aa r X i v : . [ m a t h . N T ] A p r Higher Green’s functions for modular forms
Anton Mellit October 30, 2018 [email protected] bstract Higher Green functions are real-valued functions of two variables on the upper halfplane which are bi-invariant under the action of a congruence subgroup, have log-arithmic singularity along the diagonal, but instead of the usual equation ∆ f = 0we have equation ∆ f = k (1 − k ) f . Here k is a positive integer. Properties of thesefunctions are related to the space of modular forms of weight 2 k . In the case whenthere are no cusp forms of weight 2 k it was conjectured that the values of the Greenfunction at points of complex multiplication are algebraic multiples of logarithms ofalgebraic numbers. We show that this conjecture can be proved in any particularcase if one constructs a family of elements of certain higher Chow groups on thepower of a family of elliptic curves. These families have to satisfy certain properties.A different family of elements of Higher Chow groups is needed for a different pointof complex multiplication. We give an example of such families, thereby proving theconjecture for the case when the group is P SL ( Z ), k = 2 and one of the argumentsis i. ntroduction The subject of the present thesis is higher Green’s functions. For any integer k > ⊂ P SL ( Z ) of finite index there is a unique function G H / Γ k on theproduct of the upper half plane H by itself which satisfies the following conditions:(i) G H / Γ k is a smooth function on H × H \ { ( τ, γτ ) | γ ∈ Γ , τ ∈ H } with values in R .(ii) G H / Γ k ( γ τ , γ τ ) = G H / Γ k ( τ , τ ) for all γ , γ ∈ Γ.(iii) ∆ i G H / Γ k = k (1 − k ) G H / Γ k , where ∆ i is the hyperbolic Laplacian with respect tothe i -th variable, i = 1 , G H / Γ k ( τ , τ ) = m log | τ − τ | + O (1) when τ tends to τ ( m is the order ofthe stabilizer of τ , which is almost always 1).(v) G H / Γ k ( τ , τ ) tends to 0 when τ tends to a cusp.This function is called the Green function. It is necessarily symmetric, G H / Γ k ( τ , τ ) = G H / Γ k ( τ , τ ) . Such functions were introduced in paper [GZ86]. Also it was conjectured in[GZ86] and [GKZ87] that these functions have “algebraic” values at CM points. Aparticularly simple formulation of the conjecture is in the case when there are nocusp forms of weight 2 k for the group Γ: Conjecture (1) . Suppose there are no cusp forms of weight 2 k for Γ. Then for anytwo CM points τ , τ of discriminants D , D there is an algebraic number α suchthat G H / Γ k ( τ , τ ) = ( D D ) − k log α. The main result of this thesis is a general approach for proving this conjectureand an actual proof for the case Γ =
P SL ( Z ), k = 2, τ = i . The number α in thelatter case is represented as the intersection number of a certain higher Chow cycleon the elliptic curve corresponding to the point τ and an ordinary algebraic cyclewhich has a certain prescribed cohomology class.The general approach can be formulated as follows. For an elliptic curve E weconsider the higher Chow group CH k ( E k − ,
1) on the product of E by itself 2 k − We use “i” to denote √− i ” for other purposes. X i ( W i , f i ) , where W i is a subvariety of E k − of codimension k − f i is a non-zero rationalfunction on W i such that the following condition holds in the group of cycles ofcodimension k : X i div f i = 0 . Denote the abelian group of higher cycles by Z k ( E k − , CH k ( E k − ,
1) isits quotient (the relations defining CH k ( E k − ,
1) will be explained in Section 2.3).We have the Abel-Jacobi map AJ k, : CH k ( E k − , −→ H k − ( E k − , C ) F k H k − ( E k − , C ) + 2 π i H k − ( E k − , Z ) , and there is a canonical cohomology class [ θ ] ∈ H ( E , C ), namely the one repre-sented by the form θ = ω ⊗ ¯ ω + ¯ ω ⊗ ω R E ω ∧ ¯ ω , where ω is a holomorphic differential 1-form on E . On may notice that for anyelement [ x ] ∈ CH k ( E k − ,
1) there is a perfectly defined number ℜ (cid:16) i k − ( AJ k, [ x ] , [ θ ] k − ) (cid:17) . So we hope that for those cases for which the conjecture above is formulated, for afixed CM point τ , there exists a family of higher cycles { x s } s ∈ S , x s ∈ CH k ( E k − s , { E s } s ∈ S ( S is an algebraic variety over C ) which“computes” the values of the Green function in the way described above. Thisapproach appeared from an attempt to understand the discussion of the algebraicityconjecture in [Zha97].To be more specific I introduce the following 3 classes of functions.Let A be a subgroup of C (we will usually take A = Z , A = Q or A = i k − R ).Consider a holomorphic multi-valued function f on the upper half plane which isallowed to have isolated singularities and is defined up to addition of polynomials of τ with coefficients in 2 π i A of degree not greater than 2 k −
2. This means that fora small disk U which does not contain the singularities of f one has an element of O an ( U ) / π i V A k − , where V A k − is the abelian group of polynomials with coefficientsin A of degree not greater than 2 k −
2. Suppose f transforms like a modular formof weight 2 − k with respect to Γ, i.e. f ≡ f | − k γ mod 2 π i V A k − for any γ ∈ Γ.Denote the abelian group of all such functions by M A − k (Γ). Note that if we putsome growth condition on f near the singularities, it will imply that the singularityof f at a point τ will be of the form F ( τ ) log( τ − τ ) for F ∈ V A k − .2imilarly, let M A ,k (Γ) be the space of multi-valued functions f with isolatedsingularities satisfying ∆ f = k (1 − k ) f , almost holomorphic (expressible as poly-nomials of τ − ¯ τ with holomorphic coefficients), invariant (in weight 0) with re-spect to Γ, defined up to addition of functions of the form P k − i =0 α i r i ( τ ), where α i ∈ π i i !(2 k − − i )!( k − A and the r i are defined by the equation k − X i =0 r i ( τ ) X i = (cid:18) ( X − τ )( X − ¯ τ ) τ − ¯ τ (cid:19) k − . Finally, let M A k (Γ) be the space of (single-valued) holomorphic functions f withisolated singularities which transform like modular forms of weight 2 k with respectto Γ and the V C k − -valued differential form ω f = f ( τ )( X − τ ) k − dτ satisfies thefollowing two conditions:(i) If we integrate ω f around a singularity we obtain an element of the abeliangroup 2 π i(2 k − V A k − .(ii) The cohomology class [ ω f ] ∈ H (Γ , V C / π i(2 k − A k − ) of ω f is trivial.Then there is a commutative diagram: M A − k (Γ) δ k − / / ( ddτ ) k − % % LLLLLLLLLL M A ,k (Γ) δ k (cid:15) (cid:15) M A k (Γ)Here the horizontal arrow is the operator δ − · · · δ − k and the vertical one is δ k − · · · δ ( δ w = ddτ + wτ − ¯ τ ). Also the horizontal arrow is an isomorphism and the vertical oneis surjective with kernel the finite group ( V k − /V A k − ) Γ . In particular if A is Q or R , then the horizontal arrow is also an isomorphism. When A = R we also havea fourth group in our system. Denote by C ω ,k (Γ) the space of real-analytic (single-valued) functions with isolated singularities satisfying the same differential equationas for the space M A ,k (Γ) and invariant (in weight 0) with respect to the action of Γ.Then the following diagram is commutative and all its arrows are isomorphisms: M i k − R − k (Γ) δ k − / / ( ddτ ) k − (cid:15) (cid:15) M i k − R ,k (Γ) ℜ ( · ) (cid:15) (cid:15) M i k − R k (Γ) C ω ,k (Γ) δ k o o Now we give two ways to construct elements of the spaces above. The first waytakes the Green function as input. Fix a point τ ∈ H . The function G H / Γ k ( τ, τ )belongs to the space C ω ,k (Γ). Therefore our construction produces elements in thespaces M i k − R − k (Γ), M i k − R ,k (Γ), M i k − R k (Γ). Denote by g τ = g H / Γ τ ,k the correspondingelement in M i k − R k (Γ). We prove that g τ is meromorphic, zero at the cusps, and its3rincipal part at τ is m ( − k − ( k − Q τ ( τ ) − k (we denote Q τ ( X ) = ( X − τ )( X − ¯ τ ) τ − ¯ τ ).The integral around τ of ω g τ = g τ ( τ )( X − τ ) k − dτ is 2 π i (2 k − k − Q τ ( X ) k − . There-fore the product ( k − D k − g τ satisfies all the requirements of M Z k (Γ) except,possibly, the last one. The last requirement is automatically satisfied in the case S k (Γ) = { } , at least up to torsion, i.e. there exists an integer N (depending onlyon k and Γ) such that N D k − g τ ∈ M Z k (Γ). The lift of N D k − g τ to M Z − k (Γ)and M Z ,k (Γ) is defined up to a finite group, therefore its product with certain integernumber N (which depends only on Γ and k ) is perfectly defined. Let N = N N . Wesee that the construction canonically gives a function b G H / Γ k,τ ∈ N D − k M Z ,k (Γ) suchthat δ k b G H / Γ k,τ = g τ . If, moreover, τ is a CM point of discriminant D , then b G H / Γ k,τ ( τ ) isdefined up to 2 π i N ( D D ) − k Z . This is called “the lifted value of the Green function”and we conjecture that it equals N ( D D ) − k log α for some α ∈ Q × . We empha-size that the Green function produces elements of spaces M i k − R − k (Γ), M i k − R ,k (Γ), M i k − R k (Γ), and only if τ is a CM point we can “lift” these elements to spaces M Z − k (Γ), M Z ,k (Γ), M Z k (Γ).There is another way to obtain functions as above, which always produces ele-ments of spaces M Z − k (Γ), M Z ,k (Γ), M Z k (Γ). Suppose we have a family of ellipticcurves { E s } s ∈ S over an algebraic variety S defined over C and an algebraic family ofhigher cycles x = { x s ∈ Z k ( E k − s , } s ∈ S . Let x s = P i ( W i,s , f i,s ) (the dimension of W i,s is k − P i div f i,s = 0). Suppose also a map ϕ : S → H / Γ is given whichis dominant, i.e. its image is H / Γ without a finite number of points, and supposethat ϕ makes the following diagram commutative ( j is the j -invariant): S ϕ / / j % % JJJJJJJJJJJ H / Γ (cid:15) (cid:15) H /P SL ( Z ) . Definition.
A triple X = ( { E s } s ∈ S , { x s } s ∈ S , ϕ ) as above is called modular if for anytwo points s , s such that ϕ ( s ) = ϕ ( s ) there exists an isomorphism ρ : E s → E s with the following properties:(i) ρ k − ( W i,s ) = W i,s ,(ii) ρ k − ( f i,s ) = β i f i,s for β i ∈ C × .If we have a modular triple then we construct an element of M Z − k (Γ) as follows.Let τ ∈ H be such that its projection to H / Γ belongs to the image of ϕ , say τ = ϕ ( s ).Choose a differential form ω on E s and suppose its period lattice is generated by Ω and Ω with Ω Ω = τ . Then put A X ( τ ) = 1Ω k − h AJ k, [ x s ] , [ ω ⊗ k − ] i . In this case the Abel-Jacobi map reduces to the following integral: A X ( τ ) = 2 π i 1Ω k − Z ξ ω ⊗ k − , ξ is a smooth 2 k − P i f ∗ i,s [0 , ∞ ] and [0 , ∞ ] is apath from 0 to ∞ on C P . It is clear that A X ( τ ) thus defined does not depend onthe choice of s with ϕ ( s ) = τ , and A X ∈ M Z − k (Γ). We will show that δ k − A X ( τ ) = (2 k − k − (cid:16)R E s ω ∧ ω (cid:17) k − h AJ k, [ x ] , [ ω ⊗ k − sym ⊗ ω ⊗ k − ] i . If τ is a CM point with minimal equation aτ + bτ + c = 0, D = b − ac and ϕ ( s ) = τ , then there are endomorphisms of E s which act on the tangent space as themultiplications by aτ and a ¯ τ . Denote their graphs by Y aτ and Y a ¯ τ correspondingly.Then one can check that the Poincar´e dual cohomology class of the difference Y aτ − Y a ¯ τ is represented by the form − √ D ω sym ⊗ ω R Es ω ∧ ω . Therefore we can construct a varietywhose Poincar´e dual class is represented by( − k − D k − (2 k − k − (cid:16)R E s ω ∧ ω (cid:17) k − ω ⊗ k − sym ⊗ ω ⊗ k − , namely, we take the product of k − Y aτ − Y a ¯ τ for each possible splittingof the product E k − s into pairs and add them up. Note that there are precisely (2 k − k − k − such splittings. Denote the variety obtained in this way by Z s . Then weobtain δ k − A X ( τ ) = ( − k − D − k log ( x s · Z s ) , where the intersection number x s · Z s is defined as x s · Z s = Y i Y p ∈ W i,s ∩ Z s f i,s ( p ) ord p W i,s · Z s if Z s does not intersect the divisors of f i,s , singularities of W i,s and intersects W i,s properly. If this is not the case one can still define the intersection number, forexample, by “shifting” Z s using the the addition law.The second construction gives examples of functions in M Z ,k whose values at CMpoints are “algebraic”. We prove that for such A X , obtained via the second con-struction, the derivative (cid:0) ddτ (cid:1) k − A X ( τ ) is meromorphic. In fact we give a methodto compute this derivative purely algebraically. Definition.
A meromorphic modular form f which belongs to M Z ,k is called geomet-rically representable if f = (cid:0) ddτ (cid:1) k − A X ( τ ) for some modular triple ( { E s } s ∈ S , { x s } s ∈ S , ϕ )with { E s } , ϕ and { x s } are defined over Q .Then we have Lemma.
Suppose Γ is a congruence subgroup, k > and f ∈ M Z ,k (Γ) is such that ( − k − N δ k f , for some integer N , is geometrically representable by a modulartriple ( { E s } s ∈ S , { x s } s ∈ S , ϕ ) . Suppose τ ∈ H is a CM point which belongs to theimage of ϕ . Let D be the discriminant of τ . Then D k − f ( τ ) ≡ N log α mod π i N ,where α is an algebraic number, which can be computed as the intersection x s · Z s asabove with ϕ ( s ) = τ . Here N is the exponent of the group H (Γ , V Q / Z k − ) , N = N N .
5n particular, if a multiple of D k − g k,τ is geometrically representable, then theconjecture is true for τ = τ and τ — any other CM point except, possibly, a finitenumber of points (those which do not belong to the image of ϕ ). We give an examplewhen this occurs. Let Γ = P SL ( Z ), k = 2. Let S be the open subset of C × C of pairs ( a, b ) satisfying 4 a + 27 b = 0 and b = 0. Let { E s } s ∈ S be the Weierstrassfamily, which is defined by the projective version of the equation y = x + ax + b .Let W s be the subvariety of E s × E s which consists of ( x , y , x , y ) with x + x = 0.Let f s be the function y − i y . It is easy to check that ( W s , f s ) ∈ Z ( E s × E s , ϕ be the j -invariant. One can verify that this gives a modular triple. Denoteit by X − (since the discriminant of the point τ = i, which is the only point whichdoes not belong to the image of ϕ , is − Theorem.
Consider the following modular form of weight : − √− δ G H /P SL ( Z )2 ( τ, i) = (2 π i) √− E ( τ ) j ( τ ) − . This form is geometrically representable by the modular triple X − . This implies the conjecture for G H /P SL ( Z )2 ( τ, i) (keep in mind that G = 2 ℜ b G ),namely Corollary.
For any CM point τ which is not equivalent to i one has √− D b G H /P SL ( Z )2 ( τ, i) ≡ W s , f s ) · ( Y Aτ − Y A ¯ τ )) mod π i Z , where s = ( a, b ) , the curve y = x + ax + b corresponds to τ , Y Aτ and Y A ¯ τ are thegraphs of the endomorphisms of this curve which act on the tangent space as Aτ and A ¯ τ , Aτ + Bτ + C is the minimal equation of τ with A > , and D = B − AC . As an example we verify
Corollary. G H /P SL ( Z )2 (cid:18) − √− , i (cid:19) = 8 √ − √ . The text is organized in five chapters. Each chapter is provided with a moredetailed introduction and reading the introduction is highly recommended for un-derstanding the chapter, especially in the case of Chapter 3. We briefly discusscontents of each chapter here. The first chapter studies various functions on theupper half plane and differential operators. The main result of this chapter is thelifting of the values of the Green function at CM points from real numbers to theelements of C / π i( D D ) − k Q , and, related to this, the refined version of the algebraicity conjecture (see Corol-lary 1.5.6 and Conjecture (2)). Also in this chapter we prove that the k -th non-holomorphic derivative of G H / Γ k is a meromorphic modular form characterized by6ertain properties (Theorem 1.5.3) and discuss ways to compute the Green functionfrom this modular form (see the discussion in the end of Section 1.5). The resultsof this chapter where known to experts but do not exist in print.The second chapter contains a definition of the higher Chow groups and a con-struction of the Abel-Jacobi map as in [GL99] and [GL00], as well as a proof thatthe definition of the Abel-Jacobi map is correct. We prove a formula (Theorem2.5.1) which in certain cases relates the value of the Abel-Jacobi map with certainintersection number which takes values in the multiplicative group.The third chapter provides a way to compute the derivative of the Abel-Jacobimap (by this we mean (cid:0) ddτ (cid:1) k − A X ( τ )) for a family of higher cycles. The answer isexpressed as a certain extension of D -modules. The result is more general than weneed and works for any families of products of curves. In the case of the power of afamily of elliptic curves we obtain an invariant of this extension which for modulartriples coincides with (cid:0) ddτ (cid:1) k − A X ( τ ) and is a meromorphic modular form.In the fourth chapter we study various objects on the Weierstrass family of ellipticcurves. We provide representatives for cohomology classes which are necessary forlater computations.The fifth chapter contains the main result, namely, the construction of a familyof higher cycles which computes the values of the function G H / Γ2 ( τ, i) and the proofof the algebraicity conjecture in this case (Theorem 5.2.1). There we show that themeromorphic modular form (2 π i) √− E ( τ ) j ( τ ) − , which equals − √− δ G H / Γ2 ( τ, i),is geometrically representable.The construction of the family of higher cycles used in the proof was inspired byother constructions of higher cycles on products of elliptic curves in [GL99], [GL00].It seems, however, that our family (see Section 5.1.3 and the construction of X − above) was not known before, though its construction is surprisingly simple.This text is submitted as a PhD thesis to the University of Bonn. The workwas done at the Max-Planck-Institute for Mathematics. The author is grateful tothe institute for hospitality and good working atmosphere. Also I wish to thank S.Bloch, J. Bruinier, N. Durov, A. Goncharov, G. Harder, D. Huybrechts, C. Kaiser,Yu. Manin, R. Sreekantan with whom I had interesting discussions on the subjectof this thesis. Special thanks to M. Vlasenko who read drafts of this text and madeuseful remarks, and to D. Zagier who introduced me to number theory, proposed theproblem, provided me with inspiration and support, and was a very good supervisor.7 ontents SL ( R ) . . . . . . . . . . . . . . . . . . . 111.1.2 Differential operators . . . . . . . . . . . . . . . . . . . . . . . 121.1.3 CM points and quadratic forms . . . . . . . . . . . . . . . . . 141.2 Eigenfunctions of the Laplacian . . . . . . . . . . . . . . . . . . . . . 141.3 Integrating modular forms . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Local study of Green’s functions . . . . . . . . . . . . . . . . . . . . 221.5 Global study of Green’s functions . . . . . . . . . . . . . . . . . . . . 25 D -modules . . . . . . . . . . . . . . . . . . . . 773.2.5 Products of elliptic curves . . . . . . . . . . . . . . . . . . . . 803.2.6 Analytic computations . . . . . . . . . . . . . . . . . . . . . . 81 ω ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.3 The class [ η ] . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4.4 Gauss-Manin derivatives . . . . . . . . . . . . . . . . . . . . . 964.4.5 Computation of the Poincar´e pairing . . . . . . . . . . . . . . 96 θ . . . . . . . . . . . . . . . . . . . . . . . . 1055.2.4 The hyperform θ . . . . . . . . . . . . . . . . . . . . . . . . 1065.2.5 The extension of D -modules . . . . . . . . . . . . . . . . . . . 1065.2.6 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.7 Special values of the Green function . . . . . . . . . . . . . . 1095.3 The torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 hapter 1 Modular forms
In this chapter we first fix some notation for the representations of SL ( R ) onthe space of functions on the upper half plane. We obtain a representation for eachinteger w which is called “weight”. We introduce three differential operators on theserepresentations which are intertwining: the non-holomorphic derivative δ or the“raising operator”, the “lowering operator” δ − and the Laplacian ∆. We also recallthe construction of the standard finite-dimensional representations V m of SL ( R )and vectors in these representations which correspond to complex multiplication(CM) points. This is done in Section 1.1.Next, in Section 1.2, we study in more detail functions which are eigenfunctionsfor the Laplacian with eigenvalue of the form k (1 − k ), k ∈ Z . The situation withthese particular eigenvalues is special. In particular, to each eigenfunction f withsuch eigenvalue corresponds a certain “extended function” e f which is harmonic, buttakes values in the space V k − . The derivatives of these extended functions areproportional to holomorphic functions of weight 2 k which we also call “derivatives”.In Section 1.3 we study the inverse problem of recovering e f from its derivativesin the case when they are invariant under the action of a congruence subgroup of P SL ( Z ). We obtain that the obstructions to solving this problem lie in certaincohomology groups.Then, in Section 1.4, we define the Green functions G H k for the upper half plane(without a congruence subgroup), the so-called “local Green functions”. These aredefined as functions of two variables z , z in the upper half plane which are SL ( R )-invariant (diagonal action) eigenfunctions for the Laplacian with eigenvalue k (1 − k ) and have only a logarithmic singularity on the diagonal. We show how thesefunctions can be explicitly written using Legendre’s functions. We also evaluate theactions of powers of the operator δ on them. We obtain particularly nice expressionsfor the k -th power which also gives formulae for the derivatives of the correspondingextended Green functions.Section 1.5 studies the “global Green functions” G H / Γ k for a quotient of the upperhalf plane. Since these Green functions can be obtained by averaging the local ones,results of Section 1.4 give us information about the singularities of the global Greenfunctions and their derivatives. We obtain a characterisation of the derivative g H / Γ k,z ofthe extended global Green function as a meromorphic modular form with described10ype of poles and having trivial class in a certain Eichler-Shimura cohomology group.Then the original Green function can be obtained from the modular form using theintegration procedure described in Section 1.3.We suppose that there are no cusp forms of weight 2 k for the group Γ. Thenbeing applied to CM points the integration procedure allows us to lift the value ofthe Green function, which is real, to a complex number defined up to an algebraicmultiple of 2 π i (see Corollary 1.5.6). This supports and refines the algebraicityconjecture formulated in papers [GZ86] and [GKZ87], which says that in this casethe value must be an algebraic multiple of the logarithm of an algebraic number (seeConjectures (1) and (2)). In the end of the section we discuss ways to compute thelifted Green function, and we show that it is a period (Theorem 1.5.7). We will consider the group SL ( R ). Elements of this group will be usually denotedby γ , and matrix elements by a , b , c , d : γ = (cid:18) a bc d (cid:19) ∈ SL ( R ) . This groups acts on the upper half plane H . SL ( R ) The group SL ( R ) naturally acts on the following linear spaces: • V = C — the space of column vectors of length 2, • V m — the symmetric m -th power of V .We explain the way we view elements of V m . Let e , e be the natural basison V . Then V m is the space of homogeneous polynomials in e , e of degree m . Ifwe substitute e by a new variable X and e by 1 we obtain a non-homogeneouspolynomial in one variable of degree less or equal m . We represent elements of V m as polynomials in the variable X of degree less or equal m , i.e. p ∈ V m , p ( X ) = p + p X + · · · + p m X m . The group acts on V m on the right by( pγ )( X ) = ( p | − m γ )( X ) = p ( γX )( cX + d ) m , γ = (cid:18) a bc d (cid:19) ∈ SL ( R ) . There is the corresponding action on the left( γp )( X ) = ( pγ − )( X ) = p ( γ − X )( − cX + a ) m . Now consider the dual space to V m , V ∗ m . Any v ∈ V ∗ m is a functional on V m .Suppose its value on p is v p + v p + . . . v m p m , v as a raw vector ( v , v , v , . . . , v m ). For any two numbers x, y ∈ C we form a vector v x,y = ( y m , y m − x, y m − x , . . . , x m ) ∈ V m . Then for any p ∈ V m ( v x,y , p ) = p (cid:18) xy (cid:19) y m =: p ( x, y ) , hence γv x,y = v γ ( x,y ) = v ax + by,cx + dy . In this way the action of SL ( R ) is given on a subset of V ∗ m which spans V ∗ m .The isomorphism between V m and V ∗ m can be given as follows. Define an invariantpairing between elements of the form v x,y ∈ V m :( v x,y , v x ′ ,y ′ ) = ( xy ′ − x ′ y ) m = m X i =0 ( − i (cid:18) mi (cid:19) x m − i y i x ′ i y ′ m − i . Since it is a homogeneous polynomial of degree m in each pair of variables, thisinduces an equivariant linear map from V ∗ m to V m ( v , v , . . . , v m ) m X i =0 ( − i (cid:18) mi (cid:19) v m − i X i . This is an isomorphism of representations. It induces invariant pairings on V m m X i =0 p i X i , m X i =0 p ′ i X i ! = m X i =0 ( − i p i p ′ m − i (cid:0) mi (cid:1) and on V ∗ m (( v i ) , ( v ′ i )) = m X i =0 ( − i (cid:18) mi (cid:19) v i v ′ m − i . The pairings above are symmetric for m even and antisymmetric for m odd. Thefollowing identities hold:(( z − X ) m , p ) = p ( z ) = ( p, ( X − z ) m ) ( p ∈ V m , z ∈ C ) . Let S be a discrete subset of the upper half plane H and f ( z ) be a function on H − S with values in C and w ∈ Z . Define differential operators δ w f = ∂f∂z + wz − ¯ z f,δ − w f = ( z − ¯ z ) ∂f∂ ¯ z , ∆ w f = ( z − ¯ z ) ∂∂z ∂∂ ¯ z f + w ( z − ¯ z ) ∂∂ ¯ z .
