such that is essentially Mumford–Tate unliftable and Mumford–Tate decomposed.

Proofs of 2.9 and 2.12.These results can be derived from the work of Satake, see for example [Del79]. The same argument can be found in [Vas03], see §4 and paragraphs 6.3 and 6.4 in particular. The strategy of the proof is as follows. It is first shown that the condition 2.9.2 implies the condition 2.9.1. We then prove that any abelian variety admits a weak Mumford–Tate lift satisfying 2.9.2 and which is Mumford–Tate decomposed. Thanks to the fact that 2.9.2 implies 2.9.1, this M-T lift is also M-T unliftable. Finally, the proof that the condition 2.9.1 implies 2.9.2 is a formality.

First assume that verifies the conditions of 2.9.2. We will show that it is essentially M-T unliftable, so let be a weak Mumford–Tate lift of and let _{} be its Mumford–Tate group. It has to be proved that the map _{}^{der} _{}^{der} is an isomorphism. It suffices to prove this after extension of scalars to C and as the only non-simply connected factors of _{C}^{der} are the factors of type _{}^{H}, we only need to consider these factors.

Consider a factor _{C} of _{C}^{der} of type _{}^{H} to which the Hodge cocharacter projects non-trivially, assuming at first that 5. As we saw, the conjugacy class of the projection of the Hodge cocharacter to _{C} is dual to one of the vertices _{1} or _{}of the Dynkin diagram. Let { \atop _{C}} be the factor of _{C}^{der} mapping onto _{C}. An appropriate direct factor of the representation of { \atop _{C}} on H_{B}^{1}( (C) Q) C provides a faithful representation on which ˜ acts with two weights 1 2. As explained in 2.5, it follows from the tables in [Del79, 1.3.9], that the highest weight of every irreducible direct factor of is _{1} and hence that { \atop _{C}} is isomorphic to _{C}. This proves that 2.9.2 implies 2.9.1 if 5.

In the case where = 4, first note that _{}^{der} is the product of its Q-simple factors. Let be a simple factor of type _{4}^{H} and let be the factor of _{}mapping onto . Consider H_{B}^{1}( (C) Q) as a representation of and let be a direct factor which is a faithful representation of . Let { \atop _{C}} be any factor of { \atop _{C}} and let _{C}be any irreducible direct factor of the restriction of C to { \atop _{C}} . Then the lifting to { \atop _{C}} of the Hodge cocharacter either acts trivially on Cor with exactly two weights. It follows from 2.7 that is the -maximal cover of and as was -maximal by hypothesis it follows that = . This proves that 2.9.2 implies 2.9.1 in case = 4.

We next show that if is any abelian variety, then there exists a weak Mumford–Tate lift of with Mumford–Tate group _{}satisfying the conditions of 2.9.2 and which is Mumford–Tate decomposed. This fact readily follows from [Del79, 2.3] and the discussions in 2.3, 2.5 and 2.7, we recall the argument.

Fix a Q-simple factor ^{ad} of _{}^{ad}. It is of the form Res_{_{0} Q} ^{ad} for some totally real number field _{0} and some absolutely simple adjoint group ^{ad} over _{0}. As usual, decompose _{C}^{ad} = _{_{}}_{C}, where _{}, _{}and _{}are as before.

Unless _{C}^{ad}is of type _{}^{H}, we let { \atop ^{der}} be the simply connected cover of ad. In the remaining case we let { \atop ^{der}} be the -maximal cover of ^{ad}, in the sense of 2.5 resp. 2.7. In all cases, put

{ \atop ^{der}} = Res_{_{0} Q} { \atop ^{der}}

For each _{}, the projection of the Hodge cocharacter to _{C}^{ad} lifts to a quasi-cocharacter _{} ˜ of { \atop _{C}^{der}} . The faithful representation of { \atop ^{der}} constructed in 2.3 and 2.5 resp. 2.7 is of the form = Res_{_{0} Q} ^{}for some faithful Q-linear representation ^{}of { \atop ^{der}} . These data have the following properties.

– There is a map { \atop ^{der}} _{}^{der} such that ˜ = _{_{}}_{}˜ lifts the Hodge cocharacter.

– If _{}, and if is an irreducible factor of _{C}, then the cocharacter ˜ acts either trivially on or with two rational weights and + 1.

There exist a torus over Q acting { \atop ^{der}} -linearly on and a quasi-cocharacter of _{C} such that the product _{} ˜ acts trivially on the _{C}for _{}and with weights 1 2 for _{}. In fact, is characterized by the condition that C is the group of the automorphisms of C acting by scalar multiplication on each isobaric component of C as { \atop _{C}^{der}} -representation. The existence of follows from the fact that if ˜ acts non-trivially on an isobaric component then it acts with two weights and + 1. Let ^{}be the image of { \atop ^{der}} in GL( ) and let ^{}= _{} ˜ .

We choose a quadratic and totally imaginary extension of _{0} and consider the Q-algebraic group ^{}. The natural action of ^{}on gives rise to a Q-linear representation of ^{}. One has

C = _{}G_{C}^{2}

and one defines a quasi-cocharacter of ^{}_{C} by _{}( ) = ( ^{1 2} ^{1 2}) for and _{}( ) = ( 1) for _{}. The action of ^{}^{}on _{_{0}} defines a faithful representation of a quotient of ^{}^{}with derived group { \atop ^{der}} in which ( ˜ ) acts with weights 0 and 1. Let ˜ : { \atop _{R}} be defined by

˜ ( ¯ ) = ( ^{})( )\overline{( ^{})(¯ )}

Shrinking the centre of , we may assume that the image of ˜ is Zariski dense in . We claim that this defines the Shimura datum ( ˜ ) associated to an abelian variety _{1}. To see why this is the case, note that ˜ defines a Q-Hodge structure of type (1 0) (0 1) on _{_{0}} . In order to establish that this Hodge structure comes from an abelian variety it is sufficient to show that it is polarizable, cf. [Del72, 2.3]. First use that, by loc. cit. 2.11, the element ad( _{}( )) defines a Cartan involution of _{}^{der} and hence that ad(˜ ( )) is a Cartan involution of { \atop _{R}^{der}} . Next, one checks that the weight _{}: G_{}is central, defined over Q and that the quotient of the center of by _{}(G_{}) is compact. The last statement is deduced from the fact that this center is contained in a product of CM tori. It now follows that ad(˜ ( )) is a Cartan involution of (G_{}) and by [Del79, 1.1.18(b)] this implies that the Hodge structure on _{_{0}} is polarizable.

The group { \atop ^{ad}} is Q-simple, { \atop ^{der}} is its -maximal cover and _{1} is Mumford– Tate decomposed by construction. Since _{}is the Mumford–Tate group of an abelian variety, _{}^{der} is a quotient of the -maximal cover of _{}^{ad}, so there is an isogeny { \atop ^{der}} _{}^{der} lifting { \atop ^{ad}} _{}^{ad}.

Applying this to all Q-simple factors of _{}^{ad} we obtain Mumford–Tate decomposed abelian varieties _{}such that = _{}is a weak M-T lift of verifying the conditions of 2.9.2. By construction, if _{}(resp. _{_{}}) denotes the Mumford–Tate group of (resp. _{}), then _{}^{der} = _{_{}}^{der}. The first part of this proof implies that is essentially M-T unliftable. This terminates the proof of theorem 2.12.

Finally, let be essentially M-T unliftable, i. e. the condition 2.9.1 is satisfied. The theorem 2.12 implies that has a weak M-T lift with Mumford–Tate group _{}satisfying the condition of 2.9.2. As is essentially M-T unliftable, we must have _{}^{der} = _{}^{der}, which implies that _{}also verifies 2.9.2.

2.13 Remarks.

2.13.1 In the above proof, fix a Q-simple factor ^{ad} of _{}^{ad}, let _{}and consider an irreducible factor of the representation _{C}of { \atop _{C}^{der}} . It follows from [Del79, 1.3] that the highest weight of is a fundamental weight of { \atop _{C}^{der}} . More precisely, according to the type of { \atop _{C}^{der}} and the quasi-cocharacter _{} ˜ , it is the weight given by the tables 1.3.9 of Deligne’s paper or [Pin98, Table 4.2].

2.13.2 With the same notations, assume that ^{ad} is not of type _{}with 2. In the above construction of the essentially M-T unliftable variety _{}corresponding to this factor, we then have = 1 2. This means that the construction of the intermediate group ^{}can be shunted in this case. It follows that the Mumford– Tate group of is contained in the image of { \atop ^{der}} ^{}in GL( _{_{0}} ).

This argument is also valid for factors of type _{}for which all the numbers are equal to 1 2.

2.13.3 Instead of using [Del72] to prove the fact ˜ defines a polarizable Hodge structure on _{_{0}} , one may also explicitly construct a polarization. This is not difficult, using the autoduality of as representation of { \atop ^{der}} .

2.14 Examples.

2.14.1 Let C be an abelian variety arising from Mumford’s construction, see [Mum69]. In this case one has _{\overline{Q}}^{der} = SL_{2}^{3} , where

= ( _{1} _{2} _{3}) _{}= 1 for = 1 2 3 and _{1} _{2} _{3} = 1

The Mumford–Tate group of the M-T unliftable and M-T decomposed weak M-T lift of satisfies _{\overline{Q}}^{der} = SL_{2}^{3}. This is the example studied in detail in [Noo01].

2.14.2 There exist simple abelian varieties C for which _{}^{ad} is not simple. One can construct such an example where _{}^{der} = _{1} _{1} with

1 \overline{Q} = _{2 \overline{Q}} = SL_{2}^{2}

and _{\overline{Q}} = ( 1 1) SL_{2}^{2} = _{}embedded diagonally into _{1} _{2}. In this case, the M-T unliftable and M-T decomposed weak M-T lift of is a product = _{1} _{2} and _{_{}}^{der} = _{}.

2.14.3 There exist simple abelian varieties C for which _{}^{ad} is absolutely simple of type _{}^{R}, with even, 𝒢_{Q} acting trivially on the Dynkin diagram and where = H_{B}^{1}( (C) Q) decomposes over \overline{Q} as a multiple of the representation of _{}^{der} with highest weight _{}. Since _{}^{der} acts faithfully on , it is not simply connected. Let be the M-T unliftable and M-T decomposed weak M-T lift of . Then _{}^{der} is the universal cover of _{}^{der} and is a product = ^{}, where = H_{B}^{1}( ^{}(C) Q) decomposes over \overline{Q} as a multiple of the representation of _{}^{der} with highest weight _{1} .

