Families of Motives and the Mumford–Tate Conjecture

We give an overview of some results and techniques related to the Mumford–Tate conjecture for motives over finitely generated fields of characteristic 0. In particular, we explain how working in families can lead to non-trivial results.


Introduction
The goal of this article is to give an account of some results and techniques in the study of algebraic cycles, with special focus on the Mumford-Tate conjecture. This conjecture was stated in Mumford's paper [60] from 1966, in which he reported on joint work with Tate, introducing, in particular, what we now call the (special) Mumford-Tate group of an abelian variety. Not long before, Serre [75] had begun the study of Galois representations on the Tate modules T of abelian varieties over a number fields, and had introduced the Lie algebra g ⊂ gl(T ⊗ Q ) of the image of such a representation. Mumford's paper ends with the conjecture that this Lie algebra, or more precisely, its intersection with sl(T ⊗Q ), should be equal to the Lie algebra of the special Mumford-Tate group tensored with Q . In a more tentative form ("on peut même espérer que...") the same problem is stated in Serre's Résumé des cours de 1965-1966 [76].
The MTC (Mumford-Tate Conjecture) is nowadays formulated in much greater generality and has no special relation to abelian varieties, other than that the strongest known results are about abelian motives. The context for it is that we consider a complete nonsingular variety X (or more generally a motive) over a finitely generated field K of characteristic 0. If we choose a complex embedding σ : K → C, we may consider the singular ("Betti') cohomology H σ = H B (X σ , Q) of X σ , which carries a Hodge structure. On the other hand, for a prime number, we have an Of course we want to get further than conjectures, and this is where one of the main themes of this article comes in. Namely, in Section 4 we discuss how working with families of varieties provides us with additional tools that in some cases lead to very nontrivial results. It will come as no surprise that monodromy is among these tools. What perhaps is surprising is that, even though this notion has been omnipresent in geometry since the 19th century, in recent years people have found still better ways to exploit monodromy; as we will see, it lies at the heart of some recent developments. In the study of families of varieties, we will in particular discuss how the Mumford-Tate groups, -adic algebraic Galois groups and motivic Galois groups of the fibres vary. Though the variation is of an erratic nature, there is now a reasonably good understanding of the "jump loci", and it is nice to see, in Theorem 4.3.1 and Corollary 4.3.9 for instance, how monodromy plays a unifying role.
Though much of what we discuss in this section is based on work of other people, and in particular on the theory developed by André in [5], we hope that our presentation will make these results more accessible, and will in particular clarify the mutual relationship between motives and their Hodge and -adic realizations. As an illustration of the power of these techniques, we end this section with a quick proof of the Tate conjecture for divisor classes on algebraic surfaces in some particular families.
The geometer may complain about our strong focus on abstract notions such as Tannakian categories and properties of algebraic groups. Indeed, many results on which we report are based on a good understanding of formal structures as much as on geometry. What geometric intuition is there behind the fact that the Hodge conjecture and the Tate conjecture are "trivially" true 1 for simple abelian varieties of prime dimension, whereas already for simple abelian fourfolds it is a deep open 260 Ben Moonen Vol.85 (2017) 4 Ben Moonen problem? To the author's mind, the fact that conjectures about algebraic cycles have a direct relation to representation theory of reductive groups is one of the wonders of the subject.
Notation and conventions. (a) Many of the categories we consider involve a base field or base variety (often called K or S), and a coefficient ring or field Q. As a general rule, the base field or variety is given in parenthesis and the coefficient field as a subscript. Example: Mot(K) Q is the category of motives over K with coefficients in Q.
(b) For us, an algebraic group G over a field K is a special case of a group scheme over K. It is therefore understood that by a homomorphism G 1 → G 2 between such algebraic groups we mean a homomorphism over K, and by a representation of G we mean a (finite dimensional) representation on a K-vector space. The category of such representations is denoted by Rep(G).

The Mumford-Tate conjecture 2.1. Mumford-Tate groups
We start by reviewing some abstract aspects of Hodge theory. Later in this section we will discuss how this is relevant to Algebraic Geometry.
Pure Hodge structures. Let HS Q be the category of pure Q-Hodge structures. (In what follows we shall mostly work with Q-coefficients.) By definition, an object of HS Q is a finite dimensional graded vector space H = ⊕ n∈Z H (n) such that each H (n) is given a Hodge structure of weight n. This category HS Q is a neutral Tannakian category; in particular we have direct sums, tensor products and duals; on the underlying Q-vector spaces they are given by the usual constructions. In addition we have Tate twists: Q(n) is the 1-dimensional Q-vector space (2πi) n · Q with Hodge structure purely of type (−n, −n), and if H is a Hodge structure then we write H(n) for H ⊗ Q(n). Note that H → H(n) decreases the weight by 2n.
If H is a pure Q-Hodge structure of weight n, a polarization of H is a morphism of Hodge structures φ : H ⊗ H → Q(−n) that satisfies a certain positivity property. We refer to [31], Définition 2.1.15 or [66], Section 2.1.2 for the precise definition. What matters for us is that the subcategory HS pol Q ⊂ HS Q of polarizable Q-Hodge structures (those which admit a polarization) is semisimple, and that the Hodge structures that are of interest for us all lie in this subcategory. The subcategory HS pol Q is closed under direct sums, tensor products and duality, and every subquotient of a polarizable Hodge structure is itself again polarizable.
If H is a pure polarizable Q-Hodge structure, its endomorphism algebra D = End HS Q (H) is a finite dimensional semisimple Q-algebra. The choice of a polarization φ gives rise to an involution d → d * on D, and it can be shown that this is a positive involution. The pair (D, * ) is therefore of the type classified by Albert; we refer to [62], Chapter 21, for further details on this classification. Let us only record here that the centre of D is either a totally real field or a CM field.
Vol. 85 (2017) Families of Motives and The Mumford{Tate Conjecture 261 Families of Motives and The Mumford-Tate Conjecture 5 The Deligne torus. Let S be the algebraic torus over R obtained as S = Res C/R (G m ), where Res denotes restriction of scalars ("Weil restriction"). We have S(R) = C × and S(C) = C × × C × . The character group of S is given by X * (S) = Z · z ⊕ Z ·z, with complex conjugation acting by z ↔z. Define Nm = zz : S → G m,R ; on Rvalued points it is given by the usual norm map C × → R × . Let w : G m,R → S be the cocharacter given on R-valued points by the inclusion R × → C × ; so z • w andz • w are both the identity on G m,R .
To give a representation S → GL(V ), for V a real vector space, is the same as giving a decomposition V C = ⊕ p,q V p,q C with the property that V p,q C = V q,p C . In this correspondence, V p,q C is the subspace of elements v ∈ V C on which (z 1 , z 2 ) ∈ C × ×C × = S(C) acts as multiplication by z −p 1 z −q 2 . (The minus signs in the exponents are just a convention, which we will not try to justify here.) Therefore, a Q-Hodge structure H of weight n may be described as a finite dimensional Q-vector space, together with a representation h : S → GL(H) R such that h • w : G m → GL(H) R is given by z → z −n ·id H . To describe an arbitrary Hodge structure (a sum of pieces that are pure of some weight n), we consider a Q-vector space H with a representation h as before, such that h•w is defined over Q; in that case the cocharacter h•w : G m,Q → GL(H) gives rise to a decomposition H = ⊕H (n) such that z ∈ Q × acts on H (n) as multiplication by z −n , and on each H (n) we have a Hodge structure of weight n.