12e think about w as the weight, attached to the function f . The weight willbe always clear from context, so we will omit the subscript w . We will follow thefollowing agreement: the operator δ increases weight by 2, the operator δ − decreasesweight by 2 and the operator ∆ leaves weight untouched. Taking into account thisagreement the following identities can be proved: δ − δ − δ δ − = w,δ δ − = ∆ ,δ − δ = ∆ + w. Let the group SL ( R ) act on functions of weight w by the usual formula:( f | w γ )( z ) = f ( γz )( cz + d ) − w . This is a right action. We also define the corresponding left action( γf )( z ) = ( f | w γ − )( z ) = f ( γ − z )( − cz + a ) − w . Note that this action commutes with the operators δ , δ − , ∆, it maps functionsdefined on H − S to functions defined on H − γS .It is also convenient to modify the complex conjugation for functions with weightto make it commuting with the action of the group. For f of weight w we put f ∗ ( z ) = ( z − ¯ z ) w f ( z ) . Assign to f ∗ weight − w . We can check that f ∗∗ = ( − w f,δ ( f ∗ ) = ( δ − f ) ∗ ,δ − ( f ∗ ) = ( δf ) ∗ , ∆( f ∗ ) = ((∆ + w ) f ) ∗ ,γ ( f ∗ ) = ( γf ) ∗ . We remark that for the weight 0 the operator ∗ is the usual complex conjugation,the operator δ is the usual ∂∂z , and the operator ∆ is the usual Laplace operator forthe hyperbolic metric − y ( ∂ ∂x + ∂ ∂y ).We list several formulae, which are convenient to use in computations. Weassume that the constant function 1 has weight 0. Consider the functions X − z, X − ¯ zz − ¯ z , which are thought as functions in z with values in V of weights − δ δ − , ( X − z ) ∗ = X − ¯ zz − ¯ z , (cid:18) X − ¯ zz − ¯ z (cid:19) ∗ = − ( X − z ) ,δ ( X − z ) = − X − ¯ zz − ¯ z , δ X − ¯ zz − ¯ z = 0 ,δ − ( X − z ) = 0 , δ − X − ¯ zz − ¯ z = X − z,δ ( X − z ) a (cid:18) X − ¯ zz − ¯ z (cid:19) b ! = − a ( X − z ) a − (cid:18) X − ¯ zz − ¯ z (cid:19) b +1 ,δ − ( X − z ) a (cid:18) X − ¯ zz − ¯ z (cid:19) b ! = b ( X − z ) a +1 (cid:18) X − ¯ zz − ¯ z (cid:19) b − , ∆ ( X − z ) a (cid:18) X − ¯ zz − ¯ z (cid:19) b ! = − b ( a + 1)( X − z ) a (cid:18) X − ¯ zz − ¯ z (cid:19) b . We will frequently use the following notation: Q z = ( X − z )( X − ¯ z ) z − ¯ z . As a function of z Q z is a function with values in V of weight 0.A CM point is a point z ∈ H which satisfies a quadratic equation of degree 2with integer coefficients. Let z be a CM point. Write the minimal equation for z : az + bz + c = 0 , a, b, c ∈ Z , a > . The discriminant of the minimal equation D z = b − ac is called the discriminantof z . An elementary computation gives Q z = aX + bX + c √ D .
It is clear that a point z ∈ H is a CM point if and only if Q z is proportional toa polynomial with integer coefficients. We consider functions on H − S for discrete subsets S ⊂ H . For integers k, w wedenote by F k,w the space of functions of weight w satisfying∆ f = ( k (1 − k ) + w ( w − ) f. It is easy to check the following properties of the spaces F k,w :14 roposition 1.2.1. (i) The space F k,w is invariant under the action of the group SL ( R ) (meaning, of course, that γ ∈ SL ( R ) changes S to γS ).(ii) The operator ∗ maps F k,w to F k, − w .(iii) The operator δ maps F k,w to F k,w +2 . It is invertible for all values of w , except,possibly, k − and − k with the inverse given by δ − w = 4( w + 2 k )( w − k + 2) δ − w +2 . (iv) The operator δ − maps F k,w to F k,w − . It is invertible for all values of w ,except, possibly, k and − k with the inverse given by ( δ − w ) − = 4( w − k )( w + 2 k − δ w − . We will occasionally use negative powers of δ and δ − when they can be definedusing this proposition.Next we state some basic facts Proposition 1.2.2.
For integer numbers k , l we have ( X − z ) k − l − (cid:18) X − ¯ zz − ¯ z (cid:19) k + l − ∈ F k, l ⊗ V k − . Proof.
Use the formulae listed in the end of Section 1.1.2.
Proposition 1.2.3.
For f , g in F k, and − k ≤ l ≤ k − we have δ − l f δ l g = ( δ − ) l f ( δ − ) − l g. Proof.
Note that ( δ − ) l f = ( − l ( k + l − k − l − δ − l f, and analogously δ l g = ( − l ( k + l − k − l − δ − ) − l g, which implies the required identity.Let us introduce the following operation. For two functions f , g from F k, weput f ∗ g = k − X l =1 − k δ − l f δ l g, which has weight 0. Note that the previous proposition implies f ∗ g = k − X l =1 − k ( − l ( δ − ) − l f ( δ − ) l g, f ∗ g = f ∗ g. It is also easy to see that ∂∂z ( f ∗ g ) = ( − k − ( δ − k f δ k g + δ k f δ − k g ) . Consider the function Q k − z . By Proposition 1.2.2 Q k − z ∈ F k, ⊗ V k − as a function of z .One can compute that for 1 − k ≤ l ≤ k − δ l Q k − z = ( − l ( k − X − z ) k − − l ( k − l − (cid:18) X − ¯ zz − ¯ z (cid:19) k − l (1 − k ≤ l ≤ k − ,δ k Q k − z = 0 . For f ∈ F k, we denote e f = ( − k − (cid:18) k − k − (cid:19) f ∗ Q k − z = ( − k − (2 k − k − k − X l =1 − k ( X − z ) k + l − ( k + l − (cid:18) X − ¯ zz − ¯ z (cid:19) k − l − δ l f. It is easy to check that f γf = γ e f for γ ∈ SL ( R ),where γ acts on e f by the simultaneous action on z in weight 0 and X in weight2 − k . Also e f = ( − k − e f . One can compute the scalar product of δ i Q k − z and δ j Q k − z as follows: Lemma 1.2.4. ( δ i Q k − z , δ j Q k − z ) = ( , if i = − j ( − k − − i (cid:0) k − k − (cid:1) − if i = − j .Proof. We prove by induction on i starting from 1 − k . Since δ − k Q k − z = ( − k − ( k − k − X − z ) k − , for any polynomial p ∈ V k − we have( δ − k Q k − z , p ) = ( − k − ( k − k − p ( z ) . δ − k Q k − z , δ j Q k − z ) is not zero only for j = k − δ − k Q k − z , δ k − Q k − z ) = (cid:18) k − k − (cid:19) − . If the statement is true for i then for any j , taking into account that the weight of( δ i Q k − z , δ j Q k − z ) is 2 i + 2 j ,0 = δ ( δ i Q k − z , δ j Q k − z ) = ( δ i +1 Q k − z , δ j Q k − z ) + ( δ i Q k − z , δ j +1 Q k − z ) . Hence ( δ i +1 Q k − z , δ j Q k − z ) = − ( δ i Q k − z , δ j +1 Q k − z ) . We see that if i + j = −
1, this is zero. If i + j = −
1, this equals exactly − ( − k − − i (cid:18) k − k − (cid:19) − . Therefore the original function f can be recovered from e f as f = ( e f , Q k − z ) . Note also that ∆ e f = 0. This is true because δ − δ k f = (∆ + 2 k − δ k − f = 0 . Let us summarize.
Theorem 1.2.5.
Let f ∈ F k, for k ≥ . Then the function e f satisfies the followingproperties (note that F , is the space of harmonic functions): e f ∈ F , ⊗ V k − ,f = ( e f , Q k − z ) ,δ l f = ( e f , δ l Q k − z ) ,∂ e f∂z = ( X − z ) k − ( − k − δ k f ( k − ,∂ e f∂ ¯ z = ( X − ¯ z ) k − ¯ δ k f ( k − δg := δ ¯ g ) . In the opposite way, if g ∈ F , ⊗ V k − is such that ∂g∂z is divisible by ( X − z ) k − and ∂g∂ ¯ z is divisible by ( X − ¯ z ) k − , then the function f ( z ) = ( g ( z ) , Q k − z ) satisfies f ∈ F k, and g = e f .Proof. It is only not clear how to prove the “opposite way” part of the statement.Suppose g satisfies the conditions above. Consider the function g = ( g, δ k − Q k − z ) . δg = ( δg, δ k − Q k − z ) + ( g, δ k Q k − z ) = (cid:18) ∂∂z g, δ k − Q k − z (cid:19) . This shows that δg is a product of a holomorphic function and the function (cid:16) ( X − z ) k − , δ k − Q k − z (cid:17) , which is constant. Therefore δg is itself holomorphic. Hence∆ g = ( δ − δ − (2 k − g = (2 − k ) g , which means g ∈ F k, k − . Next we use that δ − g is divisible by ( X − ¯ z ) k − to provethat ( g, δ l Q k − z ) = δ l +1 − k g ∈ F k, l . This shows that f = δ − k g ∈ F k, and g = e f .We mention one more formula for e f , this time expressing it as a polynomial in( X − z ): Proposition 1.2.6.
Let f ∈ F k, for k ≥ . Then e f = ( − k − (2 k − k − k − X i =0 ( X − z ) i i ! (cid:18) ∂∂z (cid:19) i δ − k f, and (cid:18) ∂∂z (cid:19) k − δ − k f = δ k f. Proof.
The first formula can be verified in two steps. First, check that both sidescoincide when X = z . Second, take the derivative of both sides with respect to z and notice that derivative of the left hand side must be divisible by ( X − z ) k − .The second step of the proof also gives the second formula.Suppose we know δ k f , ¯ δ k f and we want to recover f . Using the theorem aboveone can transform the latter problem into the problem of recovering e f from ∂ e f∂z and ∂ e f∂ ¯ z . The next section explains how this can be done. Let Γ be a congruence subgroup of SL ( R ), S — a finite union of orbits of Γ in H , U = H − S . Consider a smooth closed differential 1-form ω on U with coefficientsin V k − , where V k − is the space of polynomials in one variable X of degree notgreater than 2 k − V k − is equipped with the natural action of SL ( R ) (even SL ( C )). Suppose moreover that ω is equivariant for the action of Γ and let A be aΓ-submodule of V k − . We are looking for a function I A, Γ ω : U −→ V k − /A, which satisfies the following properties: 18i) I A, Γ ω is smooth, which means that for any point z ∈ U there is a neighborhood W of z and a smooth function g : W −→ V k − , such that g coincides with I A, Γ ω modulo A .(ii) dI A, Γ ω = ω ,(iii) I A, Γ ω is equivariant for the action of Γ.Our basic example of ω is ω = f ( z )( X − z ) k − dz, for some meromorphic modular form f of weight 2 k . The module A may be thesubmodule of V k − consisting either of polynomials with real coefficients, polyno-mials with imaginary coefficients or of polynomials with coefficients in some discretesubgroup of C .It is clear that for existence of I A, Γ ω the following condition is necessary: Condition (Residue condition) . For every point s ∈ S the integral of ω along asmall loop around s belongs to A .Choose a basepoint a ∈ U . The differential ω defines a cocycle with values in V k − /A in the following way: σ a ( γ ) = Z γaa ω. It satisfies σ a ( γγ ′ ) = σ a ( γ ) + γσ a ( γ ′ ) . The integral of ω is a function with values in V k − /A : I a ( x ) = Z xa ω. If we change the basepoint from a to a ′ the integral and the cocycle change in thefollowing way: I a ′ ( x ) = I a ( x ) + Z aa ′ ω,σ a ′ ( γ ) = σ a ( γ ) + Z aa ′ ω − Z γaγa ′ ω, so if we introduce an element v aa ′ of V k − /A by v aa ′ = Z aa ′ ω, then I a ′ ( x ) = I a ( x ) + v aa ′ ,σ a ′ = σ a − δv aa ′ , δ denotes the differential ( δv )( γ ) = γv − v. Therefore σ a defines a class in σ A, Γ ω ∈ H (Γ , V k − /A ), which does not depend onthe base point.If such an I A, Γ ω as explained in the beginning of the section exists, it must differfrom I a by a constant, say v a ∈ V k − /A . Let us describe all v a ∈ V k − such that I a + v a is equivariant for the action of Γ. This means that for any γ ∈ Γ I a ( γx ) + v a = γ ( I a ( x ) + v a ) . Splitting the path of integration and using the equivariance of ω we conclude: I a ( γx ) = I γa ( γx ) + σ a ( γ ) = γI a ( x ) + σ a ( γ ) , so the last equation is equivalent to the following: σ a ( γ ) = γv a − v a = δv a . Therefore
Proposition 1.3.1.
Suppose ω satisfies the residue condition. Then the func-tion I A, Γ ω satisfying the properties (i)-(iii) exists if and only if the class σ A, Γ ω ∈ H (Γ , V k − /A ) is trivial. If such a function exists, then it is unique up to additionof an element of H (Γ , V k − /A ) and satisfies γI A, Γ ω ( z ) − I A, Γ ω ( z ) = Z γzz ω mod A for all γ ∈ Γ . For any Γ-module M we denote by C (Γ , M ) the abelian group of cocycles, i.e.maps σ : Γ −→ M such that σ ( γ γ ) = σ ( γ ) + γ σ ( γ ) , for all γ , γ ∈ Γ.Then we have the following exact sequence:0 −→ H (Γ , M ) −→ M d −−→ C (Γ , M ) −→ H (Γ , M ) → , where d is given by the usual formula( dm )( γ ) = γm − m, m ∈ M, γ ∈ ΓDenote by C (Γ , M ) the abelian group of cycles, by which we mean the quotientgroup of Z Γ ⊗ M by the subgroup generated by elements of the form γ γ ⊗ m − γ ⊗ m − γ ⊗ γ m, for m ∈ M , γ , γ ∈ Γ.We have the corresponding exact sequence for homology:0 −→ H (Γ , M ) −→ C (Γ , M ) d −−→ M −→ H (Γ , M ) → , d is given by d ( γ ⊗ m ) = γm − m, m ∈ M, γ ∈ Γ . Suppose we have an invariant biadditive pairing( · , · ) : M ⊗ M ′ −→ N, where M and M ′ are Γ-modules and N is an abelian group. Denote by C (Γ , M ) the group of cocycles which map to zero in H (Γ , M ), and by M ′ the group ofelements in M ′ which map to zero in H (Γ , M ′ ).Note that there is a canonical pairing C (Γ , M ) ⊗ C (Γ , M ′ ) −→ N, given by ( σ, γ ⊗ m ) = ( σ ( γ − ) , m ) , since( σ, γ γ ⊗ m − γ ⊗ m − γ ⊗ γ m )= ( σ ( γ − γ − ) , m ) − ( σ ( γ − ) , m ) − ( σ ( γ − ) , γ m ) = 0 . This pairing has the following property which can be easily checked:
Proposition 1.3.2.
For any m ∈ M , c ∈ C (Γ , M ′ )( dm, c ) = ( m, dc ) . This implies that ( · , · ) induces pairings H (Γ , M ) ⊗ H (Γ , M ′ ) −→ N, H (Γ , M ) ⊗ H (Γ , M ′ ) −→ N. If we have σ ∈ C (Γ , M ) and m ′ ∈ M ′ , we can either represent σ as a cobound-ary, i.e. σ = dm , and then consider ( m , m ′ ), or represent m ′ as a boundary, m ′ = dc , and then consider ( σ, c ). Proposition 1.3.3.
Either of these two approaches defines a pairing C (Γ , M ) ⊗ M ′ −→ N, moreover the resulting two pairings coincide. We put M = V k − /A , M ′ = B , where B is a Γ- invariant subgroup of V k − and N = C / ( A, B ). Theorem 1.3.4.
Let S ⊂ H be a finite union of orbits of Γ , A and B be Γ -invariantsubgroups of V k − , ω be a smooth closed invariant differential -form on U = H − S with coefficients in V k − whose integrals along small loops around points of S belongto A , z be a point in U , v be an element in B such that: i) The class of ω in H (Γ , V k − /A ) is ,(ii) the class of v in H (Γ , B ) is .Then the following two approaches lead to the same element of C / ( A, B ) , which wedenote by I A,B, Γ ( ω, z, v ) :(i) First represent ω as a differential of an invariant V k − /A -valued function I A, Γ ω , and then put I A,B, Γ ( ω, z, v ) = ( I A, Γ ω ( z ) , v ) . (ii) First represent v as v = n X i =1 ( γ i u i − u i ) for γ i ∈ Γ , u i ∈ B ,and then put I A,B, Γ ( ω, z, v ) = n X i =1 Z γ − i zz ω, u i ! . Let k be an integer, k >
1. We consider the Green function for the upper half planeof weight 2 k , which we denote by G H k (note that this is not yet the Green functionwhich is our main object of study, the property (ii) will be different). It is the uniquefunction which satisfies the following properties:(i) G H k is a smooth function on H × H − { z = z } with values in R .(ii) G H k ( γz , γz ) = G H k ( z , z ) for all γ ∈ SL ( R ).(iii) ∆ i G H k = k (1 − k ) G H k , where ∆ i denotes the Laplace operator with respect to z i .(iv) G H k = log | z − z | + O (1) when z tends to z .(v) G H k tends to 0 when z tends to infinity.In this section we obtain two formulae. The first formula is for δ n δ m G H k , and itinvolves hypergeometric series. The second formula is a particular case of the firstfor n = k , and the resulting expression for this case is a rational function of z , z , ¯ z .Note that because of the symmetry between z and z this case is similar to the case m = k .Because of the second property the function G H k is a function of the hyperbolicdistance. Denote by t ( z , z ) the hyperbolic cosine of the hyperbolic distance, i.e. t ( z , z ) = 1 + 2 ( z − z )(¯ z − ¯ z )( z − ¯ z )( z − ¯ z ) = − z − ¯ z )( z − ¯ z )( z − ¯ z )( z − ¯ z )= − z ¯ z + ( z + ¯ z )( z + ¯ z ) − z ¯ z ( z − ¯ z )( z − ¯ z ) . G H k ( z , z ) = − Q k − ( t ) , where Q k − is the Legendre’s function of the second kind. The function Q k − hasthe following integral definition (see [WW62]): Q k − ( t ) = 2 − k Z − (1 − x ) k − ( t − x ) − k dx. The function Q k − has the following two expansions at infinity, which can beobtained from the integral representation above expanding the integrand into powerseries (the first expansion can also be found in [WW62]): Q k − ( t ) = 2 k − ( k − (2 k − t − k F (cid:18) k , k + 12 ; k + 12 ; t − (cid:19) = 2 k − ( k − (2 k − t + 1) − k F (cid:18) k, k ; 2 k ; 21 + t (cid:19) , here F denotes the hypergeometric series. We are going to compute various deriva-tives of G H k using the second expansion. For this purpose we first compute: δ t = ∂t∂z = 2 (¯ z − z )(¯ z − ¯ z )( z − ¯ z ) ( z − ¯ z ) ,δ t = ∂t∂z = 2 ( z − ¯ z )(¯ z − ¯ z )( z − ¯ z )( z − ¯ z ) , noting that δ t has weight 2 in z and weight 0 in z we compute: δ t = δ t = 0 ,δ δ t = − (cid:18) ¯ z − ¯ z ( z − ¯ z )( z − ¯ z ) (cid:19) = δ t δ tt + 1 . We will use the following formula for the derivative of the hypergeometric series: ∂F ( a, b ; c ; x ) ∂x = a F ( a + 1 , b ; c ; x ) − F ( a, b ; c ; x ) x . We find that ∂ (( t + 1) − m F ( m, n ; c ; t +1 )) ∂t = − m ( t + 1) − m − F ( m + 1 , n ; c ; t +1 ) , so δ n G H k ( z , z ) = ( − n +1 k ( k − k + n − k − × ( t + 1) − k − n F ( k + n, k ; 2 k ; t +1 ) ( δ t ) n .
