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THE GAUSS–MANIN CONNECTION ON THE HODGE STRUCTURES M.ROVINSKY 1 The Gauß–Manin connection is an extra structure on the de Rham cohomology of any 0 0 algebraic variety, : H∗ Ω1 H∗ (its definition will appear below). If one 2 ∇ dR/k −→ k ⊗k dR/k believes the Hodge conjecture then for a given pure Hodge structure H there is at most n one connection such that H is a Hodge substructure of a cohomology group of a smooth a ∇ J projective complex variety with induced by the Gauß–Manin connection. Independently ∇ 8 of the Hodge conjecture, there are at most countably many connections on a given pure 1 Hodge structure H such that H is a Hodge substructure of a cohomology∇group of a smooth projective complex variety with induced by the Gauß–Manin connection (cf. Corollary ] G ∇ 2.4). A The original motivation for this paper were the properties of the forgetful functor . h graded-polarizable Hodge structures equipped with t a a connection respecting the weight filtration and Φ Hodge structures , m   −→ { } polarizations, and satisfying the Griffiths transversality [   meaning the following three questions: 2   v For a complex algebraic variety X is the Gauß–Manin connection determined by the 3 • 4 Hodge structure H∗(X(C))? 1 If yes, does there exist a functor Ψ right inverse to Φ such that for each geometric 1 • mixed Hodge structure H the pair Ψ(H) coincides with H endowed with the Gauß– 0 1 Manin connection, and Ψ(H H′) = Ψ(H) Ψ(H′)? ∼ 0 ⊗ ⊗ If the is determined by the Hodge structure in a unique way, how to express in / • ∇ ∇ h terms of the Hodge structure? (This assumes that there should be a certain supply of t a functions on Shimura varieties classifying the Hodge structures.) m In general these questions are very difficult. In this paper we consider some special cases : v of the problem. In particular, in Proposition 3.2 below we show that such a functor Ψ exists i X for the restriction of Φ to the subcategory of mixed Tate structures and it is unique. The r connection constructed there is non-integrable in general, so, if any Hodge substructure of a a geometric Hodge–Tate structure is again geometric, its integrability gives a non-trivial necessary condition for a Hodge–Tate structure to be geometric. We also compute explicitly the Gauß–Manin connection in terms of the Hodge structure for some geometric Hodge stuctures of small rank. I would like to thank P.Deligne for pointing out non-integrability of the connection constructed in Proposition 3.2. I am grateful to Andrey Levin for the inspiring discussions, to the I.H.E.S. for its hospitality and to the European Post-Doctoral Institute for its support. 1. The absolute Gauß–Manin connection: definition Let X be a smooth simplicial quasiprojective variety over a field k of characteristic zero. • For any i 0 reduction modulo the ideal generated by (i+1)-forms on the base field k gives ≥ 1 the following short exact sequences of complexes of sheaves of exterior powers of absolute Kähler differentials on X • 0 Ωi+1 Ω≥p−1 Ωi Ω≥p [ · ]i Ωi Ω≥p 0. (1) −→ k ∧ X• −→ k ∧ X• −→ k ⊗k X•/k −→ This gives rise to the sequence of homomorphisms with composition called the Gauß– ∇ Manin connection: Ωi Hq (X ) ∇ - Ωi+1 Hq (X ) k ⊗k dR/k • k ⊗k dR/k • 6 ∼= ∼= (2) ? Hq+i(Ωi Ω• )cobound-aryHq+i+1(Ωi+1 Ω•−1) [ · ]i+-1∗ Hq+i(Ωi+1 Ω•−1 ) k ⊗k X•/k k ∧ X• k ⊗k X•/k Let X be a smooth compactification