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Vol.85 (2017) 6 Ben Moonen (iv) For H 1 , . . . , H r in HS Q and positive integers m 1 , . . . , m r we have MT(H m 1 1 ⊕ · · · ⊕ H mr r ) ∼ = MT(H 1 ⊕ · · · ⊕ H r ) . As a special case of this, for H in HS Q and r ≥ 1, the Mumford-Tate group of H ⊕r is isomorphic to MT(H), acting diagonally on H ⊕r . Remark 2.1.5. Let µ S : G m,C → S C be the cocharacter such that z • µ S is the identity on G m,C andz • µ S is the trivial endomorphism of G m,C . On C-valued points µ S is the homomorphism C × → (C × × C × ) given by a → (a, 1).
Let H be a Hodge structure given by a homomorphism h : S → GL(H) R . Write µ = h C • µ S : G m,C → GL(H) C . It is not hard to show that MT(H) is the smallest algebraic subgroup of GL(H) such that the cocharacter µ factors through MT(H) C . Note that z ∈ C × acts on H p,q (via µ) as multiplication by z −p . In Section 2.4 we will further discuss how this leads to restrictions on the possibilities for the Mumford-Tate group.
It is important to also understand the Mumford-Tate group from a Tannakian perspective. If we denote by H ⊂ HS Q the Tannakian subcategory generated by H, the forgetful functor ω : H → Vec Q is a fibre functor that has MT(H) as its automorphism group. This means that H is equivalent to the category Rep MT(H) of (finite dimensional, algebraic) representations of MT(H). In view of property 2.1.4(i), this connects Hodge theory to the representation theory of reductive groups.
Definition 2.1.6. Let H be a Q-Hodge structure. An element ξ ∈ H is called a Hodge class if ξ is purely of type (0, 0) in the Hodge decomposition H C = ⊕H p,q C . Put differently, the Hodge classes are those rational classes (i.e., classes in the underlying Q-vector space H) that are purely of type (0, 0) in the Hodge decomposition of the complexification of H. Writing 1 = Q(0), which is the identity object with respect to the tensor product in HS Q , the space of Hodge classes in H can be identified with Hom HS Q (1, H). Lemma 2.1.7. Let H be a Q-Hodge structure that is pure of weight n, and let T = T r 1 ,s 1 ⊕ · · · ⊕ T r k ,s k be a tensor construction as in 2. Remark 2.1.8. If H is a vector space over a field k of characteristic 0 and G ⊂ GL(H) is a reductive subgroup, then G is completely determined by its tensor invariants; see [35], Proposition 3.1.(c). Hence the Mumford-Tate group of a polarizable Q-Hodge structure H may also, somewhat indirectly, be defined as the unique reductive subgroup of GL(H) whose tensor invariants are the Hodge classes. (Note that these spaces can be non-zero only if n 1 = n 2 .) As a special case of this, for H in HS Q we have End HS Q (H) = End(H) MT(H) .
Hodge structures on the cohomology of varieties. If Y is a compact Kähler manifold, its singular cohomology H n (Y, Q) carries a natural Hodge structure of weight n. We refer to [66] or [105] for a detailed discussion of how this is obtained.
is an isomorphism of Hodge structures. Note that the Hodge structure H n (Y, Q) is not, in general, polarizable.
If X is a projective nonsingular variety over C, the associated complex analytic variety X an is compact Kähler and we have a natural Hodge structure of weight n on the singular cohomology group H n = H n (X an , Q). In this case we do get that H n is polarizable. We can extend this to complete nonsingular varieties X, even though X an need not be Kähler. Indeed, by Chow's lemma and resolution of singularities we can find a surjective morphism f : Y → X with Y projective nonsingular. The induced map f * : H n (X, Q) → H n (Y, Q) is injective, realizing H n (X, Q) as a sub-Hodge structure (automatically polarizable) of H n (Y, Q), and the Hodge structure on H n (X, Q) thus obtained is independent of choices.
Many examples that we are going to discuss use the fact that the functor that sends a complex abelian variety X to the Q-Hodge structure H 1 (X, Q) (which is the dual of H 1 (X, Q)) is an equivalence of categories complex abelian varieties up to isogeny → polarizable Q-Hodge structures of type (−1, 0) + (0, −1) .
(There is also a version with Z-coefficients which gives an equivalence of categories between abelian varieties and integral polarizable Hodge structures of type (−1, 0) + (0, −1 Suppose H is simple of CM type, of weight n. Let Σ be the set of complex embeddings of its endomorphism algebra E. For σ ∈ Σ, let H C (σ) ⊂ H C be the subspace on which E acts through the embedding σ. Writing H p,q C (σ) = H C (σ)∩H p,q C we then have a decomposition Each H C (σ) is 1-dimensional, so for σ ∈ Σ there is a unique integer p = p(σ) with H p,q C (σ) = 0. This gives us a function p : Σ → Z with p(σ) = n − p(σ). The Hodge structure H is completely determined by the pair (E, p).
A classical example where this arises is when we have a g-dimensional complex abelian variety X of CM type, which is equivalent to saying that the Hodge structure H = H 1 (X, Q) is of CM type. For simplicity, assume X is simple, so that E = End 0 (X) = End HS Q (H) is a CM field of degree 2g. In this case the function p : Σ → Z only takes the values −1 and 0. Instead of giving the function p we may give the subset Φ ⊂ Σ of embeddings σ for which p(σ) = −1; this set has the property that Σ = Φ Φ. The pair (E, Φ) is classically called the CM type of X. For later use let us make explicit how we can recover H, and therefore also X up to isogeny, from (E, Φ): As underlying Q-vector space we take H = E; then H C = ⊕ σ∈Σ C (σ) (in which the superscript (σ) is included only for bookkeeping purposes), and we declare the summand C (σ) to be of Hodge type (−1, 0) (resp. (0, −1)) if σ ∈ Φ (resp. σ / ∈ Φ). It is easy to see that the Tannakian subcategory of HS pol Q consisting of CM Hodge structures is generated by the Hodge structures H 1 (X, Q) associated with CM types (E, Φ). See for instance [2], Section 2.
Cycle classes. If X is a proper smooth algebraic variety over C then to an algebraic cycle Z of codimension n we can attach a cohomology class cl (Z) ∈ H 2n X, Q(n) , which is a Hodge class. The cohomology class that we obtain only depends on the class of Z modulo rational equivalence, so we obtain a map cl : CH n (X) ⊗ Q → Hodge classes in H 2n X, Q(n) . (2. 2) The Hodge conjecture expresses that all Hodge classes in the cohomology should arise from algebraic cycles in this way: For n = 1 the Hodge conjecture is true; this is Lefschetz's theorem on divisor classes. For n = 0 and n = d = dim(X) the Hodge conjecture is "trivially" true, as H 0 (X, Q) ∼ = Q(0) and H 2d X, Q(d) are both 1-dimensional, spanned by the class of X (assumed to be irreducible), and the class of a point on X, respectively. Apart from these cases, the Hodge conjecture is widely open. For further reading we recommend [50], [87], [103] and [106]. To remedy this, one may consider the special Mumford-Tate group SMT(H) ⊂ SL(H) (also sometimes called the Hodge group). To define it, consider the circle group S 1 = Ker(Nm : S → G m,R ). Then SMT(H) is defined as the smallest algebraic subgroup M ⊂ GL(H) such that h| S 1 factors through M R . With notation as in 2.1.3, an element t ∈ T r,s is a rational (q, q)-class, with q = p(r − s), if and only if t is invariant under the action of SMT(H). This breaks down if we consider classes t in an arbitrary tensor construction T = T r 1 ,s 1 ⊕ · · · ⊕ T r k ,s k . The problem is that the special Mumford-Tate group does not see the weight of a Hodge structure. For instance, if ξ ∈ H is a (q, q)-class with q = 0 then (ξ, 1) ∈ H ⊕ Q(0) is invariant under the action of SMT(H) but is not a (p, p)-class.
If H has weight 0 we have SMT(H) = MT(H). If H is pure of weight n = 0, If X is complete nonsingular, the rational (p, p)-classes in H • (X, Q) form a subring B(X), called the Hodge ring of X. If this Hodge ring is generated by divisor classes, it follows from Lefschetz theorem on (1, 1)-classes that the Hodge conjecture for X is true.
Example 2.1.13 (How to use the Mumford-Tate group). The Mumford-Tate group is particularly effective as a tool to study Hodge classes on abelian varieties. For instance, it leads to the striking fact that we know the Hodge conjecture to be true for all simple abelian varieties of prime dimension; see [70]. (By contrast, already for abelian fourfolds the Hodge conjecture is still open.) For a given dimension g = dim(X) there is a finite list of possible types for End 0 (X) (see [62], Section 21). This leads to the following strategy. As before, X is an abelian variety and H = H 1 (X, Q). . If this Hodge ring is generated by divisor classes, the Hodge conjecture for X k is true.
To illustrate this method, let us carry it out in the simplest case, namely when X is an elliptic curve with End 0 (X) = Q. As before, let H = H 1 (X, Q). the Hodge ring of X k as the ring of SL(H)-invariants in ∧ • H ∨,⊕k . It is a classical result from invariant theory that if V is the standard representation of SL 2 , the ring of SL 2 -invariants in ∧ • (V ⊕k ) is generated by its elements in degree 2. This means precisely that the Hodge ring B(X k ) is generated by divisor classes, and we conclude that the Hodge conjecture is true for all powers of X.
The example just given is only the tip of the iceberg. There is a rich literature on the topic, and wonderful results have been obtained based on such methods. See for instance [58], [59], [63], [64], [70], [88], [89], [90]. The method has its limitations, though. Already for abelian fourfolds, there are cases where the Hodge ring is not generated by divisor classes (see for instance [58], [107]), and apart from a couple of exceptions (see [72], [74], [99]) the Hodge conjecture is not known to be true in these cases. Also, in general it is not possible to determine MT(X) based only on information about End 0 (X). We will further discuss what is known in Section 2.4.