23o apply δ m we rewrite the last expression as δ n G H k ( z , z ) = ( − n +1 k ( k − k + n − k − × ( t + 1) − k F ( k, k + n ; 2 k ; t +1 ) ( δ δ t ) n ( δ t ) − n , so we can again apply the same formula, since δ of δ δ t and δ t is zero: δ m δ n G H k ( z , z ) = ( − m + n +1 k ( k + m − k + n − k − × ( t + 1) − k − m F ( k + m, k + n ; 2 k ; t +1 ) ( δ δ t ) n ( δ t ) m − n . To make the formula symmetric in m and n we rewrite it as δ m δ n G H k ( z , z ) = ( − m + n +1 k ( k + m − k + n − k − × ( t + 1) − k − m − n F ( k + m, k + n ; 2 k ; t +1 ) ( δ t ) n ( δ t ) m . Let us introduce the following function of weight − z and 0 in z : Q z ( z ) = ( z − z )( z − ¯ z ) z − ¯ z , there is a corresponding function Q z ( z ). One can check: δ t = t − Q z ( z ) − ,δ t = t − Q z ( z ) − , so our first formula is δ m δ n G H k ( z , z ) = ( − m + n +1 ( k + m − k + n − k − (cid:18) t + 12 (cid:19) − k × (cid:18) t − (cid:19) m + n F ( k + m, k + n ; 2 k ; t +1 ) Q z ( z ) − n Q z ( z ) − m ( m, n ≥ − k ) . In particular, when n = k we obtain δ m δ k G H k ( z , z ) = ( − m + k +1 − m ( k + m − t + 1) − k ( t − k + m × F ( k + m, k ; 2 k ; t +1 ) Q z ( z ) − k Q z ( z ) − m , and using the identity F ( a, b ; b ; x ) = (1 − x ) − a we get the second formula δ k δ m G H k ( z , z ) = ( − m + k +1 ( k + m − (cid:18) t + 12 (cid:19) m Q z ( z ) − m Q z ( z ) − k = ( − k − ( k + m − z − ¯ z ) k − m ( z − z ) k + m ( z − ¯ z ) k − m = ( − k − ( k + m − z − z ) m Q z ( z ) m − k ( m ≥ − k ) . .5 Global study of Green’s functions Let Γ be a congruence subgroup of
P SL ( Z ) and k >
1. The Green’s function on H / Γ of weight 2 k is the unique function G H / Γ k with the following properties:(i) G H / Γ k is a smooth function on H × H − { z = γz | γ ∈ Γ } with values in R .(ii) G H / Γ k ( γ z , γ z ) = G H / Γ k ( z , z ) for all γ , γ ∈ Γ.(iii) ∆ i G H / Γ k = k (1 − k ) G H / Γ k .(iv) G H / Γ k = m log | z − z | + O (1) when z tends to z ( m is the order of thestabilizer of z , which is almost always 1).(v) G H / Γ k tends to 0 when z tends to a cusp.The series X γ ∈ Γ G H k ( z , γz )is convergent and satisfies the properties above, so G H / Γ k ( z , z ) = X γ ∈ Γ G H k ( z , γz ) . Consider the function G H / Γ k ( z, z ) for a fixed z ∈ H . We put G H / Γ k,z ( z ) = ^ G H / Γ k ( z, z )= ( − k − (cid:18) k − k − (cid:19) k − X l =1 − k ( − l δ − l ( Q k − z ) δ l G H / Γ k ( z, z )= ( − k − (2 k − k − k − X l =1 − k ( X − z ) k + l − ( k + l − (cid:18) X − ¯ zz − ¯ z (cid:19) k − l − δ l G H / Γ k ( z, z ) . recall that negative powers of δ can be defined for eigenfunctions of the Laplacian.Note that since G H / Γ k ( z, z ) has real values and Q z has imaginary values (we letcomplex conjugation act on X identically), we have the following Proposition 1.5.1.
The values of the function i k − G H / Γ k,z ( z ) are real polynomials. Since δ k ( Q k − z ) = 0 , it is easy to compute that ∂ G H / Γ k,z ( z ) ∂z = δ G H / Γ k,z ( z ) = (cid:18) k − k − (cid:19) δ − k ( Q k − z ) δ k G H / Γ k ( z, z )= ( X − z ) k − ( − k − δ k G H / Γ k ( z, z )( k − .
25n the other hand, because of the proposition above ∂ G H / Γ k,z ( z ) ∂ ¯ z = ( X − ¯ z ) k − δ k G H / Γ k ( z, z )( k − . Consider the function g H / Γ k,z ( z ) = δ k G H / Γ k ( z, z ) . This is a meromorphic modular form in z . There is a corresponding differential1-form with coefficients in V k − ( X − z ) k − g H / Γ k,z ( z ) dz. Let V R k − denote the space of polynomials in V k − which have real coefficients. Proposition 1.5.2.
The class of the differential form ( X − z ) k − g H / Γ k,z ( z ) dz in the cohomology group i k − H (Γ , V R k − ) = H (Γ , V k − / i k V R k − ) is trivial and the function ( − k − ( k − G H / Γ k,z ( z )2 is an integral of ω .Proof. Let us denote ω = ( X − z ) k − g H / Γ k,z ( z ) dz. We have proved before that( − k − ( k − d G H / Γ k,z ( z, X ) = ω + ( − k − ¯ ω, so that ( − k − ( k − k − d G H / Γ k,z ( z, X ) = i k − ω + i k − ω. It implies that the integral of i k − ω around a pole of ω is in i V R k − , so the integral of ω around a pole is in i k V R k − , this is why ω satisfies the residue condition of Section1.3 for A = i k V R k − . Hence the class of ω in H (Γ , V k − / i k V R k − ) is correctlydefined. Moreover,( − k − ( k − d G H / Γ k,z ( z, X ) = ω − ( − k ¯ ω + ω , so ( − k − ( k − G H / Γ k,z ( a, X ) − G H / Γ k,z ( b, X )2 ≡ Z ab ω mod i k V R k − , which implies that the class of ω is trivial: σ i k V R k − , Γ ω ≡ i k V R k − , and ( − k − ( k − G H / Γ k,z ( z ) is an integral of ω .26 heorem 1.5.3. For any z ∈ H the function g H / Γ k,z ( z ) is the unique function whichsatisfies the following properties:(i) It is a meromorphic modular form of weight k in z whose set of poles is Γ z which is zero at the cusps.(ii) In a neighborhood of z g H / Γ k,z ( z ) = m ( − k − ( k − Q z ( z ) − k + O (1) ( m = | Stab Γ z | ) (iii) The class of the corresponding differential form in the cohomology group H (Γ , V k − / i k V R k − ) = i k − H (Γ , V R k − ) is trivial as in Proposition 1.5.2.Proof. First we prove that the function g H / Γ k,z ( z ) actually satisfies these conditions.The first two conditions follow from the local study. In fact, we have δ k G H / Γ k ( z, z ) = X γ ∈ Γ δ k G H k ( z, γz ) , because for all 0 ≤ l ≤ kδ l G H / Γ k ( z, z ) = O ( t ( z, z ) l − k | Q z ( z ) | − l ) , and using the inequality | Q z ( z ) | ≥ t ( z, z ) − | z − ¯ z | we obtain that the series X γ ∈ Γ δ l G H k ( z, γz )is locally uniformly majorated by the series X γ ∈ Γ t ( z, γz ) − k , which converges locally uniformly in z .This already implies that the function g H / Γ k,z ( z ) is meromorphic and has polesof the specified type. The transformation property follows from the invariance of G H / Γ k ( z, z ). Since we can move any cusp to ∞ by an element of SL ( Z ) it is enoughto check the cuspidality at ∞ . This is clear from the following (recall that k > δ k G H / Γ k ( z, z ) = ( − k − ( k − z − ¯ z ) k X γ ∈ Γ γz − z ) k ( γz − ¯ z ) k ( cz + d ) k . k which has either purely real or purelyimaginary cohomology depending on whether k is odd or even. This contradicts theEichler-Shimura theorem, which says that H parabolic (Γ , V k − ) ∼ = S k ⊕ S k , so any nontrivial parabolic cohomology class which is either purely real or purelyimaginary cannot be represented by a cusp form.The value of the Green function can be recovered in the following way: Theorem 1.5.4.
Let k > , z ∈ H and g H / Γ k,z ( z ) be the function that satisfies theconditions of the theorem above. Put w = ( X − z ) k − g H / Γ k,z ( z ) dz and apply one of the two approaches formulated in Theorem 1.3.4 for A = i k V R k − , B = i k − V R k − , v = Q k − z to get an element I i k V R k − , i k − V R k − , Γ ( ω, z, Q k − z ) ∈ C / i R = R , for some z ∈ H . Then G H / Γ k ( z, z ) = ( − k − ( k − ℜ I i k V R k − , i k − V R k − , Γ ( ω, z, Q k − z ) . Proof.
We note that Theorem 1.3.4 can be applied since(i) The class of ω in H (Γ , V k − /A ) is trivial by the third property of the function g H / Γ k,z ( z ).(ii) The whole homology group H (Γ , B ) is trivial because k > I A, Γ ( z ) = ( − k − ( k − G H / Γ k,z ( z )because of Proposition 1.5.2. This implies that I i k V R k − ,i k − V R k − , Γ ( ω, z, Q k − z ) ≡ ( − k − ( k − (cid:18) G H / Γ k,z ( z ) , Q k − z (cid:19) mod C / i R , so the statement follows from the identity( G H / Γ k,z ( z ) , Q k − z ) = G H / Γ k ( z, z ) , which was proved in Section 1.2. 28et us use Theorem 1.5.3 to compute the residues of the form ω = ( X − z ) k − g H / Γ k,z ( z ) dz. The residue at z = z can be computed as follows:res( X − z ) k − Q z ( z ) − k dz = res ( X − z ) k − ( z − ¯ z ) k ( z − z ) k ( z − ¯ z ) k dz = res ( X − z − ( z − z )) k − ( z − z ) k (1 + z − z z − ¯ z ) k dz, so we have to compute the residue of the following Laurent series X i,j ( − i + j (cid:18) k − i (cid:19)(cid:18) k + j − j (cid:19) ( X − z ) i ( z − z ) k − − i + j ( z − ¯ z ) − j dz. This gives (recall that m is the order of the stabilizer of z .( − k − X j (cid:18) k − k + j − (cid:19)(cid:18) k + j − j (cid:19) ( X − z ) k + j − ( z − ¯ z ) − j = ( − k − (cid:18) k − k − (cid:19) X j (cid:18) k − j (cid:19) ( X − z ) k + j − ( z − ¯ z ) − j = ( − k − (cid:18) k − k − (cid:19) ( X − z ) k − ( X − ¯ z ) k − ( z − ¯ z ) k − = ( − k − (cid:18) k − k − (cid:19) Q k − z . Therefore we obtain res ω = m (2 k − k − Q k − z . This suggests that if z is a CM point we should put A = 2 π i (2 k − k − D − k V Z k − , where D is the discriminant of z . Note that A ⊂ i k V R k − and the integrals of ω along small loops around z are still in A .Suppose z is also a CM point. Then we may refine B in the following way: B = D − k V Z k − . We still have Q k − z ∈ B .Consider the groups H (Γ , B ) and H (Γ , V k − /A ). Since H (Γ , V k − ) = 0 and B is finitely generated over Z and spans V k − over C , H (Γ , B ) is a finite group.The form ω defines certain class in H (Γ , V k − /A ), which is cuspidal. We have adirect sum decomposition V k − = i k V R k − ⊕ i k − V R k − and A ⊂ i k V R k − . Therefore V k − /A = i k V R k − /A ⊕ i k − V R k − . By Proposition 1.5.2 the second component ofthe class of ω is trivial, but there is no reason, in general, for the class in i k V R k − /A to be trivial. However the following proposition says29 roposition 1.5.5. Suppose there are no cusp forms for the group Γ of weight k .Then the group of classes in i k V R k − /A which are cuspidal is a finite group.Proof. The Γ-module i k V R k − /A can be obtained from the Γ-module V Z k − by ex-tension of scalars. Since the corresponding cuspidal cohomology group of V Z k − isfinite, the statement follows.Suppose there are no cusp forms for the group Γ of weight 2 k . Let N A be theexponent of the finite group in the proposition above. Let N B be the exponent of H (Γ , B ). Then applying Theorem 1.3.4 to N A ω and N B Q k − z we obtain an element I A,B, Γ ( N A ω, z, N B Q k − z ) ∈ C / ( A, B ) . Note that (
A, B ) ⊂ π i( D D ) − k m (2 k − k − h V Z k − , V Z k − i . It is easy to see that h V Z k − , V Z k − i ⊂ ( k − k − Z . Therefore (
A, B ) ⊂ π i( D D ) − k Z . Let us denote N = N A N B ( k − b G H / Γ k ( z, z ) = ( − k − N I
A,B, Γ ( N A ω, z, N B Q k − z ) ∈ C / π i( D D ) − k N Z . This construction proves the following
Corollary 1.5.6.
Let Γ be a congruence subgroup of P SL ( Z ) , k > an integersuch that there are no cusp forms of weight k for Γ . Let N A be the exponent of thegroup H parabolic (Γ , V Q / Z k − ) , and N B be the exponent of the group H (Γ , V Z k − ) , N = N A N B ( k − . Then for any two non-equivalent CM points z , z of discriminants D , D the pair N A ( X − z ) k − g H / Γ k,z dz , N B Q k − z satisfies the conditions of Theorem1.3.4 for A = 2 π i (2 k − k − D − k V Z k − , B = D − k V Z k − . The corresponding value b G H / Γ k ( z , z ) = ( − k − N I
A,B, Γ ( N A ( X − z ) k − g H / Γ k,z ( z ) dz, z , N B Q k − z ) is defined modulo π i N Z and its real part equals G H / Γ k ( z , z ) . The algebraicity conjecture was formulated in [GZ86] (p. 317) and extended in[GKZ87] (p. 556). In the case when there are no cusp forms it says the following:
Conjecture (1) . Suppose there are no cusp forms of weight 2 k for Γ. Then for anytwo CM points z , z of discriminants D , D there is an algebraic number α suchthat G H / Γ k ( z , z ) = ( D D ) − k log α.
30e can refine the conjecture now:
Conjecture (2) . Suppose there are no cusp forms of weight 2 k for Γ. Then for anytwo CM points z , z of discriminants D , D there is an algebraic number α suchthat ( D D ) k − b G H / Γ k ( z , z ) = log α mod 2 π i N Z . The second version of the conjecture is stronger since it gives not only the abso-lute value of α , but also its argument. Remark . The conjecture as it is formulated in [GZ86] and [GKZ87] is moregeneral since it also deals with the case when there are cusp forms of weight 2 k for Γ.It is not difficult to extend Corollary 1.5.6 and Conjecture (2) to the more generalcase.Both ways to define I A,B, Γ (see Theorem 1.3.4) are useful. The first way, whichwas also explained in the introduction, shows that for a given CM point z for anyinvariant V k − / N A A -valued function g such that dg = ( X − z ) k − g H / Γ k,z ( z ) dz, one has b G H / Γ k ( z, z ) ≡ ( − k − ( k − g, Q k − z ) mod 2 π i( D D ) − k N Z . This gives us a way to prove the conjecture. Namely, the idea is to construct somefunction g such that g satisfies the condition above and ( g, Q k − z ) can be computedalgebraically. Later we will indeed construct such g for the point z = i usingAbel-Jacobi maps for higher Chow groups.The second way is to represent Q k − z as a sum Q k − z = n X i =1 ( γ i u i − u i ) for γ i ∈ Γ, u i ∈ N B B ,and then put b G H / Γ k ( z, z ) ≡ ( − k − ( k − n X i =1 Z γ − i zz ( X − z ) k − g H / Γ k,z ( z ) dz, u i ! mod 2 π i( D D ) − k N Z . This approach provides a way to compute b G H / Γ k ( z, z ) approximately by numericalintegration and proves the following important fact: Theorem 1.5.7.