of X by a divisor with normal crossings D . Then • • • the de Rham cohomology of X is identified canonically with the hypercohomology of the • simplicial logarithmic de Rham complex Ω• (logD ) on X , so one can define the Hodge X•/k • • filtration on the de Rham cohomology groups by FpHq (X ) = image Hq(X ,Ω≥p (logD )) Hq (X ) . dR/k • • X•/k • −→ dR/k • One has the following short exact(cid:16)sequences of complexes of sheaves of ex(cid:17)terior powers of absolute Kähler differentials on X which is a logarithmic version of the sequence (1): • 0 Ωi+1 Ω≥p−1(logD ) Ωi Ω≥p(logD ) [ · ]i Ωi Ω≥p (logD ) 0. (3) −→ k ∧ X• • −→ k ∧ X• • −→ k ⊗k X•/k • −→ Replacing the complex Ω• on X by the complex Ω≥p(logD ) on X in the commutative X• • X• • • diagram (2), we get the Griffiths transversality property: Ωi FpHq (X ) ∇ Ωi+1 Fp−1Hq (X ). (4) k ⊗k dR/k • −→ k ⊗k dR/k • 2. Basic properties of the Gauß–Manin connection Though the most of what follows is presumably valid for arbitrary smooth simplicial schemes, we restrict ourselves to the case of smooth proper varieties. The following properties of the Gauß–Manin connection are almost immediate. As coincides with the first differential in the Leray spectral sequence Es,t = Ωs • ∇ 1 k ⊗k Ht (X ) converging to Hs+t (X ) (associated to the filtration by powers of the ideal dR/k • dR/Q • Ω≥1 in Ω•), one has 2 = 0. k k ∇ Since each morphism f : X Y of smoothsimplicial schemes induces a morphism of • • • −→ theLerayspectralsequences forX/k andY/k, theGauß–Maninconnection isfunctorial with respect to pull-backs. It is easy to see from the Ku¨nneth decomposition Ω• = pr∗ Ω• pr∗ Ω• for a • X×Y X X ⊗OX×Y Y Y pair of smooth proper schemes X, Y, that the Gauß–Manin connection on H∗ (X dR/k ×k Y) coincides with the tensor product of the Gauß–Manin connections on H∗ (X) and dR/k H∗ (Y). dR/k Combining the latter with the functoriality with respect to pull-backs applied to the • diagonal embedding, we get the Leibniz rule for cup-products. 2 In fact, one has a stronger version of the Gauß–Manin connection. Namely, it is defined • also on the entire “conjugate spectral sequence” : Ea,b = Ha(X, b ) Ω1 ∇ 2 HdR/k −→ k ⊗k Ea,b. It comes from the composition of morphisms of sheaves: 2 b = b(Ω• ) b+1(Ω1 Ω•−1) Ω1 b , HdR/k H X/k −→ H k ∧ X −→ k ⊗k HdR/k where the first map is the coboundary in the cohomological sequence for (1) with i = p = 0 and the second is evident. This implies that the coniveau filtration is also respected by the Gauß–Manin connection. In particular, all elements of the subgroups CHq(X)/CHq (X) Eq,q are horizontal. alg ⊂ 2 The above properties together with Poincaré duality give the functoriality with respect • to Gysin maps. There is an increasing filtration W on the complex Ω• (logD ) given by limiting the • • X•/k • number of logarithmic poles, and inducing a functorial increasing weight filtration W • on the cohomology groups Hq (X ). The weight filtration is obviously respected by dR/k • the Gauß–Manin connection: (W Hq (X )) Ω1 W Hq (X ). ∇ w dR/k • ⊆ k ⊗k w dR/k • One can also state some properties of the gauge-equivalence class of the Gauß–Manin • connection using the properties of variations of Hodge structures. Proposition 2.1. If the Hodge conjecture for H2q is true then for a given pure effective Hodge structure H of weight q there is at most one connection such that H is a Hodge ∇ substructure of the q-th