Galois representations
We now turn to the arithmetic cousin of Hodge theory. In what follows, K is a field, K ⊂ K s is a separable closure, and we write Γ K = Gal(K s /K). Further, denotes a prime number different from char(K).
Algebraic Galois groups. We write Rep(Γ K ) Q for the category of continuous representations of Γ K on finite dimensional Q -vector spaces. An object in this category is given by a homomorphism (with H a Q -vector space of finite dimension) which is continuous with respect to the -adic topology on the target and the Krull topology on Γ K . There is an obvious tensor product, and this makes Rep(Γ K ) Q a neutral Tannakian category. The forgetful functor Rep(Γ K ) Q → Vec Q is a fibre functor. With ρ as above, the image Im(ρ) is an -adic Lie subgroup of GL(H). We define to be the Zariski closure of Im(ρ). By definition, this is a linear algebraic group over Q , and it is the smallest algebraic subgroup G ⊂ GL(H) such that the representation ρ factors through G(Q representation χ n ·ρ. Moreover, the Galois representations in which we are interested are pure and have a weight. We will not give details on this; see Deligne's seminal paper [36] or [48], Section 2. Remarks 2.2.2. (i) Let K ⊂ L be a finitely generated field extension and L ⊂ L s a separable closure that contains K s . Then K = L ∩ K s is a finite extension of K and we have Γ L Γ K ⊂ Γ K . If we have a Galois representation ρ as in (2.3), we may restrict it to Γ L . The associated group G (H) will in general become smaller, but the identity component G 0 (H) does not change.
(ii) If we denote by 0 = 0 G (H) the (finiteétale) group scheme of connected components of G (H), the kernel of the composition corresponds to a finite Galois extension K ⊂ K conn . It has the property that, for K ⊂ L as in (i), the algebraic group associated with ρ| Γ L is connected if and only if K conn ⊂ L. In what follows we will sometimes assume that K = K conn ; this means that the group G (H) is connected, and does not change if we replace K by a finitely generated field extension. If the context requires it, we use the notation K conn (ρ).
Galois representations on -adic cohomology. We are primarily interested in Galois representations coming from -adic cohomology of algebraic varieties. Keeping the above notation, let X be a proper smooth scheme over K. The -adic cohomology H m X K s , Q (n) is defined as the cohomology in degree m of a sheaf Q (n) on the pro-étale site (X K s ) pro-ét of X K s . See [12]. (Before the pro-étale topology was introduced, one would work on theétale site of X K s and define H m X K s , Z (n) as the limit over all H m X K s ,ét , (Z/ i Z)(n) ; then set This gives the same cohomology groups.) Fixing m and n, let us abbreviate H = H m X K s , Q (n) . The Galois group Γ K = Gal(K s /K) acts on X K s , and by functoriality we obtain an action on the -adic cohomology, i.e., a representation This representation is continuous with respect to the -adic topology on the target and the Krull topology on Γ K , making it an object of The Galois representations of the form ρ ,H m (X)(n) constitute only a small part of the category Rep(Γ K ) Q ; we refer to [97] for an in-depth discussion of the properties they enjoy.
Let X be a complete nonsingular variety over a separably closed field Ω. Fix integers m and n, and write H = H m X, Q (n) . We associate with H a connected algebraic group G 0 (H) ⊂ GL(H), as follows. Choose any subfield K ⊂ Ω that is of finite type over the prime field, and a model X K of X over K. (This means: a Kscheme X K together with an isomorphism X K ⊗ K Ω ∼ = X.) Let K s be the separable 268

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Vol.85 (2017) 12 Ben Moonen closure of K in Ω. Writing X K s = X K ⊗ K K s , we have a canonical isomorphism H m X K s , Q (n) ∼ −→ H = H m X, Q (n) . Taking this as an identification, we get a representation ρ as in (2.4), and, as before, we define G 0 (H) ⊂ GL(H) to be the identity component of the Zariski closure of its image. By the Remark 2.2.2(i), the group thus obtained is independent of how we choose the model X K /K. Definition 2.2.3. With notation as above, an element ξ ∈ H is called a Tate class if x is invariant under G 0 (H). Cycle classes. Still with X a complete nonsingular variety over Ω, to every algebraic cycle Z of codimension n we can associate a cohomology class cl (Z) ∈ H 2n X, Q (n) . The cohomology class that we obtain is a Tate class that only depends on Z modulo rational equivalence; so we obtain a map cl : CH n (X) ⊗ Q → Tate classes in H 2n X, Q (n) . (2.5) The Tate conjecture expresses that all Tate classes in the cohomology should arise from algebraic cycles in this way. Tate Conjecture). Let X be a complete nonsingular algebraic variety over a separably closed field Ω. Then for every n ≥ 0 the group G 0 (H) associated with H = H 2n X, Q (n) is reductive and the cycle class map (2.5) is surjective.

Conjecture 2.2.5 (The
If a specific value for n is chosen, we refer to this conjecture as the Tate conjecture for cycles in codimension n on X. Remarks 2.2.6. (i) IfΩ is a separably closed field containing Ω, the Tate conjecture for X is equivalent to the Tate conjecture for XΩ. (ii) We have formulated the conjecture for (complete, nonsingular) varieties over a separably closed field. This should not obscure the fact that the Tate conjecture is really a statement about varieties over fields that are finitely generated over their prime field. The TC in this form is equivalent to the assertion that for every smooth projective X over a finitely generated field K, every class ξ ∈ H 2n X K s , Q (n) that is invariant under the action of Γ K is in the image of the -adic cycle class map CH n (X) ⊗ Q → H 2n X, Q (n) .
(iii) The group G 0 (H) is reductive if and only if the representation ρ is completely reducible (=semisimple).
(iv) It is proven in [57] that if for all complete nonsingular varieties over Q and all n ≥ 0 the cycle class map (2.5) is surjective, then for every complete nonsingular X over a finitely generated field K of characteristic 0 the Galois representation ρ of (2.4) is completely reducible (for all m and n).
Vol. 85 (2017) Families of Motives and The Mumford{Tate Conjecture 269 Families of Motives and The Mumford-Tate Conjecture 13 For n = 0 and n = d = dim(X) the Tate conjecture is "trivially" true: choosing a model X/K as before, H 0 (X K s , Q) and H 2d X K s , Q(d) are both isomorphic to Q (0) as Galois representations (assuming X to be irreducible), and are spanned by the class of X, and the class of a point on X, respectively. Already for n = 1 the Tate conjecture is not known in general. We refer to [96] for a much more detailed discussion of the Tate conjecture.
For abelian varieties, we have the following deep result, due to Tate [95] (over finite fields), Zarhin [110], [111] (over finitely generated fields of characteristic p > 2), Mori (the case p = 2), and Faltings [38] (finitely generated fields of characteristic 0). In the statement of the theorem we consider, for X/K an abelian variety, theadic Tate module T (X), which is the first integral -adic homology group. We have H 1 (X K s , Q ) ∼ = Hom(T (X), Q ) as Galois-modules, and since H • (X K s , Q ) is the exterior algebra on H 1 (X K s , Q ) the entire cohomology ring of X is determined by its Tate module.
Theorem 2.2.7. Let X and Y be abelian varieties over a finitely generated field K, and let be a prime number different from char(K). Then T (X)⊗Q and T (Y )⊗Q are semisimple as representations of Γ K = Gal(K s /K) and the natural homomorphism is an isomorphism.
Corollary 2.2.8. Let X be an abelian variety over a separately closed field Ω. Let be a prime number different from char(Ω). Then the -adic Tate conjecture for divisor classes on X is true.
To deduce this from Theorem 2.2.7 is a little exercise in the theory of abelian varieties. Consider an abelian variety X over a finitely generated field K. If X t is the dual abelian variety, we have a perfect Galois-equivariant pairing E : The isomorphism of Theorem 2.2.7 restricts to an isomorphism where on the left we consider the homomorphisms F with F t = F . On the other hand, it is known that the map NS(X) → Hom sym (X, X t ) that sends the class of a line bundle L to the "Mumford homomorphism" ϕ L : is the map that associates to L its first Chern class, this gives the corollary.
Remark 2.2.9. Over fields of characteristic p > 0 there is also a p-adic version of the Tate conjecture. Over finite fields this is due to Tate. See de Jong's paper [29], Section 2 for the general case.
Remark 2.2.10. As already remarked, the Galois representations that are of interest have many good properties. One aspect of this is that, for X/K smooth proper, the collection of Galois representations ρ ,H m (X)(n) forms a strictly compatible system of -adic representations in the sense defined by Serre in [77], Chapter I. This has given rise to numerous important results, especially about the independence of of properties of the associated -adic algebraic Galois groups G . For the key results, most of which are due to Serre or to Larsen and Pink, we refer to [26], [81], [82], [51], [52]. Let us already note that for motives in the sense of André (or for motives for absolute Hodge classes) it is not known in general if the associated -adic Galois representations form a strictly compatible system; see Remark 3.2.1.

The Mumford-Tate conjecture for complex varieties
Let X be a complete nonsingular complex algebraic variety. Fixing integers m and n, We refer to the conjectural equality of MT(H)⊗Q and G 0 (H ) as the Mumford As an example of how this can be useful, recall that the Hodge conjecture is known in codimension 1 (Lefschetz's theorem on divisor classes), whereas the Tate conjecture is not, in general, known for divisor classes. Hence the MTC for H 2 (X)(1) implies the TC in codimension 1 for X.
There is a partial converse to the proposition. We keep the above notation, with m = 2n. If the HC and TC on X are true in codimension n, it follows that In general this does not suffice to conclude that the MTC is true for H 2n (X)(n). The reason is that the groups MT(H) and G 0 (H ) control all tensor spaces built from H and H respectively. These tensor spaces occur in the cohomology of powers of X. One can show that if for all k ≥ 1 the HC and TC are true for X k in codimension kn then the MTC is true for H 2n (X)(n).