Let Γ be a congruence subgroup of P SL ( Z ) , k > an integer suchthat there are no cusp forms of weight k for Γ . Then for any two non-equivalentCM points z , z the value b G H / Γ k ( z , z ) is a period (see [KZ01]). hapter 2 Higher Chow groups andAbel-Jacobi maps
In this chapter we assume X is a smooth projective variety over C and k is aninteger. We give definition of the first higher Chow group CH k ( X,
1) as the groupof cycles modulo the group of boundaries. The cycles are formal sums X i ( W i , f i )where each W i is a subvariety of codimension k − X and each f i is a non-zero rational function on W i such that the sum of divisors of f i is 0 as a cycle ofcodimension k . The boundaries are cycles which can be constructed from pairs offunctions ( f, g ) as (div f, g ) − (div g, f ) (see Section 2.3 for details).In Section 2.4 we give a construction of the Abel-Jacobi map from the firsthigher Chow group to the quotient of the cohomology group H k − ( X, C ) by acertain subspace and a certain lattice. The subspace is the k -th step in the Hodgefiltration and the lattice is the integral cohomology. We prove that the definitiondoes not depend on various choices in the remaining part of Section 2.4.It is clear that it makes sense to consider the pairing of the value of the Abel-Jacobi map with the cohomology class of an ordinary cycle Z (linear combinationof irreducible subvarieties) of the “complementary dimension” (dimension is k − π i is the logarithm of the intersectionnumber, which can be computed explicitly as the product over the intersection pointsof Z and W i of values of f i over all i (taking multiplicities into account). Althoughthis statement follows from compatibility of Abel-Jacobi maps with intersectionproducts, we give two independent proofs. One uses currents, and another usesrelative cohomology and a construction of logarithmic representatives of fundamentalclasses in relative cohomology. The symbol “i” denotes √− i ”, which will usually be usedas an index. 32 .2 The Hodge theory Let X be a smooth projective variety over C . For each k we denote the sheaf ofsmooth k -forms by A kX . We have the usual decomposition A kX = M p + q = k A p,qX , where A p,qX is the sheaf of smooth ( p, q )-forms on X . We have the Hodge filtrationon A kX defined as F j A kX := M p + q = k, p ≥ j A p,qX . One can compute the cohomology groups of X by taking the cohomology of thecomplex of global forms: H k ( X, C ) = H k ( A • ( X )) , where for a sheaf F and an open set U F ( U ) denotes the sections of the sheaf over U and we omit the subscript X when we write A • ( X ). One obtains the Hodgefiltration on H k ( X, C ) as the one induced by the filtration on A kX , i.e. F j H k ( X, C ) = Ker( d : A k ( X ) −→ A k +1 ( X )) ∩ F j A k ( X )Im( d : A k − ( X ) −→ A k ( X )) ∩ F j A k ( X ) . By the Hodge theory there is a canonical decomposition H k ( X, C ) = M p + q = k H p,q ( X, C )with F j H k ( X, C ) = M p + q = k, p ≥ j H p,q ( X, C ) . We will frequently use the following consequence of the Hodge theory:
Proposition 2.2.1. If ω ∈ F j A k ( X ) is exact, i.e. there exists η ∈ A k − ( X ) with dη = ω , then ω = dη ′ for some η ′ ∈ F j A k − ( X ) .Proof. Let ω = X p + q = k, p ≥ j ω p,q , η = X p + q = k − η p,q , ω p,q , η p,q ∈ A p,q ( X ) . Put η = X p + q = k − , p ≥ j η p,q . Then the following form is exact: ω = ω − dη = ω j,k − j − ¯ ∂η j,k − j − . dω = 0 and ω ∈ F j A k ( X ) we have ¯ ∂ω j,k − j = 0. Therefore ¯ ∂ω = 0, so ω gives some class in the Dalbeaut cohomology group H j,k − j ¯ ∂ ( X ) ∼ = H j,k − j ( X, C ).This class is trivial because of the Hodge decomposition and the fact that the classof ω is trivial in H k ( X, C ). Hence ω = ¯ ∂η ′ for some η ′ ∈ A p,q − ( X ). This impliesthat ω − d ( η + η ′ ) ∈ F j +1 A k ( X ), so we can use induction on j to complete theproof.Note that there is a canonical homomorphism ι : H k ( X, Z ) −→ H k ( X, C ) . To simplify the notation we will sometimes write H k ( X, Z ) ∩ F j H k ( X, C ) , H k ( X, Z ) + F j H k ( X, C ) , respectively, instead of H k ( X, Z ) ∩ ι − ( F j H k ( X, C )) , H k ( X, Z ) + ι ( F j H k ( X, C )) . Let X be a smooth complex projective variety of dimension n . Recall that theordinary Chow group CH k ( X ) of codimension k cycles is the quotient group CH k ( X ) := Z k ( X ) /B k ( X ) , where Z k ( X ) is the free abelian group generated by irreducible algebraic subvarietiesof X of codimension k and B k ( X ) is the subgroup generated by principal divisorson subvarieties of X of codimension k − CH k ( X,
1) (see [GL00]) consider the group C k ( X,
1) which is the free abelian group generated by pairs (
W, f ), where W is anirreducible algebraic subvariety of X of codimension k − f is a non-zero rationalfunction on W , modulo the relations( W, f f ) = ( W, f ) + ( W, f ) , where f and f are two rational functions on W . The group Z k ( X,
1) is defined tobe the kernel of the map C k ( X, −→ B k ( X )sending ( W, f ) to div f , the divisor of f . Define the group B k ( X,
1) as the subgroupof Z k ( X,
1) generated by elements of the form(div g, h | div g ) − (div h, g | div h ) , where g , h are non-zero rational functions on some V ⊂ X of codimension k − W, f )to linear combinations of subvarieties, i.e. if W = P j n j W j is a linear combination ofirreducible subvarieties with integer coefficients and f is a non-zero rational function34n some bigger subvariety which restricts to a non-zero rational function on each W j , then ( W, f ) := X j ( W j , f n j | W j ) . We put CH k ( X,
1) := Z k ( X, /B k ( X, CH k ( X,
1) has the form X i ( W i , f i ) , where X i div f i = 0 . Example . If k = 1 then the corresponding higher Chow group is simply themultiplicative group of complex numbers, CH ( X,
1) = C × . Example . The group CH n +1 ( X,
1) is generated by pairs (
W, f ) where W isa point and f is a non-zero complex number. In fact one can see that there is asurjective homomorphism of abelian groups CH n ( X ) ⊗ C × −→ CH n +1 ( X, . Using the Weil reciprocity law one can also check that there exists a homomorphismfrom CH n +1 ( X,
1) to C × which sends ( W, f ) to f . Recall that for the ordinary Chow group we have the cycle class mapcl k : CH k ( X ) −→ H k ( X, Z ) ∩ F k H k ( X, C ) , which sends V , a subvariety of codimension k , to its class[ V ] ∈ H n − k ( X, Z ) ∼ = H k ( X, Z ) . Denote by CH k ( X ) the kernel of cl k . We have the Abel-Jacobi map AJ k : CH k ( X ) −→ H k − ( X, C ) F k H k − ( X, C ) + H k − ( X, Z ) . This is defined as follows. Let γ be an algebraic cycle of codimension k whosehomology class is 0. It follows that γ = ∂ξ for some 2 n − k + 1-chain ξ . Choosingsuch ξ we obtain a linear functional on the space of 2 n − k + 1-forms given byintegrating a form against ξ . 35et us show that this defines a linear functional on F n − k +1 H n − k +1 . Indeed, if ω is an exact form from F n − k +1 A n − k +1 then ω = dη for η ∈ F n − k +1 A n − k byHodge theory. Hence Z ξ ω = Z γ η is zero.Choosing another ξ ′ such that ∂ξ ′ = γ we have ξ − ξ ′ closed, so the correspondingfunctionals for ξ and ξ ′ differ by the functional induced by the corresponding elementof H n − k +1 ( X, Z ). So we obtain a map AJ k : CH k ( X ) −→ ( F n − k +1 H n − k +1 ( X, C )) ∗ H n − k +1 ( X, Z ) ∼ = H k − ( X, C ) F k H k − ( X, C ) + H k − ( X, Z ) . Let x represent an element of CH k ( X, k ∈ Z i ( X,
1) i.e. x = X i ( W i , f i )with X i div f i = 0 . We denote the corresponding element in CH k ( X,
1) by [ x ]. We choose a path[0 , ∞ ] ⊂ CP . Let us denote γ i = f ∗ i [0 , ∞ ] , which is a 2 n − k + 1-chain on X whose boundary is − div f i . This implies that thechain γ = X i γ i is a cycle. By the Poincar´e duality γ has a class [ γ ] ∈ H k − ( X, Z ). Proposition 2.4.1.
The map x → [ γ ] defines a cycle class map cl k, : CH k ( X, −→ H k − ( X, Z ) ∩ F k H k − ( X, C ) . Proof.
We have to check two things:(i) For any x ∈ Z k ( X,
1) the image of [ γ ] belongs to F k H k − ( X, C ).(ii) If x = (div g, h | div g ) − (div h, g | div h ) for V , g and h as in the definition of B k ( X,
1) in Section 2.3, then γ is homologically trivial.36or (i) it is enough to prove that the pairing of γ with any element of F n − k +1 H n − k +1 ( X, C )is zero. Take a closed form ω ∈ F n − k +1 A n − k +1 ( X ). Recall that Z γ ω = X i Z γ i ω. Let n ( γ i ) be a small tubular neighborhood of γ i in W i . Then, up to terms whichtend to 0 as the radius of the neighborhood tends to 0, we have Z γ i ω = 12 π i Z ∂n ( γ i ) ω log f i = − π i Z W i − n ( γ i ) d ( ω log f i )= − π i Z W i − n ( γ i ) df i f i ∧ ω. The form in the last integral belongs to F n − k +2 A n − k +2 and W i has complexdimension n − k + 1, so the integral is zero.For proving (ii), if x = (div g, h | div g ) − (div h, g | div h ), then the chain − ( g × h ) ∗ ([0 , ∞ ] × [0 , ∞ ]) has boundary γ .On the other hand since complex conjugation acts trivially on the group H k − ( X, Z )and F k H k − ( X, C ) ∩ F k H k − ( X, C ) = { } , we have the following Proposition 2.4.2.
For any [ x ] ∈ CH k ( X, the class cl k, [ x ] is torsion. Thus the cycle class map is a mapcl k, : CH k ( X, −→ H k − ( X, Z ) tors . Let us denote the kernel of this map by CH k ( X, . To construct the Abel-Jacobimap AJ k, : CH k ( X, −→ H k − ( X, C ) F k H k − ( X, C ) + 2 π i H k − ( X, Z )we first identify H k − ( X, C ) F k H k − ( X, C ) + 2 π i H k − ( X, Z ) ∼ = ( F n − k +1 H n − k +2 ( X, C )) ∗ π i H k − ( X, Z ) . If [ x ] ∈ CH k ( X, with x ∈ Z k ( X,
1) then γ = ∂ξ for some 2 n − k + 2-chain ξ .Then for any ω ∈ F n − k +1 A n − k +2 ( X ) with dω = 0 we take the following number: h AJ k, [ x ] , [ ω ] i = X i Z W i \ γ i ω log f i + 2 π i Z ξ ω, (*)where the logarithm on CP is defined using the cut along the chosen path [0 , ∞ ].To prove that this correctly defines a map AJ k, : CH k ( X, −→ F n − k +1 H n − k +2 ( X, C ) ∗ π i H k − ( X, Z )we need to show that the construction does not depend on the following choices:37 the choice of the path [0 , ∞ ]; • the choice of the branch of the logarithm on CP − [0 , ∞ ]; • the choice of the representative of x , which is defined up to an element of B k ( X, • the choice of ξ , which is defined up to a 2 n − k + 2-cycle; • the choice of ω , which is defined up to a coboundary.We prove this in a series of propositions. Proposition 2.4.3.
The value of (*) does not depend on the choice of the path [0 , ∞ ] .Proof. Let p and p ′ be two different paths on CP from 0 to ∞ . Let γ i = f ∗ i p, γ ′ i = f ∗ i p ′ ,γ = X i γ i , γ ′ = X i γ ′ i . Choose a 2-chain q on CP whose boundary is p ′ − p , let η i = f ∗ i q, η = X i η i . We choose ξ such that ∂ξ = γ and put ξ ′ = ξ + η so that ∂ξ ′ = γ ′ . Let l be a branchof the logarithm on CP − p . Then the function l ′ ( t ) = l ( t ) − π i q ( t )is a branch of the logarithm on CP − p ′ , where q is the characteristic function of q . Then we compare X i Z W i − γ ′ i ω l ′ ( f i ) − X i Z W i − γ i ω l ( f i ) = − π i Z η ω, Z ξ ′ ω − Z ξ ω = Z η ω, so the value of (*) does not change. Remark . In fact this proof also shows that we can even vary each γ k as longas its class in the homology H n − k +1 ( W i , | div f i | ) remains constant. Proposition 2.4.4.
Changing the branch of the logarithm changes the value of (*)by an element from π i H k − ( X, Z ) .Proof. Changing the branch of the logarithm amounts to adding 2 π i m for m ∈ Z ,which changes the value of (*) by 2 π i m X i Z W i ω, which is a functional induced by the image of mW i in H k − ( X, Z ).38 roposition 2.4.5. If x = div g ⊗ h | div g − div h ⊗ g | div h , for V , g and h as in the definition of B k ( X, in Section 2.3, then the value of (*)is zero.Proof. We may take ξ = − ( g × h ) ∗ ([0 , ∞ ] × [0 , ∞ ])since ∂ ([0 , ∞ ] × [0 , ∞ ]) = [0 , ∞ ] × ([0] − [ ∞ ]) − ([0] − [ ∞ ]) × [0 , ∞ ] . Inside V we consider div g which has real codimension 2 and γ h = h ∗ [0 , ∞ ] whichhas real codimension 1. Consider a small neighborhood of the divisor of g which wedenote by n (div g ) and a small neighborhood of γ g which we denote by n ( γ g ). Wewill use corresponding notation for h , i.e. n (div h ) and n ( γ h ). Then we may rewrite Z div g − γ h ω log h = 12 π i Z ∂n (div g ) ∩ n ( γ h ) c dgg ∧ ω log h. Take a chain S = V ∩ n (div g ) c ∩ n ( γ h ) c . Then ∂S = − ∂n (div g ) ∩ n ( γ h ) c − ∂n ( γ h ) ∩ n (div g ) c , so using the Stokes formula we obtain − Z ∂n (div g ) ∩ n ( γ h ) c dgg ∧ ω log h − Z ∂n ( γ h ) ∩ n (div g ) c dgg ∧ ω log h = Z S dhh ∧ dgg ∧ ω. Since ω ∈ F n − k +1 A n − k +2 ( X ), dhh ∧ dgg ∧ ω ∈ F n − k +3 A n − k +4 ( S ) and its integralis zero because S ⊂ V , which is a n − k + 2-dimensional complex variety. Hence Z div g − γ h ω log h = − π i Z ∂n ( γ h ) ∩ n (div g ) c dgg ∧ ω log h = − Z γ h ∩ n (div g ) c dgg ∧ ω. We apply the Stokes formula again for T = γ h ∩ n ( γ g ) c and ω log g . We have ∂T = − div h ∩ n ( γ g ) c − ( ∂n ( γ g ) ∩ γ h ) , so − Z div h ∩ n ( γ g ) c ω log g − Z ∂n ( γ g ) ∩ γ h ω log g = Z γ h − n (div g ) dgg ∧ ω. This implies that Z div g − γ h ω log h − Z div h − γ g ω log g = Z ∂n ( γ g ) ∩ γ h ω log g = 2 π i Z γ g ∩ γ h ω. Since ξ = − γ g ∩ γ h the statement follows.39 roposition 2.4.6. Changing ξ by a n − k + 2 -cycle changes the value of thefunctional (*) by an element of π i H k − ( X, Z ) .Proof. This is clear.
Proposition 2.4.7.
Changing ω by a coboundary does not change the value of thefunctional (*).Proof. Indeed, if ω is a coboundary, then ω = dη with η ∈ F n − k +1 A ( X ) by theHodge theory. This implies( ∗ ) = X i Z W i − γ i d ( η log f i ) − X i Z W i − γ i df i f i ∧ η + 2 π i Z γ η. Since df i f i ∧ η ∈ F n − i +2 A ( X ) and W i is a n − k + 1-dimensional complex variety,the second summand is zero. Applying the Stokes formula to the first summand weobtain X i Z W i − γ i d ( η log f i ) = − X i Z ∂n ( γ i ) η log f i = − π i X i Z γ i η, where n ( γ i ) denotes a small neighborhood of γ i inside W i . Hence the statement. Let x = P i ( W i , f i ) represent an element [ x ] ∈ CH k ( X, . Then AJ k, [ x ] ∈ F n − k +1 H n − k +2 ( X, C ) ∗ π i H k − ( X, Z ) . Given a subvariety Z ⊂ X of codimension n − k + 1 we may consider cl n − k +1 Z ∈ H n − k +2 ( X, Z ) ∩ F n − k +1 H n − k +2 ( X, C ). Then( AJ k, [ x ] , cl n − k +1 Z ) ∈ C / π i Z , so this number can be written as log α for a unique α ∈ C . We now show how toconstruct this number in a different way. Consider the cycle S ⊂ X which is theunion of all | div f i | and singular parts of W i . We say that Z intersects x properlyif Z properly intersects S and all W i . This means that Z does not intersect S and intersects each W i in several points. Note that by the Moving Lemma for anygiven Z there exists Z ′ which is rationally (hence homologically) equivalent to Z and intersects x properly. One defines the intersection number x · Z = Y i Y p ∈ W i ∩ Z f i ( p ) ord p ( W i · Z ) . Theorem 2.5.1.
Let x = P i ( W i , f i ) be a representative of [ x ] ∈ CH k ( X, . Let Z ⊂ X be a smooth subvariety of dimension k − intersecting x properly. Then ( AJ k, [ x ] , cl n − k +1 Z ) ≡ log( x · Z ) mod 2 π i . roof. By the definition (*) of AJ k, we have( AJ k, [ x ] , cl n − k +1 Z ) = X i Z W i \ γ i ω log f i + 2 π i Z ξ ω, where ω ∈ F n − k +1 A n − k +2 ( X ) is a form whose class [ ω ] equals to the Poincar´edual of the class [ Z ] ∈ H k − ( X, Z ).The current ω − δ Z . is homologically trivial since both ω and δ Z represent the same class in the coho-mology. Hence there exists a current η ∈ F n − k +1 D n − k +1 ( X ), smooth outside | Z | ,such that dη = ω − δ Z . If we denote by η the corresponding form on X \ | Z | , η ∈ F n − k +1 A n − k +1 ( X \ | Z | ) , we obtain an identity dη = ω, which is true outside | Z | .In the definition of the Abel-Jacobi map we choose γ k and ξ to be transversal to | Z | . This means that γ k does not intersect | Z | for each k and ξ intersects | Z | onlyin several points.Choosing small neighborhoods of | div f i | , | Z |∩ W i , γ i inside W i and | Z |∩ ξ inside ξ and denoting them by n (div f i ), n ( | Z | ∩ W i ), n ( γ i ), n ( | Z | ∩ ξ ) respectively we maywrite X i Z W i \ ( n ( γ i ) ∪ n ( | Z |∩ W i )) ω log f i + 2 π i Z ξ \ n ( | Z |∩ ξ ) ω = X i Z W i \ ( n ( γ i ) ∪ n ( | Z |∩ W i )) dη log f i + 2 π i Z ξ \ n ( | Z |∩ ξ ) dη . We transform the second term into2 π i Z γ η − π i Z ∂n ( | Z |∩ ξ ) η and the i -th summand in the first term into − Z ∂n ( γ i ) η log f i − Z ∂n ( | Z |∩ W i ) η log f i − Z W i \ ( n ( γ i ) ∩ n ( | Z |∩ W i )) df i f i ∧ η , where the last term equals to 0 because df i f i ∧ η ∈ F n − k +2 A n − k +2 .Consider the integral Z ∂n ( | Z |∩ W i ) η log f i . p be an intersection point of | Z | and W i . Then there is a neighborhood U of p which is analytically isomorphic to the product of open balls B W and B Z , and W i ∩ U maps to B W × { } , | Z | ∩ U maps to { } × B Z . Let n ( | Z | ∩ W i ) have only oneconnected component in U and this is D W × { } where D W ⊂ B W is a closed ball.We extend f i to U by means of the projection U −→ B W . Let χ ε ( t ) be a family ofsmooth functions on R which approximate δ as in [GH78]. For any current T on U we put T ε = T ∗ ( χ nε )where ∗ denotes the convolution and χ nε ( w , . . . , w n − k +2 , b , . . . , b k − )= χ ε ( w ) . . . χ ε ( w n − k +2 ) χ ε ( b ) . . . χ ε ( b k − ) . Then Z ∂D W ×{ } η log f i = lim ε → Z U ( δ ∂D W ×{ } ) ε ∧ η log f i since η is smooth outside { } × B Z . Applying the identity( δ ∂D W ×{ } ) ε = − d ( δ D W ×{ } ) ε the last expression equals to − lim ε → Z U d ( δ D W ×{ } ) ε ∧ η log f i = lim ε → ( η log f i , d ( δ D W ×{ } ) ε ) , where we treat η log f i as a current and d ( δ D W ×{ } ) ε as a form. This is, by thedefinition of the differential for currents,lim ε → ( d ( η log f i ) , ( δ D W ×{ } ) ε ) . Now we expand d ( η log f i ) = ( ω − δ Z ) log f i + df i f i ∧ η , thus obtaininglim ε → (( ω − δ Z ) log f i + df i f i ∧ η , ( δ D W ×{ } ) ε )= Z D W ×{ } ω log f i − log f i ( p ) · ord p ( Z · W i ) + lim ε → ( df i f i ∧ η , ( δ D W ×{ } ) ε ) , the last summand being zero because η ∈ F n − k +1 D n − k +1 , ( δ D W ×{ } ) ε ∈ F k − A k − . Note that when the radius of the ball D W tends to zero the first summand tends tozero, so can be neglected. Therefore the limit value of Z ∂n ( | Z |∩ W i ) η log f i − X p ∈ W i ∩| Z | log f i ( p ) · ord p Z. The sum X i Z ∂n ( γ i ) η log f i annihilates (in the limit) the integral2 π i Z ∂ξ η . The remaining summand − π i Z ∂n ( | Z |∩ ξ ) η tends to 2 π i times the intersection number of Z and ξ according to a reasoningsimilar to the one used above. We would like to produce another proof of this theorem without the use of currents.For this we recall some cohomology constructions. What follows can be done forany smooth variety X over C .Let Ω • X be the holomorphic de Rham complex of X . For any integer j denoteby F j Ω • X the subcomplex F j Ω iX = ( i < j ,Ω iX if i ≥ j .There are natural maps F j Ω • X → Ω • X and F j Ω • X → Ω jX .Let Y ⊂ X be a smooth subvariety of codimension j . Recall the construction ofthe fundamental class of Y in H jY ( X, F j Ω • X ) (see [Gro62], Expos´e 149 and [Blo72]).We first construct the Hodge class c H ( Y ) ∈ H jY ( X, Ω jX ). There is a spectralsequence E p,q = H p ( X, H qY (Ω jX )) ⇒ H p + qY ( X, Ω jX ) . Since Ω jX is locally free H qY (Ω jX ) = 0 for q < j . This implies that H jY ( X, Ω jX ) = Γ( X, H jY (Ω jX )) . Let V be an open subset of X on which Y is a complete intersection, so thereexist regular functions f , f ,. . . , f j on V which generate the ideal of Y ∩ V . Put V i = V \ { f i = 0 } . Then V i form a covering of V \ ( V ∩ Y ). So we can consider theˇCech cohomology and the section(2 π i) − j df f ∧ . . . df j f j ∈ Γ( \ V i , Ω jX )43roduces an element of H j − ( V \ ( V ∩ Y ) , Ω jX ). We obtain an element of H jV ∩ Y ( V, Ω jX )by applying the boundary map of the long exact sequence . . . −→ H j − ( V \ ( V ∩ Y ) , Ω jX ) −→ H jV ∩ Y ( V, Ω jX ) −→ H j ( V, Ω jX ) −→ . . . . This element is a section of the sheaf H jY (Ω jX ) over V . These local sections gluetogether to produce a global section c H ( Y ) ∈ H jY ( X, Ω jX ) . The differential d : Ω jX → Ω j +1 X induces the differential on cohomology d : H jY ( X, Ω jX ) → H jY ( X, Ω j +1 X ). We have dc H ( Y ) = 0 since for V , f i as above d (cid:18) (2 π i) − j df f ∧ . . . df j f j (cid:19) = 0 . There is a spectral sequence for the hypercohomology E p,q = H p ( H qY ( X, F j Ω • X )) ⇒ H p + qY ( X, F j Ω • X ) , which shows that H jY ( X, F j Ω • X ) = H j ( H jY ( X, F j Ω • X ))= Ker( d : H jY ( X, Ω jX ) → H jY ( X, Ω j +1 X )) . Therefore the natural map H jY ( X, F j Ω • X ) → H jY ( X, Ω jX ) is an injection and c H ( Y )lifts to a unique c F ( Y ) ∈ H jY ( X, F j Ω • X ). The natural map H jY ( X, F j Ω • X ) −→ H jY ( X, Ω • X ) ∼ = H jY ( X, C )sends c F ( Y ) to an element c DR ( Y ) ∈ H jY ( X, C ).We will prove now that c DR ( Y ) is the Thom class of Y , i.e. that its valueon a class in H j ( X, X \ Y ) equals to the intersection number of this class with Y . Since the real codimension of Y is 2 j , H pY ( C ) = 0 for p < j . Therefore H jY ( X, C ) = Γ( X, H jY ( C )) and c DR ( Y ) can be described locally. Let V be aneighborhood of X which is isomorphic to a product of unit disks, V ∼ = D n , andsuch that V ∩ Y is given by equations z i = 0 for i = 1 , . . . j , where z i is the coordinateon i -th disk. Then H jY ∩ V ( V, C ) = H j ( D n , ( D j \ { } j ) × D n − j ; C ) = C . So to evaluate the restriction c DR ( Y ) | V it is enough to evaluate c DR ( Y ) | V on thegenerator of the homology H j ( D n , ( D j \ { } j ) × D n − j ; C ), the transverse class D j ×{ } n − j . Since the class c H ( Y ) | V is coming from the boundary map H j − (( D j \{ } j ) × D n − j , Ω j ) → H j ( D n , ( D j \ { } j ) × D n − j ; Ω j ) h c DR ( Y ) | V , D j × { } n − j i = h c H ( Y ) | V , D j × { } n − j i = h c , ∂ ( D j × { } n − j ) i , c is the corresponding class in H j − (( D j \ { } j ) × D n − j , Ω j ). The class c ,in its turn, comes from the map H ( D j × D n − j , Ω j ) −→ H j − (( D j \ { } j ) × D n − j , Ω j ) . This map comes from ˇCech cohomology and can also be constructed using successiveapplication of Mayer-Vietoris exact sequences. Hence we have the dual map inhomology H j − (( D j \ { } j ) × D n − j , C ) −→ H j ( D j × D n − j , C )and one can see that the image of ∂ ( D j × { } n − j ) under this map is S j × { } n − j ,where S is the unit circle. Indeed, let U k = D k × ( D j − k \ { } j − k ) × { } n − j . Then U = ( D j \ { } j and U j − = D j × D n − j . Put X k = D k × D j − k × { } n − j , X k = D k − × D × ( D j − k \ { } j − k ) × { } n − j . Then X k ∩ X k = U k , X k ∪ X k = U k − , so there is a boundary map in the Mayer-Vietoris sequence associated to X k , X k which goes from H j − k ( U k − ) to H j − k − ( U k ). We are going to show, by induction,that the k -th iterated image of ∂ ( D j × { } n − j ) under these maps is S k × ∂ ( D j − k ) × { } n − j ∈ H j − k − ( U k , C ) . The boundary map in the Mayer-Vietoris sequence can be decomposed (see [Dol80],p. 49) as H j − k ( U k − ) −→ H j − k ( U k − , X k ) ∼ = H j − k ( X k , U k ) −→ H j − k − ( U k ) . Writing S k − × ∂ ( D j − k +1 ) × { } n − j = S k × D j − k × { } + S k − × D × ∂ ( D j − k ) × { } n − j we see that the second summand is contained in X k , so is trivial in homology group H j − k ( U k − , X k ). The first summand belongs to H j − k ( X k , U k ), so it remains totake its boundary, which is exactly S k × ∂ ( D j − k ) × { } n − j .Therefore h c , ∂ ( D j × { } n − j ) i = Z S j (2 π i) − j dz z ∧ . . . dz j z j = 1 , which means that the constructed class c DR ( Y ) is indeed the Thom class of Y .It is clear now that the following theorem is true: Theorem 2.5.2.