cohomology group of a smooth projective complex variety with ∇ induced by the Gauß–Manin connection. Proof. Suppose that H as a pure Hodge structure is isomorphic to Hodge substructures of both Hq(X) and of Hq(Y) for some smooth projective complex varieties X and Y. By Lefschetz hyperplane section theorem we may suppose that dimX = q. Then there is a morphism of Hodge structures α : Hq(X) Hq(Y) commuting with embeddings H ֒ −→ → Hq(X) and H ֒ Hq(Y). The class of α is an element in H2q(X Y)(q) of Hodge type → × (0,0), and thus, is presented by an algebraic cycle γ. By a standard argument γ induces a morphism of pairs (Hq(X), ) (Hq(Y), ). 2 X Y ∇ −→ ∇ h∇ · , · i The image Ω of the Q-linear map H H∨ Ω1, which is a Q-subspace (of H C ⊗ −−−−→ dimension (rkH)2) in Ω1, is one of basic invariants of the connection . Here H∨ is C H ≤ ∇ the Hodge structure dual to H. It follows from the compatibility of with polarizations, ∇ duality and tensor products that Ω = Ω , Ω = Ω for any integer M 1 and H H∨ H H⊗M ≥ Ω Ω +Ω . H1⊗H2 ⊆ H1 H2 If X is a complex algebraic variety and H = Hq(X) is the group of singular cohomology of ∼ X(C)withZ-coefficientsforsomeintegerq 0,onehasthedeRhamisomorphismH C ≥ ⊗ −→ H , andpolarizationsQ : grWH grWH Z( w), Q (a,b) = ( 1)wQ (b,a) foreach dR/C w w × w −→ − w − w integer w. Denote by , or simply by , the composition H ֒ H C ∼ H ∇ H dR/C ∇ ∇ → ⊗ −→ −→ Ω1 H ∼ Ω1 H. After a choice of a basis of H one can view this map as a C ⊗C dR/C −→ C ⊗ (rkH rkH)-matrix with entries in Ω1. × C The Gauß–Manin connection respects the polarizations in the sense Q (a,b) = Q ( (a),b)+Q (a, (b)). Z(−w) w w grWH w grWH ∇ ∇ w ∇ w 3 If w is even one can choose such a basis e ,...,e of grWH that the matrix of the { 1 N} w polarization is diagonal: Q (e ,e ) = (2π√ 1)wλ δ for some rational λ ’s. If Ω = (ω ) is w i j i ij i ij − the matrix of in this basis then ω = wdπ and ω = λiω for i = j. ∇grwWH ii −2 π ij −λj ji 6 If w is odd one can choose a basis e ,...,e of grWH Q where the matrix of Q is { 1 N} w ⊗ w equal to 0 I (2π√ 1)−w − · I 0 (cid:18) − (cid:19) If Ω = (Ω ) is the matrix of in this basis then Ω = Ωt , Ω = Ωt and ij 1≤i,j≤2 ∇grwWH 12 12 21 21 Ω +Ωt = wdπ I. This gives a (very rough) estimate 11 22 − π · N(N−1) if H is a pure structure of weight 0 dim Ω 2 Q H ≤ N(N−(−1)w) +1 if H is a pure structure of weight w. (cid:26) 2 We will need the following Proposition 2.2 (Katz, [K], [D1]). Let S be a pathwise connected and locally simply connected topological space and (H,F,W) is a family of mixed Hodge structures on S such that each of the families (grWH,F,W) is polarizable. Suppose that the Hodge filtration F is lo- n cally constant, i.e., comes from a filtration of the complexification H by local subsystems. C Then there exists a finite étale cover π : S′ S such that π∗(H,F,W) is a constant family of mixed Hodge structures on S′. −→ 2 Lemma 2.3 (Rigidity). Let f : X T be a morphism of complex smooth algebraic −→ varieties. Denote by f the natural morphism of topological spaces X(C) T(C). Suppose −→ that Rqf Z is a local system of isomorphic Hodge structures. ∗ Then each path γ : [0,1] T(C) gives rise to an isomorphism −→ ((Rqf Z) , ) ∼ ((Rqf Z) , ). ∗ γ(0) γ(0) ∗ γ(1) γ(1) ∇ −→ ∇ Proof. Since the Hodge filtration on the stalks of Rqf Z defines a filtration of Rqf C by ∗ ∗ local subsystems, the Proposition 2.2 implies that the local system Rqf Z becomes trivial ∗ on a finite cover of T. Without a loss of generality we replace T with such a cover, and moreover, assume T affine and connected. Then for any point s of T(C) the restriction Hq (X) rs Hq (X ) is surjective and its dR/C −→ dR/C s kernel is independent of s. Applying the functoriality of the Gauß–Manin connection to r , s one gets that ∇(kerrs) ⊂ Ω1C ⊗kerrs, and thus, ((Rqf∗Q)s,∇s) ∼= (Hq(X)/kerrs,∇X). 