Some known results (1)
This is the first of three sections (see 3.3 and 4.4 for the other) in which we briefly discuss what is known about the Mumford-Tate conjecture and give some pointers to the literature. These sections have no pretense at completeness, and we apologize in advance to authors whose contributions we fail to give the attention they deserve.
Representation-theoretic constraints. The method that was outlined in Example 2.1.13 lies at the basis of many interesting results about abelian varieties. Let (X, λ) be a g-dimensional polarized complex abelian variety, and write H = H 1 (X, Q). One should like to determine MT(X) based on information about the endomorphism algebra End 0 (X). The polarization λ (in the sense of the theory of abelian varieties) gives rise to a polarization φ : H × H → Q(1) (in the sense of Hodge theory), which, viewed as an element of (H ∨ ) ⊗2 (1), is a Hodge class. This gives that MT(X) is a connected reductive subgroup of the group of symplectic similitudes CSp(H, φ), and as discussed, End 0 (X) is the algebra of MT(X)-invariants in End(H), i.e., the algebra of endomorphisms of H that commute with MT(X). This can be refined in the following way.
In general, if H is a polarizable Q-Hodge structure, write M = MT(X) and choose a maximal torus T ⊂ M . We have a root datum with underlying lattice X * (T ), and the tautological representation M C → GL(H ⊗ C) is given by a multiset P of weights in X * (T )⊗R. As in Remark 2.1.5, we have a cocharacter µ : G m → M C . Some conjugate ν of it factors through T C and hence gives rise to a linear map ν * : X * (T ) ⊗ R → R. We then know that the image of P under ν * is precisely the set of integers −p counted with multiplicity the Hodge number h p,q . Moreover, ν is defined over Q and M Q is generated by ν together with its Gal(Q/Q)-conjugates. In this way, knowing the Hodge numbers gives restrictions on the representation.
For a g-dimensional abelian variety this means that we have a number of projections X * (T ) ⊗ R → R, corresponding to ν and its Galois-conjugates, under each of which P has as image the multiset {0 g , 1 g }. This leads to the following result.

its highest weight (after choosing a basis for the root system) is a minuscule weight.
For the notion of a minuscule weight, see Bourbaki [16], Chap. VIII, § 7, no. 3. The theorem can be found in [33], Section 1.2 or, with more details, in [80], Section 3. For Hodge structures of arbitrary level, a detailed analysis of the representationtheoretic constraints that are obtained from the Hodge numbers can be found in the work of Zarhin [112].
When we try to obtain analogous results about the groups G associated with the Galois representation on H 1 (X, Q ) = T (X) ⊗ Q , it is not clear a priori how to proceed. While the Mumford-Tate group can be described as the smallest algebraic group receiving the cocharacter µ, this has no direct analogue in the -adic setting. However, Pink [67] has proven that such representation-theoretic constraints also hold in the -adic setting, by making very clever use of cocharacters associated to local Galois representations via Hodge-Tate theory. Specifically, [67], Theorem 3.18, is precisely what is needed to obtain analogues of the results of Deligne, Serre and Zarhin in the -adic setting. For abelian varieties, this is discussed in detail in [67], Sections 4 and 5. In particular, for abelian varieties X with End(X) = Z satisfying some numerical conditions on dim(X), Pink proves that the HC, TC and MTC are true for all powers of X; this improves earlier results of Serre [82], [83] and Tankeev [94]. For some other types of endomorphism algebras, analogous results have been obtained for instance in [9] and [10]. See [55], Section 2, for an application of Pink's results in the context of motives of K3 type.
Some results. Let us now briefly mention some other results about the HC, TC and MTC for abelian varieties. Throughout, X denotes a complex abelian variety and H (resp. H ) denotes the first homology group (singular, resp. -adic) of X. In some cases we only state a precise result in the Hodge-theoretic setting, leaving it to the reader to formulate the -adic analogue.

Taking the isomorphism
This is an immediate consequence of a much deeper result, Theorem 3.3.1; we will return to this in Section 3.3.   [68]; as is nicely explained in [109], it is in fact a consequence of the results about complex multiplication due to Shimura-Taniyama and Weil. In 3.3.2 we will discuss a much more general result.
2.4.5. It was proven by Larsen and Pink that the Mumford-Tate conjecture for abelian varieties is independent of : if it is true for one , it is true for all. See [52], Theorem 4.3. We will return to this in Section 3.3.
2.4.6. LetL(X) denote the algebraic subgroup of Sp(H, φ) consisting of elements that commute with the action of End 0 (X). Let L(X) be the identity component ofL(X). On the other hand, we can define an invariant rdim(X), called the reduced dimension of X. If X is simple and k is the Schur index of the central simple algebra End 0 (X), the reduced dimension is given by For a general X, let X ∼ X n 1 1 × · · · × X nr r be the decomposition (up to isogeny) of X into simple factors with X i ∼ X j for i = j; then one defines rdim(X) = rdim(X 1 ) + · · · + rdim(X r ). Combining results of Murty [64] and Hazama [41], we find that the following properties are equivalent: (a) the Hodge ring of X n is generated by divisor classes for all n ≥ 1, (b) X has no simple factors of Type III and MT(X) = L(X), and (c) the special Mumford-Tate group SMT(X) has rank equal to rdim(X). The -adic analogue of this is also true. If (a)-(c) are true for X 1 and X 2 , they are also true for the product X 1 × X 2 .
2.4.7. Let X be a simple abelian variety of dimension 1 or of prime dimension p. Then the HC, TC and MTC are true for all powers of X. For elliptic curves this is due to Serre; the essential ingredient is [76], Théorème 1. (Once one has this result, one can argue as in Example 2.1. 13.) The result about simple abelian varieties of prime dimension is due to Tankeev; see [70], [90], [91]. Note that in this case there are only four possibilities for the endomorphism algebra End 0 (X): it is Q, a totally real field of degree p, an imaginary quadratic field, or a CM field of degree 2p. Especially in the first two cases we see Theorem 2.4.1 coming into action. For instance, if End 0 (X) = Q, the first homology group H is an absolutely simple symplectic representation of the special Mumford-Tate group, of dimension 2p. We find that SMT(X) is absolutely simple of Lie type A, B, C or D, and that H is defined by a minuscule highest weight. Inspecting the table of minuscule weights and their dimensions (see [80], Annexe, for instance), one then concludes that SMT(X) is necessarily the full symplectic group Sp(H, φ). In the -adic setting, the same arguments apply. One concludes by using the results of Hazama and Murty mentioned in 2.4.6.
For simple abelian varieties, the case of dimension 4 is the next one to consider. (We will discuss the non-simple case in Section 3.3.) For X simple of dimension 4 the MTC is known, except when End(X) = Z. In that case, a construction of Mumford (see [61], § 4) implies that knowing the endomorphism algebra does not suffice to determine the Mumford-Tate group MT(X) or the -adic algebraic Galois groups G 0 (X). This leaves open the possibility that G 0 (X) is strictly contained in MT(X) ⊗ Q , and it is not known how to exclude this. Still with X simple of dimension 4, the HC and TC are known for X for some types of endomorphism algebras, but not in general. See [58].

2.4.8.
A complex abelian variety X gives rise to a Shimura datum (G, X ) with G = MT(X). The associated adjoint Shimura datum is a product of simple adjoint Shimura data (H j , Y j ). Vasiu [102] has proven a very general result that if certain Lie types do not occur among the (H j , Y j ), the Mumford-Tate conjecture for X is true.