Let X be a smooth variety over C of dimension n . Let Y ⊂ X bea smooth closed subvariety of X of codimension j . Then there is a class c F ( Y ) ∈ H jY ( X, F j Ω • X ) which satisfies the following conditions: i) The image c DR ( Y ) of c F ( Y ) in H jY ( X, C ) is the Thom class of Y .(ii) The class c H ( Y ) , which is the image of c F ( Y ) in H jY ( X, Ω jX ) , is logarithmic,i.e. for any open set V ⊂ X and a holomorphic function f on V which is zeroon Y ∩ V the product f · c H ( Y ) | V is . We extend the definition of c F , c DR , c H to formal linear combinations of subva-rieties in the obvious way. We show how to interpret the results of the previous section using smooth forms. Forthis we show how local cohomology can be computed using Dolbeault resolutions.Let X be a smooth variety over C and Y be a subvariety of codimension j , U = X \ Y .Let be the inclusion U → X . Proposition 2.5.3.
Let S be a soft sheaf on X which locally has no nonzero sectionssupported on Y . Then H pY ( X, S ) and H pY ( S ) are zero unless p = 1 , H Y ( X, S ) = Coker(Γ(
X, S ) −→ Γ( U, S )) , H Y ( S ) = Coker( S −→ ( S | U )) . Proof.
For any open set V ⊂ X we have the long exact sequence for local cohomol-ogy: . . . −→ H pV ∩ Y ( V, S ) −→ H p ( V, S ) −→ H p ( V ∩ U, S ) −→ . . . . The groups H p ( V, S ) and H p ( V ∩ U, S ) vanish for p >
0. Hence H pV ∩ Y ( V, S ) vanishfor p >
1. This group also vanishes for p = 0 by the condition on S . For V = X this implies the statement about the groups H pY ( X, S ). Since H pY ( S ) is the sheafassociated to the presheaf ( V → H pV ∩ Y ( V, S )), this implies the statement for thegroups H pY ( S ).Therefore one can compute local cohomology in the following way: Proposition 2.5.4.
Let F • be a bounded complex of sheaves on X and F • → S • abounded soft resolution, such that each S i locally has no nonzero sections supportedon Y . Then H iY ( X, F • ) ∼ = H i ( S • ( X, U )) , H iY ( F • ) ∼ = H i ( S • X,U ) , where S • X,U is the complex of sheaves defined as follows: S iX,U = S i ⊕ ∗ ( S i − | U ) , d ( a, b ) = ( da, − db − a | U ) , where a , b are sections of S i , ∗ ( S i − | U ) , and S • ( X, U ) = Γ(
X, S • X,U ) .Proof. Consider the following spectral sequence: E pq = H p ( H qY ( X, S • )) ⇒ H p + qY ( X, S • ) .
46y the proposition above the spectral sequence degenerates and H i ( X, S • ) ∼ = H p − (Coker(Γ( X, S • ) −→ Γ( U, S • )))= H p (cone(Γ( X, S • ) −→ Γ( U, S • ))[ − H p ( S • ( X, U )) . The statement about H iY ( F • ) can be proved similarly.In particular the proposition works for the following resolutions:Ω jX −→ ( A j • X , ¯ ∂ ) , F j Ω • −→ ( F j A • X , d ) . We now give the second proof of Theorem 2.5.1.
Proof.
Let j = n − k + 1, the codimension of Z and the dimension of W i . By thedefinition (*) of AJ k, we have( AJ k, [ x ] , cl n − k +1 Z ) = X i Z W i \ γ i ω log f i + 2 π i Z ξ ω, where ω ∈ F n − k +1 A n − k +2 ( X ) is a form whose class [ ω ] equals to the Poincar´edual of the class [ Z ] ∈ H k − ( X, Z ).In the definition of the Abel-Jacobi map we choose γ k and ξ to be transversal to | Z | . This means that γ k does not intersect | Z | for each k and ξ intersects | Z | onlyin several points. Let U = X \ | Z | .Let ( ω, η ) ∈ F j A j ( X, U ) be a representative of c F ( Z ). Then ( ω, η ) is also arepresentative of c DR ( Z ), the Thom class of Z . This means that ω ∈ F j A j ( X ), η ∈ F j A j − ( U ), ω = − dη on U , and for any 2 j -chain c on X with boundary in U Z c ω + Z ∂c η = [ c ] · Z. It is clear that we may choose ω as a representative of [ Z ].Since ( ω, η ) represent the Thom class of Z , we have the following identity: Z ξ ω + Z ∂ξ η = [ ξ ] · Z, where [ ξ ] ∈ H j ( X, U ) is the class of ξ . Therefore2 π i Z ξ ω ≡ − π i Z γ η mod 2 π i Z and we have the following decomposition:( AJ k, [ x ] , cl n − k +1 Z ) = X i ( Z W i \ γ i ω log f i − π i Z γ i η ) . n ( γ i ) be a small neighborhood of γ i in W i .Then, up to terms which tend to zero when the radius of the neighborhood tends tozero, we may write 2 π i Z γ i η = Z ∂n ( γ i ) η log f i and Z W i \ γ i ω log f i − π i Z γ i η = Z W i \ n ( γ i ) ω log f i + Z ∂ ( W i \ n ( γ i )) η log f i . We see that this is nothing else than the pairing of classes[ W i \ n ( γ i )] ∈ H j ( W i \ γ i , W i \ ( γ i ∪ Z i )) , [( ω | W i log f i , η | W i \ Z i log f i )] ∈ H jZ i ( W i \ γ i ) , where Z i = | Z | ∩ W i . Indeed, d ( η | W i \ Z i log f i ) = − ω | W i \ Z i log f i since η ∈ F j ( U ), so the pair ( ω | W i log f i , η | W i \ Z i log f i ) defines a class in the localcohomology.For each point p in the finite set W i ∩ | Z | we choose a small neighborhood n ( p ) of p in W i in such a way that the n ( p ) do not intersect each other and do not intersect γ i . Then Z W i \ n ( γ i ) ω log f i + Z ∂ ( W i \ n ( γ i )) η log f i = X p ∈| Z |∩ W i ( Z n ( p ) ω log f i + Z ∂n ( p ) η log f i ) . Since the dimension of n ( p ) is j , F j Ω • n ( p ) is just Ω jn ( p ) in degree j . Consider themultiplication by the function log f i − log f i ( p ), which acts on Ω jn ( p ) . Since the class[( ω, η ) | n ( p ) ] ∈ H jp ( n ( p ) , Ω j ) is the restriction of the class c H ( Z ), which is logarithmic,it is logarithmic itself, so the multiplication by log f i − log f i ( p ) kills it. Hence Z n ( p ) ω log f i + Z ∂n ( p ) η log f i = Z n ( p ) ω + Z ∂n ( p ) η ! log f i ( p )= ord p ( Z · W i ) log f i ( p )and the assertion follows. Remark . One could also consider the product CH k ( X, × CH n − k +1 ( X ) → CH n +1 ( X, P i ( W i , f i ) , Z ) to X i X p ∈ W i ∩| Z | ord p ( Z · W i ) ( p, f i ( p )) . CH n +1 ( X, −→ H ( X, C ) ∗ π i H ( X, Z ) ∼ = C / π i Z sending ( p, a ), p ∈ X , a ∈ C × to log a , so the theorem proved above simply meansthat the following diagram commutes: CH k ( X, × CH n − k +1 ( X ) −−−−→ CH n +1 ( X, AJ k, × cl n − k +1 ( X ) y AJ n +1 , y ( F n − k +1 H n − k +2 ( X, C )) ∗ π i H k − ( X, Z ) ×× ( F n − k +1 H n − k +2 ( X, C ) ∩ H n − k +2 ( X, Z )) −−−−→ C / Z There is a more general statement that the regulator map from higher Chow groupsinto the Deligne cohomology is compatible with products. This is mentioned in[Blo86]. The construction of higher Chow groups there is different from the oneabove, but it can be proved that they are canonically isomorphic and the regulatormap corresponds to the Abel-Jacobi map (see [KLMS06]).49 hapter 3
Derivatives of the Abel-Jacobimap
This chapter consists of two parts. In the first part we explain how we representcohomology classes using algebraic forms.For any open cover of a topological space X and any sheaf F of abelian groupson X one has the ˇCech complex, which is a complex formed by the groups of sectionsof F over all possible intersections of the sets of the cover. We use a slightly moregeneral approach which involves hypercovers. Roughly speaking, hypercover is asystem of open subsets of X organised in a certain way so that for any sheaf ofabelian groups F we obtain a complex of abelian groups formed by the groups ofsections of F over the sets of the hypercover. Note that the open subsets in thedefinition of the ˇCech complex are naturally parametrised by the faces of a simplex.Similarly, the open subsets of a hypercover are parametrised by the cells of a certaincell complex σ . In our examples σ will be a cube.If X is an algebraic variety with a hypercover U for the Zariski topology, oneobtains a complex Ω i ( U ) • of groups corresponding to the sheaf Ω iX of algebraic deRham forms of degree i for each i . Thus we obtain a bicomplex. The total complex ofthis bicomplex is denoted Ω • ( U ) and the elements of Ω i ( U ) are called “hyperforms”of degree i . Thus a hyperform of degree i is given by several forms of degree i oncertain open subsets of X , several forms of degree i −
1, etc. There is a canonicalhomomorphism from H i (Ω • ( U )) to the algebraic de Rham cohomology H i ( X, Ω X ) of X , which is an isomorphism if U is composed of affine open sets. Dually one obtainsa notion of “hyperchains” so that one can integrate a hyperform along a hyperchain.We obtain the Stokes theorem for hyperforms, and certain structures on hyperforms,namely the Hodge filtration, products, and the Gauss-Manin connection if X variesin a family. We emphasize that the hyperchains are composed of topological chains onthe manifold X ( C ) with its analytic topology, while the hyperforms are composed of algebraic forms. This way of representing cohomology classes is extremely convenientwhen X is a product of curves because one can easily write down representatives forclasses obtained by external multiplication of classes on curves, while the classes oncurves can be represented by differentials of second kind .The definition and properties of hyperforms are given in Sections 3.1 –3.1.7 and50.1.13.In Sections 3.1.8 –3.1.12 we study certain residues and trace maps associatedwith hypercovers, which are first defined as certain integrals, and then, in the casewhen our variety is embedded in a product of curves and the hypercover is theproduct hypercover, we give a way to compute these residues and trace maps in analgebraic way using iterated residues.Our trace maps are defined in the case when X is a product of curves and on eachcurve a Zariski cover is given. Suppose a finite family M of irreducible subvarieties of X is given. The elements of M will be called special . All further constructions do notchange if one adds varieties to M , so one possible way to think is to consider limitsover all possible M , but we will not pursue this point of view in order to simplifythe exposition. Let Z be a special variety and let Z denote the complement in Z ofall proper special subvarieties of Z . Then we construct a family of classes h a ( Z ) ofdimension dim Z in the homology of Z ( a runs over the set of dim Z -dimensionalcells of σ ). Then the residues res int Z,a are defined for forms from Ω dim Z ( Z ) as theintegrals along h a ( Z ) (“int” stands for “integral”). The trace map Tr int Z is a linearfunctional on the space of hyperforms Ω Z ( U ∩ Z ), obtained by summing up theresidues of the corresponding components of a hyperform (see Section 3.1.8).Then we compute these residues algebraically in terms of iterated residues. Weintroduce Tr Z (without “int”) as a certain combination of iterated residues so thatTr int Z is a simple multiple of Tr Z (see Section 3.1.12). In the end of Section 3.1.12we summarize by formulating a recipe for computation of the trace map.The second part of this chapter is motivated by the following idea. Supposewe have a family of varieties { X s } s ∈ S depending on certain parameters. Supposewe have a family of higher cycles { x s ∈ Z k ( X s , } s ∈ S . Let us denote the value ofthe Abel-Jacobi map on x s by AJ s . The element AJ s belongs to the quotient of acohomology group by a subspace and a lattice. One notices that one can “kill” thelattice by differentiating AJ s with respect to parameters.The objects AJ s are defined in terms of their pairing with cohomology classesfrom certain step in the Hodge filtration. In fact for any hyperform ω of right degree(without the condition that ω belongs to a certain step in the Hodge filtration) wemay define ( AJ s , ω ), but this value depends on some choices. We obtain two naturalformulae (Propositions 3.2.3, 3.2.4). The first one gives d ( AJ s , ω ) − ( AJ s , ∇ ω ) ( ∇ is the Gauss-Manin derivative), and the second one gives ( AJ s , dη ), where η is ahyperform of degree one less. Note that our Gauss-Manin derivative is actually alift of the ordinary Gauss-Manin derivative, which is defined on cohomology classes,to hyperforms, and it depends on some choices.In Section 3.2.4 we define two maps Ψ and Ψ , which give values of ( AJ s , dη )and d ( AJ s , ω ) − ( AJ s , ∇ ω ) correspondingly. Although the definition of AJ s is purelytranscendental, the maps Ψ and Ψ can be computed algebraically using the tracemap. The map Ψ is defined on hyperforms of degree 2 n − k + 2 ( n is the dimensionof X ) and takes values in the sheaf of differential 1-forms on the base S . The mapΨ is defined on hyperforms of degree 2 n − k + 1 and takes values in the sheafof functions on the base S . The maps Ψ and Ψ satisfy a list of axioms (seePropositions 3.2.5, 3.2.6) which allows us to construct a sheaf of D -modules, whichis an extension of the cohomology sheaf H n − k +2 ( X s , C ) by the structure sheaf of S .51his extension contains all the information about the derivative of the Abel-Jacobimap.The results mentioned above are stated only for the case when X s is a productof curves because they depend on our definition of trace maps.Next, in Section 3.2.5 we specialise to the case when X s is the product of an ellip-tic curve by itself 2 k − D -modules mentionedabove corresponds a certain invariant Ψ ′ an ( B k − ), which becomes a meromorphicmodular form of weight 2 k if the family is an open subset of a modular family.Finally, in Section 3.2.6 we show that if the modular form obtained from theextension of D -modules obtained from a family of higher cycles is proportional tothe modular form g k, H / Γ z , obtained by taking derivatives of the Green function, then(in the case when there are no cusp forms) we obtain a formula for the values of theGreen function, which proves the algebraicity conjecture (Section 1.5) for the given k , Γ, z , and arbitrary second CM point z (see Theorem 3.2.9). Definition 3.1.1. An abstract cell complex is a graded set σ = [ i ≥ σ i together with homomorphisms d i : Z [ σ i ] −→ Z [ σ i − ] such that d i ◦ d i +1 = 0 and ε ◦ d = 0, where ε is the augmentation map Z [ σ ] −→ Z . The elements of the set σ i are called cells of dimension i and the homomorphisms d i are called boundary maps . · · · d −−−−→ Z [ σ ] d −−−−→ Z [ σ ] d −−−−→ Z [ σ ] ε y Z If an abstract cell complex is given we denote it usually by σ and encode theboundary maps by the coefficients D ab : da = X b ∈ σ i − D ab b, ( a ∈ σ i ) . The basic example is the standard simplex of dimension n , ∆ n . In this case σ i isthe set of all subsets of { , , . . . , n } of size i + 1 and, writing subsets as increasingsequences, d i ( j , . . . , j i ) = i X k =0 ( − k ( j , . . . , ˆ j k , . . . , j i ) . Given two abstract cell complexes σ , σ ′ the product σ ′′ = σ × σ ′
52s again an abstract cell complex in the following way: σ ′′ i = i [ j =0 σ j × σ ′ i − j ,d i ( a × b ) = ( d j a ) × b + ( − j a × d i − j b for a ∈ σ j , b ∈ σ ′ i − j .For example, the cube is defined as (cid:3) n = ∆ × · · · × ∆ , the product has n terms. Let X be a topological space and σ an abstract cell complex. Definition 3.1.2. A hypercover of X (indexed by σ ) is a system of open subsets U = ( U a ) indexed by elements a ∈ σ such that:(i) Whenever a cell b belongs to the boundary of a cell a , U a ⊂ U b .(ii) For any point x ∈ X consider all cells a such that x ∈ U a . These cells form asubcomplex of σ by the first property, call it σ x . We require the complex ofabelian groups coming from σ x to be a resolution of Z with the augmentationmap induced by ε . Example . Put σ = ∆ n and let X be a space covered by n + 1 open subsets U , . . . U n . For any sequence ( j , . . . , j i ) ∈ σ i we put U ( j ,...,j i ) = U j ∩ · · · ∩ U j i . This clearly gives a hypercover which is called the ˇCech hypercover.If σ and σ ′ are two abstract chain complexes, ( U a ) and ( U ′ a ′ ) are hypercovers ofspaces X , X ′ indexed by σ and σ ′ correspondingly, by making all products ( U a × U ′ a ′ )one gets a hypercover of X × X ′ indexed by σ × σ ′ , the product hypercover. Let X be a topological space, σ an abstract chain complex and U = ( U a ) a hyper-cover.Let Z a denote the constant sheaf with fiber Z on U a extended by zero to X . Let Z U i = M a ∈ σ i Z a . If U a ⊂ U b , there is a canonical morphism r ab : Z a → Z b . We define d i : Z U i → Z U i − .If a ∈ σ i then the morphism from Z a to Z U i − is X b ∈ σ i − D ab r ab , where53 a = X b ∈ σ i − D ab b. Denote the corresponding sequence of sheaves by Z U . The augmentation map ε : Z U → Z X is defined as the sum of the canonical morphisms ε a : Z a → Z X for a ∈ σ . · · · d −−−−→ Z U d −−−−→ Z U d −−−−→ Z U ε y Z X Proposition 3.1.1.
The sequence Z U is a resolution of Z X .Proof. For any point x ∈ X the stalk of Z U at x is simply the complex of abeliangroups corresponding to σ x . So the statement follows from the second condition ofhypercover.In other words, we have obtained a quasi-isomorphism ε : Z U → Z X . For any sheaf or complex of sheaves F on X this gives a morphism of complexes ε ∗ : Hom ( Z X , F ) → Hom ( Z U , F ) . Note that
Hom ( Z X , F ) = Γ( X, F ) and for any a ∈ σ Hom ( Z a , F ) = Γ( U a , F ).We denote the complex Hom ( Z U , F ) by Γ( U , F ) and call elements of this complex hypersections .The complex of hypersections can be defined more precisely as follows. Let F be a complex of sheaves on X written as F → F → . . . . Then we put Γ( U , F ) i = Y j ≥ ,a ∈ σ j Γ( U a , F i − j ) . The coboundary of a hypersection s = ( s a ) ∈ Γ( U , F ) i is a hypersection ds = ( s ′ a ) ∈ Γ( U , F ) i +1 , where s ′ a = ds a + ( − i − j +1 X b ∈ σ j − D ab s b | U a , a ∈ σ j . (3.1)One can check directly that d = 0. Let d s = ( s ′′ a ), a ∈ σ j . s ′′ a = ds ′ a + ( − i − j X b ∈ σ j − D ab s ′ b | U a = ( − i − j +1 X b ∈ σ j − D ab ds b | U a + ( − i − j X b ∈ σ j − D ab ds b | U a + X b ∈ σ j − X c ∈ σ j − D ab D bc s c | U a = 0 . ε ∗ : Γ( X, F ) → Γ( U , F ) is defined by sending asection s ∈ Γ( X, F i ) to the hyperchain ( s ′ a ) where s ′ a is not zero only for a ∈ σ andequals to the restriction of s to U a . Remark . In the case of the ˇCech hypercover (corresponding to an open cover)we obtain the ˇCech complex.
