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Conjectural properties of motivic Galois groups and \(\ell\)-adic representations. (Propriétés conjecturales des groupes de Galois motiviques et des représentations \(\ell\)-adiques.) (French) Zbl 0812.14002
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. 1, 377-400 (1994).
Let \(k\) be a field of characteristic zero, embeddable into the complex numbers by \(\sigma : k \hookrightarrow \mathbb{C}\). Also, let \(\overline \sigma : \overline k \hookrightarrow \mathbb{C}\) extend \(\sigma\) to an algebraic closure of \(k\). Assume Grothendieck’s standard conjectures and the Hodge conjecture hold. Write \({\mathcal M} = {\mathcal M}_ k\) for the category of Grothendieck motives. For such a motive \(E\) write \({\mathcal M} (E)\) for the smallest Tannakian subcategory of \({\mathcal M}\) that contains \(E\). For two motives \(E\) and \(E'\) one has a relation of domination: \(E' \prec E\) when \(E'\) is contained in \({\mathcal M} (E)\). The category \({\mathcal M}\) can now be considered as the inductive limit of the \({\mathcal M} (E)\). \({\mathcal M}\) admits a fibre functor, viz. the Betti realization \(H_ \sigma : {\mathcal M} \to \text{Vect}_ \mathbb Q\), where \(\text{Vect}_ \mathbb{Q}\) denotes the category of finite dimensional \(\mathbb Q\)-vector spaces. The automorphism group scheme \(\operatorname{Aut}^ \otimes (H_ \sigma)\) is called the motivic Galois group, also written \(G\) or \(G_{\mathcal M}\). Similarly, one has \(G_{\mathcal M} (E)\) for a motive \(E\). A priori \(G\) depends on the embedding \(\sigma\), but for two embeddings the corresponding motivic Galois groups are inner forms of each other. The Tannakian formalism says that \({\mathcal M}\) should be equivalent to \(\text{Rep}_ \mathbb Q (G)\), the category of finite dimensional \(\mathbb Q\)-representations of \(G\). As \({\mathcal M}\) will be semisimple \(G\) is proreductive.
Several (conjectural) properties of \(G\) and/or \(G_{{\mathcal M} (E)}\) for a motive \(E\) are discussed. First, \(G_{{\mathcal M} (E)}\) can be characterized in several ways: (i) by invariant tensors; (ii) by \(\ell\)- adic representations; (iii) in terms of Mumford-Tate groups. E.g. in (iii) one is led to conjecture that the component of the identity \(G^ 0_{{\mathcal M} (E)}\) of \(G_{{\mathcal M} (E)}\) is the Mumford-Tate group associated with the Hodge structure on \(H_ \sigma (E)\). As a matter of fact, most assertions concern the identity components of \(G_{{\mathcal M} (E)}\) and \(G_{\mathcal M}\). One has an exact sequence \(1 \to G^ 0_{\mathcal M} \to G_{\mathcal M} \to \text{Gal} (\overline k/k) \to 1\). One has \(G^ 0_{{\mathcal M},k} = G_{{\mathcal M}, \overline k}\). Writing \(C\) for the identity component of the center of \(G^ 0_{\mathcal M}\) and \(D\) for the derived group of \(G^ 0_{\mathcal M}\) one should have \(G^ 0_{\mathcal M} = C \cdot D\). \(C\) is a pro-torus and \(D\) is a pro-semi-simple group. Let \(S : = (G^ 0_{\mathcal M})^{\text{ab}} = G^ 0_{\mathcal M} / D = C / (C \cap D)\). \(S\) is sometimes called the Serre group. For \(k = \mathbb Q\) the quotient \(T : = G_{\mathcal M}/D\) is called the Taniyama group. The corresponding Tannakian category is the category of motives potentially of CM-type. One has an exact sequence \(1 \to S \to T \to \text{Gal} (\overline{\mathbb Q}/ \mathbb Q) \to 1\). For the character group \(X(S)\) or \(S\) one has a description in terms of locally constant functions \(f : \text{Gal} (\overline{\mathbb Q}/ \mathbb Q) \to \mathbb Z\) satisfying two conditions related to complex conjugation. For the group of characters \(X(C)\) of \(C\) one obtains a similar description where now the locally constant functions take their values in \(\mathbb Q\). The dual \(X(C \cap D)\) of the group \(C \cap D\) admits a description in terms of locally constant functions \(f : \text{Gal} (\overline{\mathbb Q}/ \mathbb Q) \to \mathbb Q/ \mathbb Z\). As a matter of fact, the groups \(S\), \(C\) and \(C \cap D\) do not depend on \(k\). This is not true for \(D\) which remains rather mysterious. It is not known whether \(D\) is simply connected. In general, one does not known which semi-simple (or even connected reductive) groups can occur as \(G_{{\mathcal M} (E)}\). It is expected that, for any polarizable Hodge structure \(H\) with Mumford-Tate group \(G\) such that \(\text{Lie} (G)\) is of type \(\{(1,-1), (0,0), (-1,1)\}\), there exists a motive over \(\mathbb C\) with Betti realization equal to \(H\). One can ask whether there exists a motive \(E\) such that \(G_{{\mathcal M} (E)}\) is simple of type \(G_ 2\) (or \(E_ 8)\).
