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Shimura varieties and motives. (English) Zbl 0816.14022
Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 447-523 (1994).
Let \(k\) be an algebraically closed field of characteristic zero, and write \(\text{Mot} (k)\) for the category of (absolute Hodge or Deligne) motives over \(k\). The tensor subcategory generated by the motives \(h_ 1 (A)\) of abelian varieties \(A\) over \(k\) will be denoted \(\text{Mot}^{\text{ab}} (k)\). Its objects are called abelian motives. In particular, the Tate motive (which is isomorphic to \(\bigwedge^ 2 h_ 1(E)\) for any elliptic curve) is an abelian motive. \(\text{Mot}^{\text{ab}} (k)\) is a Tannakian category over \(\mathbb{Q}\) and Betti cohomology provides a fibre functor \(\omega_ B\) over \(\mathbb{Q}\). Let \(G_{\text{Mab}} = \operatorname{Aut}^ \otimes (\omega_ B)\). Then the functor \(M \mapsto \omega_ B (M) \otimes \mathbb{R} : \text{Mot}^{\text{ab}} (\mathbb{C}) \to \text{Hdg}_ \mathbb{R}\), where \(\text{Hdg}_ \mathbb{R}\) is the Tannakian category of real Hodge structures corresponding to representations of the group \(\mathbb{S} : = \text{Res}_{\mathbb{C}/ \mathbb{R}} \mathbb{G}_ m\), defines a homomorphism \(h_{\text{Mab}} : \mathbb{S} \to G_{\text{Mab}}\). One may also consider the Tannakian category of polarizable rational Hodge structures \(\text{Hdg}_ \mathbb{Q}\) with group \(G_{\text{Hdg}}\). Such a polarizable rational Hodge structure \((V,h)\) determines its Mumford-Tate group \((G,h)\), where \(G = \operatorname{Aut}^ \otimes (\omega)\) \((\omega\) the forgetful functor on the tensor category generated by \((V,h))\) can be identified with a subgroup of \(\text{GL} (V)\) and where \(h\) can be considered as a homomorphism \(h : \mathbb{S} \to G_ \mathbb{R}\). \((V,h)\) is said to be of CM-type if its Mumford-Tate group is commutative. Let \(\text{Hdg}_ \mathbb{Q}^{\text{cm}}\) be the corresponding Tannakian category with group \(S\). \(S\) is the so-called Serre group. It is known that every Hodge structure of CM-type is the Betti realization of an abelian motive. One has \(\text{Hdg}_ \mathbb{Q}^{\text{cm}} \hookrightarrow \text{Mot}^{\text{ab}} (\mathbb{C}) \hookrightarrow\text{Hdg}_ \mathbb{Q}\) and \(G_{\text{Hdg}} \twoheadrightarrow G_{\text{Mab}} \twoheadrightarrow S\). Of particular interest are polarizable rational Hodge structures \(h : \mathbb{S} \to G\) satisfying one or more of the following conditions:
(SV1) The Hodge structure on \({\mathfrak g} : = \text{Lie} (G)\) defined by \(\text{Ad} \circ h : \mathbb{S} \to \text{GL} ({\mathfrak g})\) is of type \(\{(1,-1), (0,0), (-1,1)\}\);
(SV2) \(\text{ad} h(i)\) is a Cartan involution of \(G^{\text{ad}}\);
(SV2*) \(\text{ad} h(i)\) is a Cartan involution of \(G/w_ h (\mathbb{G}_ m)\), where \(w_ h : \mathbb{G}_ m \to G_ \mathbb{R}\) is the weight homomorphism;
(SV3) The weight homomorphism \(w_ h : \mathbb{G}_ m \to G_ \mathbb{R}\) is defined over \(\mathbb{Q}\) and maps into the center of \(G\).
A polarizable rational Hodge structure is called special if its Mumford- Tate group satisfies (SV1). These form a Tannakian subcategory of \(\text{Hdg}_ \mathbb{Q}\). An example is provided by the Betti realization of an abelian motive. It is expected that every special Hodge structure is the Betti realization of a motive. Now let \(H\) be a semisimple group over \(\mathbb{Q}\).
\(H\) satisfies condition \((*)\) if here is an isogeny \(H' \to H\) with \(H'\) a product of simple groups \(H_ i'\) such that
(a) \(H_ i'\) is simply connected of type \(A\), \(B\), \(C\) or \(D^ \mathbb{R}\), or,
(b) \(H_ i'\) is of type \(D^ \mathbb{H}_ n\) \((n \geq 4)\) and equals \(\text{Res}_{F/ \mathbb{Q}} H_ 0\) \((F\) a totally real field) for \(H_ 0\) the double covering of an adjoint group of the form \(\text{SO} (2n)\).