2.15 Corollary. For every abelian motive C there exists an essentially Mumford– Tate unliftable and Mumford–Tate decomposed abelian variety C which provides a weak Mumford–Tate lift for .

Proof. There exist an abelian variety C and a surjection of the corresponding Mumford–Tate groups _{}_{}commuting with the maps _{}and _{}. Let C be the essentially Mumford–Tate unliftable and Mumford–Tate decomposed weak Mumford–Tate lift for provided by theorem 2.12. This gives a morphism _{^{}}: _{^{}}^{der} _{}^{der}. The fact that ^{}is Mumford–Tate decomposed implies that there is an isogeny ^{}_{}_{}, where the _{}are M-T decomposed abelian varieties such that the groups _{_{}}^{der} are Q-simple, and such that _{^{}}^{der} = _{}_{_{}}^{der}. Let be the subset of indices such that _{_{}}^{der} is not in the kernel of _{^{}}and let = _{}_{}. Then _{}= _{}_{_{}}and _{^{}}_{_{}^{der}} is an isogeny from _{}^{der} onto _{}^{der}, so verifies the condition of the corollary.

3 Mumford–Tate liftings and motives

3.1 Proposition. Suppose that _{C} and _{C} are abelian varieties over C, let ( _{}_{}) and ( _{}_{}) be the associated Mumford–Tate data and assume that there exists an isomorphism _{}^{ad} = _{}^{ad} such that _{}^{ad} _{}= _{}^{ad} _{}.

Let C be an algebraically closed field. Then there exists an abelian variety such _{}C = _{C} if and only if there exists an abelian variety such that C = _{C}.

Proof. This is proved as in [Noo01, 4.5]. Let _{}:  _{R} and _{}:  _{R}

be the maps defining the Hodge structures on H_{B}^{1}( (C) Q) and H_{B}^{1}( (C) Q) respectively. Let _{}and _{}be the _{}(R)- and _{}(R)-conjugacy classes of _{} and _{}. The main theorem of [Del79] implies that for all compact open subgroups _{}_{}(A_{}) and _{}_{}(A_{}), one can construct quasi-canonical models _{_{}}_{\overline{Q}}( _{}_{})^{0} and _{_{}}_{\overline{Q}}( _{}_{})^{0} over \overline{Q} of the corresponding connected Shimura varieties. Here \overline{Q} is the algebraic closure of Q in C.

For _{}and _{}sufficiently small, there exist “universal” abelian schemes 𝒜 _{_{}}_{\overline{Q}}( _{}_{})^{0} and ℬ _{_{}}_{\overline{Q}}( _{}_{})^{0} and points

\overline{Q}( _{}_{})^{0}(C) _{_{}}_{\overline{Q}}( _{}_{})^{0}

such that = 𝒜_{}and = ℬ_{}.

Let ^{ad} = _{}^{ad} = _{}^{ad}. One can choose _{}, _{}and ^{ad} ^{ad}(A_{}) such that, in addition to the above conditions, there is a diagram

\xymatrix{_{_{}}_{\overline{Q}}( _{}_{})^{0} \ar@{-->}[dr] &  & _{_{}}_{\overline{Q}}( _{}_{})^{0} \ar@{-->}[dl] \\  & _{^{ad}} _{\overline{Q}}( ^{ad} ^{ad})^{0}}

in which each arrow is a quotient map for the action of a finite group and hence is a finite morphism (cf. [Del79, 2.7.11(b)]) and such that and have the same image in_{^{ad}} _{\overline{Q}}( ^{ad} ^{ad})(C)^{0}. The proposition follows.

3.2 Corollary. The statement of the proposition is true in particular if _{C} provides a (weak) Mumford–Tate lift of _{C}.

3.3 Proposition. Let C be an algebraically closed field and let and abelian varieties over such that _{C} provides a Mumford–Tate lift of _{C}. Then the motive 1( ) belongs to the category ^{1}( ) Q(1) .

The map _{}_{}induced by the Betti realization of this inclusion is the map given by the structure of as a Mumford–Tate lift of .

Proof. This generalises [Noo01, 4.8–4.11].

As explained in 1.5, it follows from [DM82, 6.25] that the Betti realization induces an equivalence of ^{1}( ) Q(1) with H_{B}^{1}( (C) Q) Q(1) , the tannakian subcategory of the category of Hodge structures generated by H_{B}^{1}( (C) Q) and the Tate Hodge structure Q(1). To prove the proposition, it therefore suffices to show that the Hodge structure H_{B}^{1}( (C) Q) belongs to H_{B}^{1}( (C) Q) Q(1) . For the rest of the proof, we write _{}= H_{B}^{1}( (C) Q) and _{}= H_{B}^{1}( (C) Q).

By definition of the Mumford–Tate group, the underlying vector space of any object of the category _{}Q(1) of Hodge structures naturally carries the structure of a representation of _{}. This gives a -equivalence of _{}Q(1) with the subcategory _{}Q(1) _{Rep} of Rep_{Q}( _{}). As we saw in 1.5, for any object of the latter category, the Hodge structure is given by composing the morphism _{}: _{R} with the action of _{}on .

Let ( _{}_{}) be the Mumford–Tate datum associated to . By hypothesis, there exists a central morphism : _{}_{}such that _{}= _{R} _{}. This makes every Q-linear representation of _{}into a representation of _{}and, for any such , it carries the Hodge structure on defined by _{}into the Hodge structure defined by _{}. To prove the proposition it is therefore sufficient to show that _{}, considered as representation of _{}, belongs to _{}Q(1) _{Rep}. This follows immediately, because _{}is a faithful representation of _{}and Q(1) is its dual, so one has

Q(1) _{Rep} = Rep_{Q}( _{})

cf. [DM82, 2.20] and its proof.

3.4 Proposition. Let and be abelian varieties over an algebraically closed field C such that _{C} provides a weak Mumford–Tate lift of _{C}. Then ^{1}( ) belongs to ^{1}( ) (CM)_{}.

Taking the Betti realization, this inclusion induces a map between the corresponding Mumford–Tate groups. On the derived group, this map is the map ^{der} : _{}^{der} _{}^{der} given by the structure of as weak Mumford–Tate lift of .

Proof. Let _{}and _{}be the connected components of the centres of _{}and of _{}respectively. We write _{}= H_{B}^{1}( (C) Q) and _{}= H_{B}^{1}( (C) Q) as in the proof of 3.3 and consider all spaces we encounter as representations of = _{}_{}_{}^{der}. In the case of _{}, the group acts via

_{}^{der} ^{id ^{der}}  _{}_{}^{der}  _{}
and in the case of _{}it acts via  _{}_{}^{der}  _{}.

The groups _{}and _{}act on _{}and _{}. Composing these representations with the projections _{}and _{}respectively we obtain representations _{}^{}and _{}^{}of . As = _{}_{}^{}_{}^{}is a faithful representation of , it follows that _{}belongs to the subcategory of Rep_{Q}( ) -generated by and its dual. As there is an isomorphism of Hodge structures _{}^{}= _{}Q(1), this in turn implies that _{}belongs to the tannakian subcategory of Rep_{Q}( ) generated by _{}and the abelian representations of , i. e. the representations where acts through a commutative quotient.

We fix an irreducible direct factor _{2} of _{}in Rep_{Q}( ). There is an irreducible abelian Q-linear representation _{3} of such that _{2} is isomorphic to a subobject of _{3} _{}^{}. Replace _{}^{}by an irreducible direct factor _{1} such that 2 still is a subobject of _{3} _{1}. The projection of _{1} _{1}^{}onto the trivial representation induces a surjection _{3} _{1} _{1}^{}↠ _{3}. The composite of this surjection with the map

2 _{1}^{}_{3} _{1} _{1}^{}

deduced from the inclusion _{2} _{3} _{1} is a map _{2} _{1}^{}_{3}. It is not difficult to check that this map is non-trivial and as _{3} was assumed to be irreducible, this implies that the map is surjective. This proves that _{3} belongs to the tannakian subcategory of Rep_{Q}( ) which is -generated by _{1}^{}and _{2}.

It follows that the action of on _{3} factors through _{}_{}and as _{3} is an abelian representation of , it is also an abelian representation of _{}_{}. This means that _{3} carries a Hodge structure of CM-type and hence that _{} belongs to _{}(CM)-Hodge as required.

3.5 Corollary. Let and be abelian varieties over and let _{}and _{}be their respective Mumford–Tate groups. Assume that provides a weak Mumford–Tate lift of and that _{}^{der} = _{}^{der}. Then the categories ^{1}( ) (CM)_{}and ^{1}( ) (CM)_{} coincide.

3.6 Corollary. Let C be an algebraically closed field and let be an object of (AV)_{}. Then there exist an essentially Mumford–Tate unliftable and Mumford–Tate decomposed abelian variety over such that belongs to

1( ) (CM)_{}

4 Motivic liftings of Galois representations

4.1 Assume that C is a number field, an abelian motive over and ( _{}_{}) the Mumford–Tate datum associated to . As we recalled in 1.6, the -adic Galois representation associated to the étale realization of factors through a morphism

: 𝒢_{}_{}(Q_{})

There exists a finite extension ^{}such that the Mumford–Tate group of is connected. The Mumford–Tate group of _{^{}}is then equal to the connected component of _{}. In what follows, we will assume that _{}is already connected (replacing by a finite extension if necessary).

It follows from theorems of Tsuji, [Tsu99] and De Jong, [dJ96] that, for every , the representation _{}of 𝒢_{}on H_{ét}( Q_{}) is geometric in the sense of Fontaine and Mazur, [FM95, §1]. Here a representation of 𝒢_{}on a finite dimensional Q_{}-vector space is called geometric if

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

– for each valuation ¯ of ¯ of residue characteristic , the restriction to the inertia group ℐ_{}¯ is potentially semi-stable (cf. [Fon94]).

More generally, for a linear algebraic group over Q_{}and a continuous morphism _{}: 𝒢_{}(Q_{}), we will say that _{}is geometric if there exists a faithful representation _{}of such that the resulting representation of 𝒢_{}on _{}is geometric. This is the case if and only if the representation of 𝒢_{}on _{}is geometric for any representation _{}of .

4.2 Definition. Assume that and are linear algebraic groups over Q or over Q_{}and that _{}: 𝒢_{}(Q_{}) and _{} ˜ : 𝒢_{}(Q_{}) are geometric Galois representations. We will say that _{} ˜ is a geometric lift of _{}if there exists a central isogeny : such that _{}= _{} ˜ . If _{}does not admit any geometric lift with ker non-trivial, it will be called geometrically unliftable.