Motives and motivic Galois groups
It is time to bring motives into the discussion. This will give us a better understanding of why we expect the MTC to be true, and will also lead to a stronger variant of the MTC.
Motivated cycles. In the past two decades the theory of motives has seen spectacular developments. As an excellent starting point for further reading we recommend the book [6]. In this article we shall only consider pure motives as defined by André in [5]. His definition is modelled after the classical construction of Grothendieck. In Grothendieck's approach one is confronted with the problem that we do not know how to construct enough algebraic cycles. André's key insight is that one obtains a theory with almost all expected properties by formally adjoining the Lefschetz operator * L for every smooth projective variety X equipped with an ample class L.
We briefly review André's construction, referring to the original paper [5] for many more details and for generalizations. Let K be a field, and denote by SmPr(K) the category of smooth projective K-schemes. We fix a Weil cohomology theory H on smooth projective K-schemes, with coefficient field Q of characteristic 0. We further assume that the hard Lefschetz theorem holds for this theory; by this we mean that for every irreducible X in SmPr(K) of dimension d and every ample class This holds, for instance, if for H we take -adic cohomology for some prime number = char(K).
A motivated cycle on X is defined to be a class ξ ∈ H(X) that can be obtained via the following procedure: • Let Y be another variety in SmPr(K) and choose ample classes L X on X and which may formally be written as L dim(X×Y )−j (x). • Let α and β be algebraic cycle classes on X × Y with Q-coefficients, and take ξ = pr X, * α ∪ * L (β) .
The set A mot (X) of motivated cycles is a graded Q-subalgebra of H even (X) that contains all classes of algebraic cycles.
Remarks 3.1.1. (i) The algebra of motivated cycles is constructed as a Q-subalgebra of H(X) for some "reference cohomology theory" H. However, the algebras A mot (X) that are obtained do not depend on the chosen cohomology theory; see [5], Section 2.3. (Note that the coefficient field Q of H may be bigger than Q.) (ii) In the above definition we have ignored Tate twists. If we include them we find that A r mot (X) is a subspace of H 2r (X)(r).
In the rest of the discussion, we assume char(K) = 0.
André's category of motives. With algebraic cycle classes replaced by motivated cycles, the construction of the category Mot(K) is the classical one. For X = ν X ν and Y in SmPr(K) with X ν irreducible of dimension d ν , one defines the graded vector space Corr • mot (X, Y ) of motivated correspondences by the rule that Corr r mot (X, Y ) = ⊕ ν A dν +r mot (X ν × Y ). One shows that the usual rule for the composition of correspondences gives a graded map Corr Z). A motive over K is defined to be a triple (X, e, n) with X a smooth projective Kscheme, e an idempotent in Corr 0 mot (X, X) and n an integer. One thinks of (X, e, n) as the motive that is cut out from X by the projector e, and then Tate twisted by n. A morphism from (X, e, n) to (Y, e , n ) is defined to be an element of the subspace e • Corr n −n mot (X, Y ) • e ⊂ Corr n −n mot (X, Y ) . In this way we obtain the category Mot(K) of motives (in the sense of André) over K.
In what follows we usually denote motives by a single bold letter, and if M = (X, e, n) then we let M(m) = (X, e, n + m). If M = (X, e, n) and M = (Y, e , n ) are motives, we define their tensor product to be M ⊗ M = (X × K Y, e × e , n + n ). We call 1 = (Spec(K), [∆], 0) the unit motive.
We have a contravariant functor H : SmPr(K) op → Mot(K), sending X to the motive H(X) = (X, [∆ X ], 0) and sending a morphism f : X → Y to the class of the transpose graph [ t Γ f ] ∈ Corr 0 mot (Y, X). An important point is that Corr 0 mot (X, X) contains all Künneth components of the diagonal; see [5], Prop. 2.2. As a result we have a decomposition H(X) = ⊕ i≥0 H i (X) that in cohomological realizations gives the usual grading. Also, the Künneth projectors are used in giving the tensor product the correct commutativity constraint; see [5]  To refer to the grading on Mot(K) one says weight rather than degree. For instance, for X in SmPr(K) the motive H i (X)(n) is pure of weight i − 2n. Motivic Galois groups. As we will discuss now, point 3.1.2(iii) leads to the introduction of motivic Galois groups. These will be important for us because of their (conjectural) relation with Mumford-Tate groups and the -adic algebraic Galois groups G 0 that we have discussed in the previous sections. As before, let H be one of the classical cohomology theories, and let Q be the coefficient field. (More generally we could work with any Weil cohomology that gives the same notion of homological equivalence and in which the hard Lefschetz theorem holds.) In what follows we will simply write H for the realization functor H mot on motives; this should not lead to confusion.
Let Mot(K) Q = Mot(K) ⊗ Q be the Q-linear extension of Mot(K), i.e., the category of Q-modules in Mot(K). The objects of Mot(K) Q can be described as triples M = (X, e, n) where now e is a projector in Corr 0 mot (X, The fact that Mot(K) Q is neutral Tannakian means that we can think of motives in two very different ways: as triples (X, e, n), or as algebraic representations of the motivic Galois group. The latter point of view, though much less geometric, turns out to be very useful. Just as the Mumford-Tate group of a polarizable Q-Hodge structure is characterized by the fact that its tensor invariants are the Hodge classes (see Remark 2.1.8), the motivic Galois group G mot (M) is characterized by the fact that its tensor invariants are the motivated cycles.
Remark 3.1.5. One of Grothendieck's standard conjectures, called Conjecture B, states that, for L an ample class on a smooth variety X, the Lefschetz involution * L is given by an algebraic cycle; if this is true for all (X, L), the motivated cycles are precisely the cohomology classes of algebraic cycles. Let us note that in characteristic 0, Conjecture B is known to imply all other standard conjectures; see [6], Corollaire 5.4.2.2.  Behaviour under field extension. Let H be a cohomology theory on SmPr K with coefficient field Q. To simplify notation, write G mot,K for G mot,K,H . We then have a short exact sequence where, as before, Γ K = Gal(K/K). See for instance [37], Section 6. On the corresponding tensor categories, the maps a and b correspond to the base extension functor Mot(K) Q → Mot(K) Q , respectively to the inclusion Mot Art (K) Q → Mot(K) Q of the subcategory of Artin motives. On the other hand, for M in Mot(K) we have a surjective homomorphism G mot,K G mot (M). The image of the kernel of this homomorphism in Γ K is an open subgroup that corresponds to a finite Galois extension K ⊂ K (M), and we obtain a diagram with exact rows The homomorphism G mot (M K ) → G mot (M) is an isomorphism on identity components. The extension K ⊂ K (M) may be characterized by its property that, for K ⊂ The extension K ⊂ K (M) is independent of the chosen cohomology theory. In fact, to test whether G mot (M K ) → G mot (M L ) is an isomorphism we may extend scalars (in the coefficient field, not the base field) to an overfield Q ⊂ Ω and the claim follows using Remark 3.1.6. Conjecturally, G mot (M K ) is connected, and hence is the identity component of G mot (M). (Assuming K to be finitely generated over Q, this follows from Conjecture 3.2.2 below.) This is not known to be true, however. (Caution: the proof that is given in [37], Proposition 6.22 is incorrect.) See [49] for some partial results in this direction. If indeed G mot (M K ) were connected, it would be natural to write K conn (M) instead of K (M), in analogy with the notation K conn (ρ) that was introduced in Remark 2.2.2(ii).
Remark 3.1.7. Let K ⊂ L be an extension of algebraically closed fields of characteristic 0, and for H let us takeétale cohomology with Q -coefficients. If M is a motive over K, we have a canonical isomorphism G mot (M L ) ∼ −→ G mot (M). This follows from the Scolie in [5], Section 2.5, together with the fact that for any X in SmPr(K) we have Hé t (X, Q ) ∼ −→ Hé t (X L , Q ).

The motivated Mumford-Tate conjecture
The title of this section is slightly misleading, in that we will not discuss one single conjecture but rather a package of several conjectures that naturally fit together. Throughout the discussion, K denotes a finitely generated field extension of Q. We fix an algebraic closure K ⊂ K and write Γ K = Gal(K/K). Betti and -adic realization functors. For every complex embedding σ : K → C we have an exact faithful tensor functor H σ : Mot(K) → HS pol Q , sending a motive M = (X, e, n) to the polarizable Hodge structure e · H(X σ , Q)(n). Composing this with the forgetful functor we obtain a fibre functor Mot(K) → Vec Q . We again denote the latter by H σ ; the context will make it clear whether by H σ (M) we mean the Q-Hodge structure or the underlying Q-vector space.
Associated with H σ we have a motivic Galois group G mot,Hσ for which we use the simpler notation G mot,σ . For M in Mot(K) we denote by    24 Ben Moonen M = (X, e, n) then each K conn (ρ ,M ) is a subfield of K conn (ρ ,H(X) ), and the latter is a finite Galois extension of K that by by a result of Serre (see [81] or [51], Proposition 6.14) is independent of . On the other hand, if by G K, we denote the projective limit of all G (M) for M in Mot(K) then we have a diagram with exact rows The Betti and -adic realization functors are related via comparison isomorphisms. Letσ : K → C be a complex embedding, σ its restriction to K, and let be a prime number. For X/K smooth projective we have comparison isomorphisms . This gives rise to an isomorphism of fibre functors on Mot(K) Q ,  This conjecture may of course be stated directly for a motive over C, without any choice of a model over a finitely generated field. Our choice to start with an M over K is motivated (no pun intended) by the desire to show the similarities with other conjectures, such as the next.  [104] that if motivated cycles on (smooth projective) varieties over Q are algebraic, the same is true on smooth projective varieties over C. Voisin in fact works with absolute Hodge classes rather than motivated cycles, but the argument works in either setting. She also proves that if the HC is true for varieties over Q, the Hodge conjecture on complex varieties can be reduced to showing that certain Hodge loci are defined over Q. See also [103], Section 3. (2) "Hodge classes are motivated" for abelian motives. One of the highlights in our present knowledge about the motivated Mumford-Tate conjecture, is the following result of Deligne and André. This result was proven by Deligne [35], working in the category of motives for absolute Hodge cycles. André, who already simplified part of Deligne's proof in [2], proved the result for his category of motives in [5], Section 6. Let us also note that Deligne's result was extended to 1-motives by Brylinski (see [17], Théorème 2.2.5), and that it was strengthened to include a p-adic comparison property by Blasius, Ogus and Wintenberger (see [13] or [108] and the references contained therein). The latter strengthening also follows from André's version of the result.