2 Corollary 2.4. For a given Hodge structure H there is at most a countable set of ’s such ∇ that (H, ) is a cohomology group of a smooth projective complex variety with the Gauß– ∇ Manin connection. Proof. There exists such a countable set of families of smooth projective complex varieties that each variety is isomorphic to an element of at least one of the families.1 It follows from [S], that the fibers of the period map are algebraic, and thus, we can apply Lemma 2.3 to 2 conclude the proof. 1Foreachpairofintegers1 r <N,acollectionofintegers2 d d d ,anintegers 0anda 1 2 r collectionofelementsP ,...,P≤ Sym•(Symd1QN Sym≤drQN≤)one≤de·fi·n·e≤s afamilybyhom≥ogeneous 1 s ∈ ⊕···⊕ equations of degrees d ,d ,...,d whose coefficients are zeroes of polynomials P ,...,P . 1 2 r 1 s 4 Proposition 2.5. Suppose that for a smooth proper algebraic variety X over C and an integer q 0 the Hodge filtration on Hq (X is horizontal, i.e., FjHq (X) Ω1 ≥ dR/C ) ∇ dR/C ⊆ C ⊗C FjHq (X). Then Hq(X) is the Hodge structure of the q-th cohomology group of a variety dR/C defined over Q. Proof. Choose a smooth surjective morphism X π S of varieties over Q and a generic −→ point s S(C), i.e., an embedding of fields Q(S) ֒ C, such that the fiber of X over s is 0 0 ∈ → X. Then, shrinking S, if necessary, we get from Proposition 2.2, that the variation Rqπ Z ∗ of mixed Hodge structures is a locally constant local system in étale topology. Then, by Lemma 2.3, for any point s S(Q) we have an isomorphism of mixed Hodge structures 1 equipped with connections (H∈q(X (C), ) = (Hq(X (C), ). 2 s0 ∇0 ∼ s1 ∇1 Example 2.6. For any integer m 2 those elements of Ext1 (Z,Z(m)) corresponding ≥ HS to cohomology groups of algebraic varieties, correspond, in fact, to cohomology groups of algebraic varieties defined over Q. Proof. In our case F1−m = = F0, so, by the Griffiths transversality (4), for any integer ··· 1 m p 0 one has − ≤ ≤ Fp = F0 Ω1 F−1 = Ω1 Fp. C C C C ∇ ∇ ⊆ ⊗ ⊗ 2 This means, that the assumptions of the Proposition 2.5 are the case. Proposition 2.7. If H is a geometric pure Hodge structure then (H C)∇ C coincides ⊗ ⊗Q with H′ C for a Hodge substructure H′ with horizontal Hodge filtration. ⊗ Proof. This follows from the exactness of the sequence H∗ (X) H∗ (X) ∇ Ω1 H∗ (X) dR/Q −→ dR/C −→ C ⊗k dR/C 2 shown in Proposition 4 of [EP]. Conjecture 2.8. For any geometric Hodge structure H the horizontal subspace H∇ is a Hodge substructure isomorphic to a power of Q(0). If H∇ is a Hodge substructure of weight w, there is a non-degenerate pairing H∇ ⊗ H∇ Q(w) compatible with the connections, so there are no rational horizontal elements −→ in geometric Hodge structures of non-zero weight. By Propositions 2.5 and 2.7 a horizontal Hodge structure comes from a variety defined over Q, and thus the Conjecture 2.8 is equivalent to transcendence of certain periods of varieties over Q. 3. Hodge–Tate structures 3.1. Calculation of the connection on the logarithmic structures. Consider the first relative cohomology group of G modulo 1,a for some a k. To calculate this group we m { } ∈ present G as the complement of P1 to the divisor (0)+( ) and then m ∞ H1 (G , 1,a ) = H1( ( (1) (a)) Ω1 ((0)+( ))). dR m { } OP1 − − −→ P1 ∞ We can choose a covering P1 = U U , say, with U = P1 a and U = P1 1 . 0 1 0 1 ∪ \{ } \{ } 5 1-cocycles in the Cˇech–de Rham complex are collections (f ,ω ) with i,j 0,1 and ij i ∈ { } df = ω ω , where ij i j − z 1 z a f (U ) = C − , − and ω Ω1 ((0)+( ))(U ). ij ∈ O 01 z a z 1 i ∈ P1/C ∞ i (cid:20) − − (cid:21) Note, however, that adding a coboundary, we may assume the 1-forms ω to be regular at the i points 1 and a, and therefore the 1-form df to be also regular at the points 1 and a. Since ij df is regular everywhere on the projective line, df = 0, so f is a constant and ω = ω . ij ij ij 0 1 The latter means that dz ω = ω F1H1 (G , 1,a ) = Γ(P1,Ω1 ((0)+( ))) = . 0 1 ∈ dR/C m { } P1/C ∞ z (cid:28) (cid:29)C Finally, dz H1 (G , 1,a ) = b,c b,c C , dR/C m { } z | ∈ (cid:26)(cid:18) (cid:19) (cid:27) where (1,0) denotes the 1-cocycle presented by the function 1 on U . 01 The group H1 (G , 1,a ) fits into the exact sequence dR/C m { } 0 H0 (G ) H0 ( 1,a ) H1 (G , 1,a ) H1 (G ) 0, −→ dR/C m −→ dR/C { } −→ dR/C m { } −→ dR/C m −→ where H0 ( 1,a ) = C C, the first map is diagonal and the second map is given by dR/C { } ⊕ (s,t) (s t,0). In particular, denote by e the image of (1,0), equivalently, 0 7−→ − e = (1,0) ˇ1( ( (1) (a))) ˇ0(Ω1 ( (1) (a))). 0 ∈ C OP1 − − ⊕C P1/C − − Note, that e lifts tautologically to a 1-cocycle in the first term 0 ˇ1( ( (1) (a))) ˇ0(Ω1 (log((0)+(a)+(1)+( )))( (1) (a))) C OP1 − − ⊕C P1/Q ∞ − − of the absolute Cˇech–de Rham complex, and therefore, e = 0. 0 ∇ To calculate (dz) we lift the (relative) form dz to a section η of the sheaf of absolute ∇ z z j 1-forms dz d(z/a) ( (1) (a)) Ω1 Ω1 ((0)+( )) OP1 − − ⊗OP1 P1 ⊕ z ·OU0 ⊕ z/a ·OU1 −→→ P1/C ∞ over each element U of the covering, say, η = dz, η = d(z/a). Then the coboundary of the j 0 z 1 z/a 1-cochain (η ), a 2-cocycle in j da ˇ1(Ω1 ( (1) (a))) ˇ0(Ω1 Ω1 ( (1) (a))), is ( ,0). C C ⊗C OP1 − − ⊕C C ⊗C P1/C − − a This gives the formula (dz) = da e . ∇ z a ⊗ 0 e is an integral generator of W H1(G , 1,a ). Another generator e of H1(G , 1,a ) is 0 0 m 2 m { } { } e = [ 1 dz] logae foranarbitrarychoice ofloga C. Then e = a−1da−d(loga) e dπ e . 2 2πi z − 2πi 0 ∈ ∇ 2 2πi ⊗ 0− π⊗ 2 Finally, for Hodge structure H of rank 2 and weights 0 and 2, and any element ξ of H = H1(G , 1,a ;Q) we get ∼ m { } e dπ ξ = d(πz )+ie−πiz0deπiz0 0 ξ, (5) 0 ∇ ⊗ π − π ⊗ where e = 0 is an element of W(cid:0) and ξ z e F1.(cid:1)It is easy to see that (5) is independent 0 0 0 0 6 − ∈ of e . 0 6 3.2. The Gauß–Manin connection on arbitrary Hodge–Tate structures. Lemma 3.1. 2 For any Hodge–Tate structure H inclusion maps induce the decomposition Fk (W C) ∼ H C. (6) k 2k ⊕ ∩ ⊗ −→ ⊗ Proof. Note that Fk (W C) = 0, so the subspaces Fk (W C) and Fl (W C) 2k−2 2k 2l ∩ ⊗ ∩ ⊗ ∩ ⊗ intersect triviallywhenk = l, andthecanonicalprojectionϕ : Fk (W C) grW Cis 6 k ∩ 2k⊗ −→ 2k⊗ an isomorphism. Comparison of dimensions of both sides of (6) ensures it is an isomorphism. 