Remark . If I took the definition of
Hom from [Har66], p. 64 the sign in(3.1) would be ( − i +1 . I assume that switching from a chain complex Z U to thecorresponding cochain complex adds the multiplier ( − j . My choice of sign is madein order to get the sign correct in the case of the ˇCech hypercover. We assume some class of topological chains on X is given, i.e. semialgebraic chains.For any open set U ⊂ X we denote by C i ( U ) the group of chains with support on U of dimension i , which is the free abelian group generated by maps of the chosen classfrom the standard simplex of dimension i to U . Suppose an abstract cell complex σ and a hypercover ( U a ) is given.The complex of hyperchains is defined similarly to the complex of hypersections,but with a change of sign. We put C i ( U ) = M j ≥ C i − j ( U , j ) , where C i ( U , j ) = M a ∈ σ j C i ( U a ) . Given a hyperchain ξ ∈ C i − j ( U a ) ⊂ C i ( U ) for a ∈ σ j its boundary is ∂ h ξ := ∂ξ + ( − i − j X b ∈ σ j − D ab j b ( ξ ) . Here ∂ξ ∈ C i − − j ( U a ) is the ordinary boundary of ξ and j b ( ξ ) denotes the samechain as ξ , but considered as an element of C i − j ( U b ) if U a ⊂ U b . The definition ofthe boundary map is then extended to C i ( U ) by linearity.The augmentation morphism ε : C ( U ) → C ( X ) sends all chains in C i − j ( U a ) with j > j = 0 to themselves.We have the following lifting property for lifting chains to hyperchains. Herethe word subdivision of a chain or of a hyperchain means some iterated barycentricsubdivision of all of its simplices. It is clear that the operation of taking subdivisioncommutes with the boundary operation. Lemma 3.1.2.
Let ξ be a chain on X , η = ∂ξ , ¯ η a hyperchain such that ε ¯ η = η and ∂ h ¯ η = 0 . Then, after possibly replacing η , ¯ η , ξ with a subdivision, there existsa hyperchain ¯ ξ such that ∂ h ¯ ξ = ¯ η and ε ¯ ξ = ξ . ¯ ξ ∈ C i ( U ) ∂ h −−−−→ ¯ η ∈ C i − ( U ) ε y ε y ξ ∈ C i ( X ) ∂ −−−−→ η ∈ C i − ( X )55 roof. The complex C • ( U ) is the total complex of the bicomplex C • ( U , • ) with thehorizontal differential induced by the boundary map of chains in space X and verticalone induced by the boundary map of the complex σ : C i − ( U , −−−−→ C i − ( U , y y C i ( U , −−−−→ C i − ( U , −−−−→ C i − ( U , ε y ε y C i ( X ) −−−−→ C i − ( X )Clearly, it is enough to prove that the vertical complexes are exact (up to a subdi-vision). That is, we need to prove that if γ = X a ∈ σ j γ a ∈ C i ( U , j ) is such that X a ∈ σ j D ab γ a = 0 for any b ∈ σ j − ,then there exists ¯ γ = X c ∈ σ j +1 ¯ γ c ∈ C i ( U , j + 1) such that X c ∈ σ j +1 D ca ¯ γ c = γ ′ a for any a ∈ σ j ,where γ ′ a is a subdivision of γ a .It is enough to prove the statement for multiples of a single simplex. Let s be asimplex in X which enters γ a with coefficient s a ∈ Z . Then X a ∈ σ j D ab s a = 0 for any b ∈ σ j − .Therefore the cycle P a s a a of σ is closed. Since the simplex s must belong to U a for all a for which s a = 0, the mentioned cycle is a closed cycle of σ x for all x in theclosure of s . For each x one can therefore represent it as a boundary of some cycleof σ x , say t x , t x = X c ∈ σ j +1 t xc c. Here t xc is zero unless x ∈ U c . Let V x be the open set which is the intersection of all U c for which t xc is not zero. Then V x form a cover of the support of s , so there existsa finite subcover. Let it be V = V x , V = V x , . . . , V k = V x k . One can subdividethe simplex s so that each simplex of the subdivision belongs to one of the chosenopen sets. Let us do this and denote the subdivision by s ′ so that s ′ = k X l =1 s ′ l , | s ′ l | ⊂ V k . γ l = X c ∈ σ j +1 t x l c j c ( s ′ l )because s ′ l belongs to U c for every c for which t x l c is not zero. Then X c ∈ σ j +1 D ca t x l c s ′ l = s a s ′ l for any a ∈ σ j ,therefore ¯ γ := P kl =1 ¯ γ l satisfies the required condition.This immediately implies Corollary 3.1.3. If ξ is a closed chain on X then there exists a closed hyperchain ¯ ξ whose augmentation is a subdivision of ξ . The hyperchain ¯ ξ is unique up to aboundary of a hyperchain whose augmentation is .Remark . One extends the construction of hyperchains to the cases of chainson an open subset of X and to the relative chains. It is clear that the boundarymaps commute with the natural projections from chains to relative chains. We again fix a space X , an abstract cell complex σ and a hypercover ( U a ). Supposesome class of chains and some class of differential forms are given such that onecan integrate a form along a chain. We have the de Rham complex of sheaves ofdifferential forms Ω • : Ω → Ω → . . . . Also we have the complex of chains on X : . . . → C ( X ) → C ( X ) . The hypersections of Ω • will be called the hyperforms . Given a hyperform ω ofdegree d and a hyperchain ξ of degree d we define the integral : Z ξ ω = X a ∈ σ Z ξ a ω a . Here ω a ∈ Γ( U a , Ω d − dim a ) and ξ a ∈ C d − dim a ( U a ) are the components of ω and ξ .The main property is the Stokes theorem: Proposition 3.1.4.
For a hyperform ω of degree d and a hyperchain ξ of degree d + 1 one has Z ∂ h ξ ω = Z ξ dω. roof. By the definition Z ξ dω = X a ∈ σ Z ξ a dω a + ( − d − dim a +1 X b ∈ σ dim a − D ab ω b . Applying the Stokes formula this is further equal to X a ∈ σ Z ∂ξ a ω a + ( − d − dim a +1 X b ∈ σ dim a − D ab Z ξ a ω b = X a ∈ σ Z ∂ξ a ω = Z ∂ h ξ ω. The Hodge filtration on hyperforms is induced by the Hodge filtration on the deRham complex: F i Ω • ( U ) = Ω •≥ i ( U ) , so a hyperform belongs to F i if all of its components which are forms of degree lessthan i are zero. For two spaces X , X ′ , two abstract complexes σ , σ ′ , two hypercovers ( U a ), U ′ a ′ wehave the product hypercover. There is exterior product on chains and on sheaves.For example, consider the exterior product on chains. For two chains c ∈ C i ( X ), c ′ ∈ C i ′ ( X ) we obtain the product chain c × c ′ ∈ C i + i ′ ( X × X ′ ). This operationsatisfies the property ∂ ( c × c ′ ) = ( ∂c ) × c ′ + ( − i c × ∂c ′ . Introduce an exterior product on hyperchains. Let c ∈ C i − j ( U a ) ⊂ C i ( U ), c ′ ∈ C i ′ − j ′ ( U ′ a ′ ) ⊂ C i ′ ( U ′ ). Here dim a = j , dim a ′ = j ′ . Then put c × h c ′ := ( − j ( i ′ − j ′ ) c × c ′ ∈ C i + i ′ − j − j ′ ( U a × U ′ a ′ ) ⊂ C i + i ′ ( U × U ′ ) . The superscript h stands for “hyper”.One can check that this satisfies Proposition 3.1.5. ∂ h ( c × h c ′ ) = ( ∂ h c ) × h c ′ + ( − i c × h ∂ h c ′ . On hypersections the exterior product is defined in a similar way. If s = ( s a ) ∈ Γ( U , F • ) i and s ′ = ( s ′ a ′ ) ∈ Γ( U ′ , F ′• ) i ′ , then( s × s ′ ) a × a ′ = ( − dim a ( i ′ − dim a ′ ) s a × s ′ a ′ . One has a similar formula
Proposition 3.1.6. d ( s × s ′ ) = ( ds ) × s ′ + ( − i s × ds ′ . .1.8 Residues Let X be a projective algebraic variety over C , σ an abstract cell complex and U = ( U a ) a hypercover of X in the Zariski topology. Definition 3.1.3.
By a refinement of U we understand any hypercover U ′ = ( U ′ a )of X in the analytic topology such that U ′ a ⊂ U a for any a ∈ σ .Suppose a finite family M of irreducible subvarieties of X is given. These sub-varieties will be called special . Definition 3.1.4.
A refinement U ′ = ( U ′ a ) is called nice if for any cell a ∈ σ andany special subvariety Z such that dim Z < dim a one has U ′ a ∩ Z = ∅ .If we have a nice refinement U ′ = ( U ′ a ) we can make the following construction.Fix a special subvariety Z . Take the fundamental class of Z , represent it by a chain(of dimension 2 dim Z ) and lift it to a closed hyperchain ( c a ) with respect to thehypercover ( U ′ a ∩ Z ). Note that for a ∈ σ dim Z +1 the set U ′ a ∩ Z is empty. Thereforefor every a ∈ σ dim Z the chain c a is a closed chain of dimension dim Z . Moreover, ifwe choose a different representation of the fundamental class or a different lift, thehyperchain will differ by a boundary. This implies that the chain c a for a ∈ σ dim Z will change by a boundary. Thus the class of c a in the homology of Z ∩ U ′ a does notdepend on the choices made. Denote this class by h a ( Z, U ′ ) ∈ H dim Z ( Z ∩ U ′ a ). Definition 3.1.5.
For any meromorphic differential form ω on Z ∩ U a of degreedim Z which is regular outside the special subvarieties of Z , for a cell a of dimensiondim Z , we define its residue asres int a,Z, U ′ ω = Z h a ( Z, U ′ ) ω. Correspondingly, if we have a meromorphic hyperform of degree 2 dim Z on Z ,then it is just a family of meromorphic forms of degree dim Z indexed by the cellsof dimension dim Z . Thus a choice of the refinement gives a trace map: Definition 3.1.6.
For any meromorphic hyperform ω on Z of degree 2 dim Z whichis regular outside the special subvarieties of Z we define its trace asTr int Z, U ′ ω = X a ∈ σ dim Z res int a,Z, U ′ ω a . In the next section we construct a nice refinement for the case when X is aproduct of curves and a finite Zariski cover is chosen for each of these curves. So wewill omit the subscript U ′ . We will construct some nice refinements in certain special situation and relate thecorresponding residues with the ordinary residues.Let X be a product of smooth projective curves, X = X × · · · × X n . Supposeon each curve a finite open cover (in the Zariski topology) is given. As we have seen59efore, this gives a hypercover of X k indexed by the standard simplex of dimension,say m k , the ˇCech hypercover. By taking products we obtain an abstract cell complex σ = ∆ m × ∆ m × · · · × ∆ m n and the product hypercover U of X indexed by σ . For any finite family M ofirreducible subvarieties of X we construct a nice refinement of U . Here we describethe construction.Without loss of generality we may assume that the set of special subvarieties M satisfies the following conditions:(i) For any open set U a belonging to the hypercover the irreducible componentsof its complement are in M .(ii) For any two sets in M the irreducible components of their intersection is alsoin M .(iii) If M contains a set Z , then it also contains the singular locus of Z .(iv) If M contains a set Z , then for any subset L ⊂ { , . . . , n } M contains theirreducible components of the set where the projection from Z to the product × k ∈ L X k is not ´etale (this may be the whole set Z ).Let X k be one of the curves above. Choose a Riemannian metric on X k . Let n ( S, ε ) denote the ε -neighborhood of a set S . For any k we give a refinement of thehypercover of the curve X k . This depends on two real numbers ε > ε ′ > X k is covered by the open sets U k, , . . . , U k,m k . Denote the complement X k \ U k, by S k . The set S k is a finite set of points p , p , . . . , p r .Consider those points among p , p , . . . , p r which are covered by U k, . Supposethey are p , p , . . . , p r . Then consider those points among the remaining ones whichare covered by U k, . Suppose they are p r +1 , . . . , p r , etc. In this way we obtain adecomposition of the set S k into m k subsets, some of them empty: S k = m k [ i =1 S k,i . Let f S k = S k ∪ { η k } , where η k is the generic point of X k . Let S k, = { η k } . For p ∈ f S k put U pk ( ε, ε ′ ) = ( n ( p, ε ) if p is a closed point, p ∈ S k , X k \ n ( S k , ε ′ ) if p is the generic point.For p ∈ S k put R pk ( ε, ε ′ ) = U pk ( ε, ε ′ ) ∩ U η k k ( ε, ε ′ ) , R k ( ε, ε ′ ) = [ p ∈ S k R pk ( ε, ε ′ ) ,U Sk ( ε, ε ′ ) = [ p ∈ S k U pk ( ε, ε ′ ) .
60e define the refinement in the following way. Put U ′ k,i ( ε, ε ′ ) = [ p ∈ S k,i U pk ( ε, ε ′ ) , i = 1 , . . . , m k . This is a cover of X k . We obtain U ′ k ( ε, ε ′ ) as the ˇCech hypercover associated to thiscover. If ε is small enough, this is a refinement of the original cover, i.e. U ′ k,i ⊂ U k,i .Moreover, if ε is small enough, the open sets U pk for p ∈ S k are non-intersectingdisks. Let q > ε < q these two conditions aresatisfied.For p ∈ f S k we define by a k ( p ) the 0-cell a such that p ∈ S k,a . For p ∈ S k wedefine by a k ( η k , p ) the 1-cell a which joins a k ( η k ) and a k ( p ). Then for any cell a ofthe standard simplex ∆ m k the elements of the hypercover are given as follows: U ′ k,a ( ε, ε ′ ) = S p ∈ f S k ,a k ( p )= a U pk ( ε, ε ′ ) if dim a = 0, S p ∈ S k ,a k ( p,η k )= a R pk ( ε, ε ′ ) if dim a = 1, ∅ otherwise.We consider vectors of real numbers ~ε = ( ε , . . . , ε n , ε ′ , . . . , ε ′ n ) such that for each k , q > ε k > ε ′ k >
0. Denote U ( ~ε ) = × nk =1 U k ( ε k , ε ′ k ) . This is a refinement of U . We denote for any L ⊂ { , . . . , n } R L ( ~ε ) = × k ∈ L R k ( ε k , ε ′ k ) , U SL ( ~ε ) = × k ∈ L U Sk ( ε k , ε ′ k ) , S L = × k ∈ L S k . For any p ∈ S L we denote p k = π k p and put R pL ( ~ε ) = × k ∈ L R p k k ( ε k , ε ′ k ) . We will simply write R L , U SL , R pL when there is no confusion. We put X L = × k ∈ L X k , π L : X → X L the projection. Definition 3.1.7.
We say that the choice of real numbers ε m , . . . , ε n , ε ′ m , . . . , ε ′ n is good if q > ε k > ε ′ k > k = m, . . . , n and there is an increasing sequence ofpositive real numbers δ m , δ ′ m , δ m +1 , δ ′ m +1 , . . . , δ n , δ ′ n such that the following conditions are satisfied for any special set Z , an index k ( m ≤ k ≤ n ) and p ∈ S k :(i) If x ∈ Z is such that dist( π k ( x ) , p ) ≤ ε k , then dist( x, Z ′ ) < δ k .(ii) If x ∈ X is such that dist( x, Z ) < δ ′ k − and dist( π k ( x ) , p ) < ε k , then dist( x, Z ′ ) <δ k .(iii) If x ∈ X is such that dist( x, Z ′ ) < δ ′ k − , then dist( π k ( x ) , p ) ≤ ε ′ k .61nd the following condition is satisfied for any two special sets Z , Z and an index k , m ≤ k ≤ n :(iv) If x ∈ X is such that dist( x, Z ) < δ k and dist( x, Z ) < δ k , then dist( x, Z ∩ Z ) < δ ′ k .We have denoted by π k the projection X → X k and by Z ′ the intersection Z ∩ π − k p .The second and the third conditions are required only for k > m .Next we prove that good choices indeed exist. For this it is enough to show Lemma 3.1.7.
For every good choice of real numbers ε m +1 , . . . , ε n , ε ′ m +1 , . . . , ε ′ n there exists a number t > such that for any ε m , ε ′ m satisfying t > ε m > ε ′ m > thechoice ε m , . . . , ε n , ε ′ m , . . . , ε ′ n is good. Therefore good sequences exist.Proof. Suppose there is a sequence δ m +1 , . . . , δ ′ n as in the definition above. Consider Z ∈ M and Z ′ = Z ∩ π − m +1 S m +1 . We first show that we can choose δ ′ m such thatthe second and the third conditions are satisfied for k = m + 1.Consider the set V = { x ∈ X : dist( x, Z ′ ) < δ k or dist( π m +1 ( x ) , S m +1 ) > ε m +1 } . This is an open neighborhood of Z by the first condition for k = m + 1. Thereforeif δ ′ m > δ ′ m -neighborhood of Z is also contained in V .Therefore the second condition is satisfied for k = m + 1.Since Z ′ is a compact set and π m +1 ( Z ′ ) ⊂ S m +1 , if δ ′ m > δ ′ m > δ ′ m < δ m +1 and both the second andthe third conditions are satisfied for all Z and k = m + 1.Next we choose δ m such that the fourth condition is satisfied. To see that thiscan be done for Z , Z we consider the compact set X \ n ( Z ∩ Z , δ ′ m ). The twocontinuous functions dist( • , Z ) and dist( • , Z ) do not attain simultaneously valuezero on this set. Therefore if δ m is small enough, these functions cannot attainsimultaneously value less than δ m . This is equivalent to the fourth condition.Next consider Z ∈ M and Z ′ = Z ∩ π − m S m . The set { x ∈ Z : dist( x, Z ′ ) ≥ δ m } is compact, hence the continuous function d ( x ) = dist( π m ( x ) , S m ) attains its min-imum. Suppose ǫ m is less than the minimal value of this function. It followsthat if x ∈ Z and d ( x ) ≤ ǫ m , then x cannot belong to the set above. Thereforedist( x, Z ′ ) < δ m and the first condition is satisfied. This implies existence of t > Recall that X = X × · · · × X n , each X k is covered by a finite number of open sets U k, , . . . , U k,m k , S k is the complement to U k, , f S k = S k ∪{ η k } , where η k is the genericpoint of X k . For a set L ⊂ { , . . . , n } we denote S L = × k ∈ L S k . Recall that a k ( p )62s defined as the minimal number such that p ∈ U k,a k ( p ) for p ∈ S k and a k ( p ) = 0for p = η k . The number a k ( p ) is viewed as a vertex of the m k -dimensional simplex.Also we have a k ( η k , p ) for p ∈ S k defined as the pair a k ( η k ) , a k ( p ), which gives anedge of the m k -dimensional simplex and corresponds to the open set U k, ∩ U k,a k ( p ) .We consider flags of subvarieties of X . A flag of length m is a sequence ofirreducible subvarieties Z • = ( Z ⊃ Z ⊃ · · · ⊃ Z m ). We say that a flag Z • starts with Z and ends with Z m . We require Z m to be not empty. Definition 3.1.8.
Let L ⊂ { , . . . , n } , L = { k , . . . , k l } . Let p ∈ S L . We say that aflag Z • = Z ⊃ · · · ⊃ Z l is L, p -special if(i) Z i is special for 0 ≤ i ≤ l ,(ii) Z i is an irreducible component of Z i − ∩ π − k i p k i for 1 ≤ i ≤ l . Definition 3.1.9.
We say that a flag Z • = Z ⊃ · · · ⊃ Z l is strict at index i if Z i = Z i − . We say that a flag is strict if it is strict at all indices.For a d -dimensional irreducible subvariety Z of X and a subset L ⊂ { , . . . , n } ofsize d the finite set of all strict L, p -special flags starting with Z for all p is denotedFl L ( Z ).If Z • ∈ Fl L ( Z ), L = { k , . . . , k d } , we construct a dim Z -dimensional cell a L ( Z • )of σ (recall that σ is the product of simplices ∆ m × · · · × ∆ m n ) in the following way.For each k ∈ L let i be such that k i = k and peek the edge a k ( η k , π k ( Z i )) (note that π k ( Z i ) ∈ S k ). For k / ∈ L peek the vertex a k ( p k ( Z • )), where p k ( Z • ) ∈ f S k is definedas π k ( Z i ) if this is a point or η k otherwise, where i is the maximal number such that k i < k or 0 if k > k . The product of these edges and vertices gives a cell in σ ,which we denote by a L ( Z • ).Now we prove that the refinements constructed in the last section are nice. Sup-pose ~ε is good for a sequence of numbers ~δ = ( δ , δ ′ , . . . , δ n , δ ′ n ). The ~δ -neighborhood of an L -special flag Z • is defined to be the set n ( Z • , ~δ ) = { x ∈ X : dist( x, Z i ) < δ k i , i ≥ } . Proposition 3.1.8.
Suppose Z ⊂ X is special, L ⊂ { , . . . , n } , x is a point forwhich either x ∈ Z or dist( x, Z ) < δ ′ min L − (if min L > ). Suppose p ∈ S L and π L ( x ) ∈ U pL ( ~ε ) . Then there exists an L, p -special flag Z • starting with Z such that x belongs to its ~δ -neighborhood.Proof. We construct the flag step by step. Ifdist( x, Z i − ) < δ ′ k i − ≤ δ ′ k i − , dist( π k i x, p k i ) < ε k i , by the second property of good sequences we obtaindist( x, Z i − ∩ π − k i p k i ) < δ k i . Therefore on each step we can choose Z i as an irreducible component of Z i − ∩ π − k i p k i so that dist( x, Z i ) < δ k i . 63 roposition 3.1.9. Let x , Z , L and p be as in the proposition above. Take an L, p -special flag Z • such that x belongs to the ~δ -neighborhood of Z • . If L ′ ⊂ L issuch that π L ′ ( x ) ∈ R L ′ ( ~ε ) , then Z • is strict at all indices i for which k i ∈ L ′ .Proof. If k i ∈ L ′ , then dist( π k i , p k i ) > ε ′ k i . By the third property of good sequenceswe obtain dist( x, Z k i ) ≥ δ ′ k i > δ ′ k i − . Since dist( x, Z k i − ) < δ k i − , Z k i − = Z k i . Corollary 3.1.10. If ~ε is a good choice of numbers, then for any special subvariety Z of dimension less than d and L ⊂ { , . . . , n } of size d the intersection Z ∩ π − L R L ( ~ε ) is empty. Therefore the refinement U ′ ( ~ε ) is nice.Proof. If the intersection was not empty, by Proposition 3.1.9 there would exist astrict
L, p -special flag which starts with Z . Therefore the dimension of Z would beat least d . Let X , σ , U , M be as in the previous section. Let Z be a subvariety of X , Z ∈ M .Let d = dim Z and ω be a meromorphic differential form on Z of degree d which isregular outside the special subvarieties of Z . Let a ∈ σ d . Let U ′ = U ( ~ε ) be the nicerefinement of U corresponding to a good choice of numbers ~ε . We will show how tocompute the residue res a, U ′ ω .Recall that the residue was defined as the integral of ω along the class h a ( Z, U ′ ) ∈ H d ( Z ∩ U ′ a ). Since σ = n Y k =1 ∆ m k , the cell a is given by a sequence of cells a ∈ ∆ m , . . . , a n ∈ ∆ m n . The open set U ′ a is the product of sets U k,a k ( ε k , ε ′ k ) for k = 1 , . . . , n . Note that for k = 1 , . . . , n theset U k,a k ( ε k , ε ′ k ) is not empty only if either a k is a point or a k is the edge joining the0 vertex and some other vertex. Therefore we get a non-zero residue only if there isa set L ⊂ { , . . . , n } of size d anddim a k = ( k ∈ L ,0 otherwise.Moreover in this case we have U ′ a ⊂ π − L R L .We have seen that any point of Z ∩ π − L R L belongs to the ~δ -neighborhood of aflag from Fl L ( Z ). In fact we have Proposition 3.1.11.