Some known results
Let M be an abelian motive over a finitely generated field K of characteristic 0. By Theorem 3.3.1, Conjecture HM σ is then true for M, for all σ; this implies that  Faltings) we also know that G 0 (M) is reductive, as it is a quotient of the G 0 of an abelian variety. Moreover, the Deligne-André theorem has as obvious consequence that we get one of the inclusions predicted by the Mumford-Tate conjecture: with notation as in Conjecture 3.2.4 we have This result was also obtained by Piatetksi-Shapiro and Borovoi; see [15]. Another consequence is that for abelian motives the Mumford-Tate conjecture is true on connected centres. With notation as in Section 3.2, the precise statement is the following.
This result, which of course generalizes the result mentioned in 2.4.4, is due to Vasiu; see Vasiu [102], Theorem 1.3.1. A different proof is given by Ullmo and Yafaev in [98], Corollary 2.11. In these papers the result is stated only for abelian varieties, but it is easy to deduce from this the same conclusion for abelian motives.
Which motives are abelian? Consider motives over an algebraically closed field K of characteristic 0. It is clear from the definition of the category of abelian motives that any submotive of a product of curves and abelian varieties lies in this category. (Note that the H 1 of a curve is isomorphic to the H 1 of its Jacobian and hence lies in Mot Ab (K).) We refer to [73] for a beautiful study of how big this class is. Let us note, for instance, that ruled surfaces, unirational varieties and Fermat hypersurfaces are all dominated by products of curves. On the other hand, "most" motives do not lie in Mot Ab (K). As remarked by Deligne at the end of his paper [32], from the structure of the Mumford-Tate group one can sometimes see that a Hodge structure does not lie in the Tannakian subcategory of HS Q generated by all abelian varieties. (This observation lies at the basis of the results in [73].) For instance, for n ≥ 2 the motive of a very general hypersurface X ⊂ P n+1 C of degree ≥ n + 3 is known not to lie in Mot Ab (C).
On the positive side, there are some non-trivial examples of abelian motives. For instance, [4], Theorem 1.5.1, contains as particular instances the fact that the motive of a complex K3 surface, and more generally the H 2 of any complex hyperkähler variety with second Betti number B 2 > 3, lies in Mot Ab (C). We will say more about the results obtained in [4] in Section 4.4. Dependence on σ and . Let us now address the question how the conjectures stated in Section 3.2 depend on σ (orσ) and . The brief answer is that these conjectures are known to be independent of σ and for abelian motives, and that for general motives not much seems known. First let us consider the dependence on σ orσ. As we have already seen, for abelian motives this is not an issue. For more general motives, the problem that we run into is the following. Suppose we have a nonsingular projective variety Y over C and an automorphism γ of C. We may then form γ Y , the pull-back of Y via Spec(γ) : Spec(C) → Spec(C), and a morphism of schemes γ Y → Y , which however is not a morphism of schemes over C, unless γ is the identity. As explained for instance in [25], on de Rham cohomology the latter map induces a γ-linear isomorphism dR(γ) If we start with a motivated cycle α in H(Y, Q), and if we write α dR for its image in H dR (Y /C), then dR(γ) α dR is again a motivated cycle, i.e., dR(γ) α dR = β dR for a motivated cycle β on γ Y . This property just means that motivated cycles are absolute Hodge classes. (Cf. [5], Proposition 2.5.1.) Now suppose Y = X σ for X in SmPr K and σ : K → C, in which case γ Y = X τ with τ = γ • σ. If we assume that conjecture HM σ is true for X then all Hodge classes α on X σ are motivated cycles; hence, by the recipe just explained, they can be transported to motivated cycles on X τ . This gives a collection of Hodge classes on X τ of which we know they are motivated. But a priori there could be more Hodge classes on X τ , which prevents us from concluding that HM σ implies HM τ . For Conjecture MTCσ , a similar problem occurs. (If K is a number field, it is easy to see that MTCσ , only depends on σ and not on the choice of an embeddingσ extending it.) Next let us discuss the dependence of conjectures TM and MTCσ , on . For abelian motives, it is known that these conjectures (which, as discussed above, for abelian motives are equivalent) are independent of . This uses some facts that are not known for arbitrary motives. To explain what is going on, let us sketch the argument for a motive M = H 1 (X), where X is an abelian variety. There are three key ingredients. (1) We know that the system of -adic Galois representations {ρ ,X } is a strictly compatible system. (Cf. Remark 2.2.10.) By a result of Serre ([81], Section 3, or [51], Proposition 6.12) this implies that the rank of G 0 (X) is independent of . (2) As discussed above, G 0 (X) ⊂ MT(X σ ) ⊗ Q . (3) As we have seen in Theorem 3.3.2, Z ⊂ (Z σ ⊗ Q ). Now suppose the MTC is true for some . Then by (1) and (2), for every the rank of G 0 (X) equals the rank of MT(X σ ), and by an application of the Borel-de Siebenthal theorem it follows from (3) that G 0 (X) = MT(X σ ) ⊗ Q . (A modern reference for the Borel-de Siebenthal theorem is [65].) Even for motives of the form M = H n (X) with X a nonsingular projective variety (for which (1) is known), attempts to generalize this get stuck on the fact that properties (2) and (3) are not known in general. If we restrict our attention to abelian motives M then (2) and (3) are still valid but property (1) is not known in general.

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Vol.85 (2017) 28 Ben Moonen Fortunately, Commelin [27] has been able to prove, for an abelian motive, that the system of Galois representations {ρ ,M } satisfies a slightly weaker compatibility property, which suffices to conclude that the rank of G 0 (M) is independent of . (There are related results of Laskar [53].) This gives the desired conclusion for abelian motives; see [28] For -adic algebraic Galois groups and motivic Galois groups, analogous remarks can be made. This has as consequence that if we know one of the conjectures 3.2.2-3.2.4 for motives M and N, in general there is no easy way to deduce that same conjecture for M ⊕ N.
For abelian varieties, under additional hypotheses more can be said. We refer the reader to [42], [69], [54], [59] and the references contained therein, and also to [102], Theorem 1.3.7. As an example of what comes out, let us mention that the HC, TC and MTC are true for any complex abelian variety whose simple factors all have dimension ≤ 2. Commelin [27] recently proved the Mumford-Tate conjecture for any product of abelian motives of K3 type. It is hoped that his methods can be extended to handle many more product situations, assuming the MTC for the factors.

Behaviour in families 4.1. Variation of Hodge structure
Algebraic monodromy groups. Let S be an irreducible nonsingular complex algebraic variety. We denote by S an the associated complex manifold. If Q is a coefficient field of characteristic 0, let LS(S an ) Q denote the category of local systems of Q-vector spaces on S an . There is an obvious tensor product on LS(S an ) Q , making it a Q-linear neutral Tannakian category. If b ∈ S is a base point, the functor is a fibre functor. We have a monodromy representation that completely determines V . The automorphism group of the fibre functor Fib b on the Tannakian subcategory V ⊂ LS(S an ) Q generated by V is the algebraic monodromy group obtained as the Zariski closure of the image of ρ. This is an algebraic group over Q. By Tannakian theory we have an equivalence of tensor categories between V and the category Rep G mono (V /S) of (finite dimensional, algebraic) representations of G mono (V /S) over Q.
Let us also recall that LS(S an ) C is equivalent to the category MIC(S an ) of flat holomorphic vector bundles (V , ∇) on S an . (MIC is for module with integrable connection.) The latter, in turn, is equivalent to the category MIC reg (S) of algebraic flat vector bundles with regular singularities; see [30], Théorème II.5.9.
The algebraic monodromy group G mono (V /S) is not connected, in general. If f : S → S is a generically finite dominant morphism, the algebraic monodromy group of f * (V ) over S may be smaller than G mono (V /S) but the two have the same identity component. The inverse image of the identity component G 0 mono (V /S) ⊂ G mono (V /S) in π 1 (S, b) corresponds to a connectedétale cover ν : S conn (V ) → S such that the algebraic monodromy group of ν * (V ) is connected. This cover plays a role analogous to the field extension K ⊂ K conn (ρ) associated with a Galois representation ρ.  (S an , b)). Conversely, if we have an algebraic subgroup G b ⊂ GL(V ) that is normalized by G mono (V /S), it is the fibre at b of a uniquely determined local system of algebraic groups G ⊂ GL(V ). We may, for instance, view the algebraic monodromy group itself as a local system G mono (V /S) ⊂ GL(V ) whose fibre at any point s is the image of π 1 (S an , s) in its monodromy representation on V s .
The generic Mumford-Tate group of a VHS. Let us recall that, with S as above, a Q-variation of Hodge structure (abbreviated VHS) of weight n over S is given by a Q-local system V on S an together with a finite descending filtration Fil • of V ⊗ Q O S by holomorphic subbundles such that: • for every s ∈ S, the filtration Fil • s on the fibre V s ⊗ C defines a Hodge structure of weight n; • for all indices i we have ∇(Fil i ) ⊂ Ω 1 S ⊗ Fil i−1 (Griffiths transversality). We refer to [66], Chapter 10, for a much more detailed discussion. In what follows we denote by V the local system underlying the VHS V . We denote the category of Q-VHS over S by VHS(S) Q . As in the case of pointwise Hodge structures, we have Tate twists and the notion of a polarization. All variations that are of interest 286

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Vol.85 (2017) 30 Ben Moonen to us are polarizable, and the subcategory VHS pol (S) Q of polarizable variations is semisimple. There is an obvious tensor product on VHS pol (S) Q , making it a neutral Q-linear Tannakian category. The functor Fib b given by V → V b is again a fibre functor. We shall describe the algebraic group corresponding with V ⊂ VHS pol (S) Q after Theorem 4.1.3 below.
For the purpose of this paper we mostly care about variations of Hodge structure V with Q-coefficients. There are some results, however, in which it is important that V comes from a variation with Z-coefficients. For b ∈ S a base point and V = V b , this is equivalent to the condition that there is a lattice V Z ⊂ V that is stable under the action of π 1 (S an , b).
If V is a Q-VHS over S then for every s ∈ S we have a Hodge structure on the fibre V s and an associated Mumford- Tate   It is easiest to describe the exceptional locus by working in a situation where the underlying local system V is trivialized. As Mumford-Tate groups control tensor invariants, we need to consider not only V but also all tensor spaces W = T (V ) built from it. If u :S → S is the universal cover andb ∈S lies above b ∈ S, we have a trivialization u . For everys ∈S we have a Hodge structure on W × {s}. Any ξ(s) ∈ W × {s} uniquely extends to a section ξ overS. Given such ξ ∈ Γ(S, u * W ), one may consider the subset Σ(ξ) ⊂S of those pointst ∈S for which ξ(t) is a Hodge class. By construction, ξ(t) is an element of the rational vector space W × {t}, so it is a Hodge class if and only if its image in the vector bundle u * W ⊗ Q OS lies in the holomorphic subbundle Fil 0 . In this way we see that Σ(ξ) is a countable union of irreducible analytic subvarieties ofS. If we start with a Hodge class ξ(s), it may happen that Σ(ξ) = S; this is equivalent to saying that at every pointt ∈S, the class ξ(t) ∈ W × {t} that is obtained from ξ(s) by horizontal transport is again a Hodge class. One obtains the exceptional locus Exc(V ) ⊂ S as the image in S of the union of all loci Σ(ξ) (for all tensor contructions W and all ξ ∈ Γ(S, u * W )) for which Σ(ξ) is strictly contained inS. The fact that the components of the exceptional locus are in fact algebraic subvarieties of S is a deep result of Cattani, Deligne and Kaplan [24]. We now turn to the relation with the algebraic monodromy group, which, as we will see, is one of the main reasons why working with families of varieties (or motives) is so effective. The proof of part (ii) of the theorem is not hard to understand and closer inspection of the argument in fact gives something a little stronger. Namely, if Z is the centre of MT(V /S) and M 1 , . . . , M r are its Q-simple factors, ordered in such a way that we have G mono (V /S) = M t+1 · M t+2 · · · M r for some t ≥ 0, then for every s ∈ S the Mumford-Tate group of V s contains a normal subgroup isogenous to Z · M 1 · · · M t . See also the first assertion in Theorem 4.3.8 below.