2 As a consequence of this Lemma, we get a new Q-structure (2πi)kϕ−1(grW) (7) ⊕k k 2k for complexification of arbitrary mixed Tate structure. Let H be an abelian category of Hodge–Tate structures containing all logarithmic structures, invariant under Tate twists and containing each Hodge substructure of each its object. Consider the category whose objects are objects of H equipped with a connection satisfying the Griffiths transversality and morphisms are morphisms of Hodge structures commuting with connections. Proposition 3.2. There is a unique functor H (H, ) right inverse to the for- H • 7−→ ∇ getful functor objects of H equipped with a connection H satisfying the Griffiths transversality −→ (cid:26) (cid:27) such that for a logarithmic structure H the connection coincides with the Gauß– H ∇ Manin connection calculated above, and induces the same connection on H C = H ∇ ⊗ H(1) C as . H(1) ⊗ ∇ The above connection on a Hodge–Tate structure H is integrable if and only if the above • connections on W /W (k 3) are integrable for all integer k. 2k 2k−6 − Proof. It follows from the functoriality, applied to the morphism W H, that k −→ ∇ respects the weight filtration, and therefore, from the calculation for the logarithmic structures, that the connection on the structure Z(0) is zero. This implies that : W (k) 2k ∇ −→ Ω1 W . Combining these with the Griffiths transversality : Fk Ω1 Fk−1, we C ⊗ 2k−2 ∇ −→ C ⊗ get (2πi)kϕ−1(grW) ∇ Ω1 (Fk−1 (W ) ). k 2k −→ C ⊗C ∩ 2k−2 C It is clear fromthe decomposition (6) that it suffices to construct the latter maps, and that is integrable if and only if the restrictions of 2 to Fk (W ) vanish for all integer k. As 2k C ∇ ∇ Fk−2 (W ) = 0, the vanishing of the latter restrictions is equivalent to the vanishing 2k−6 C T of the induced maps Fk (W ) Ω2 (W /W ). T 2k C −→ C ⊗ 2k 2k−6 BythefunctorialityandcompatibilitywiththeTatetwists, toconstruct thatmapforsome T k wemayidentify thespaceFk (W ) withthespaceF1H′ (W H′) , andFk−1 (W ) 2k C 2 C 2k−2 C ∩ ∩ ∩ with F0H′ (W H′) , where H′ = (W /W )(k 1). In fact, we may suppose that there 0 C 2k 2k−4 ∩ − is an exact sequence 0 Z(0)s H′ Z( 1) 0 −→ −→ −→ − −→ 2This is a very particular case of Deligne’s construction of a functorial splitting of Hodge structures (see, e.g., [BZ], Definition and Proposition 2.6). 7 for some non-negative integer s. Then H′ can be identified with a Hodge substructure of a 2 direct sum of logarithmic Hodge structures, where we have fixed the connection. Remarks. 1. It is easy to see that the connection constructed in Proposition 3.2 on the tensor product of two Hodge–Tate structures coincides with the tensor product of the connections on that Hodge–Tate structures. 2. It follows from Section 3.1 that for a Hodge–Tate structure H of rank 3 with weights 0, 2, 4 the connection is integrable if and only if eπiz0 and eπiz2 are algebraically dependent, H ∇ where z and z are determined by the conditions e W 0 , e W W , e W W , 0 2 0 0 2 2 0 4 4 2 ∈ \{ } ∈ \ ∈ \ e z e F1 and e z e F2(W /W ) . This implies that for each element a C× 2 0 0 4 2 2 4 0 C − · ∈ − · ∈ ∈ there is a natural embedding C×/(Q(a))× ֒ Ext2(Z(0),Z(2)), where Ext2 is calculated in → the category of flat Hodge–Tate structures. 4. Examples: some Hodge structures of rank 3 ≤ Up to the Tate twists and the duality the options for a Hodge structure of rank 3 are: ≤ it is Hodge–Tate; • it is pure of weight 0, or 1; • − it is an extension of Q(0) by a pure structure of negative weight. • We have calculated the “Gauß–Manin” connection for Hodge–Tate structures in the pre- vious section, so we eliminate them in what follows. 4.1. Pure Hodge structures of rank 2. For any square-free positive integer D denote by E an elliptic curve with complex multiplication in Q(√ D). Since 2H1(E ) = Q( 1), D D one has a surjection ndH1(E ) ∼ H1(E ) H1(E )(−1) Sym2∧H1(E )(1) with−the D D D D E −→ ⊗ −→→ kernel Q id. The classes of endomorphisms of the curve generate a Hodge substructure in · ndH1(E ) isomorphic to Q(0) Q(0), so its image in Sym2H1(E )(1) is isomorphic to D D E ⊕ Q(0). Let α Sym2H1(E )(1) be the element corresponding to 1 Q(0). D ∈ ∈ Denote by Mk the cokernel of the injection Symk−2H1(E )( 1) ·α SymkH1(E ). This D D − −→ D is a geometric Hodge structure of weight k, rank 2 and with Hodge numbers h0,k = hk,0 = 1. Alternatively,onecandefineMk(k)asaHodgesubstructure inH (E )⊗k asfollows. Letτ D 1 D beacomplexmultiplicationofE ,τ theinducedendomorphismofthefirsthomologygroup, D ∗ and γ a non-trivial element of H (E ). Set γ = τ γ , γ(k) = (γ τγ )⊗k +(γ τγ )⊗k 1 1 D 2 ∗ 1 1 2 − 1 2 − 1 and γ(k) = (τ τ)−1 (γ τγ )⊗k (γ τγ )⊗k . Then γ(k) and γ(k) are integral elements, 2 − 2 − 1 − 2 − 1 1 2 generating a Hodge substructure Mk(k) with induced polarization (cid:0) D (cid:1) 1+( 1)k 1 ( 1)k γ(k),γ(k) = (1+( 1)k)ck, γ(k),γ(k) = − ck, γ(k),γ(k) = − − ck, h 1 1 i − h 2 2 i − (τ τ)2 h 1 2 i τ τ − − where c = (τ τ) γ ,γ . In particular, when k is even the polarization is of type x2+Dy2. 1 2 − h i Claim 4.1. If γ(k),γ(k) is a basis of Mk(k) with γ(k) +(τ τ)γ(k) F0 then { 1 2 } D 1 − 2 ∈ ∇γ1(k) = dπ γ(k) +(τ τ) dπ w −D dΓ(n/m) γ(k); k 2π ⊗ 1 − π − 4h n∈(Z/m)× n Γ(n/m) ⊗ 2 ∇γ2(k) = 1 dπ w (cid:16) −PD dΓ(n/m) (cid:0) γ(cid:1)(k) + dπ (cid:17) γ(k), k τ−τ π − 4h n∈(Z/m)× n Γ(n/m) ⊗ 1 2π ⊗ 2 8 (cid:16) (cid:17) P (cid:0) (cid:1) where w = 4 if D = 1, w = 6 if D = 3, and w = 2 otherwise; h is the number of classes of ideals in the ring of integers in Q(√ D); and − D, if D 1 mod 4 m = ≡ 4D, otherwise. (cid:26) Proof. It will follow from Example 4.2 that 2πi γ = (ω dη η dω ) γ + (η dω 1 2 1 2 1 1 1 1 ∇ − ⊗ − ω dη ) γ and 2πi γ = (ω dη η dω ) γ +(η dω ω dη ) γ . Rewriting the latter 1 1 2 2 2 2 2 2 1 1 2 1 2 2 ⊗ ∇ − ⊗ − ⊗ using γ = τ γ as 2πi γ = 2πi(id τ ) γ = ττ(η dω ω dη ) γ + (...) γ , we 2 ∗ 1 2 ∗ 1 1 1 1 1 1 2 ∇ ⊗ ∇ − ⊗ ⊗ get η dω ω dη = τ(η dω ω dη ), or equivalently, d(η τη ) = (η τη )dω1. From 2 1 − 2 1 1 1 − 1 1 2 − 1 2 − 1 ω1 the Legendre identity ω η ω η = 2πi and ω = τω we get d(η τη ) = d 2πi . As a 1 2 − 2 1 2 1 2 − 1 ω1 corollary of these formulas, 2πi (γ τγ ) = ω d 2πi (γ τγ ); and 2πi ((cid:16)γ (cid:17)τγ ) = ∇ 2− 1 1 ω1 ⊗ 2− 1 ∇ 2− 1 2πidω1 (γ τγ ). (cid:16) (cid:17) ω1 ⊗ 2 − 1 Finally, γ(k) dπ dω1 dπ (τ τ) γ(k) 1 = k 2π ω1 − 2π − 1 ∇ γ(k) !  1 dω1 dπ (cid:16) dπ(cid:17)  γ(k) ! 2 τ−τ ω1 − 2π 2π 2  (cid:16) (cid:17)  Due to the Chowla–Selberg formula (cf., e.g., [W]) ω is an algebraic multiple of 1 π3/2 m−1Γ(n/m)−wχ(n)/4h, where w is the number of roots of unity in Q(√ D), h is the n=1 − number of classes of ideals in the ring of integers in Q(√ D), m is the discriminant of Q − Q(√ D) and χ(n) = −m is the Jacobi symbol. 