Any point x ∈ Z ∩ π − L R L belongs to the ~δ -neighborhood of aunique flag from Fl L ( Z ) .Proof. Suppose x belongs to the ~δ -neighborhood of two flags Z • and Z ′• . Let i bethe minimal index for which Z i = Z ′ i . Since dist( x, Z i ) < δ k i and dist( x, Z ′ i ) < δ k i ,by the fourth property of good sequences we have dist( x, Z i ∩ Z ′ i ) < δ ′ k i . Let Y bean irreducible component of Z i ∩ Z ′ i such that dist( x, Y ) < δ ′ k i . Let K be the subsetof elements of L which are greater than k i . By Proposition 3.1.9 one can construct64 strict K -special flag starting from Y whose ~δ -neighborhood contains x . But onecan see that the length of the flag equals to the dimension of Z i and is at leastone more than the dimension of Y . Hence such flag does not exist and we obtain acontradiction.This gives us a possibility to decompose Z ∩ π − L R L according to the set Fl L ( Z ). Corollary 3.1.12.
The set Z ∩ π − L R L is the union of non-intersecting open sets R L,Z • = Z ∩ π − L R L ∩ n ( Z • , ~δ ) , one for each Z • ∈ Fl L ( Z ) . Correspondingly we obtainthe decomposition res int a ω = X Z • ∈ Fl L ( Z ) res int a,L,Z • ω, where res int a,L,Z • ω = Z h a ( Z, U ′ ) ∩ R L,Z • ω. On the other hand, the set Z ∩ π − L R L is a union of non-intersecting open sets U ′ a for a running over the products of cells × nk =1 a k with dim a k = 1 for k ∈ L anddim a k = 0 for k / ∈ L . It appears that each R L,Z • is contained in exactly one such U ′ a and we will see that a = a L ( Z • ).The last variety in the flag, Z d , is a point from S L . Proposition 3.1.13.
Suppose x ∈ R L . If x ∈ R L,Z • and π L ( x ) ∈ R pL , then p = Z d .Proof. Since x ∈ Z ∩ π − L R pL , one has an L, p -flag whose ~δ -neighborhood contains x . By Proposition 3.1.11 this flag must be Z • . Hence Z • is L, p -special. Thus Z d = p .If x ∈ Z ∩ R L then π k ( x ) for any k ∈ L c belongs to U p ′ k k ( ε, ε ′ ) for exactly one p ′ k ∈ f S k . If it was not true, then x would belong to R L ∪{ k } , which is a contradiction.This defines a point p ′ k ( x ) ∈ f S k for each k ∈ L c . Proposition 3.1.14.
For any x ∈ Z ∩ R L if x ∈ R L,Z • , then p k ( Z • ) = p ′ k ( x ) for all k ∈ L c .Proof. Let K be the set of such k that either k ∈ L or p ′ k ∈ S k (i.e. p ′ k is not thegeneric point). For k ∈ L let p ′ k be such that dist( π k ( x ) , p ′ k ) < ε k . This defines apoint p ′ ∈ S K . Then for every k ∈ K dist( π k ( x ) , p ′ k ) < ε k . Thus π K ( x ) ∈ U p ′ L .By Proposition 3.1.8 there exists a K, p ′ -special flag Z ′ starting from Z whose ~δ -neighborhood contains x . This flag must be strict for all indices which correspondto elements of L . Since the number of this indices equals the dimension of Z the flagmust be not strict at all other indices. Therefore this flag defines a L, p ′ -special flag,which must be Z by Proposition 3.1.11. At the same time this shows that p k = p ′ k for all k ∈ K .Let k / ∈ K . Then dist( π k ( x ) , S k ) > ε ′ k . Suppose i is the maximal index for which k i < k or 0 if k < k . If i = 0, by the third property of good sequencesdist( x, Z k i ∩ π − k S k ) ≥ δ ′ k − ≥ δ k i ≥ dist( x, Z k i ) . Therefore Z k i = Z k i ∩ π − k S k which means that p k = η k . The case i = 0 is obvious.65e see that Corollary 3.1.15.
For any cell a of dimension d the set Z ∩ U ′ a is the disjoint unionof open sets R L,Z • where Z • runs over all Z • ∈ Fl L ( Z ) with a L ( Z • ) = a .Remark . Therefore we may omit a in the notation res int a,L,Z • . Note that Z and X L have the same dimension. Proposition 3.1.16.
If the restriction of π L to Z is not surjective, then the set Fl L ( Z ) is empty.Proof. If not, then dim π L ( Z ) < d . Let L = { k , . . . , k d } . Let Z • ∈ Fl L ( Z ). Considerthe corresponding flag of irreducible subvarieties of X L : π L ( Z ) = π L ( Z ) ⊃ π L ( Z ) ⊃ · · · ⊃ π L ( Z d ) . Because of the dimension reasoning there must be an index i with π L ( Z i ) = π L ( Z i − ).This implies π k i ( Z i − ) = π k i ( Z i ) ⊂ S k i . Therefore Z i = Z i − , so the flag is not strict, which is a contradiction.We may therefore suppose without loss of generality that π L : Z → X L issurjective. Hence the set of points on Z where this map is not ´etale is a properclosed subvariety. The irreducible components of this subvariety have dimension d − π − L R L ∩ Z belongs to its complement. Thismeans the following is true. Proposition 3.1.17.
The projection π L : π − L R L ∩ Z → R L is an unramified covering. Let p = π L Z d ∈ S L . Since for a flag Z • ∈ Fl L ( Z ) the set R L,Z • is open andclosed in π − L R L ∩ Z we get Proposition 3.1.18.
The projection π L : R L,Z • → R pL is an unramified covering.Remark . By the proposition we see that since R pL is a product of annuli, R L,Z • is a disjoint union of products of annuli.We are going to determine the cycle h a ( Z, U ′ ) ∩ R L,Z • ∈ H d ( R L,Z • ). The d -thhomology group of a product of d annuli is Z . Hence there is a canonical generator h c of H d ( R pL ). In fact h c can be defined as the product c × · · · × c d , where c i is thecircle in R p ki k i going around p k i counterclockwise.66 roposition 3.1.19. The cycle h a ( Z, U ′ ) ∩ R L,Z • is the pullback of ( − d ( d − h c viathe projection π L : R L,Z • → R pL .Proof.Remark . It is clear that we can decrease numbers ε k and increase numbers ε ′ k .The sequence obtained in this way will be also good. Moreover the open subsetsof the new hypercover are contained in the corresponding open subsets of the oldone. Therefore the residues computed with respect to these hypercovers are equal.Therefore one can assume that the projection π L : R L,Z • → R pL extends to anunramified covering for the closures of R L,Z • in Z , R pL in X L .Consider the commutative diagram: H d ( X ) −−−−→ H d ( R L,Z • , ∂R L,Z • ) π ∗ L ←−−−− H d ( R pL , ∂R pL ) h a y h a y h a y H d ( U ′ a ) −−−−→ H d ( R L,Z • ) π ∗ L ←−−−− H d ( R pL )We see that it is enough to prove that the image of the fundamental class of H d ( R pL , ∂R pL ) by the map h a in H d ( R pL ) is ( − d ( d − h c . There is a direct productdecomposition R pL = × k ∈ L R p k k . The hypercover on R pL is the product hypercover.For k ∈ L let φ k be the fundamental class of H ( R pL , ∂R pL ). Let us lift it to ahyperchain f φ k .The hypercover of R pL is the ˇCech hypercover associated to the cover with twoopen sets: V = n ( p k , ε k ) \ n ( p k , ε k ) corresponding to the cell a ( η k ) ,V = n ( p k , ε k ) \ n ( p k , ε k ) corresponding to the cell a ( p k ) ,V ∩ V = n ( p k , ε k ) \ n ( p k , ε k ) corresponding to the cell a ( η k , p k ) . Let ε ′ < r < ε . Consider topological chains c k = n ( p k , ε k ) \ n ( p k , r ) ∈ C ( V ) , c k = n ( p k , r ) \ n ( p k , ε ′ k ) ∈ C ( V ) ,c k = ∂n ( p k , r ) ∈ C ( V ∩ V ) . They define a hyperchain c k . We have c + c = φ k . The hyperchain is closed,therefore h ( φ k ) = c k , which is the canonical generator of H ( R p k k ).Since the product of φ k is the fundamental class of H d ( R pL , ∂R pL ), the prod-uct of the hyperchains c = × k ∈ L c k lifts the fundamental class. The term of c at × k ∈ L ( η k , p k ) is, by the definition of the product for hyperchains, ( − d ( d − × k ∈ L c k .This is exactly ( − d ( d − h c .Let Z • be a strict L, p -special flag, L = { k , . . . , k d } , p = ( p k , . . . , p k d ). Let Z ′ = Z , k = k , L ′ = { k , . . . , k d } . Let Z ′• be the flag Z ⊃ · · · ⊃ Z d , Z ′• ∈ Fl L ′ ( Z ′ ).Let t i be a local parameter on X k i at the point p k i = π k i Z d . Let p ′ = π L ′ p . For Z ′ we have the projection π L ′ : R L ′ ,Z ′• → R p ′ L ′ , e Z = Z ′ × X L ′ Z = ( X k × Z ′ ) × X L Z. We obtain the canonical diagrams e Z ρ −−−−→ Z e Z ρ −−−−→ Z τ y y e τ y y Z ′ −−−−→ X L ′ ( X k × Z ′ ) −−−−→ X L The diagonal embedding Z ′ → Z ′ × X L ′ Z ′ induces a morphism ∆ : Z ′ → e Z , whichis a section to the natural projection τ : e Z → Z ′ . Let R Z = Z ∩ π − L ( U p k k × R p ′ L ′ ) ∩ n ( Z • , ~δ ) . Consider the fiber products over U p k k × R p ′ L ′ : e R ρ −−−−→ R Z e τ y π L y U p k k × R L ′ ,Z ′• −−−−→ U p k k × R p ′ L ′ Again, we have a section ∆ : R L ′ ,Z ′• → e R . Let e R ′ be the union of connectedcomponents of e R which intersect the image of ∆. Let ρ ′ be the restriction of ρ to e R ′ . Remark . The set R Z is open and closed in Z ∩ π − L ( U p k k × R p ′ L ′ ) because Propo-sition 3.1.11 and the first sentence of Corollary 3.1.12 work if we replace R L by U p k k × R p ′ L ′ . Remark . All special subsets of Z which intersect R Z are contained in Z ′ .Therefore the map π L on the diagram is an unramified covering outside { p k } × R p ′ L ′ and π − k p k ∩ R Z = R L ′ ,Z ′• . Proposition 3.1.20.
The map ρ ′ is an analytic isomorphism.Proof. Since ρ ′ is a base change of the unramified covering π L ′ : R L ′ ,Z ′• → R p ′ L ′ , itis an unramified covering itself. Therefore it is enough to construct a continuoussection s ′ : R Z → e R ′ to ρ ′ which extends the diagonal map. This is equivalent toconstructing a retraction s : R Z → R L ′ ,Z ′• which respects the projection π L ′ .Take a compact connected set A ⊂ R p ′ L ′ such that π − L ′ ( A ) ∩ R L ′ ,Z ′• = A ∪· · ·∪ A m is a disjoint union of spaces isomorphic to A .One can choose disjoint open subsets V , . . . , V m in π − L ′ ( A ) ∩ R Z such that A i ⊂ V i . The set C = π − L ′ ( A ) ∩ R Z \ ( V ∪ · · · ∪ V m ) is closed. Therefore π L ( C ) is closed.Take α > n ( p k , α ) × A does not intersect π L ( C ). This means that theopen set π − L ( n ( p k , α ) × A ) ∩ R Z is contained in the union of the sets V i .Let V ′ i = π − L ( n ( p k , α ) × A ) ∩ V i . Let us prove that V ′ i is connected for each i . Ifnot, then V ′ i = B ⊔ B with B j open, closed and nonempty. Since A i is connected,68or some j B j does not intersect A i . Therefore π L B j is open, closed and nonempty.Hence it must be the whole U p k k × A . This contradicts the assumption that B j doesnot intersect A i .One can construct a deformation retract retracting π − L ′ ( A ) ∩ R Z inside π − L ( n ( p k , α ) × A ) ∩ R Z . Therefore there are exactly m connected components of π − L ′ ( A ) ∩ R Z , eachcontaining exactly one A i . Therefore there is a unique map s A : π − L ′ ( A ) ∩ R Z → π − L ′ ( A ) ∩ R L ′ ,Z ′• which is identity on π − L ′ ( A ) ∩ R L ′ ,Z ′• and makes the diagram belowcommutative. π − L ′ ( A ) ∩ R Z −−−−→ U p k k × A s A y y π − L ′ ( A ) ∩ R L ′ ,Z ′• −−−−→ A Patching these s A together gives s as required proving the first statement of theproposition.Take a meromorphic form ω on Z which is holomorphic outside the specialsubvarieties of Z . The form ω can be written as ω = f dt ∧ dt ∧ · · · ∧ dt n , where f is a rational function on Z . Let K be the field of fractions of Z ′ . Then Z × Spec K is a curve and ∆(Spec K ) is a point. Therefore the algebraic residueres ∆(Spec K ) ρ ∗ f dt is defined. We have Proposition 3.1.21.
Put ω ′ = (res ∆(Spec K ) ρ ∗ f dt ) dt ∧ · · · ∧ dt n . Then res int
L,Z • ω = ( − d − π i res int L ′ ,Z ′• ω ′ . Proof.
By the definition res U ,L,Z • ω = Z h a ∩ R L,Z • ω. Since ρ ′ is an isomorphism, Z h a ∩ R L,Z • ω = Z ρ ′∗ ( h a ∩ R L,Z • ) ρ ∗ ω. We have ρ ′∗ ( h a ∩ R L,Z • ) = ( − d ( d − e R ′ ∩ ρ ∗ π ∗ L h c = ( − d ( d − e R ′ ∩ e τ ∗ ( h ck × ( R L ′ ,Z ′• ∩ π ∗ L ′ h ′ c )) , where h ck is the circle in R p k k and h ′ c is the product of circles in R p ′ L ′ . By Fubini’stheorem Z e R ′ ∩ f τ ∗ ( h ck × ( R L ′ ,Z ′• ∩ π ∗ L ′ h ′ c )) ρ ∗ ω = Z R L ′ ,Z ′• ∩ π ∗ L ′ h ′ c g ( z ′ ) dt ∧ · · · ∧ dt d , z ′ ∈ R L ′ ,Z ′• g ( z ′ ) = Z e R ′ ∩ τ − z ′ ∩ π − k h ck ρ ∗ f dt . The last integral is nothing else than2 π i res ∆( z ′ ) ρ ∗ f dt . Therefore g ( z ′ ) dt ∧ · · · ∧ dt d = 2 π i ω ′ Taking into account that R L ′ ,Z ′• ∩ π ∗ L ′ h ′ c = ( − ( d − d − R L ′ ,Z ′• ∩ h a ′ ,Z ′ , where a ′ is the cell obtained from a by replacing the component a ( η k , p k ) with thecomponent a ( p k ), we obtain the statement.Let us denote by res L,Z • ω the iterated algebraic residue of ω with respect to theflag Z • . This is defined by induction on the dimension of Z by the formulares L,Z • f dt ∧ · · · ∧ dt d = res L ′ ,Z ′• (res ∆(Spec K ) ρ ∗ f dt ) dt ∧ · · · ∧ dt d . Thus we obtain a formula for our residue.
Corollary 3.1.22. res int
L,Z • ω = ( − d ( d − (2 π i) d res L,Z • ω. For any subvariety Z ⊂ X and any hyperform using the decomposition of theresidue according to flags we also can state a formula for the trace: Corollary 3.1.23.
For a meromorphic hyperform ω of degree d on Z ⊂ X × · · · × X n which is holomorphic outside the special subvarieties of Z , dim Z = d , Tr int Z ω = ( − d ( d − (2 π i) d X L ⊂{ ,...,n } , | L | =dim Z X Z • ∈ Fl L ( Z ) res L,Z • ω a L ( Z • ) . Dropping the coefficient ( − d ( d − (2 π i) d we may also define the algebraic versionof the trace: Tr Z ω := X L ⊂{ ,...,n } , | L | =dim Z X Z • ∈ Fl L ( Z ) res L,Z • ω a L ( Z • ) . Then the latter corollary can be reformulated as follows:
Corollary 3.1.24.
For a meromorphic hyperform ω of degree d on Z ⊂ X × · · · × X n which is holomorphic outside the special subvarieties of Z , dim Z = d , Tr int Z ω = ( − d ( d − (2 π i) d Tr Z ω. Z ω . The necessary notationis explained in Section 3.1.10.(i) List all the subsets L ⊂ { , . . . , n } of size dim Z .(ii) For each L list all the flags Z • ∈ Fl L ( Z ).(iii) For each flag find the cell a L ( Z • ) of ∆ m × · · · × ∆ m n , which is a product ofseveral vertices and edges (vertices for k ∈ L c and edges for k ∈ L ).(iv) Compute the iterated residue corresponding to the flag Z • of the form whichis the component of ω on the open set U a L ( Z • ) = × k U k , where U k is an openset from the cover of X k for k ∈ L c , or an intersection of two open sets of thecover of X k for k ∈ L .(v) Add these residues. Suppose we have a morphism of algebraic varieties X → S . Let σ be an abstractcell complex and U = ( U a ) a hypercover on X with respect to Zariski’s topology.We assume that S = Spec R is affine and all the open sets of the hypercover areaffine.Associated to the de Rham complex on X we have the complex of hypersectionsΩ X ( U ) → Ω X ( U ) → · · · . Consider the complex of hypersections corresponding to the relative de Rham com-plex. Ω X/S ( U ) → Ω X/S ( U ) → · · · . These are both complexes of R -modules. Proposition 3.1.25.
The following natural sequence is exact for all k ≥ : Ω S ( S ) ⊗ R Ω k − X ( U ) → Ω kX ( U ) → Ω kX/S ( U ) → . Proof.
Since the corresponding sequence of sheaves is exact, it induces an exactsequence over any affine set.We define Gauss-Manin connection as follows. Let ω ∈ Ω kX/S ( U ) be closed. Liftit to ω ∈ Ω kX . Then dω ∈ Ker(Ω k +1 X ( U ) → Ω k +1 X/S ( U )). Choose η ∈ Ω S ( S ) ⊗ R Ω kX ( U )which maps to dω . Let η be the projection of η in Ω kX/S ( U ) ⊗ R Ω S ( S ). We say that η is a Gauss-Manin derivative of ω . Of course, this construction depends on severalchoices. Although η is not well-defined, by abuse of notation we will write η = ∇ ω if η is a Gauss-Manin derivative of ω .Suppose we have a family of hyperchains c s ∈ C k ( U ), s ∈ S . This means that c s is a linear combination of simplices c is with each c is being a map from ∆ × S to X X → S gives the projection ∆ × S → S . Here ∆ denotes astandard simplex. We require the maps c is to be of the same type as we require forsimplices, i.e. semi-algebraic. For ω ∈ Ω kX/S ( U ) consider the integral f ( s ) = Z c s ω, s ∈ S. For a path s s in S let c s s ∈ C k +1 ( U ) denote the corresponding hyperchain overthis path. It provides a homotopy: ∂ h c s s + ( ∂ h c ) s s = c s − c s . Lemma 3.1.26.
Let ω ∈ Ω kX/S ( U ) be a closed relative hyperform and c = ( c s ) s ∈ S be a family of hyperchains, c s ∈ C k ( U ) . If η = ∇ ω with η and ω as in the definitionof ∇ , then we have d Z c s ω = Z c s ∇ ω + R ( ∂ h c, ω ) . Here R is the bilinear operator which is constructed as follows. The value of R ona vector in S represented by a path s t is h [ s t ] , R ( ∂ h c, ω ) i = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Z ( ∂ h c ) s st ω. Proof.