From now on \(k\) will be of finite type over \(\mathbb Q\), eventually \(k\) will be a number field. Let \(\ell\) be a prime number. Then, for a motive \(E\) over \(k\), its \(\ell\)-adic cohomology \(V_ \ell(E)\) (identified with \(\mathbb Q_ \ell \otimes H_ \sigma (E))\) defines an \(\ell\)-adic representation (associated with \(E)\), \(\rho_{\ell,E} : \text{Gal} (\overline k/k) \to G_{{\mathcal M} (E)} (\mathbb Q_ \ell)\). It is conjectured that \(\text{Im} (\rho_{\ell,E})\) is open in \(G_{{\mathcal M} (E)} (\mathbb Q_ \ell)\), and furthermore, it should meet every connected component of \(G_{{\mathcal M} (E)}\). This would be equivalent to the semi- simplicity of \(V_ \ell (E)\) as \(\text{Gal} (\overline k/k)\)-module, and the Zariski density of \(\text{Im} (\rho_{\ell,E})\) in \(G_{{\mathcal M} (E)/ \mathbb Q_ \ell}\). Letting \(\ell\) run over the set \(P\) of all prime numbers, one gets a homomorphism \(\rho_ E : \text{Gal} (\overline k/k) \to \prod_{\ell \in P} G_{\ell,E}\), where \(G_{\ell,E} = \text{Im} (\rho_{\ell, E}) \subset G_{{\mathcal M} (E)} (\mathbb Q_ \ell)\). The \(\rho_{\ell,E}\) are said to be independent over \(k\) if \(\rho_ E\) is surjective. One conjectures the existence of a finite field extension \(k'\) of \(k\) (depending on \(E)\) such that the \(\rho_{\ell,E}\) are independent over \(k'\). For \(\ell \in P\) one defines a \(\mathbb Z_ \ell\)- lattice \(L_ \ell = \mathbb Z_ \ell \otimes L\) in \(V_ \ell (E)\), where \(L\) is a lattice in \(H_ \sigma (E)\). \(L\) is called a \(\mathbb Z\)-form of \(E\) if for all \(\ell\) the lattices \(L_ \ell\) are stable under the \(\text{Gal} (\overline k/k)\)-action via \(\rho_{\ell,E}\). It is conjectured that, modulo the action of \(\operatorname{Aut} (E)\), there are only finitely many \(\mathbb Z\)-forms for \(E\). When \(E\) is the motive attached to an abelian variety this was proved by Faltings: The number of abelian varieties defined over \(k\) which are \(k\)-isogenous to a given abelian variety \(A\) is finite. Several conjectures for the \(G_{\ell,E}\) are stated in terms of \(\mathbb Z\)-forms for \(E\) and similarly for the reduction mod \(\ell\) of \(\rho_{\ell,E}\). Still another conjecture on \(\text{Im} (\rho_{\ell, E})\) can be stated in terms of ultraproducts. \(\rho_ E\) can also be considered as an adelic representation, \(\rho_ E : \text{Gal} (\overline{k}/k) \to G_{{\mathcal M} (E)} (\mathbb A^ f)\), and one may ask whether its image is open. It is conjectured, at least when \(G_{{\mathcal M} (E)}\) is connected, that this is equivalent to the fact that \(E\) is maximal, where \(E\) is called maximal if \(\text{Ker} (G^ 0_{\mathcal M} \to G_{{\mathcal M} (E)})\) is connected. A somewhat stronger (conjecturally, equally strong) notion of maximality is \(H\)-maximality: \(E\) is called \(H\)-maximal if the Hodge homomorphism \(h_ E : \mathbb G_ m \times \mathbb G_ m \to G_{{\mathcal M} (E)/ \mathbb C}\) cannot be lifted to any nontrivial connected covering \(G' \to G_{{\mathcal M} (E)}\). One should have that, for \(H\)-maximal \(E\), \(\text{Im} (\rho_ E)\) is open in \(G_{{\mathcal M} (E)} (\mathbb A^ f)\).