If there exists a central isogeny ^{der} : { \atop ^{der}} ^{der} such that there is an equality ^{ ˜ ad} _{} ˜ = ^{ad} _{}then _{} ˜ is said to be a weak geometric lift of _{}. As in 2.1, the maps ^{ad} and ^{ ˜ ad} are the projections ^{ad} and { \atop ^{ad}} . The central isogeny ^{der} induces an isomorphism ^{ad} = { \atop ^{ad}} , giving a sense to the equality ^{ ˜ ad} _{} ˜ = ^{ad} _{}. We will say that _{}is essentially geometrically unliftable if it does not admit a weak geometric lift with ker ^{der} non-trivial.

4.3 Proposition. Suppose that  C is a number field,  an abelian variety

with connected Mumford–Tate group _{}and that is an abelian variety over a finite extension ^{}such that _{C} provides a Mumford–Tate lift of _{C}.

The number field ^{}can be chosen such that, for every prime number , the Galois representation _{}of 𝒢_{^{}}on H_{ét}^{1}( ¯ Q_{}) belongs to the subcategory of Rep_{Q_{}}(𝒢_{^{}}) -generated by H_{ét}^{1}( ¯ Q_{}) and Q_{}(1). For every , the morphism _{}_{}then realises the Galois representation _{}as a geometric lift of _{}.

Proof. By proposition 3.3, the motive ^{1}( ¯ ) belongs to ^{1}( ¯ Q(1) and, for large enough, this is already the case over ^{}. Taking the -adic étale realizations, this implies the corresponding statement for the Galois representations. The inclusion of ^{1}( _{^{}}) in ^{1}( _{^{}}) Q(1) corresponds to a morphism and taking the -adic étale realizations this gives rise to a commutative diagram

\xymatrix{ & _{}(Q_{}) \ar[d] \\ 𝒢_{^{}} \ar[ur]^{_{}} \ar[r]^{_{}} & _{}(Q_{})}

proving the proposition.

4.4 As above, assume that is a number field contained in C. Following the notations of [FM95, §6], let (CM)-Rep_{Q_{}}(𝒢_{}), or (CM)-Rep if no confusion is likely, denote the tannakian subcategory of Rep_{Q_{}}(𝒢_{}) consisting of the potentially abelian geometric representations, in other words, the geometric representations such that the restriction to a subgroup of 𝒢_{}of finite index has abelian image. It follows from [FM95, §6] that (CM)-Rep is the tannakian subcategory of Rep_{Q_{}}(𝒢_{}) generated by the representations factoring through finite groups and the representations of the form H_{ét}^{1}( ¯ Q_{}) for an abelian variety which is potentially of CM-type. Thus, (CM)-Rep is the tannakian category of the -adic étale realizations of the objects of (CM)_{}.

4.5 Proposition. Suppose that  C is a number field,  an abelian variety

with connected Mumford–Tate group _{}and that is an abelian variety over a finite extension ^{}such that the Mumford–Tate group _{}is connected and _{C} provides a weak Mumford–Tate lift of _{C}.

For every prime number , the representation _{}of 𝒢_{^{}}on H_{ét}^{1}( ¯ Q_{}) is an object of the subcategory H_{ét}^{1}( ¯ Q_{}) (CM)-Rep_{Q_{}}(𝒢_{^{}}) of Rep_{Q_{}}(𝒢_{^{}}). Via the map _{}^{der} _{}^{der}, the Galois representation _{}provides a weak geometric lift of 𝒢_{^{}}.

Proof. It follows from proposition 3.4 that there is a finite extension ^{}of ^{}such that the motive ^{1}( _{^{}}) belongs to ^{1}( _{^{}}) (CM)_{^{}}. As the Mumford–Tate group of ^{1}( _{^{}}) (CM)_{^{}}is a pro-algebraic group with derived group _{}^{der}, this gives rise to an isogeny : _{}^{der} _{}^{der} and induces an isomorphism ad : _{}^{ad} _{}^{ad}. Taking -adic étale realizations, there is a commutative diagram

\xymatrix{ & _{}(Q_{}) \ar@{-->}[dr] \\ 𝒢_{^{}} \ar[ur]^{_{}} \ar[dr]^{_{}} &  & _{}^{ad}(Q_{}) = _{}^{ad}(Q_{}) \\  & _{}(Q_{}) \ar@{-->}[ur]}  (4.5. )

This proves all statements of the proposition with ^{}instead of ^{}.

Faltings’ theorem, [Fal83, Satz 4] implies that if is the commuting algebra of _{}in End(H_{}^{1}( (C) Q)), then Q_{}is the commuting algebra of _{}(𝒢_{^{}}) in End(H_{ét}^{1}( (C) Q_{})). This implies that that the centralizer of the image of 𝒢_{^{}}in _{}^{ad}(Q_{}) is trivial. It follows from lemma 4.6 that the diagram 4.5. also commutes with 𝒢_{^{}}replaced by 𝒢_{^{}}. Lemma 4.7 implies that the representation of 𝒢_{^{}}on H_{ét}^{1}( ¯ Q_{}) is an object of H_{ét}^{1}( ¯ Q_{}) (CM)-Rep_{Q_{}}(𝒢_{^{}}) . 4.6 Lemma. Assume that is a prime number and that is a connected linear algebraic group over Q_{}. Let be a number field and let _{1} _{2}: 𝒢_{}(Q_{}) be Galois representations coinciding on 𝒢_{^{}}for some finite extension ^{}of . Also assume that the centralizer of _{1}(𝒢_{^{}}) = _{2}(𝒢_{^{}}) in is trivial. Then _{1} = _{2} on 𝒢_{}. Proof. It is sufficient to treat the case where ^{}is a Galois extension of .

Let : 𝒢_{}(Q_{}) be defined by ( ) = _{1}( ) _{2}( )^{1} . This map satisfies the cocycle condition ( ) = ( ) _{2}( ) ( ) _{2}( )^{1} . This implies that is constant on the classes 𝒢_{^{}}for 𝒢_{}. As 𝒢_{^{}}is normal, it follows that is also constant on the classes 𝒢_{^{}}and it follows that for all 𝒢_{}and 𝒢_{^{}}one has ( ) = _{2}( ) ( ) _{2}( )^{1} . Therefore ( ) lies in the centralizer of _{2}(𝒢_{^{}}) and we conclude that is trivial. 4.7 Lemma. Let be a prime number, _{1} and _{2} connected linear algebraic groups over Q_{}and ^{der} : _{1}^{der} _{2}^{der} a central isogeny. Let _{1} be a faithful Q_{}-linear representation of _{1} and let _{2} be any Q_{}-linear representation of _{2}.

Let be a number field and let _{}: 𝒢_{}_{}(Q_{}) (for = 1 2) be geometric Galois representations. Assume that _{1}^{}lies in the subcategory of Rep_{Q_{}}(𝒢_{}) - generated by _{1} and (CM)-Rep_{Q_{}}(𝒢_{}) and that _{1}^{ad} _{1} = _{2}^{ad} _{2}, where the _{}^{ad} : _{}_{}^{ad} are the canonical projections and _{1}^{ad} = _{2}^{ad} is the identification induced by ^{der}. Then _{2} is an object of the subcategory of Rep_{Q_{}}(𝒢_{}) -generated by _{1} and (CM)-Rep_{Q_{}}(𝒢_{}).

Proof. Fix an object _{3} of (CM)-Rep_{Q_{}}(𝒢_{}) such that _{1}^{}lies in the subcategory of Rep_{Q_{}}(𝒢_{}) -generated by _{1} and _{3}. Let _{3} be the Zariski closure of the image of 𝒢_{}in GL( _{3}) and let _{3}: 𝒢_{}_{3}(Q_{}) be the morphism giving the action of 𝒢_{}on _{3}.

Let _{1} and _{2} be the connected components of the centres of _{1} and of _{2} respectively. Put = _{1} _{2} _{3} _{1}^{der} and consider the natural maps _{1} and _{2}. Via these maps, we consider _{1} and _{2} as representations of . It can be shown exactly as in the proof of proposition 3.4 that _{2} belongs to the subcategory of Rep_{Q_{}}( _{3} _{}_{}) generated by _{1} and the abelian representations. For any abelian representation _{3} of _{3} _{}_{}, the induced Galois representation belongs to (CM)-Rep (𝒢_{}) so this implies that _{2} belongs to the subcategory _{1} (CM)-Rep (𝒢_{}) of Rep_{Q_{}}(𝒢_{}).

4.8 We keep the above notations, i. e. C is a number field, an abelian variety and _{}its Mumford–Tate group, which is assumed to be connected. For each prime number , we denote by _{}: 𝒢_{}_{}(Q_{}) the -adic Galois representation associated to .

Fix an algebraic group over Q and a central isogeny { \atop ^{der}} _{}^{der} and suppose that for every in a set of prime numbers _{} ˜ : 𝒢_{}(Q_{}) is a weak geometric lift of _{}.

4.9 Theorem. Assume that any simple factor of { \atop _{C}^{der}} lying over a factor of _{C}^{der} of type _{}^{H} is a quotient of the -maximal cover, cf. 2.5. Let ^{}be a finite extension of and an essentially Mumford–Tate unliftable abelian variety over ^{}with connected Mumford–Tate group _{}such that _{C} provides a weak Mumford–Tate lift of _{C}. For each prime number , let _{}= H_{ét}^{1}( ¯ Q_{}) be the -adic representation of 𝒢_{^{}} associated to .

Then the map _{}^{der} _{}^{der} lifts to _{}^{der} { \atop ^{der}} . For every

– the representation _{}is a weak geometric lift of the restriction _{}_{ ˜ 𝒢_{^{}}}and

– for every representation { \atop _{}}of { \atop _{Q_{}}}, the Galois representation on { \atop _{}}is an object of _{}(CM)-Rep .

Proof. We fix a faithful self-dual representation of and for each prime number we write { \atop _{}}= _{Q} Q_{}. As generates the tannakian category of representations of , it is sufficient to prove the corollary for the representations { \atop _{}}.

Since is essentially Mumford–Tate unliftable, it follows from theorem 2.9 that _{}^{der} is -maximal. It follows that the map _{}^{der} _{}^{der} lifts to a map _{}^{der} { \atop ^{der}} . Let _{}: 𝒢_{^{}}_{}(Q_{}) be the map giving the Galois representation on _{}. Write _{}^{ad} : _{}_{}^{ad}, _{}^{ad} : _{}_{}^{ad} and ^{ ˜ ad} : { \atop ^{ad}} for the projections. For any , the proposition 4.5 and the fact that _{} ˜ is a weak geometric lift of _{}imply that ^{ ˜ ad} _{ 𝒢̃_{^{}}}= _{}^{ad} _{𝒢_{^{}}}= _{}^{ad} _{}. The remaining statement of the theorem follows from lemma 4.7.

4.10 Important remark. Concerning the abelian variety ^{}which appears in propositions 4.3 and 4.5 and in theorem 4.9, it follows from theorem 2.12 and proposition 3.1 that there exist a number field ^{}and an essentially M-T unliftable and M-T decomposed weak Mumford–Tate lift ^{}of as in the propositions and in the theorem. The condition that _{}is connected can be forced by replacing ^{}by a finite extension.

4.11 Corollary. Let notations be as in 4.8 with { \atop ^{der}} _{}^{der} satisfying the hypotheses of the theorem. Then, for every and every representation { \atop _{}}of { \atop _{Q_{}}}, the induced representation of 𝒢_{}on { \atop _{}}occurs in the -adic étale realization of an abelian motive.

Proof. The remark and the theorem imply that there is is a finite extension ^{}of such that the representation of 𝒢_{^{}}on { \atop _{}}occurs in the -adic étale realization of an object ^{}of (AV)_{^{}}. The representation of 𝒢_{}on { \atop _{}}then occurs in the -adic étale realization of the Weil restriction Res_{^{}}^{}, which is also in (AV)_{}.

5 Abelian varieties with Mumford–Tate group of type _{}^{H}

5.1 The following notation and hypotheses will be in force until definition 5.10. We let C be a simple abelian variety and ( _{}_{}) the associated Mumford– Tate datum. This implies that _{}is connected. We assume throughout that is essentially Mumford–Tate unliftable, Mumford–Tate decomposed and that _{} is of type _{}^{H} with 4.

It follows that there exist a totally real number field _{0} and an absolutely simple algebraic group ^{}_{0} such that _{}^{der} = Res_{_{0} Q} ^{}. By assumption, the group ^{}_{0} is -maximal in the sense of 2.5. The representation of _{}^{der} on H_{B}^{1}( (C) Q) decomposes over \overline{Q} as a multiple of the direct sum of the standard (orthogonal) representations of the different factors of _{\overline{Q}}^{der} . As in [Del79] and in §2 of this paper, denote by = : _{0} C the set of complex embeddings of _{0}. As _{0} is totally real, is also the set of real embeddings of _{0}. The Dynkin diagram of _{C}^{der} is a disjoint union, indexed by , of diagrams of type . The Hodge cocharacter _{}: G_{C} _{C} associated to projects trivially on some factors of _{C}^{ad}. On the other factors, the conjugacy class of the projection is dual to one of the vertices _{1} or _{}(or possibly _{1} if = 4) of the corresponding component of the Dynkin diagram. Without loss of generality, we will henceforth assume that, on these factors, it corresponds to _{}. As in section 2, let _{}be the set of embeddings corresponding to the factors onto which _{}projects trivially and let _{}= _{}. Recall that _{}is the set of embeddings : _{0} R such that the factor of _{R}^{der} corresponding to is compact and _{}is the set of real embeddings of _{0} for which the corresponding factor of _{R}^{der} is non-compact.

As the Hodge filtration and its complex conjugate are opposite filtrations, the complex conjugate of is conjugate to ^{1} , up to a central cocharacter. This implies that complex conjugation acts on the Dynkin diagram by the main involution and hence that it acts trivially if is even and exchanges _{1} and _{} on every factor if is odd. It follows that 𝒢_{Q} (or Aut(C)) acts on the Dynkin diagram through Gal( Q) for a number field _{0} with [ : _{0}] = 1 or 2 and which is totally real if is even, a CM field if is odd. In particular [ : _{0}] = 2 if is odd. Note that the statement is also true for = 4, because it follows from the definition of the case _{4}^{H} (see 2.5) that the vertices _{1} of the connected components of the Dynkin diagram form a 𝒢_{Q}-orbit, so the stabilizer of a connected component is of order at most 2.

5.2 Construction of and ˜ . We aim to construct an algebraic group Q such that { \atop ^{der}} is simply connected and that { \atop ^{ad}} = _{}^{ad}, together with a cocharacter ˜ of { \atop _{C}} such that ^{ ˜ ad} ˜ = ^{ad} . The argument is strongly inspired by [Del79], see also the proof of theorems 2.9 and 2.12. Here, as before, ^{ad} and ˜ ad are the projections _{}_{}^{ad} and _{}^{ad} respectively.

Let { \atop ^{}}_{0} be the simply connected cover of ^{}. Consider the direct sum of the representations of { \atop _{\overline{Q}}^{}}with highest weights _{1} and _{}. A multiple of that representation can be defined over _{0}, let ^{}be the resulting representation of { \atop ^{}}. By construction, there is a decomposition ^{}_{_{0}} = _{1}^{}_{2}^{}, where _{1}^{}(resp. _{2}^{}) is a multiple of the representation with highest weight 1 (resp. _{}). A non-trivial element of Gal( _{0}) exchanges the factors _{1}^{}and _{2}^{}of this decomposition. If [ : _{0}] = 2, then the composite map 0 ↠ _{1}^{}is an isomorphism of _{0}-vector spaces and endows ^{} with a structure of -vector space.

Put { \atop ^{der}} = Res_{_{0} Q} { \atop ^{}}and let be the rational representation of { \atop ^{der}} deduced from ^{}. Since is the underlying Q-vector space of ^{}, it carries a structure of -vector space. The cocharacter ^{ad} of _{C}^{ad} lifts to a quasicocharacter of { \atop _{C}^{der}} , cf. 1.1.

5.3 Lemma. There is a decomposition

Q C = _{}_{1}^{( )} _{2}^{( )}  (5.3. )

such that, for each , the representation _{1}^{( )} (resp. _{2}^{( )}) is a multiple of the irreducible representation with highest weight _{1} (resp. _{}) of the factor of _{C}^{der} corresponding to .

The weights of on the _{}^{( )} are trivial for = 1 2 if _{}. For _{}, the weights of on _{1}^{( )}are of the form \frac{2}{4} _{1} and those on _{2}^{( )}are of the form \frac{}{4} _{2} with _{1} _{2} Z. If is even _{1} runs from 0 to \frac{2}{2} and _{2} from 0 to \frac{}{2}. For odd, both _{1} and _{2} run from 0 to \frac{1}{2} .

Proof. The decomposition of ^{}induces the decomposition (5.3. ). The quasicocharacter projects trivially to the factors of { \atop _{C}^{der}} corresponding to the _{} and non-trivially to the other factors. The highest weights of on _{1}^{( )} and _{2}^{( )} can easily be deduced from the information collected in [Del79, Table 1.3.9], it is the rational number corresponding to _{1} resp. _{}in that table. The lowest weight of on _{1}^{( )} is the opposite of the number corresponding to _{1} if is even and the opposite of the number corresponding to _{}if is odd. For _{2}^{( )}, the converse is the case.

We continue the construction of and ˜ . The case where is even and the case where is odd will be treated separately.

5.4 The case where is even. In this case, is totally real and either = _{0} or [ : _{0}] = 2. Let ^{}be a totally imaginary quadratic extension of _{0}, put = ^{}and define the algebraic torus _{}over Q by _{}= ker( _{}) ^{}, where _{}: ^{}^{}is the field norm. The group _{}naturally acts on and this gives rise to a Q-linear representation _{}of _{}. We have a natural structure of -vector space on _{}. For each embedding : C we choose a complex embedding of extending and this gives an identification

C = _{}G_{C}

where is the set of complex embeddings of .

From now on, we will further distinguish the cases where [ : _{0}] = 1 and where [ : _{0}] = 2. The subcase where = _{0}. Consider the decomposition (5.3. ) of _{Q} C. Since ^{}= _{1}^{}_{2}^{}is a decomposition of ^{}as a direct sum of _{0}-vector spaces, we obtain a Q-linear decomposition = _{1} _{2}. For _{}, the weights of on _{1}^{( )} are in \frac{1}{2} + Z if 0 (mod 4) and in Z if 2 (mod 4). For the weights on _{2}^{( )}, the converse is the case.

Let = _{}_{}and define _{3} (resp. _{4}) be the representation of given by the action of the first (resp. the second) factor _{}on _{}. We define a quasicocharacter

: G_{C} _{C} = _{}G_{C}^{2}

by

( )_{}=  ;  ;  ;  ; (1 1) if  _{} ( \overline{}1) if  _{}and 0 (mod 4) (1 \overline{}) if  _{}and 2 (mod 4)

where _{}( )_{}is the component of _{}( ) in the factor of _{C} corresponding to . Finally, let be the representation of { \atop ^{der}} defined by

= _{1} _{_{0}} _{3} _{2} _{_{0}} _{4}

and let be the image of { \atop ^{der}} in GL(). This will not cause any confusion, since { \atop ^{der}} acts faithfully on , so the derived group of is { \atop ^{der}} . The weights of ( _{}) on are in Z, so the projection of the quasi-cocharacter ( _{}) from ({ \atop ^{der}} )_{C} to { \atop _{C}} is a true cocharacter ˜ of { \atop _{C}} . By construction,

˜ ad ˜ = ^{ ˜ ad} = ^{ad}

so we have constructed the couple ( ˜ ) in this particular subcase.

The subcase where [ : _{0}] = 2. The structure of -vector space on gives rise to a decomposition

Q C = _{˜ : C} ^{(˜ )}  (5.4. )

Recall that is the set of complex embeddings of . Put { \atop _{}}= ˜ _{_{ ˜ 0}} _{} and { \atop _{}}= ˜ _{_{ ˜ 0}} _{}. For ˜ { \atop _{}}, the weights of on ^{(˜ )} are all 0, for ˜{ \atop _{}}, these weights are either in Z or in \frac{1}{2} + Z. Let { \atop _{1}} be the set of ˜ { \atop _{}} where the former possibility occurs and { \atop _{2}} its complement in { \atop _{}}. Note that for any _{}one of the embeddings C restricting to lies in { \atop _{1}}and the other one lies in { \atop _{2}}.

This time we define a quasi-cocharacter

: G_{C} _{C} = _{˜}G_{C}

by

( )˜ = \begin{matrix}1 & if ˜ { \atop _{}}{ \atop _{1}} \\ \overline{} & if ˜ { \atop _{2}} \end{matrix}

As above, _{}( )˜ is the component of _{}( ) in the factor of _{C} corresponding to ˜ . Finally, let = _{}_{}as representation of { \atop ^{der}} and let be the image of { \atop ^{der}} in GL(). Once again, there is no risk of confusion, because the derived group of is { \atop ^{der}} . Since the weights of ( _{}) on are in Z, the projection of ( _{}) from ({ \atop ^{der}} )_{C} to { \atop _{C}} is a true cocharacter ˜ of { \atop _{C}} . By construction, ^{ ˜ ad} ˜ = ^{ ˜ ad} = ^{ad} , so this achieves the construction of the couple ( ˜ ) in the case where is even.