Families of Galois representations
We now discuss some -adic analogues of the above results about variations of Hodge structure. Consider a nonsingular and geometrically connected variety S over a field K of characteristic 0. Fix an algebraic closure K ⊂ K and a geometric base point b of S K . We use the same notation b for the induced base points of S and of Spec(K). In what follows, π 1 (S, b) denotes the Grothendieck fundamental group.
In their barest form, the families of Galois representations that we will consider are just the lisse Q -sheaves on S that admit an integral structure. Such a sheaf V is given by a continuous representation Note that by compactness of the fundamental group, V contains a Z -lattice that is stable under the action of π 1 (S, b). We define G (V /S) ⊂ GL(V ) to be the Zariski closure of the image of ρ . For S = Spec(K) we recover the groups G (H) that we have studied in Section 2.2. We denote by V the restriction of V to S K . It is sometimes useful to view G (V /S) as an -adic local system of algebraic subgroups of GL(V ), whose fibre at any geometric points is the image of π 1 (S,s) in its monodromy representation on V ,s . (Remark 4.1.1 has an obvious -adic analogue.) For reasons to become clear (see Remark 4.2.3), we refer to G (V /S) as the generic -adic algebraic Galois group of V over S.
Galois-generic points. Recall that we have a short exact sequence with Γ K = Gal(K/K) = π 1 (Spec(K), b). We define G mono (V /S K ) ⊂ GL(V ) to be the Zariski closure of ρ π 1 (S K , b) . By construction it is a normal algebraic subgroup of G (V /S). Note that this monodromy group only depends on V /S K , so the notation is justified. It is sometimes more convenient for us to work with the corresponding local systems on S K of algebraic groups For a point s ∈ S with residue field κ(s) we can complete (4.1) to a commutative diagram Γ κ(s) in which σ s is independent of choices only up to conjugacy by an element of π 1 (S K , b). For the discussion that follows, it is important to note that Γ κ(s) → Γ K has open image. Some useful basic properties concerning this notion can be found in Section 3.2 of [20]. Apart from the fact that the generic point η of S is Galois-generic, it is not clear a priori if there are any other Galois-generic points. One can say more if the field K is hilbertian, which is the case, for instance, if K is finitely generated over Q.
(See [85], Section 9.5.) In that case, it follows from the results in [85], Section 10.6, that for d large enough there are infinitely many Galois-generic points s ∈ S with Vol. 85 (2017) Families  (Cadoret-Tamagawa). Let S be a geometrically connected nonsingular curve over a field K that is finitely generated over Q. As above, let V be a lisse Qsheaf on S that admits a Z -structure. Let g be the Lie algebra of ρ π 1 (S K , b) , and assume that g ab = 0. Then for every d ≥ 1 the set This result is Theorem 1.1 in [23]. To see this one needs the following. If K is a number field then the -adic Galois representation associated with a motive over K has the property that it is a Hodge-Tate representation at all places above ; this follows from a result of Faltings (see [46], Corollaire 2.1.3) together with the fact that any subquotient of a Hodge-Tate representation is again Hodge-Tate. By a result of Bogomolov [14], this property implies that the image of such a representation is open in the Qpoints of its Zariski closure. This extends to fields K that are finitely generated over Q by using Hilbert's irreducibility theorem, as discussed above; see again [85], Section 10.6, or [81].
For such -adic sheaves it follows that s ∈ S is Galois-generic if and only if the inclusion G 0 (V ,s ) ⊂ G 0 (V /S) is an equality. This property, together with the abundance of Galois-generic points (assuming the base variety S to be defined over a finitely generated field K) justifies calling G 0 (V /S) the generic -adic Galois group of V /S.