A result of Deligne says that \(\omega_ B : \text{Mot}^{\text{ab}} (\mathbb{C}) \to \text{Hdg}_ \mathbb{Q}\) is fully faithful and thus defines a surjective homomorphism \(G_{\text{Hdg}} \to G_{\text{Mab}}\). The following theorem is proved: Let \(G\) be an algebraic group over \(\mathbb{Q}\), and let \(h : \mathbb{S} \to G_ \mathbb{R}\) generate \(G\) and satisfy (SV1), (SV2*) and (SV3). Then \((G,h)\) is the Mumford-Tate group of an abelian motive iff the derived group \(G^{\text{der}}\) satisfies \((*)\).
For moduli problems one is naturally led to consider ‘families’ of motives and Hodge structures over some base space. E.g. one may consider the category \(\text{Hdg}_ \mathbb{Q} (S)\) of polarizable rational Hodge structures over a connected complex manifold \(S\). This category is semisimple Tannakian. A fundamental result due to Griffiths says that for a smooth projective map of complex algebraic varieties \(\pi : Y \to S\), \(R^ i \pi_ * \mathbb{Q}\) is a polarizable variation of Hodge structures on \(S\) of weight \(i\) for any \(i\). To come to terms with the moduli problem for motives one may first try to solve the moduli problem for their Hodge structures. It turns out to be more reasonable to study polarizable Hodge structures with some additional structure such as an integral structure and a level \(N\)-structure. A first result is that, for \(N\) sufficiently divisible, there exists a complex manifold \(S(N)\) and a holomorphic family \(\mathbb{V} (N)\) of integral Hodge structes on \(S(N)\) giving a solution to a moduli problem for polarized Hodge structures (of a given weight) \(\dots\). If one member \((V, h_ 0)\) of the family is special, then \(\mathbb{V} (N)\) is a polarizable variation of Hodge structures on \(S(N)\) and \(S(N)\) has a unique algebraic structure compatible with its complex structure. Conversely, if \(\mathbb{V} (N)\) is a variation of Hodge structures, then \((V, h_ 0)\) is special.
For a variation of Hodge structures \(\mathbb{V}\) over a smooth complex variety \(S\) one may introduce the notions of algebraicity, motivicity and abelian motivicity, in increasing order of strength. E.g. \(\mathbb{V}\) is called abelian-motivic if there exist a dense open subset \(U\) of \(S\), an integer \(m\), a projective smooth morphism \(\pi : Y \to U\), where \(Y\) is an abelian scheme over \(U\), such that \(\mathbb{V} | U\) is realized by an absolute Hodge tensor as a direct summand of \({\mathcal H}_ B (Y/U) (m) : = \bigoplus R^ i \pi_ * \mathbb{Q} (m)\). One proves the following result: If \((V, h_ 0)\) is the Betti realization of an abelian motive, then the variation of Hodge structures \(\mathbb{V} (N)\) on \(S(N)\) is abelian-motivic. For a smooth connected \(k\)-variety \(S\) one introduces the notion of a motive over \(S\) and one obtains a Tannakian category \(\text{Mot} (S)\). One also defines abelian motives over \(S\) and one gets a Tannakian subcategory \(\text{Mot}^{\text{ab}} (S)\) of \(\text{Mot} (S)\). There is an exact tensor functor from \(\text{Mot} (S)\) to the Tannakian category of local systems of \(\mathbb{A}_ f\)-modules on \(S_{\acute e t}\). This makes it possible to define a moduli problem for motives. One proves that if \((V,h_ 0)\) is the Betti realization of an abelian motive, then \(S(N)\) (combined with \(H_ B)\) gives a solution of this moduli problem.
One can reverse the above process by the introduction of Shimura varieties. One starts from a reductive group \(G\) over \(\mathbb{Q}\) and a \(G(\mathbb{R})\)-conjugacy class \(X\) of homomorphisms \(h : \mathbb{S} \to G\) which satisfy the conditions (SV), and the Shimura variety \(\text{Sh} (G,X)\) can be defined, in a well-known way, as the projective limit of algebraic varieties \(\text{Sh}_ K (G,X)\), where the \(K\) are compact open subgroups of \(G(\mathbb{A}_ f)\). \(\text{Sh} (G,X)\) is said to be of abelian type if \(G^{\text{der}}\) satisfies \((*)\). One has the notion of a canonical model of \(\text{Sh} (G,X)\) over some number field. It is shown that Shimura varieties of abelian type with rational weight are moduli varieties for abelian motives. Another result of the paper can be stated: Every Shimura variety of abelian type admits a canonical model over its reflex field. Some applications are briefly discussed.
Thus, for a Shimura variety of abelian type whose weight is defined over \(\mathbb{Q}\) one may give a motivic description of its points with coordinates in any field containing the reflex field. In the last section it is shown that if one assumes the existence of a good theory of abelian motives in mixed characteristic, this description extends to the points of the Shimura variety in finite fields. This is closely related to the conjecture of Langlands and Rapaport.
For the entire collection see [Zbl 0788.00054].

14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14A20 Generalizations (algebraic spaces, stacks)
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
11F55 Other groups and their modular and automorphic forms (several variables)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)