5.5 The case where is odd. In this case, is a totally imaginary quadratic extension of _{0}. In order to unify this case with the previous one as much as possible, we put = . Let _{}be the Q-algebraic torus ker( _{_{0}} ), where 0 : ^{}_{0}^{}is the field norm. As in the beginning of 5.4, the action of _{} on by multiplication on the left gives rise to a representation _{}of _{}.

We again consider the decomposition (5.4. ) and the subsets { \atop _{}}and { \atop _{}}of the set of complex embeddings of = . By lemma 5.3, the weights of on ^{(˜ )} are either in \frac{1}{4} + Z or in \frac{1}{4} + Z for ˜ { \atop _{}}and trivial for ˜ { \atop _{}}. Let { \atop _{1}} the set of ˜ { \atop _{}}for which the weights of on ^{(˜ )} are in \frac{1}{4} + Z and let { \atop _{2}} be the complement of { \atop _{1}} in { \atop _{}}. For each _{}, one of the embeddings C restricting to lies in { \atop _{1}} and the other one lies in { \atop _{2}}. Thus, the map : given by ˜ _{_{ ˜ 0}} induces a bijection of { \atop _{1}} with _{}. For _{}, we arbitrarily fix a subset { \atop _{1}} { \atop _{}}such that induces a bijection of { \atop _{1}}with _{}. Putting { \atop _{1}} = { \atop _{1}} { \atop _{1}}, we get an identification

C = _{˜{ \atop _{1}}} G_{C}

Under this identification the factor G_{C} corresponding to ˜ acts on the factor of _{C} corresponding to ˜ by multiplication in case ˜ { \atop _{1}} and by multiplication with the inverse if ˜ { \atop _{1}}.

The quasi-cocharacter _{}: G_{C} _{C} is defined by

( )˜ = \begin{matrix}1 & if ˜ { \atop _{1}} \\ ^{1 4} & if ˜ { \atop _{1}} \end{matrix}

We put = _{}_{}as representation of { \atop ^{der}} and again let be the image of { \atop ^{der}} in GL(). As before, the derived group of is { \atop ^{der}} and the projection of ( _{}) from ({ \atop ^{der}} )_{C} to { \atop _{C}} is a true cocharacter ˜ of { \atop _{C}} . We obviously have equalities ^{ ˜ ad} ˜ = ^{ ˜ ad} = ^{ad} , so this settles the construction of the couple ( ˜ ) in this case.

The preceding discussion establishes the following theorem.

5.6 Theorem. Let C be a simple abelian variety, ( _{}_{}) its Mumford–Tate datum and _{}the Hodge cocharacter associated to . Assume that is essentially Mumford– Tate unliftable and that _{}is of type _{}^{H} with 4. Then there exist an algebraic group with { \atop ^{der}} simply connected, an identification { \atop ^{ad}} = _{}^{ad} and a cocharacter ˜ : G_{C} { \atop _{C}} such that ^{ ˜ ad} ˜ = ^{ad} _{}.

5.7 Remark. In all cases above, the group _{R} occurring in the construction of is compact. It thus follows that { \atop _{R}^{ab}} is compact. It is left to the reader to construct an isomorphism { \atop ^{ab}} = _{}and to compute the composite

{ \atop ^{ab}} = _{}.

5.8 Suppose that C is a number field and that is an abelian variety over with connected Mumford–Tate group such that the conditions of the theorem are verified for _{C} and its Mumford–Tate datum ( _{}_{}). Let and ˜ be as in the conclusion of the theorem.

Let { \atop ^{ab}} = { \atop ^{der}} and define ^{ab} to be the composite of the cocharacter ˜ : G_{C} { \atop _{C}} with the natural projection { \atop _{C}} { \atop _{C}^{ab}} . Define

ab:  { \atop _{R}^{ab}}
ab( )\overline{^{ab}(¯ )}

The remark 5.7 implies that { \atop _{R}^{ab}} is compact. As the weight _{}: G_{R} { \atop _{R}^{ab}} is trivial, [Del79, 1.1.18(b)] implies that ^{ab} defines a polarizable Hodge structure on any Q-linear representation ^{ab} of { \atop ^{ab}} .

Fix a faithful rational representation ^{ab} of { \atop ^{ab}} . It follows from proposition A.1 of [Del82b] and the remark preceding it that the Hodge structure on ^{ab} is the Betti realization of an absolute Hodge motive _{\overline{Q}}^{ab} belonging to (CM)\overline{Q}. Note that using the isomorphism { \atop ^{ab}} = _{}from remark 5.7, this motive can also be constructed explicitly. Replacing by a finite extension and fixing an embedding \overline{Q} we can assume that _{\overline{Q}}^{ab} descends to a motive ^{ab} over . At the cost of a further finite extension of , we can also assume that the Mumford–Tate group of ^{ab} is connected and hence contained in { \atop ^{ab}} . This implies that for every , the -adic realization of ^{ab} factors through a map _{}^{ab}: 𝒢_{}{ \atop ^{ab}} (Q_{}). Note that the _{}^{ab} form a compatible system of representations (see 6.7) and that the _{}^{ab} do not depend on the choice of the representation ab of { \atop ^{ab}} .

For each prime number , let Σ_{}be the set of -adic places of . There is a finite set Σ of places of such that and ^{ab} have good reduction outside Σ. For every prime number , the representations _{}and ˜ _{}^{ab} are unramified at all Σ Σ_{}. For any finite extension ^{}of , let Σ_{}^{}be the set of -adic places of ^{}and let Σ^{}be the set of places lying over the Σ.

Let ^{}= _{}^{ad} { \atop ^{ab}} and let ^{}: ^{}be defined by the projections of onto { \atop ^{ab}} and _{}^{ad} = { \atop ^{ad}} . For each , define _{}^{ad} = ^{ad} _{}: 𝒢_{}_{}^{ad}(Q_{}) and put

_{}^{}= ( _{}^{ad} _{}^{ab}): 𝒢_{}^{}(Q_{})

5.9 Corollary. There exist a finite extension ^{}of and a system of geometric Galois representations _{} ˜ : 𝒢_{^{}}(Q_{}) lifting the restrictions _{}^{}: 𝒢_{^{}}^{}(Q_{}). The system ( _{} ˜ ) can be chosen in such a way that each _{} ˜ is unramified at for each Σ^{}Σ_{}^{}.

For each prime number , the representation _{} ˜ is a weak geometric lift of _{}and it is essentially geometrically unliftable.

Proof. Since _{}is the Mumford–Tate group of , the adjoint representation of gives rise to a motive ^{ad} belonging to the subcategory of Mot_{AH}( ) which is -generated by ^{1}( ) and the Tate motive. The Mumford–Tate group of ^{ad} is the adjoint group _{}^{ad} = { \atop ^{ad}} .

Consider the object ^{}= ^{ad} ^{ab} of Mot_{AH}( ). Its Mumford–Tate group is contained in ^{}and the Hodge cocharacter ^{}: G_{C} _{C}^{}associated to its Betti realization is the product ( ^{ad} ^{ab}). It follows that ^{}= _{C}^{}˜ . On the other hand, for each , the Galois representation on the -adic realization of the motive ^{}is the above map _{}^{}. The motive ^{}is an abelian motive, so it follows from [Bla94, Theorem 0.3] that the -adic comparison maps linking the -adic and the DeRham realizations of ^{}are compatible with absolute Hodge classes. The main theorem 2.1.7 of [Win95] implies that there is a finite extension ^{}such that every ( _{}^{})_{𝒢_{^{}}}lifts to a representation _{} ˜ : 𝒢_{^{}}(Q_{}) satisfying the conditions of the corollary.

It is clear from the above that, for each , the map _{} ˜ provides a weak geometric lift of _{}. As { \atop ^{der}} is simply connected, it is also clear that the _{} ˜ are essentially geometrically unliftable.

5.10 Definition. The representations _{} ˜ constructed as in the corollary from Mumford–Tate decomposed abelian varieties will be called representations of lifted abelian _{}^{H}-type.

5.11 Corollary. Let  C be a number field and  an abelian variety with

connected Mumford–Tate group _{}and associated system of Galois representations ( _{}). Then there exist an linear algebraic group over Q such that { \atop ^{der}} is the universal cover of _{}^{der}, a finite extension ^{}of and a system of weak geometric lifts

˜ : 𝒢_{^{}}(Q_{})

of the restrictions to 𝒢_{^{}}of the _{}.

Proof. By theorem 2.12 and proposition 3.1, there exist a number field ^{}and an essentially M-T unliftable and M-T decomposed weak Mumford–Tate lift ^{} of with connected Mumford–Tate group _{}. It follows from proposition 4.5 that the system of Galois representations ( _{}) associated to is a system of weak geometric lifts of the system ( _{}).

Let _{C} _{C} be the isogeny decomposition as in definition 2.10. After enlarging ^{}, we can assume that this decomposition exists over ^{}. In that case, each _{}is Mumford–Tate decomposed with _{_{}}^{ad} simple and _{}^{ad} is the product of the _{_{}}^{ad}. For each , let _{_{}}: 𝒢_{^{}}_{_{}}(Q_{}) be the -adic Galois representation associated to _{}. For every such that _{_{}}is not of type _{}^{H}, put { \atop _{}}= _{_{}}and _{} ˜ = _{_{}}. For each such that _{_{}}is of type _{}^{H} let _{} ˜ : 𝒢_{^{}}{ \atop _{}}(Q_{}) be the representation of lifted abelian _{}^{H}-type resulting from corollary 5.9, replacing ^{}by a finite extension again if necessary. Let = { \atop _{}}and let _{} ˜ : 𝒢_{^{}}(Q_{}) be the product of the maps _{} ˜ .

For each prime number , the representation _{} ˜ is a weak geometric lift of and therefore also of _{}. As { \atop ^{der}} is simply connected, the corollary follows.

5.12 Corollary. Let be a number field, an abelian motive and ^{}: 𝒢_{}^{}(Q_{}) a weak geometric lift of the associated -adic Galois representation. Then, after replacing by a finite extension and for any linear representation _{}^{}of _{Q_{}}^{}, the representation of 𝒢_{}on _{}^{}deduced from _{}^{}lies in the tannakian subcategory of Rep_{Q_{}}(𝒢_{}) generated by the -adic representations associated to abelian motives and the representations of lifted abelian _{}^{H}-type.