Families of motives
Again let S be a nonsingular geometrically connected variety over a field K of characteristic 0. Let f : X → S be a projective smooth morphism. Fixing i ≥ 0 and n ∈ Z, consider the lisse Q -sheaf H = R i f * Q (n) . If K = C we may also consider H = R i f * Q(n) , which is a variation of Hodge structure on S an . Let ξ ∈ H 0 (S K , H ) (respectively ξ ∈ H 0 (S, H )) be a global section. If b ∈ S(K) is a geometric base point, ξ may also be given as a π 1 (S K , b)-invariant class in the is a Hodge class for some s ∈ S(C) then ξ(t) is a Hodge class for every t ∈ S(C).  (S, b). This implies that the same is true at every closed point t, hence ξ(t) is again a Tate class. Part (iii), which is much deeper, is one of the main results (Théorème 0.5) of André's article [5].
Remark 4.3.2. Take K = C. If for "P" we take the stronger property of being the cohomology class of an algebraic cycle, we arrive at the Variational Hodge Conjecture (VHC) as formulated by Grothendieck in [40], footnote 13 on page 103. With ξ ∈ H 0 S, R i f * Q(n) as above, this is the assertion that if ξ(s) ∈ H i X s , Q(n) is an algebraic cycle class for some s ∈ S(C), the same is true for all ξ(t). It follows from Theorem 4.3.1(i) that the Hodge conjecture implies the Variational Hodge Conjecture, and the latter should in fact be viewed as a key (conjectural) step towards the Hodge conjecture. It is known that the VHC for abelian schemes implies the Hodge conjecture for abelian varieties; see [1], Section 6 or [5], Section 6.
Combining Theorem 4.3.1(iii) with Remark 3.1.5, we see that the Standard Conjecture B implies the VHC. See [7] for a partial extension of this to families in positive characteristic. such that for every s ∈ S (or, equivalently, some s ∈ S) the value e(s) ∈ H 2d X s × κ(s) X s , Q (d) is a projector in Corr 0 mot (X s , X s ) = A d mot (X s × κ(s) X s ), as in Section 3.1.
Let us note that, a priori, such families are more general than those considered in [5], Section 5. s ∈ S is a motive over κ(s). If the base field is C we may also realize the projector e as a global section of R 2d (f × f ) * Q X× S X (d).
Realizations. Assume the base field K is finitely generated over Q. Let M = (X, e, n) be a family of motives over S. As in Section 3.2 we can consider the Hodge realizations and -adic realizations of this family. Given a complex embedding σ : K → C, we define H σ (M) to be the variation of Hodge structure on S an σ = (S ⊗ K,σ C) an that is obtained as where f σ : X σ → S σ is the morphism obtained from f by base change. We denote the underlying local system by H σ (M). Similarly, for a prime number, let which is a lisse Q -sheaf on S. We denote by H (M) its restriction to S K . To understand the role of this local system, note that in working with a single motive over a field K, we have used the symbol H (M) for both the associated Galois representation and its underlying vector space. One should think of the Galois representation as an -adic sheaf on Spec(K); the underlying vector space is then the pull-back of this sheaf to Spec(K). In that situation, using the same notation H (M) is not likely to cause any confusion. When working over a base variety S, however, the underlying vector space gets replaced by a local system on S K , and it seems a good idea to use a special notation for this object, namely H (M). The local systems H σ (M) on S an σ and H (M) on S K may be compared once we choose an embeddingσ : K → C withσ| K = σ. To express this, take a base point b ∈ S(K). We again write b for the induced C-valued point of S σ . The morphism S σ → S K induces an isomorphism π 1 (S K , b) ∼ = π 1 (S σ , b), and the latter group is the pro-finite completion of π 1 (S an σ , b). Writing H σ = H σ (M) b and H = H (M) b , we have a comparison isomorphism Iσ , : H σ ⊗ Q ∼ −→ H . The comparison of local systems then takes the form of a commutative diagram in which c is the natural map from π 1 (S an σ , b) to its pro-finite completion, which is π 1 (S σ , b), followed by the isomorphism π 1 (S σ , b) ∼ −→ π 1 (S K , b). The following result is well-known but as it is important for the discussion, let us make it explicit. Proof. The image of c is dense in π 1 (S K , b), so G mono H (M)/S K is the Zariski closure of the image of π 1 (S an σ , b) in GL(H ). As this image is contained in GL(H σ ) Q , the assertion follows. . This of course requires some care, as the first is a local system on S an σ whereas the second is an -adic local system on S K . To compare the two, one uses that there is a morphism of topoi (ε * , ε * ) : S an σ → S σ,ét (compare [11], Section 6.1); the assertion is then that H σ (M) ⊗ Q ∼ = ε * σ * (H (M)) , where σ * (H (M)) = σ * H (M) is the pull-back of H (M) to an -adic local system on S σ = (S K )σ.
The generic motivic Galois group. The following result of André (see [5], Théorème 5.2) gives an analogue of Theorem 4.1.2 for motivic Galois groups. We again consider a family of motives M/S with S a geometrically connected nonsingular variety over a field K that is finitely generated over Q. If σ is a complex embedding of K we write MT(M σ /S σ ) for the generic Mumford-Tate group of the VHS H σ (M) on S an σ , and we view it as a locally constant subgroup     (ii) We call G mot,σ (M/S) (Betti incarnation) or G mot, (M/S) ( -adic incarnation) the generic motivic Galois group of M/S. These two are "the same": with notation as in the discussion preceding Lemma 4.3.4, the isomorphism GL(H σ )⊗Q ∼ −→ GL(H ) on fibres at a point b restricts to an isomorphism G mot,σ (M/S) b ⊗ Q ∼ −→ G mot, (M/S) b . We can also formulate the comparison in a more global way: with notation as in Remark 4.3.5, the local system of algebraic groups G mot,σ (M/S) ⊗ Q is isomorphic to ε * G mot, (M/S) .
(iii) In our formulation of the theorem we take M/S as the primary object, whereas most assertions are about either the family of motives M σ over S σ or the family M K over S K . It should be understood that the conclusions of the theorem apply to any family of motives N/T with T a nonsingular variety over an algebraically closed field of characteristic 0, as one can always find a model M/S of N/T over a finitely generated field K. Whether to view N/T as the principal object or M/S is a matter of choice. This may be compared to the two ways in which we have presented the Mumford-Tate conjecture, in 2.3.1 and 3.2.4: apart from the fact that in Section 2.3 we had not yet generalized the conjecture to motives, the two versions are different formulations of the same mathematical problem.
With M/S a family of motives as in the above discussion, the following result gives a nice connection between the various "generic loci" (Hodge, Galois, motivic) and the monodromy action. If we choose a complex embedding σ : K → C we obtain a VHS H σ (M) on S an σ and, abbreviating H σ (M) to H σ , we have local systems of algebraic groups  For the remaining assertions, the argument is the same, using parts (ii) and (iii) of Theorem 4.3.1. Note however, that for the version about G (M s ) we need to assume reductiveness for the argument to work. (i) For s ∈ S(C) we have and if G 0 (M s ) is reductive, the reverse implication is also true. Furthermore, s is motivically generic ⇐⇒ G 0 mono (H /S K ) s ⊂ G mot, (M s ) . Proof. The direct implications follow from (4.2) and (4.3). For the reverse implications we may, passing to a finite cover of S, assume the algebraic monodromy group is connected; the assertions are then immediate from the previous theorem.  family of motives. Let us sketch how one argues to deduce from André's theorem the results as we have stated them.
Over a dense open part U ⊂ S we have a family of motives in the sense considered by André. We have the local systems of algebraic groups G mono H σ (M)/S σ on S an σ . Its restriction to U σ is the algebraic monodromy group associated with the family of motives over U , and André's theorem gives us a local system of algebraic groups G U,σ = G mot,σ (M U /U ). Because the monodromy groups on U and S are the same, this extends to a local system of algebraic groups G σ = G mot,σ (M/S) ⊂ GL H σ (M) on S σ . By using that the motivic Galois group can only go down under specialization (cf. [18], Sections 4-5), we have G mot,B (M s ) ⊂ G σ,s for every s ∈ S σ (C).
To define the motivic exceptional locus and show that it has the expected properties, let us fix an embeddingσ : K → C, as André does. We proceed by adding to U a component Z of the non-singular part of S \ U ; this step will be iterated until we have reached S. We start by including in the exceptional locus on U ∪ Z the closure of the exceptional locus of U . Arguing by induction on the dimension, we may assume the theorem is true on Z. Now compare the local system of generic motivic Galois groups of M Z /Z with the local system G σ | Z . If G mot,σ (M Z /Z) is strictly contained in G σ | Z , we add the entire Z to the exceptional locus; else we only add Exc(M Z /Z). This construction gives that, for s ∈ (U ∪ Z) C , the inclusion G mot,B (M s ) ⊂ G σ,s is strict if and only if s lies in the exceptional locus.
The next point is that we can also realize the generic motivic Galois group as a locally constant system G = G mot, (M/S) ⊂ GL H (M) on S K . This we do in two steps: (a) first go from S σ for the analytic topology to S σ with theétale topology; (b) next pass to S K with theétale topology.
Step (a) is essentially the same argument as in Lemma 4.3.4. For step (b) we use that theétale fundamental group does not change if we pass from S K to S σ . (As we are in characteristic 0, this follows from the Künneth formula for fundamental groups given in [39], Exposé XIII, Proposition 4.6.) Since we know that the "pointwise" motivic Galois groups are the same (see the isomorphism γσ , just before Conjecture 3.2.2), the properties stated in part (ii) of the Theorem 4.3.6 follow.
Finally, let us justify our claim that the exceptional locus is defined over K. For this, let α be an automorphism of K/K, let s ∈ S(K) and t = α s. We have to show that if s is motivically generic, so is t. The group G mot, (M t ) is an inner form of G mot, (M s ): the latter is associated with the fibre functor on Mot(K) given by N → H (N), whereas G mot, (M t ) is isomorphic to the motivic Galois group of M s (sic!) associated with the fibre functor N → H (α * N). On the other hand, we have the inclusions G mot, (M t ) ⊂ G ,t and G mot, (M s ) ⊂ G ,s , and the algebraic groups G ,t and G ,s are (non-canonically) isomorphic. Because s is motivically generic, G mot, (M t ) has the same dimension and the same number of geometric components as G ,s ; the same is then true for its inner twist G mot, (M t ), and it follows that t is motivically generic, too.

Ben Moonen
Vol.85 (2017) 42 Ben Moonen With this notation, the following two refinements of the MTC were proposed by Serre; see [79], Conjectures C.3.7 and C.3.8. By Hodge-maximality of H B we mean that there is no non-trivial isogeny of connected Q-groups M → MT Q such that the homomorphism h : S → MT R that defines the Hodge structure on H B lifts to a homomorphism h : S → M R . It is known (see [21], Remark 2.6) that Hodge-maximality is a necessary condition in order for Im(ρ) to be open in MT(Ẑ).
For elliptic curves, Serre himself proved in [78] that both parts of the conjecture are true. (In this case we know the MTC and H B is Hodge-maximal.) Much more recently, part (i) was proven for arbitrary abelian varieties (for which the usual MTC is true) by Hindry and Ratazzi [43] and, independently, Cadoret and the author [21]. In [21], also (ii) is proven for abelian varieties. For (the H 2 of) K3 surfaces (for which, as discussed above, the MTC is true), it is shown in [21] that H 2 B is Hodge-maximal and that the image ofρ is an open subgroup of MT(Ẑ).
Dependence of Ggen on . If M/S is a family of motives then for every prime number we have, abbreviating H = H (M), a (Galois-)exceptional locus Exc(H ) in S. Conjecture 3.2.3 predicts that these loci do not depend on . If M is given by (the H 1 of) an abelian scheme over S, Hui [45] proved that this is indeed true. It seems difficult to extend Hui's proof to more general cases, as it makes essential use of Faltings's theorem 2.2.7. To the author's knowledge, even for abelian motives the -independence of Exc(H ) is not known in general. Cadoret [19] has shown that the Exc(H ) are independent of if the algebraic monodromy group of the family only has factors of Lie type A.

An example
We conclude with an application that illustrates how some of the techniques we have discussed can be used to prove results about algebraic cycles. We first prove this for d = 5. Throughout the argument, let E be the cyclotomic field Q[x]/(x 4 + x 3 + x 2 + x + 1) and write ζ 5 = exp(2πi/5). Let Σ be the set of complex embeddings of E. In our calculations we identify it with F × 5 , letting j ∈ F × 5 correspond to the embedding given by x → ζ j 5 .
Vol. 85 (2017) Families of Motives and The Mumford{Tate Conjecture 299 Families of Motives and The Mumford-Tate Conjecture 43 Let S ⊂ C[T 0 , T 1 , T 2 ] 5 be the affine subvariety of nonsingular homogeneous polynomials s of degree 5, and let π : Y → S be the smooth projective family of surfaces whose fibre at s ∈ S is given by the equation s(T 0 , T 1 , T 2 ) + T 5 3 = 0. Let α be the automorphism of Y /S given by (y 0 : y 1 : y 2 : y 3 ) → (y 0 : y 1 : y 2 : ζ 5 · y 3 ). We use the same notation α for the induced automorphisms of the fibres.
The subset Φ = {1,2} ⊂ Σ is a CM type. Consider the 2-dimensional abelian variety B of CM type (E, Φ); see Example 2.1.10. It is uniquely determined up to isogeny. Explicitly, B can be realized as the Jacobian of the genus 2 curve given by y 2 = x 5 − 1, with E-action induced by the automorphism β given by (x, y) → (ζ 5 · x, y). By construction, We use B to construct "half twists" of the Hodge structures H 2 (Y s , Q), as introduced by van Geemen in [100]. See also [101], Section 5. Our formulation of this construction is the one we introduced in [55], Section 7; see also [56], Section 3. 2 Namely, we consider the Q-Hodge structures It is important to note here that the abelian variety B stays constant, and that H 1 (B) is 1-dimensional as an E-vector space. By a Hom-tensor adjunction we find