Let \(k\) be a number field. One can introduce the notions of ramification (of \(\rho_{\ell,E})\) and good reduction (of \(E\)) at a (non-archimedean place \(v\) of \(k)\). For an extension \(w\) to \(\overline k\) of \(v\) one also defines an arithmetic Frobenius element \(F_{\ell,E,w}\), where it is assumed that \(E\) has good reduction at \(v\) and \(\ell \neq p_ v\), the residue characteristic. \(F_{\ell,E,w}\) is conjectured to act semi- simply on \(V_ \ell (E)\). Also, its characteristic polynomial is conjectured to have \(\mathbb Q\)-coefficients, independent of \(\ell\) \((\ell \neq p_ v)\), and for \(E\) pure of weight \(i\), its eigenvalues have absolute value \(Nv^{-i/2}\). The Frobenius element \(F_{\ell,E,w}\) defines an element, also written \(F_{\ell,E,w}\), in the variety of conjugacy classes of \(G_{{\mathcal M} (E)} (\mathbb Q_ \ell)\) and this is conjectured to be rational over \(\mathbb Q\), and independent of \(\ell \). The ramified places are briefly considered.
The final section deals with the equidistribution of the Frobenius elements. For \(E\) with good reduction at \(v\), one may consider the Frobenius class \(F_{E,v} \in G_{{\mathcal M} (E)} (\mathbb C)\) of \(v\). Normalizing by the weight of \(Nv^{1/2}\) one obtains an element \(\phi_{E,v} \in G^ 1_{{\mathcal M} (E)} (\mathbb C) : = \text{Ker} ({\mathbf t} : G_{\mathcal M} \to \mathbb G_ m)\), where \(\mathbb G_ m @>{\mathbf w}>> G_{\mathcal M} @>{\mathbf t}>> \mathbb G_ m\), \({\mathbf t} \circ {\mathbf w} = - 2\), is the basic sequence which says that the Tate motive has weight \(- 2\). For a maximal compact subgroup \(K \subset G_{{\mathcal M} (E)} (\mathbb C)\), the conjugacy class of \(\phi_{E,v}\) will lie in \(\text{Cl} (K)\), the space of conjugacy classes of \(K\). The Sato-Tate conjecture says that the classes of the \(\phi_{E,v}\) are equidistributed in \(\text{Cl} (K)\) for the normalized Haar measure of \(K\). Another conjecture is the following: For any (nontrivial) irreducible complex linear representation \(r\) of \(G^ 1_{{\mathcal M} (E)}\) one has \(\sum_{Nv \leq X} \text{Tr} (\phi_{E,v}) = \text{o}(X/ \log X)\) for \(X \to \infty\). In terms of \(L\)-functions one would have: The function \(L_ r(s) = \prod_{v\, \text{good}} {1 \over \det (1 - r(\phi_{E,v}) Nv^{-s}}\) can be meromorphically continued to the whole complex plane with a pole of order at most 1, and \(L_ r(s)\) is holomorphic and \(\neq 0\) on \(\text{Re} (s) = 1\).
Throughout the paper a series of examples clarifies some of the main ideas. These examples are concerned with the trivial motive, (sums of powers of) the Tate motive, the motive associated with an abelian variety, and the less well-known Ramanujan motive.
For the entire collection see [Zbl 0788.00053].

14A20 Generalizations (algebraic spaces, stacks)
11E72 Galois cohomology of linear algebraic groups
14D07 Variation of Hodge structures (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry