Bertolin, Cristiana

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MOTIVIC GALOIS GROUPS OF 1-MOTIVES: A SURVEYCRISTIANA BERTOLINAbstract. We investigate the structure of the motivic Galois groups of 1-motives defined over a field of characteristic 0.In this note we review the main results of [B03] and [B04].Let k be a field of characteristic 0 and let k be its algebraic closure. Let T be aTannakian category over k. The tensor product of T allows us to define the notionof Hopf algebras in the category IndT of Ind-objects of T . The category of affinegroup T -schemes is the opposite of the category of Hopf algebras in IndT .The fundamental group pi(T ) of T is the affine group T -scheme Sp(Λ), whoseHopf algebra Λ is endowed for each object X of T with a morphism X −→ Λ⊗Xfunctorial in X, and is universal for these properties. Those morphisms {X −→Λ ⊗X}X∈T define an action of the fundamental group pi(T ) on each object of T .For each fibre functor ω of T over a k-scheme S, ωpi(T ) is the affine group S-scheme Aut⊗S (ω) which represents the functor which associates to each S-schemeT , u : T −→ S, the group of automorphisms of ⊗-functors of the functor u∗ω.If T (k) is a Tannakian category generated by motives defined over k (in an ap-propriate category of mixed realizations), the fundamental group pi(T (k)) is calledthe motivic Galois group Gmot(T (k)) of T (k) and for each embedding σ : k −→ C,the fibre functor ωσ “Hodge realization” furnishes the Q -algebraic groupωσGmot(T ) = Spec(ωσ(Λ))= Aut⊗Q (ωσ)which is the Hodge realization of the motivic Galois group of T (k).EXAMPLES:(1) From the main theorem on neutral Tannakian categories, we know that theTannakian category Vec(k) of finite dimensional k-vector spaces is equiva-lent to the category of finite-dimensional k-representations of Spec (k). Inthis case, affine group T -schemes are affine group k-schemes and pi(Vec(k))is Spec (k).(2) Let T = Repk(G) be the Tannakian category of k-representations of anaffine group k-scheme G. The affine group T -schemes are affine k-schemesendowed with an action of G and the fundamental group pi(T ) of T is Gendowed with its action on itself by inner automorphisms (see [D89] 6.3).(3) Let T0(k) be the Tannakian category of Artin motives over k, i.e. theTannakian sub-category of the Tannakian category of mixed realizationsfor absolute Hodge cycles (see [J90] I 2.1) generated by pure realizationsof 0-dimensional varieties over k. The motivic Galois group Gmot(T0(k)) ofT0(k) is the affine group Q -scheme Gal(k/k) endowed with its action onitself by inner automorphisms. We denote it by GAL(k/k). In particular,12 CRISTIANA BERTOLINfor any fibre functor ω over Spec (Q) of T0(k), the affine group schemeω(GAL(k/k)) = Aut⊗Spec (Q)(ω) is canonically isomorphic to Gal(k/k).(4) The motivic Galois group Gmot(Z(0)) of the unit object Z(0) of T0(k) is theaffine group 〈Z(0)〉⊗-scheme Sp(Z(0)). For each fibre functor “Hodge real-ization” ωσ, we have that ωσ(Gmot(Z(0))) := Spec(ωσ(Z(0))) = Spec(Q),which is the Mumford-Tate group of Tσ(Z(0)).(5) Let 〈Z(1)〉⊗ be the Tannakian category over Q defined by the k-torus Z(1).The motivic Galois group Gmot(Z(1)) of the torus Z(1) is the affine group〈Z(1)〉⊗-scheme Gm defined by the Q -scheme Gm/Q. For each fibre func-tor “Hodge realization” ωσ, we have that ωσ(Gm)= Gm/Q, which is theMumford-Tate group of Tσ(Z(1)).(6) If k is algebraically closed, the motivic Galois group of motives of CM-typeover k is the Serre group (cf. [M94] 4.8).(7) The Tannakian category T1(k) of 1-motives over k is the Tannakian sub-category of the Tannakian category of mixed realizations (for absoluteHodge cycles) generated by mixed realizations of 1-motives over k. Re-call that a 1-motive M = [X u−→G] over k consists of• a group scheme X over k, which is locally for the e´tale topology, aconstant group scheme defined by a finitely generated free Z-module,• a semi-abelian variety G defined over k, i.e. an extention of an abelianvariety A by a torus Y (1), which cocharacter group Y ,• a morphism u : X −→ G of group schemes over k.1-motives are mixed motives of level ≤ 1: the weight filtration W∗ on Mis Wi(M) = M for each i ≥ 0, W−1(M) = G,W−2(M) = Y (1), Wj(M) = 0for each j ≤ −3. If GrWn = Wn/Wn−1, we have the quotients GrW0 (M) =X,GrW−1(M) = A and GrW−2(M) = Y (1). We will denote by W−1T1(k)(resp. GrW0 T1(k), ...) the Tannakian sub-category of T1(k) generated by allW−1M (resp. GrW0 M , ...) with M a 1-motive. With this notation we caneasily compute the following motivic Galois groups• Gmot(GrW0 T1(k)) = GAL(k/k),• Gmot(GrW−2T1(k)) = GAL(k/k)×Gm.• Gmot(GrW0 T1(k)) = Gmot(T0(k)) = Sp(Z(0))• Gmot(GrW−2T1(k)) = Gm1. Motivic Galois theoryFor each Tannakian sub-category T ′ of T , let HT (T ′) be the kernel of the faith-fully flat morphism of group T -schemes I : pi(T ) −→ ipi(T ′) corresponding to theinclusion functor i : T ′ −→ T . In particular we have the short exact sequence ofgroup pi(T )-schemes0 −→ HT (T ′) −→ pi(T ) −→ ipi(T ′) −→ 0.In [D89] 6.6, Deligne proves that the Tannakian category T ′ is equivalent, as tensorcategory, to the sub-category of T generated by those objects on which the actionof pi(T ) induces a trivial action of HT (T ′). In particular, this implies that thefundamental group pi(T ′) of T ′ is isomorphic to the group T -scheme pi(T )/HT (T ′).The group T -scheme HT (T ′) characterizes the Tannakian sub-category T ′. In factMOTIVIC GALOIS GROUPS AND 1-MOTIVES 3we have a clear dictionary between Tannakian sub-categories of T and normal affinegroup sub-T -schemes of the fundamental group pi(T ) of T :Theorem 1.0.1. There is bijection between the Tannakian sub-categories of T andthe normal affine group sub-T -schemes of pi(T ), which associates- to each Tannakian sub-category T ′ of T , the kernel HT (T ′) of the morphismof T -schemes I : pi(T ) −→ ipi(T ′) corresponding to the inclusion i : T ′ −→ T ;- to each normal affine group sub-T -scheme H of pi(T ), the Tannakian sub-category T (H) of objects of T on which the action of pi(T ) induces a trivial actionof H.2. The case of motives of level ≤ 1In order to study the category T1(k) of motives of level ≤ 1, in [B04] we haveapplied the above theorem to some sub-categories of T1(k). The weight filtrationW∗ of 1-motives induces an increasing filtration W∗ of 3 steps on the motivicGalois group Gmot(T1(k)) which we describe through the action of Gmot(T1(k)) onthe generators of T1(k) : For each 1-motive M over k, we have that• W0(Gmot(T1(k))) = Gmot(T1(k))• W−1(Gmot(T1(k))) = {g ∈ Gmot(T1(k)) | (g − id)M ⊆W−1(M),(g − id)W−1(M) ⊆W−2(M), (g − id)W−2(M) = 0},• W−2(Gmot(T1(k))) = {g ∈ Gmot(T1(k)) | (g − id)M ⊆W−2(M),(g − id)W−1(M) = 0},• W−3(Gmot(T1(k))) = 0.According to the motivic analogue of [Br83] §2.2, GrW0 (Gmot(T1(k))) is a reduc-tive group sub-T1(k)-scheme of Gmot(T1(k)) and W−1(Gmot(T1(k))) is the unipotentradical of Gmot(T1(k)). Each of these 3 steps W−i(Gmot(T1(k))) (i = 0, 1, 2) can bereconstructed as intersection of group sub-T1(k)-schemes of Gmot(T1(k)) associatedto Tannakian sub-categories of T1(k) through the bijection 1.0.1:Lemma 2.0.2. (1) W−1(Gmot(T1(k))) = ∩i=−1,−2HT1(k)(GrWi T1(k)),(2) W−2(Gmot(T1(k))) = HT1(k)(W−1T1(k)) = HT1(k)(W0/W−2T1(k)).The explicit computation of these group sub-T1(k)-schemes involved in the abovelemma will provide four exact short sequences of group sub-T1(k)-schemes ofGmot(T1(k)) :Theorem 2.0.3. We have the following diagram of affine group T1(k)-schemes0 → Resk/kGmot(T1(k)) → Gmot(T1(k)) → GAL(k/k) → 0↑ || ↑0 → Resk/kHT1(k)(〈Z(1)〉⊗) → Gmot(T1(k)) → GAL(k/k)×Gm → 0↑ || ↑0 → W−1Gmot(T1(k)) → Gmot(T1(k)) → Gmot(GrW∗ T1(k)) → 0↑ || ↑0 → W−2Gmot(T1(k)) → Gmot(T1(k)) → Gmot(W−1T1(k)) → 0where all horizontal short sequences are exact and where the vertical arrows on theleft are inclusions and those on the right are surjections.4 CRISTIANA BERTOLIN3. The case of a 1-motiveLet M = [X u−→G] be a 1-motive defined over k. The motivic Galois groupGmot(M) of M is the fundamental group of the Tannakian sub-category 〈M〉⊗ ofT1(k) generated by M i.e. the affine group 〈M〉⊗-scheme Sp(Λ), where Λ is theHopf algebra of 〈M〉⊗ universal for the following property: for each object X of〈M〉⊗, there is a morphism λX : X∨ ⊗X −→ Λ functorial in X. The morphisms{λX}, which can be rewritten in the form X −→ X ⊗ Λ, define an action of thegroup Gmot(M) on each object X of 〈M〉⊗, and in particular on itself. The mainresult of [B03] is thatTheorem 3.0.4. The unipotent radical W−1(LieGmot(M)) of the Lie algebra ofGmot(M) is the semi-abelian variety defined by the adjoint action of the gradedGrW∗ (W−1LieGmot(M)) on itself.The idea of the proof is as followed: Before recall that according to [D75](10.2.14), to have M is equivalent to have the 7-uplet (X,Y ∨, A,A∗, v, v∗, ψ) where• X and Y ∨ are two group k-schemes, which are locally for the e´tale topology,constant group schemes defined by a finitely generated free Z-module;• A and A∗ are two abelian varieties defined over k, dual to each other;• v : X −→ A and v∗ : Y ∨ −→ A∗ are two morphisms of group k-schemes;• ψ is a trivialization of the pull-back (v, v∗)∗PA by (v, v∗) of the Poincare´biextension PA of (A,A∗).Observe that the 4-uplet (X,Y ∨, A,A∗) corresponds to the pure part of the 1-motive, i.e. it defines the pure motives underlying M , and the 3-uplet (v, v∗, ψ)represents the “mixity” of M .Consider the motive E = W−1(End(GrW∗ M)) : it is a split 1-motive of weight-1 and -2 obtained from the endomorphisms of the graded GrW∗ M of M . Thecomposition of endomorphisms endowed E with a Lie algebra structure (E, [ , ]),whose crochet [ , ] corresponds to a Σ−X∨ ⊗ Y (1)-torsor B living over A⊗X∨ +A∗ ⊗ Y. The action of E on the motive GrW∗ (M) is described by a morphismE ⊗GrW∗ (M) −→ GrW∗ (M)which endowed the motive GrW∗ (M) with a structure of (E, [ , ])-module.Denote by b = (b1, b2) the k-rational point b = (b1, b2) of the abelian varietyA ⊗X∨ + A∗ ⊗ Y defining the morphisms v : X −→ A and v∗ : Y ∨ −→ A∗. LetB be the smallest abelian sub-variety of X∨ ⊗ A + A∗ ⊗ Y containing this pointb = (b1, b2). The restriction i∗B of the Σ − X∨ ⊗ Y (1)-torsor B by the inclusioni : B −→ X∨ ⊗ A × A∗ ⊗ Y is a Σ − X∨ ⊗ Y (1)-torsor over B. Denote by Z1the smallest Gal(k/k)-module of X∨⊗Y such that the torus Z1(1), that it defines,contains the image of the restriction [ , ] : B⊗B −→ X∨⊗Y (1) of the Lie crochet toB ⊗B. The direct image p∗i∗B of the Σ−X∨ ⊗ Y (1)-torsor i∗B by the projectionp : X∨ ⊗ Y (1) −→ (X∨ ⊗ Y/Z1)(1) is a trivial Σ − (X∨ ⊗ Y/Z1)(1)-torsor overB. We denote by pi : p∗i∗B −→ (X∨ ⊗ Y/Z1)(1) the canonical projection. Themorphism u : X −→ G defines a point b˜ in the fibre of B over b. We denoteagain by b˜ the points of i∗B and of p∗i∗B over the point b of B. Let Z be thesmallest sub-Gal(k/k)-module of X∨ ⊗ Y, containing Z1 and such that the sub-torus (Z/Z1)(1) of (X∨ ⊗ Y/Z1)(1) contains pi(˜b). If we put Z2 = Z/Z1, we havethat Z(1) = Z1(1)× Z2(1).MOTIVIC GALOIS GROUPS AND 1-MOTIVES 5With these notations, the unipotent radical W−1(LieGmot(M)) of the Lie algebraof Gmot(M) is the extension of the abelian variety B by the torus Z(1) defined bythe adjoint action of (B + Z(1), [ , ]) on itself. Since in the construction of B andZ(1) are involved only the parameters v, v∗ and u, the computation of the unipotentradical W−1(LieGmot(M)) of the Lie algebra of Gmot(M) depends only on the 3-uplet(v, v∗, ψ), i.e. on the “mixity” of the 1-motive M .EXAMPLES:(1) Let M be the split 1-motive Z ⊕ A ⊕ Gm. In this case all is trivial:W−1(LieGmot(M)) = B = Z(1) = 0.(2) Let M = [Z u−→E ] be a 1-motive over k defined by u(1) = P with P anon-torsion k-rational point of the elliptic curve E . We have that the torusZ(1) is trivial and the unipotent radical W−1(LieGmot(M)) is the ellipticcurve B = E .(3) Let M = [Z u−→G3m×A] be a 1-motive over k defined by u(1) = (q1, q2, 1, 0)with q1, q2 two elements of Gm(k)− µ∞ multiplicatively independents(µ∞ is the group of roots of the unity in k). In this example the abelianvariety B is trivial and the unipotent radical W−1(LieGmot(M)) is the torusZ(1) = G2m.With the above notations we have also thatProposition 3.0.5. The derived group of the unipotent radical W−1(LieGmot(M))of the Lie algebra of Gmot(M) is the torus Z1(1).Proposition 3.0.6.dim LieGmot(M) = dimB + dimZ(1) + dim LieGmot(GrW∗ M).References[B03] C. Bertolin, Le radical unipotent du groupe de Galois motivique d’un 1-motif, Math. Ann.327 (2003).[B04] C. Bertolin, Motivic Galois theory for motives of level ≤ 1, submited (2004).[Br83] Brylinski J.-L., 1-motifs et formes automorphes (the´orie arithme´tique des domaines deSiegel), Publ. Math. Univ. Paris VII, 15, (1983).[D75] P. Deligne, The´orie de Hodge III, Pub. Math. de l’I.H.E.S 44 (1975).[D89] P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groupover Q, Math. Sci. Res. Inst. Pub. 16 (1989).[J90] U. Jannsen, Mixed motives and algebraic K-theory, LN 1400 (1990).[M94] J.S. Milne, Motives over finite fields, Motives, Proc. of Symp. in Pure Math. 55 (1994).D-MATH HG G 33.4, ETHZ-Zentrum, CH-8092 Zu¨richE-mail address: bertolin@math.ethz.ch

Motivic Galois groups of 1-motives: a survey

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Motivic Galois groups of 1-motives: a survey

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