Proof. It is sufficient to prove the corollary for one fixed faithful self-dual representation _{}^{}of ^{}.

Let be the essentially Mumford–Tate unliftable and Mumford–Tate decomposed abelian variety supplied by applying corollary 3.6 to , so is an object of ^{1}( ) Q(1) . This inclusion corresponds to a map _{}_{}. Suppressing isogeny factors of , we can assume that the induced map _{}^{der} _{}^{der} is an isogeny. After replacing by a finite extension, define the group and the representation _{} ˜ : 𝒢_{}(Q_{}) as in the proof of 5.11.

As { \atop ^{der}} is the simply connected cover of _{}^{der}, the isogeny

{ \atop ^{der}}  _{}^{der}  _{}^{der}

lifts to an isogeny { \atop ^{der}} ^{der}. This results in an identification of the adjoint groups ^{ad} = _{}^{ad} = _{}^{ad} = { \atop ^{ad}} and, by construction, the projections to _{}^{ad}(Q_{}) of the representations _{}^{}, _{}, _{}and _{} ˜ all coincide. The corollary now follows from lemma 4.7 with _{1} = , _{1} any faithful self-dual representation of _{1}, _{2} = ^{}and _{2} = _{}^{}.

6 Galois representations of lifted abelian _{}^{H}-type

6.1 We revert to the notations of 5.8, in particular is essentially Mumford– Tate unliftable, _{}is of type _{}^{H} and is the group constructed in theorem 5.6. We will also assume to be Mumford–Tate decomposed. For the first part of this section, we fix a prime number and assume that there exists a weak geometric lift _{} ˜ : 𝒢_{}(Q_{}) of _{}with as in theorem 5.6. It follows from corollary 5.9 that such a lifting exists after replacing by a finite extension and that, for big enough, _{} ˜ is part of a system of -adic representations for variable , but we do not assume for the moment that such a system exists.

6.2 Theorem. Let _{} ˜ : 𝒢_{}(Q_{}) be a representation of lifted abelian _{}^{H}-type, constructed as weak geometric lift of the representation _{}as described in 6.1, with 5. Assume moreover that _{}(𝒢_{}) _{}(Q_{}) is Zariski dense.

Let { \atop _{}}be a faithful Q_{}-linear representation of { \atop _{Q_{}}}. Then there is no finite extension ^{}of such that the representation of 𝒢_{^{}}on { \atop _{}}induced by _{} ˜ belongs to the category (AV)-Rep_{Q_{}}(𝒢_{^{}}).

Proof. Replace by a finite extension and assume that the representation of 𝒢_{}on { \atop _{}}belongs to (AV)-Rep_{Q_{}}(𝒢_{}). After further enlarging and applying proposition 4.5 and remark 4.10, it follows that there is an essentially Mumford– Tate unliftable and Mumford–Tate decomposed abelian variety such that { \atop _{}}belongs to H_{ét}^{1}( ¯ Q_{}) (CM)-Rep .

Let _{=1}^{1} _{}be the decomposition from the definition 2.10, so that _{}^{der} = _{_{}}^{der}, each _{_{}}^{ad} is simple and _{}^{ad} = _{_{}}^{ad}. For = 1 1, let _{}= _{_{}}, put _{}= H_{ét}^{1}( _{}¯ Q_{}) and let _{}be the Zariski closure of _{}(𝒢_{}) in _{_{}}. Let _{}be an object of (CM)-Rep such that { \atop _{}}belongs to _{1} _{}, let _{}be the Zariski closure of the image of the Galois representation on _{}and let _{}: 𝒢_{}_{}(Q_{}) be the corresponding morphism. Finally, let _{=1}^{}_{} be the Zariski closure of the image of

( _{1} _{}): 𝒢_{}_{=1}^{}_{}(Q_{})

and : 𝒢_{}(Q_{}) the induced representation. At the cost of a further finite extension of , we can assume to be connected. The fact that { \atop _{}}belongs to the tannakian category generated by the _{}implies that there is a surjection : { \atop _{Q_{}}}such that _{} ˜ = _{}.

For any -adic place ¯ of ¯ , the cocharacter _{_{}˜}¯ : G_{C_{}}{ \atop _{C_{}}}associated to the Hodge–Tate decomposition corresponding to _{}_{ ˜ ℐ}¯ lifts to the cocharacter _{}¯ : G_{C_{}}_{C_{}}associated to the Hodge–Tate decomposition corresponding to _{ℐ}¯ . There is at least one simple factor _{C_{}}^{}of { \atop _{C_{}}^{ad}} to which _{_{}˜}¯ projects non-trivially. It follows from [Win88, Proposition 7] and the fact that [Bla94, Theorem 0.3] implies [Win88, Conjecture 1] that this projection is dual to the root _{}of this factor.

Let _{C_{}}^{}be the simple isogeny factor of _{C_{}}^{der} which surjects onto _{C_{}}^{}, such a factor exists by hypothesis. As { \atop ^{der}} is simply connected, _{C_{}}^{}is simply connected as well. The Hodge–Tate cocharacter _{}¯ lifts to a quasi-cocharacter ; ¯ of _{C_{}}^{}. This quasi-cocharacter is still dual to the vertex _{}of the Dynkin diagram. Over C_{}, any faithful representation ^{}of _{C_{}}^{}whose highest weights are fundamental weights contains a direct factor with highest weight _{1} or and it follows from lemma 5.3 applied to _{C_{}}^{}, with ; ¯ playing the role of , that ; ¯ has at least three weights on this factor. We will show that this leads to a contradiction.

As = _{=1}^{1} _{}is the Galois representation on H_{ét}^{1}( ¯ Q_{}), the Hodge– Tate cocharacter _{}¯ acts on C_{}with two weights. By construction, is a faithful representation of ^{der} so _{Q_{}}C_{}is a faithful representation of _{C_{}}^{}. Let ^{}be the direct sum of the direct factors of the representation of _{C_{}}^{}on C_{}on which _{C_{}}^{}acts non-trivially. The quasi-cocharacter ; ¯ then acts with exactly two weights on ^{}. It follows from [Del79, 1.3.7] that for every irreducible direct factor ^{}of ^{}, the highest weight is a fundamental weight of _{C_{}}^{}. We have shown above that this implies that ; ¯ has at least three weights on ^{}, so we arrive at the contradiction we were looking for.

6.3 Remark. It follows from theorems 2.9 and 2.12 and their proof, that for every 4 there exists an essentially M-T unliftable and M-T decomposed abelian variety C such that _{}is of type _{}^{H}. It follows from [Noo95, Theorem 1.7] that there also exists such an abelian variety which can be defined over a number field and for which _{}(𝒢_{}) _{}(Q_{}) is Zariski dense. Thus, for every 5, there is an abelian variety for which the hypothesis of the theorem is fulfilled.

6.4 The case = 4. We consider the case of Galois representations of lifted abelian _{4}^{H}-type, so we keep all the notations of 6.1 but fix = 4. In this case, the method of the proof of the above theorem gives a more limited statement. The reason for this lies in the facts that the representation of 𝒢_{}on { \atop _{}}may be reducible and that the group { \atop _{C}} does have faithful representations in which the Hodge cocharacter acts with only two weights.

It is left to the reader to verify the following statement, whose proof is completely analogous to the proof of theorem 6.2.

6.5 Proposition. Let notations and hypotheses be as in theorem 6.2, but with = 4.

Let be a faithful Q-linear representation of . Then there is no finite extension of such that the representation of 𝒢_{^{}}on Q_{}induced by _{} ˜ belongs to the category (AV)-Rep_{Q_{}}(𝒢_{^{}}).

6.6 Remark. In the situation of the proposition, assume that is a prime number such that the image of 𝒢_{Q_{}}in the automorphism group of the Dynkin diagram is equal to the image of 𝒢_{Q} in this automorphism group. Then the statement of the proposition (and of theorem 6.2) holds for every Q_{}-linear representation { \atop _{}}of { \atop _{Q_{}}}. If 𝒢_{Q} acts on the Dynkin diagram of \overline{Q} through a cyclic group, then the images of 𝒢_{Q} and 𝒢_{Q_{}}in the automorphism group of the Dynkin diagram coincide for infinitely many .

On the other hand, there also exist examples of Galois representations of lifted abelian _{4}^{H}-type where the proof of the theorem fails for every . One can find an example of this situation by taking an abelian variety for which 𝒢_{Q} acts on the Dynkin diagram of the Mumford–Tate group through (Z 2Z)^{2}. For every prime number , the image of 𝒢_{Q_{}}in the automorphism group of the Dynkin diagram is then of order at most 2.

6.7 Frobenius elements. Let be a valuation of . By Fr_{}𝒢_{}, we will denote a geometric Frobenius element at , i. e. an element of the decomposition group of a place ¯ of ¯ lying over such that, on the residue field _{} ¯ of ¯ at ¯ , Fr_{} induces the inverse of the map ^{_{}}, where _{}is the order of the residue field _{}of a . Note that Fr_{}is defined only up to conjugation and up to multiplication by an element of ℐ_{}¯ . This implies that the image of Fr_{}in a representation which is unramified at is defined up to conjugation and that the eigenvalues and the characteristic polynomial of the image of Fr_{}in such a representation are well defined.

Let _{}be a Q_{}-linear representation of 𝒢_{}on an étale cohomology group of a proper and smooth -variety and let be a place where this variety has good reduction. It follows from the Weil conjectures, proved by Deligne, that the eigenvalues of a Frobenius element Fr_{}at are algebraic integers and that all complex absolute values of all these eigenvalues coincide. If the Fontaine– Mazur conjecture is true, then every geometric representation of 𝒢_{}should have this property. We will verify this for the representations of lifted abelian _{}^{H}- type. It is also shown that Fr_{}acts semi-simply in any representation of lifted abelian _{}^{H}-type, a property which is conjectured for all representations coming from the cohomology of algebraic varieties.

The Weil conjectures also imply that, for varying , the -adic étale cohomology groups of a proper and smooth variety form a compatible system ( _{}) of Galois representations. This means that there is a finite set Σ of valuations of such that _{}is unramified at for all Σ with ( ) = 0 and that for all Σ there is a polynomial ^{}Q[ ] which is equal to the characteristic polynomial of Fr_{}acting on _{}for all with ( ) = 0. We will prove a result in this direction for the representations of lifted abelian _{}^{H}-type.