2 − n Now, a pure polarized Hodge structure of rank 2 of even weight w is a triplet of a lattice • (cid:0) (cid:1) H of rank 2 with a symmetric definite Q( w)-valued bilinear form on it, and an integer − k > w/2. Then the Hodge filtration is given by H = Fw−k Fk = Fw−k+1 Fk+1 = C ⊃ ⊃ 0, where Fk is the isotropic line, on which the corresponding hermitian form is positive, so for a fixed triplet of integers (k > w/2,D > 0) there is at most one isogeny class of polarized Hodge structures of rank 2, weight w with polarization of discriminant D. On the other hand, the Hodge structure M2k(k w/2) gives the example. D − To determine a polarized Hodge structure of rank 2 and odd weight w means to fix a • lattice H of rank 2 with an isomorphism 2H ∼ Z( w), an integer k > w/2 and a ∧ −→ − line Fk H , on which the corresponding hermitian form is positive. Then the Hodge C ⊂ filtration is given by H = Fw−k Fk = Fw−k+1 Fk+1 = 0. C ⊃ ⊃ If k > w+1 then the Griffiths transversality condition and Proposition 2.5 imply that 2 such geometric Hodge structures correspond to varieties defined over number fields. At least some of the examples of them are constructed in [D2]. In particular, the Hodge structure M2k−1(k w+1) gives the example. D − 2 Tensoring with Z(w+1), we reduce the case k = w+1 to the case of a Hodge structure • 2 2 H of rank 2 with Hodge numbers h0,−1 = h−1,0 = 1. Then we have an embedding H ֒ H /F0. Denote the one-dimensional C-space H /F0 by L and by Ls its s-th C C → tensor power if s > 0 and the dual of L−s otherwise. Let ζ = ζ : L H L−1 be H − −→ the meromorphic function given by 1 1 1 z ζ (z) = + + + = z−1 z2n−1 λ−2n . H z z λ λ λ2 −   λ∈H\{0}(cid:18) − (cid:19) n≥2 λ∈H\{0} X X X 9   Set A = 15 λ−4 L−4 and B = 35 λ−6 L−6. − ∈ − ∈ λ∈H\{0} λ∈H\{0} X X The Hodge structure H is naturally isomorphic to H (E(C)) for the elliptic curve 1 E(C) = L/H. Our nearest aim is to obtain an algebraic equation (8) of E, and therefore, express a basis (11) of the de Rham cohomology of E in terms of H. This enables us to calculate the connection in (12) and in Example 4.2. Set X = ℘ (z) = 1 + 1 1 = z−2 1Az2 1Bz4 +O(z6) L−2, H z2 λ∈H\{0} (z−λ)2 − λ2 − 5 − 7 ∈ Y = 1℘′ (z) = 1ζ′′(z) = 1 +(cid:16) (cid:17)1 = z−3 + 1Az + 2Bz3 +O(z5) L−3. −2 H 2 HP z3 λ∈H\{0} (z−λ)3 5 7 ∈ Since any holomorphic elliptPic function is constant, we land on the well-known relation Y2 = X3 +AX +B. (8) Fix a twelve-th root ∆ L−1 of 4A3 27B2 and define the complex numbers ∈ − − a = A/∆4 and b = B/∆6, so 4a3 +27b2 = 1. Set x = X/∆2 and y = Y/∆3. − Also, any connection on L−2 induces conections on L−4 and L−6. We set ∇ da db 2A B 3B A κ = = = ∇ − ∇ Ω1 . (9) 18b −4a2 2∆10 ∈ C/Q Starting with elementary identities in Ω1 C(x,y)/C d 1 = (3x2+a)dx; d x = (2b+ax−x3)dx; d x2 = (x4+3ax2+4bx)dx, (10) − y 2y3 y 2y3 y 2y3 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) and fixing the following differentials of the second kind on E dx xdx ω = = dz and ϕ = = dζ(z), (11) 2y − 2y we get then the following congruences modulo exact forms (x3 +ax+b)dx (2ax+3b)dx (x4 +ax2 +bx)dx (2a2 9bx)dx ω = ; ϕ = − . 2y3 ≡ 2y3 2y3 ≡ 6y3 This information is enough to find the Gauß–Manin connection: dy2 dx (xda+db) dx 4a2 dx 18b xdx ω = ∧ = ∧ = κ · − · . ∇ − 4y3 − 4y3 ∧ 4y3 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) So we get ω = 3κ [ϕ]; and similarly, ϕ = aκ [ω]. (12) ∇ ⊗ ∇ ⊗ Then from the identities ω,γ + ω, γ = d ω,γ and ϕ,γ + ϕ, γ = d ϕ,γ , h∇ i h ∇ i h i h∇ i h ∇ i h i we get ω, γ = d ω,γ 3 ϕ,γ κ and ϕ, γ = d ϕ,γ a ω,γ κ. h ∇ i h i− h i h ∇ i h i− h i Fix such a basis γ ,γ of the lattice H L that the imaginary part of γ /γ is 1 2 2 1 { } ⊂ positive. Let ω = ∆ γ and ω = ∆ γ be the complex numbers corresponding to γ 1 1 2 2 1 · · and γ , respectively, and η = ∆−1 2ζ (γ /2). 2 j H j · Using the Legendre relation ω η ω η = 2πi, one easily verifies that the following 2 1 1 2 − holds. 10