It should be pointed out that this property does not follow from the conjecture of Fontaine and Mazur. However, combined with the Mumford–Tate conjecture, the Fontaine–Mazur conjecture implies that for any geometric -adic representation _{}of 𝒢_{}, there should exist a number field and a compatible system (indexed by the primes 𝔭 of ) of _{𝔭}-linear representations of 𝒢_{}such that _{}occurs in this system. The characteristic polynomials of Fr_{}acting on the _{}would then lie in [ ] and be independent of , for all with ( ) = 0.

6.8 Proposition. Let C be a number field and let _{} ˜ : 𝒢_{}(Q_{}) be a representation of lifted abelian _{}^{H}-type, with 4. Let Σ be the set of places of bad reduction defined in 5.8 and let be an irreducible Q-linear representation of .

Then, for each Σ Σ_{}, the Frobenius element _{} ˜ (Fr_{}) acts semi-simply on { \atop _{}}= Q_{}and its eigenvalues are algebraic integers with all complex absolute values equal to 1.

Proof. It suffices to prove the proposition for the faithful representation constructed in 5.4 and 5.5 (according to the parity of ) and after replacing by a finite extension. Let _{} ˜ be lifted from the representation associated to the M-T decomposed and M-T unliftable abelian variety , with Mumford–Tate group _{}. The derived group _{}^{der} is of the form Res_{_{0} Q} ^{der} and we have { \atop ^{der}} = Res_{_{0} Q} { \atop ^{der}} where { \atop ^{der}} is the universal cover of ^{der}. By construction, the representation carries a structure of _{0}-vector space compatible with the structure of a Weil restriction on { \atop ^{der}} , so the action of on commutes with the action of _{0}.

It is enough to prove the proposition for the representation of 𝒢_{}on the tensor product { \atop _{}}= { \atop _{}}_{_{0}} { \atop _{}}, where _{0} = _{0} _{Q} Q_{}. The representation of { \atop _{Q_{}}^{der}} on { \atop _{}}factors through a representation of _{Q_{}}^{der} . In fact, { \atop _{}}is a faithful representation of _{Q_{}}^{der} , but this plays no role in what follows. By theorem 4.9, { \atop _{}}occurs in the -adic realization of an abelian motive . The proof of 4.9 even shows that we can find such a motive in the subcategory of (AV)_{}generated by ^{1}( ) and (CM)_{}so it follows that has potentially good reduction at for each Σ. After replacing by a finite extension, we can assume that has good reduction at all Σ. This implies that for any valuation Σ Σ_{}, the Frobenius element _{} ˜ (Fr_{}) acts semi-simply on { \atop _{}}and that each eigenvalue is an algebraic integer with all complex absolute values of the form N ^{( ) 2} , where N is the cardinal of the residue field of at and ( ) is an integer.

To show that all ( ) are equal to 0, it suffices to show that the Betti realization of is pure of weight 0. This realization can be determined as follows. The cocharacter ˜ constructed in 5.6 defines a map _{ C̃} : _{C} { \atop _{C}} given by _{ C̃}( ¯ ) = ˜ ( )\overline{˜ (¯ )}. This map descends to a map ˜ : { \atop _{R}} lifting _{}^{ad} : _{}^{ad} and these data define a Hodge structure on on which _{0} acts by endomorphisms. The tensor product = _{_{0}} is a representation of _{}^{der}, and the Hodge structure on derived from the Hodge structure on is the Hodge structure on the Betti realization of the abelian motive . Since ˜ and its complex conjugate are inverse to each other, it follows that the Hodge structure on is of weight 0 and thus the same thing holds for .

6.9 We keep the notations of proposition 6.8. In particular, Σ is the set of places of bad reduction defined in 5.8. For the representation however, we fix the Q-linear representation of constructed in 5.4 or 5.5 as in the proof of proposition 6.8. It follows from corollary 5.9 that, after replacing by a finite extension, there is a system of weak geometric lifts _{} ˜ : 𝒢_{}(Q_{}) of the _{}, for varying . From now on we will assume that we dispose of such a system.

The rest of the paper concerns the variation of the characteristic polynomials of the _{} ˜ (Fr_{}) on the { \atop _{}}= Q_{}, for a fixed valuation Σ of and varying with ( ) = 0. We will deduce our result on the characteristic polynomials from the statement 6.12 which is independent of the choice of a representation of .

As in the proof of proposition 6.8, assume that _{} ˜ is lifted from the representation associated to the Mumford–Tate decomposed and essentially M-T unliftable abelian variety , with Mumford–Tate group _{}. Without loss of generality, we can assume that _{}is of the type constructed in remark 2.13.2. We also keep the notations _{}^{der} = Res_{_{0} Q} ^{der} and { \atop ^{der}} = Res_{_{0} Q} { \atop ^{der}} , for a totally real number field _{0} and semi-simple groups ^{der} and { \atop ^{der}} over _{0}. As explained in 2.13.2, there is a totally imaginary quadratic extension of _{0} such that _{}is isogenous to a subgroup of _{}^{der} ^{}. It follows from loc. cit. that is contained in the centre of End^{0}( _{C}) = End( _{C}) _{Z} Q. In this case, there exists a linear algebraic group { \atop ^{}}over _{0} such that = Res_{_{0} Q} { \atop ^{}}

Let { \atop ^{der}} be the centre of { \atop ^{der}} . It is the kernel of the natural map = { \atop ^{ad}} { \atop ^{ab}} . There is an isomorphism = Res_{_{0} Q} { \atop ^{}}, where { \atop ^{}} is the centre of { \atop ^{der}} , a finite group scheme over _{0} of order 4, geometrically isomorphic to (Z 2Z)^{2} if is even and to Z 4Z if is odd. Note that, by construction, each _{} ˜ is determined up to a finite character 𝒢_{}(Q_{}) which is unramified outside Σ Σ_{}. This ambiguity explains the fact that in the next proposition the characteristic polynomials may vary with .

For each valuation of , each prime number and each (Q_{}), let { \atop _{}^{}}( ) Q_{}[ ] be the characteristic polynomial

{ \atop _{}^{}}( ) = det{ \atop _{}}_{}˜ (Fr_{}) id

We write { \atop _{}^{}}( ) = { \atop _{1}^{}}( ) for the characteristic polynomial of _{} ˜ (Fr_{}).

6.10 Proposition. Assume that for some, hence any, prime number , the rank of the Zariski closure of the image of _{}is equal to the rank of _{}. Then there exist a set Σ_{Fr} Σ of valuations of of Dirichlet density 0 and, for each Σ_{Fr}, a polynomial { \atop ^{}}Q[ ] such that for every prime number with ( ) = 0, one has { \atop _{}^{}}= { \atop ^{}}for some = _{}(Q_{}).

6.11 Corollary. Let notations and hypotheses be as in the proposition. Let ^{nor} be a normal closure of _{0} Q. For each prime number , let _{}^{nor} be the image of ^{nor} in \overline{Q}_{}and put _{}= _{}^{nor} Q_{}. Then, for every Σ_{Fr}, the characteristic polynomial { \atop _{}^{}}of _{} ˜ (Fr_{}) lies in _{}[ ].

Proofs. The -independence of the rank of the Zariski closure of the image of follows from [Ser85, 2.2.4]. The corollary easily follows from the proposition. To prove the proposition we will reformulate the result in terms of the the quotient variety of by a subgroup Aut^{}Aut({ \atop ^{der}} ).

If 5, we put Aut^{}({ \atop ^{der}} ) = Aut({ \atop ^{der}} ) and Out^{}({ \atop ^{der}} ) = Out({ \atop ^{der}} ). If = 4 then Out({ \atop ^{der}} ) = S_{3}, acting naturally on the vertices _{1} _{3} _{4} of the Dynkin diagram. Let Out^{}Out({ \atop ^{der}} ) be the stabiliser of _{1} and Aut^{}({ \atop ^{der}} ) the inverse image of Out^{}in Aut({ \atop ^{der}} ).

In either case, Aut^{}({ \atop ^{der}} ) is an extension of Out^{}({ \atop ^{der}} ) = Z 2Z by { \atop ^{ad}} . It acts on the centre of { \atop ^{der}} through its quotient Out^{}({ \atop ^{der}} ) and by going through the constructions in 5.2 to 5.5 (sub)case by (sub)case, it is not very difficult to check that this action extends to an action of Out^{}({ \atop ^{der}} ) on the centre of { \atop ^{}}. We thus obtain an action of Aut^{}({ \atop ^{der}} ) on { \atop ^{}}and, taking Weil restrictions, an action of Aut^{}= Res_{_{0} Q} Aut^{}({ \atop ^{der}} ) on .

Let Cl() be the categorical quotient of by this action, see [MF82, Chapter 1]. This means that if = Γ( 𝒪) is the affine coordinate ring of then the quotient is given by Cl() = Spec( ^{Aut^{}}). For each prime number , we denote by Cl( _{} ˜ ): 𝒢_{}Cl()(Q_{}) the map deduced from _{} ˜ . The proposition 6.10 then follows from the following, more precise, statement.

6.12 Proposition. Assume that for some, hence any, prime number , the rank of the Zariski closure of the image of _{}is equal to the rank of _{}. Then there are a set Σ_{Fr} Σ of valuations of of Dirichlet density 0 and elements Cl({ \atop Fr}_{}) Cl()(Q), for Σ_{Fr}, such that for every prime number there exists _{}(Q_{}) such that the conjugacy class of _{}_{}˜ (Fr_{}) is Cl({ \atop Fr}_{}).

The proof requires several lemmas and the following notation. The group Aut^{}defined above also acts on _{}^{der}. As Out^{}({ \atop ^{der}} ) Out( _{}^{der}) acts trivially on the centre of _{}^{der}, this action extends to an action of Aut^{}on _{}with trivial action on the centre. As above, we write Cl( _{}) for the categorical quotient of _{}by this action and let Cl( _{}): 𝒢_{}Cl( _{})(Q_{}) be the map induced by _{}: 𝒢_{}_{}(Q_{}), for each prime number .

Similarly, Aut^{}acts naturally on { \atop ^{ad}} and on { \atop ^{ab}} so we deduce an action on = { \atop ^{ad}} { \atop ^{ab}} . Let Cl({ \atop ^{ad}} ) (resp. Cl( ^{})) be the quotient of { \atop ^{ad}} (resp. ^{}) by Aut^{}. The maps _{}{ \atop ^{ad}} ^{}and ^{}are Aut^{}-equivariant and therefore induce maps Cl( _{}) Cl( ^{}) and Cl() Cl( ^{}).

Recall from 5.8 and corollary 5.9 that the _{} ˜ lift the representations

_{}^{}= ( _{}^{ad} _{}^{ab}): 𝒢_{}^{}(Q_{})

Let the Cl( _{}^{}): 𝒢_{}Cl( ^{})(Q_{}) be the induced maps.

6.13 Lemma. For each Σ, there is an element

Cl(Fr_{}) Cl( _{})(Q)

such that for every prime number with ( ) = 0 we have Cl( _{})(Fr_{}) = Cl(Fr_{}).

Proof. We use the notation of 6.9. The hypotheses that _{}arises from the construction of 2.13.2 imply that _{}^{der} ^{}acts on H_{B}^{1}( (C) Q) and that this representation is a Weil restriction of _{_{0}} ^{}, where is the representation of on by left multiplication and ^{}is a multiple of the representation of der of highest weight _{1}. It follows in particular that lies in the centre of End^{0}( _{C}) = End( _{C}) _{Z} Q.

Since Σ, the abelian variety has good reduction _{}at so we can identify End^{0}( _{C}) with a subalgebra of End^{0}( ¯ ). Here ¯ is the base extension of _{}to the algebraic closure of the residue field of at . We obtain an embedding End^{0}( ¯ ). Let _{}: _{}_{}be the Frobenius endomorphism. For each prime number different from the residue characteristic at , there is a canonical, hence -equivariant, identification H_{ét}^{1}( ¯ Q_{}) = H_{ét}( ¯ Q_{}). Under this identification, the action of _{}(Fr_{}) on the left hand side corresponds to the action of _{}on the right hand side.

Since _{}is semi-simple and lies in the centre of End^{0}( _{}), the subalgebra = [ _{}] End^{0}( _{}) is a product of number fields _{}. Each _{}is of the form Q( _{}) for some _{}End^{0}( _{}) and it is a standard fact (cf. [Mum70, §19, theorem 4]) that the characteristic polynomial of _{}acting on H_{ét}( ¯ Q_{}) has coefficients in Q and is independent of . This implies that there exists an - module _{}such that for every prime number with ( ) = 0 there is an isomorphism H_{ét}( ¯ Q_{}) = _{}_{Q} Q_{}of _{Q} Q_{}-modules. Let [ ] [ ] be the characteristic polynomial of _{}acting on _{}as an -linear endomorphism. For every with ( ) = 0, the characteristic polynomial of _{}(Fr_{}), acting _{Q} Q_{}-linearly on H_{ét}^{1}( ¯ Q_{}), is equal to . This result is due to Shimura, see [Shi67, 11.10.1].

Consider the map _{\overline{Q}}^{der} A_{\overline{Q}}^{}corresponding to the characteristic polynomial in the representation with highest weight _{1}. It factors through the quotient of ^{der} for the action of Aut^{}( ^{der}), giving a map Cl( _{}) Res_{Q} A_{}^{}. This map is easily seen to be injective on geometric points. The fact that is the characteristic polynomial of _{}(Fr_{}) for every with ( ) = 0 implies that is the image in Res_{Q} A_{}^{}of an element Cl(Fr_{}) Cl( _{})(Q) and that this element Cl(Fr_{}) verifies the condition of the lemma.

6.14 Corollary. For each Σ, there is an element

Cl(Fr_{}^{}) Cl( ^{})(Q)

such that for every prime number with ( ) = 0 we have Cl( _{}^{})(Fr_{}) = Cl(Fr_{}^{}).

Proof. By 5.8, the _{}^{ab}: 𝒢_{}{ \atop ^{ab}} (Q_{}) form a compatible system, so all _{}^{ab}(Fr_{}) are defined over Q and coincide. This gives rise to an element Fr_{}^{ab} { \atop ^{ab}} (Q). On the adjoint side, let Cl({ \atop Fr}_{}^{ad}) Cl({ \atop ^{ad}} )(Q) be the image of Cl(Fr_{}) and let Cl(Fr_{}^{}) be the image of (Cl({ \atop Fr}_{}^{ad}) Fr_{}^{ab}) in Cl( ^{})(Q).

For each , the representation _{} ˜ is a weak geometric lift of _{}so the projection ˜ _{}^{ad} : 𝒢_{}{ \atop ^{ad}} (Q_{}) coincides with the composite of _{}with the projection _{}(Q_{}) { \atop ^{ad}} (Q_{}). It follows that Cl(Fr_{}^{}) is the element promised by the corollary.

6.15 Lemma. Let be either the group scheme = Res_{_{0} Q} { \atop ^{}}(the centre of { \atop ^{der}} , cf. 6.9) or

Out^{}() = Aut^{}() { \atop ^{ad}} = Res_{_{0} Q} _{2}

and let be a co-finite set of prime numbers. Then the natural map

H_{ét}^{1}(Spec(Q) ) _{}H_{ét}^{1}(Spec(Q_{}) )

is injective.

Proof. In the lemma, denotes either or Out^{}(). We put ^{}= { \atop ^{}}in the former and ^{}= _{2} in the latter case. Let : Spec( _{0}) Spec(Q) be the natural morphism. This implies that = _{}^{}as étale sheaves and since _{}is exact by [Mil80, II, Corollary 3.6], this gives an isomorphism

H_{ét}^{1}(Spec(Q) ) = H_{ét}^{1}(Spec( _{0}) ^{})

cf. [Ser94, I. 2.5] for the interpretation in terms of Galois cohomology. Similarly, for every prime number , there is an isomorphism

H_{ét}^{1}(Spec(Q_{}) ) = _{𝔭} H_{ét}^{1}(Spec( _{0 𝔭}) ^{})

obtained from the above identification by base change to Spec(Q_{}). To prove the lemma, it is sufficient to prove the injectivity of the map

H_{ét}^{1}(Spec( _{0}) ^{}) _{𝔭 ^{}}H_{ét}^{1}(Spec( _{0 𝔭}) ^{})

where ^{}is the set of primes of _{0} lying over the rational primes in . In the case where = Out^{}(), the group scheme ^{}= _{2} is constant and as H_{ét}^{1}(Spec( ) _{2}) = Hom (𝒢_{}_{2}( )) for any field , the lemma follows.

In the case where ^{}= { \atop ^{}}, there are two possibilities, ^{}is geometrically isomorphic to either (Z 2Z)^{2} or Z 4Z. In both cases the group scheme becomes trivial over an extension of degree (at most) 2. If ^{}is already trivial over _{0}, then the above argument applies. If ^{}is geometrically isomorphic to (Z 2Z)^{2} and trivial over the quadratic extension _{0}, then ^{}is a Weil restriction and the lemma follows by the same argument as before.

We are left with the case where _{}^{}= (Z 4Z)_{}for a quadratic extension 0. The proof in this remaining case, which also applies in the other cases where ^{}= { \atop ^{}}, makes use of the long exact cohomology sequences associated to the short exact sequence 1 _{2} ^{}_{2} 1 of étale sheaves on Spec( _{0}). This gives a commutative diagram with exact rows,

2( _{0})  H^{1}( _{0} _{2}) ^{}H^{1}( _{0} ^{})  H^{1}( _{0} _{2})
 ;  ;  ;  ; ^{loc}^{_{2}}   ;  ; ^{loc}_{^{}} ;  ; ^{loc}^{_{2}}
2( _{0 𝔭})  H^{1}( _{0 𝔭} _{2}) ^{_{𝔭}}  H^{1}( _{0 𝔭} ^{})  H^{1}( _{0 𝔭} _{2})

Here all H^{1} are étale cohomology groups over the spectrum of the specified field. These groups can be identified with the corresponding Galois cohomology groups. The products in the second row are over all 𝔭 ^{}.

The image of identifies with _{0}^{}_{0}^{2}  for some  _{0}^{}, where _{0}^{2}

is the subgroup of _{0}^{}generated by and the squares in _{0}^{}. Similarly, the image of each _{𝔭} is identified with _{0 𝔭}^{}_{0 𝔭}^{2} . If ker loc_{^{}}, then the injectivity of loc_{_{2}} implies that lies in the image of . Let _{0}^{}represent the preimage of in _{0}^{}_{0}^{2} . Since ker loc_{^{}}, the element lies in _{0 𝔭}^{2} for each 𝔭 ^{}, so is a square in _{0}( \overline{}) locally at each place above a 𝔭 ^{}. It follows that is a square in _{0}( \overline{}), hence it maps to 1 in _{0}^{}_{0}^{2} .

Proof of proposition 6.12.The map ^{}induces a map pr : Cl() Cl( ^{}). Obviously,

Cl( _{}^{}) = pr_{Q_{}}Cl( _{} ˜ ): 𝒢_{}Cl( ^{})(Q_{})

for each .

For the groups and ^{}, let Conj() and Conj( ^{}) be the varieties of geometric conjugacy classes, i. e. the quotients for the actions of { \atop ^{ad}} . Note that Cl() (resp. Cl( ^{})) is the quotient of Conj() (resp. Conj( ^{})) by Out^{}() and that there is a commutative diagram

Conj()  Cl()
 ;  ;  ;  ; ^{pr}
Conj( ^{})  Cl( ^{})

There is a Zariski closed subset Cl( ^{}) such that Conj( ^{}) Cl( ^{}) and pr are both unramified outside . Let the Cl(Fr_{}^{}) be as in corollary 6.14. The hypothesis on the rank of the Zariski closure of the image of _{}and the Chebotatev density theorem imply that there is a set Σ_{Fr} of places of of Dirichlet density 0 such that Cl(Fr_{}^{}) (Q) if and only if Σ_{Fr}. To prove the proposition, it suffices to show that for every Σ_{Fr}, the fibre of pr over Cl(Fr_{}^{}) contains an element Cl({ \atop Fr}_{}) Cl()(Q).

Fix Σ_{Fr}. The fibre of Conj( ^{}) Cl( ^{}) over Cl(Fr_{}^{}) is a Out^{}-torsor which admits a Q_{}-valued point Conj( _{}^{}(Fr_{})) for almost every . It follows from lemma 6.15 that there exists Conj(Fr_{}^{}) Conj( ^{})(Q) above Cl(Fr_{}^{}). The fibre of Conj() Conj( ^{}) over this element is a -torsor. As this fibre identifies with pr^{1} (Cl(Fr_{}^{})), this last fibre is also a -torsor. It has a Q_{}-valued point Cl( _{} ˜ (Fr_{})) for almost every , so the promised existence of Cl({ \atop Fr}_{}) Cl()(Q) follows from another application of lemma 6.15.

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Rutger Noot I. R. M. A. Université Louis Pasteur and CNRS 7 rue René Descartes 67084 Strasbourg, France

rutger.noot@math.u-strasbg.fr
http://www-irma.u-strasbg.fr/~noot