For a fixed s ∈ S let s t be a path in S starting from s . We have f ( s t ) − f ( s ) = Z ∂ h c s st +( ∂ h c ) s st ω. The first summand can be transformed as Z ∂ h c s st ω = Z c s st η. Over the path we are considering Ω S is generated by dt . Therefore η = dt ∧ η . Weobtain Z c s st η = Z c s st dt ∧ η = Z c s st ( d ( tη ) − tdη ) = Z ∂ h c s st tη − Z c s st tdη . Examining the second term we see ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Z c s st tdη = 0 . The first one transforms to Z ∂ h c s st tη = Z c st − c s tη − Z ( ∂ h c ) s st tη . The second term gives ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Z ( ∂ h c ) s st tη = 0 . ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Z c st − c s tη = lim t → Z c st η = Z c s η . For example, if c s is closed: Corollary 3.1.27. If c s is a closed family of hyperchains and ω is a closed relativehyperform, then d Z c s ω = Z c s ∇ ω. By a curve we mean a smooth projective curve over C . Let X , X ,. . . , X n be curves.Put X = X × X × · · · × X n . Let x ∈ Z k ( X,
1) be a higher cycle, x = X i ( W i , f i ) . Recall that dim C W i = n − k + 1, f i ∈ C ( W i ), γ i = f ∗ i [0 , ∞ ], γ = P i γ i , ∂ξ = γ . TheAbel-Jacobi map was defined as h AJ k, [ x ] , [ w ] i = X i Z W i \ γ i ω log f i + 2 π i Z ξ ω (3.2)for w ∈ F n − k +1 A n − k +2 ( X ), dw = 0. Remark . Since X is a product of curves its cohomology has no torsion. Thisimplies that the class of γ is trivial. In fact we should justify the construction of γ i and the integration in (3.2). Theproblems are that W i are not necessarily smooth, the rational functions f i do notnecessarily define maps to P and log f i is not bounded. To define all the objects wemay embed X as a semi-algebraic set into some R N and consider the semi-algebraicsubsets W i , | div f i | , f − i [0 , ∞ ]. We may apply [Hir75] to get a triangulation of R N which is compatible with all the sets mentioned above. This triangulation issemi-algebraic and smooth on the interiors of simplices. Therefore we can integratesmooth forms over simplices. Moreover we have the necessary bounds on the growthof log f i restricted to any simplex which belongs to W i and does not belong to | div f i | .In fact any such simplex intersects | div f i | only along the boundary and log f i growsnot faster than some multiple of the logarithm of the distance to the boundary.We will only consider simplices obtained by a linear subdivision of simplices of73he constructed triangulation. Formal linear combinations of simplices of equaldimension are called chains.For each i let us consider the space V i constructed from W i by cutting out γ i and attaching two copies of γ i glued together along the boundary. Let us denotethe two copies of γ i by γ + i and γ − i and suppose that they are attached in such away that the function log f i extends to γ + i and γ − i , ∂V i = γ + i − γ − i , and the value oflog f i on γ − i is 2 π i plus the value on γ + i . Denote by ι + i , ι − i the natural isomorphisms γ i −→ γ + i , γ i −→ γ − i . Denote by p i the natural projection V i −→ W i . The space V i is naturally endowed with the triangulation coming from the triangulation of W i .Let l ≥ l ,∆ = { ( x , x , . . . , x l ) | l X j =0 x j = 0 , x j ≥ } . For ǫ > ǫ = { ( x , x , . . . , x l ) ∈ ∆ | x j ≥ ǫ } . Let σ : ∆ −→ V i be a simplex in V i . Let ω be a smooth l -form on a neighborhoodof σ (∆). Definition 3.2.1.
Suppose σ (∆) is not contained in | div f i | . Put Z σ ω log f i = lim ǫ −→ Z ∆ ǫ σ ∗ ω log f i . Definition 3.2.2.
A simplex σ is called good if it is not contained in | div f i | for all i and any simplex of its boundary is not contained in | div f i | for all i . A chain iscalled good if it is a linear combination of good simplices.One can check that the Stokes formula holds: Proposition 3.2.1. If σ is a good simplex and ω is a smooth l − -form on aneighborhood of σ (∆) , then Z σ dω log f i = Z ∂σ ω log f i − Z σ df i f i ∧ ω. Let us choose a Zariski affine cover on each of the curves X j this gives a hypercover U j on X j indexed by σ j , hence a product hypercover U on X indexed by σ . Thisinduces hypercovers on V i . Let us lift chains div f i to hyperchains ^ div f i , then liftthe chains γ i to hyperchains e γ i , then lift V i to e V i and ξ to e ξ . Lifting γ i determinesa lifting of γ + i , γ − i . These liftings are denoted e γ + i , e γ − i . We choose a nice analyticrefinement U ′ of U and assume that all the hyperchains we are considering are infact hyperchains on U ′ . The following relations hold: ∂ h e γ i = − ^ div f i , ∂ h e ξ = X i e γ i , ∂ h e V i = e γ + i − e γ − i . ω ∈ F n − k +1 A n − k +2 ( X ), dω = 0 we can denote by the same letter ω thecorresponding hyperform. We get h AJ k, [ x ] , [ ω ] i = X Z e V i ω log f i + 2 π i Z e ξ ω. Let L ⊂ { , . . . , n } , L ′ ⊂ { , . . . , n } such that L ∩ L ′ = ∅ , | L | = n − k + 1, | L ′ | = n − k + 1. Let ω k be a holomorphic 1-form on X k for k ∈ L (which isconsidered as a hyperform) and a closed algebraic 1-hyperform on X k for k ∈ L ′ .Recall that 1-hyperform on X k is a collection of 1-forms on the open sets of thecover of X k and functions on the pairwise intersections. Saying algebraic we requirethe functions and the 1-forms to be regular. For k ∈ L ′ by the Hodge theory onecan choose a smooth 0-hyperform g k such that ω ′ k = ω k − dg k is a smooth 1-form.Put ω ′ k = ω k for k ∈ L . We obtain a smooth closed form on Xω ′ = × k ∈ L ∪ L ′ ω ′ k ∈ F n − k +1 A n − k +2 ( X ) , and an algebraic closed hyperform ω = × k ∈ L ∪ L ′ ω k ∈ Ω n − k +2 ( U )which is zero on U a if dim a > n − k + 1. Moreover they differ by a coboundary, ω ′ − ω = dg, g ∈ Ω n − k +1smooth ( U ) , with g zero on U a if dim a > n − k . Proposition 3.2.2.
One can compute AJ k, using algebraic forms: h AJ k, [ x ] , [ ω ] i = X i Z e V i ω log f i + 2 π i Z e ξ ω. Proof.
By the definition h AJ k, [ x ] , [ ω ′ ] i = X i Z e V i ω ′ log f i + 2 π i Z e ξ ω ′ = X i Z e V i ω log f i + 2 π i Z e ξ ω + X i Z e V i ( dg ) log f i + 2 π i Z e ξ dg. The summand in the third term gives Z e V i ( dg ) log f i = Z e γ + i − e γ − i g log f i − Z e V i df i f i ∧ g. The second integral is zero. Indeed, if a ∈ σ is such that dim a > n − k , then ( df i f i ∧ g ) a is zero as it was mentioned above. Otherwise ( df i f i ∧ g ) a ∈ Ω n − k +2 − dim a ( U a ).Therefore the degree of this form is at least n − k + 2, which is greater than thedimension of V i . The first integral can be further transformed to Z e γ + i − e γ − i g log f i = − π i Z e γ i g Z e ξ dg = X i Z e γ i g. Suppose we have families of curves X i → S with S affine and X = X × S X × S · · · × S X n . Suppose we have a family of cycles x s ∈ Z k ( X s , ω ∈ Ω n − k +2 X/S ( U ). Suppose ω a = 0 if dim a > n − k + 1. Let us compute d h AJ k, [ x s ] , [ ω ] i . We first generalize it. Let for any ω ∈ Ω n − k +2 X/S ( U ) I ( ω ) = X i Z e V i ω log f i + 2 π i Z e ξ ω. We also define a new operation on hyperforms. Let ω ∈ Ω n − k +2 X ( U ). For any index i consider the hyperform df i f i ∧ ω . It is zero on all U a ∩ W i with dim a < n − k + 1and is a form of maximal degree on U a ∩ W i with dim a = n − k + 1. Therefore thereexists an element φ f i ω ∈ Ω ( S ) ⊗ Ω n − k +2 X ( U ) such that the difference df i f i ∧ ω − φ f i ω is zero on all U a ∩ W i with dim a ≤ n − k + 1. Its image in Ω ( S ) ⊗ Ω n − k +2 W i /S ( U )will be denoted by φ f i ω . Proposition 3.2.3.
We have dI ( ω ) = I ( ∇ ω ) + Z f W i φ f i ω. Proof.
Let η = ∇ ω . Then (see Lemma 3.1.26) d Z e ξ ω = Z e ξ η + X i R ( e γ i , ω ) . Consider the hyperform ω log f i on U ∩ V i . The restriction of ω log f i to U ′ ∩ V i isclosed simply because Ω n − k +3 W i /S ( U ′ ∩ W i ) = 0. We have d ( ω log f i ) = η log f i + df i f i ∧ ω. The sum η log f i + φ f i ω gives a Gauss-Manin derivative of ω log f i on V i with respectto the hypercover U ′ . Therefore d Z e V i ω log f i = R ( e γ + i − e γ − i , ω log f i ) + Z e V i η log f i + Z e V i φ f i ω. R ( e γ + i − e γ − i , ω log f i ) = − π i R ( e γ i , ω ) . The third summand equals to Z e V i φ f i ω = Z f W i φ f i ω. We also can compute I ( ω ) for an exact hyperform. Proposition 3.2.4.
Let ω = dη with η ∈ Ω n − k +1 X/S ( U ) . Then I ( ω ) = − X i Z W i df i f i ∧ η Proof.
By the definition I ( ω ) = X i Z e V i ω log f i + 2 π i Z e ξ ω. We have ω log f i = d ( η log f i ) − df i f i ∧ η . Applying the Stokes formula we obtain theresult.Let us denote J i ( ω ) = (2 π i) − n + k − Z e V i ω. for any ω ∈ Ω n − k +2 W i /S ( U ). We see that the following relations hold: dI ( ω ) = I ( ∇ ω ) + (2 π i) n − k +1 X i J i ( φ f i ω ) , I ( dη ) = − (2 π i) n − k +1 X i J i ( df i f i ∧ η ) . Note that the value of J i can be expressed as a certain sum of iterated residues.Therefore it is possible to compute J i . D -modules Recall that S is an affine variety. Suppose S = Spec R for a commutative ring R .Suppose S is smooth and Ω( R ) is free. Let D be the ring of differential operatorson S . Consider R -modules Ω iX ( U ). We have two filtrations on Ω iX ( U ). The firstis the Hodge filtration. Elements of F j Ω iX ( U ) are those hyperforms which have ascomponents only forms of rank at least j . We have another filtration defined as G j Ω iX ( U ) := Ω j ( R ) ∧ Ω i − jX ( U ). The exterior derivative respects these filtrations. Wehave Ω iX/S ( U ) = Ω iX ( U ) /G Ω iX ( U ) .
77o understand the results of the previous section we introduce two operationson hyperforms. The first one is Ψ : Ω n − k +1 X → R defined asΨ ( η ) = (2 π i) − n + k − X i Z f W i df i f i ∧ η ( η ∈ Ω n − k +1 X ( U )) . The second one is Ψ : Ω n − k +2 X → Ω( R ) defined asΨ ( ω ) = (2 π i) − n + k − X i Z f W i φ f i ω ( ω ∈ Ω n − k +2 X ( U )) , the operation φ being defined in the previous section. Remark . The integral with respect to f W i can be expressed as a sum of iteratedresidues as is proved in the previous chapter. Therefore the operations Ψ and Ψ can be defined purely algebraically Proposition 3.2.5.
The first operation satisfies the following properties:(i) Ψ is R -linear.(ii) Ψ | G Ω n − k +1 X ( U ) = 0 .(iii) Ψ | F n − k +1 Ω n − k +1 X ( U ) = 0 .(iv) Ψ ( η ) = − (2 π i) − n + k − I ( dη ) for η ∈ Ω n − k +1 X ( U ) .(v) If dη ∈ G Ω n − k +2 X ( U ) , then Ψ ( η ) = 0 . Proposition 3.2.6.
The second operation satisfies the following properties:(i) Ψ is R -linear.(ii) Ψ | G Ω n − k +2 X ( U ) = 0 .(iii) Ψ | F n − k +2 Ω n − k +2 X ( U ) = 0 .(iv) Ψ ( u ∧ η ) = − u Ψ ( η ) for u ∈ Ω( R ) , η ∈ Ω n − k +1 X ( U ) .(v) Ψ ( ω ) = (2 π i) − n + k − ( dI ( ω ) − I ( ∇ ω )) for ω ∈ Ω n − k +2 X ( U ) such that dω ∈ G Ω n − k +3 X ( U ) , and ∇ ω is defined as the class of dω in Ω( R ) ⊗ Ω n − k +2 X ( U ) .(vi) Ψ ( dη ) = d Ψ ( η ) for η ∈ Ω n − k +1 X ( U ) . Let B n − k +2 ⊂ Z n − k +2 ⊂ Ω n − k +2 X/S ( U ) be defined as B n − k +2 = Im d X/S , Z n − k +2 = Ker d X/S . Let H n − k +2 = Z n − k +2 /B n − k +2 . F i Ω n − k +2 X/S ∩ B n − k +2 = d X/S ( F i Ω n − k +1 X/S ) . Let M be defined as M = ( R ⊕ Z n − k +2 ) / Im(Ψ , d X/S ) . We then have the following commutative diagram with exact rows:0 −−−−→ B n − k +2 −−−−→ Z n − k +2 −−−−→ H n − k +2 −−−−→ Ψ y i y (cid:13)(cid:13)(cid:13) −−−−→ R i −−−−→ M −−−−→ H n − k +2 −−−−→ s : F n − k +1 H n − k +2 → M which is definedby sending a class [ ω ] ∈ F n − k +1 H n − k +2 ( ω ∈ Z n − k +2 ) to (0 , ω ).Both R and H n − k +2 are modules over D . On M we have the following D -module structure: v ( r, ω ) = ( v ( r ) − h v, Ψ ( ω ) i , h v, ∇ ω i ) , where r ∈ R , v ∈ Der( R ), ω ∈ Z n − k +2 , ω ∈ Ω n − k +2 X ( U ) represents ω . One cancheck that this indeed gives a D -module structure. The section s is compatible withthe D -module structure so that the following diagram commutes: F n − k +2 H n − k +2 ∇ −−−−→ Ω( S ) ⊗ F n − k +1 H n − k +2 s y id S ⊗ s y M −−−−→ Ω( S ) ⊗ M. Let us choose S smaller so that H n − k +2 is free and F n − k +1 H n − k +2 is a directsummand. Extend the section s to a section e s : H n − k +2 → M . This providesan isomorphism of R -modules M ∼ = R ⊕ H n − k +2 . We define a homomorphismΨ ∈ Hom R ( H n − k +2 , Ω( R )) in the following way. For any [ ω ] ∈ H n − k +2 putΨ([ ω ]) = ∇ e s ([ ω ]) − (id ⊗ e s )( ∇ [ ω ]) ∈ Ω( R ) . One can see that Ψ | F n − k +2 H n − k +2 = 0 and Ψ is correctly defined up to the dif-ferential of an element of Hom R ( H n − k +2 , Ω( R )). The structure of D -module on R ⊕ H n − k +2 induced from M can be recovered as follows: v ( r, h ) = ( v ( r ) + Ψ( h ) , ∇ v h ) ( h ∈ H n − k +2 , r ∈ R, v ∈ Der( R )) . Let N be the kernel of the multiplication homomorphism N := Ker( D ⊗ R H n − k +2 → H n − k +2 ) . It inherits the filtration F • from the one on H n − k +2 . We have the canonicalhomomorphism of D -modules Ψ ′ alg : F n − k +1 N → R defined as follows.Ψ ′ alg ( X j α j ⊗ h j ) = X j α j s ( h j ) ∈ R ( X j α j ⊗ h j ∈ N, h j ∈ F n − k +1 N, α j ∈ D ) .
79n the other hand we could have defined Ψ ′ an : F n − k +1 N → R Ψ ′ an ( X j α j ⊗ h j ) = X j α j h AJ k, [ x ] , h j i , but the properties of Ψ , Ψ imply: Corollary 3.2.7.
We have Ψ ′ an = (2 π i) n − k +1 Ψ ′ alg . Therefore Ψ ′ an can be computedalgebraically. Suppose we have a family of elliptic curves π : E → S and X i = E . We have S = Spec R . We suppose that R is a 1-dimensional domain and its field of fractionsis denoted as R . Let V = R π ∗ Ω • E/S . This is a locally free module over R of rank 2. The Hodge filtration has two pieces,each a rank 1 locally free R -module. Denote M = F V. This is a line bundle of modular forms of weight 1. We denote M j = ( M ) ⊗ j for j ∈ Z . We have the canonical pairing F V ⊗ R ( V /F ) → R. With the help of this pairing we identify
V /F ∼ = M − . Consider the Kodaira-Spencer map KS : M → Ω( R ) ⊗ M − . Suppose that the family E is not constant and S is small enough, so that KS isan isomorphism. Therefore we have Ω( R ) ∼ = M . Let H = R π ∗ Ω • E/S , which is a D -module of rank 2 over R .Suppose n = 2 k −
2. Then, using the notations of the previous section ( H n = R n π ∗ Ω • E n /S ) M n = F n H n − k +2 = F n H n . Let H n sym denote the direct summand of H n which corresponds to H n sym = Sym n H . We have the following fact:
Proposition 3.2.8.
There is a unique element B n ∈ M n +2 ⊗ R F n N which containsonly differential operators of degree at most n + 1 and the corresponding symbol in M n +2 ⊗ R Der( R ) ⊗ ( n +1) ⊗ R M n ∼ = R is . roof. Let us prove uniqueness. Let D n denote the differential operators of orderat most n . The following map is a monomorphism: m : D n ⊗ R F n H n → H n . To prove this look at the filtration by the order of differential operator on D n ⊗ R F n H n and the Hodge filtration on H n . The graded pieces of the map are theKodaira-Spencer maps which are injective.Existence follows from the fact that the image of m is Sym n H . So we can pickany element of M n +2 ⊗ R D n +1 ⊗ R M n with symbol 1 and then subtract an elementof M n +2 ⊗ R D n ⊗ R M n which maps to the same element in M n +2 ⊗ R Sym n H .One can apply Ψ ′ to B = B n and get an element(2 π i) − n + k − Ψ ′ an ( B ) = Ψ ′ alg ( B ) ∈ M n +2 . Therefore we have a canonical modular form constructed from a family of cycles { x s ∈ Z k ( X s , } s ∈ S . We still have n = 2 k −
2. Suppose we have a family of elliptic curves π : E → S overan affine smooth irreducible curve. We now translate the notions from Section 3.2.5to the analytic language. The elliptic curve over a point s ∈ S is denoted E s . Let U be an analytic subset in S homeomorphic to a disk. Choose families of 1-cycles c , c over U such that c ( s ) and c ( s ) generate H ( E s , Z ) and the intersection numberis c · c = 1.Any other choice c ′ , c ′ can be obtained from the choice c , c by a transformation γ = (cid:18) a bc d (cid:19) ∈ SL ( Z ) : c ′ = a c + b c , c ′ = c c + d c . Let ω be a closed relative differential 1-form on E . We denoteΩ ( ω ) = Z c ω, Ω ( ω ) = Z c ω. The cup product provides a pairing:( ω , ω ) = Z E s ω ∧ ω . Let us integrate ω = df over the universal cover of E s . Then( ω , ω ) = Z ∂ e E s f ω = Ω ( ω )Ω ( ω ) − Ω ( ω )Ω ( ω ) , where e E s denotes a fundamental domain of the universal cover of E s .If ω is holomorphic we put z = Ω ( ω )Ω ( ω ) . S and the upper half plane. Indeed,Im z = Ω ( ω )Ω ( ω ) − Ω ( ω )Ω ( ω )2iΩ ( ω )Ω ( ω ) = − Im R E s ω ∧ ω ( ω )Ω ( ω ) . If we represent E s as a quotient C / Λ with ω = dx + i dy , then ω ∧ ω = − dx ∧ dy, therefore Im z = vol ω E s | Ω ( ω ) | > , where vol ω E s is the volume of E s defined with the help of the form ω . For anotherchoice c ′ , c ′ we obtain z ′ = az + bcz + d . We define a canonical isomorphism of the analytic version of the sheaf M of modularforms of weight 1, as defined in the previous section, and the pullback via z of theusual sheaf of modular forms of weight 1 on the upper half plane: ω → Ω ( ω ) . Let X be a formal variable. We identify H ( E s , C ) with the space of polynomialsof degree not greater than 1 in X in the following way: h , c i = − , h , c i = 0 , h X, c i = 1 , h X, c i = 0 . Let ω be a closed differential 1-form. Then the corresponding polynomial is[ ω ] c = Ω ( ω ) X − Ω ( ω ) . In particular, if ω is holomorphic,[ ω ] c = Ω ( ω )( X − z ) . If c ′ = γ c , then one can check that[ ω ] c ′ = γ (Ω ( ω ) X − Ω ( ω )) , where the action on polynomials is defined as γ ( p ) = p | − γ − ( p = p X + p ) . Therefore the map ω → f ω = Ω ( ω ) X − Ω ( ω ) X − z = Ω ( ω ) + Ω ( ω ) − z Ω ( ω ) z − X defines an isomorphism between the sheaf V = H ( E, C ) and the pullback of thesheaf of quasi-modular forms of weight 1 and depth 1. Quasi-modular forms ofweight w and depth d are functions of the form f ( z, X ) = d X i =0 f i ( z ) i ! ( z − X ) i w in z and weight 0 in X: f ( γ ( z ) , γ ( X )) ( cz + d ) − w = f ( z, X ) . One can see that the pairing can be written as( aX + b, a ′ X + b ′ ) = − ab ′ + a ′ b. Therefore if a ( X − z ) ∈ F V and a ′ X + b ′ ∈ V , then( a ( X − z ) , a ′ X + b ′ ) = − a ( a ′ z + b ′ ) . If ω is a holomorphic differential and η an arbitrary closed differential, then( ω, η ) = f ω ( f η ) , where ( f η ) denotes the coefficient at ( z − X ) − of f η and is a modular form ofweight −
1. Therefore the isomorphism
V /F V → M − is given by sending f to f . The Gauss-Manin derivative with respect to the parameter z sends the coho-mology class with periods Ω , Ω to the cohomology class with periods ∂ Ω ∂z , ∂ Ω ∂z .Therefore on the level of quasi-modular forms it can be written as ∂∂z + 1 z − X .
If we take a modular form f of weight 1, the Gauss-Manin derivative of the corre-sponding family of cohomology classes will be given by (cid:18) ∂f∂z + fz − X (cid:19) dz. Therefore the
Kodaira-Spencer map sends f → f dz. If t is a local parameter on S then the Kodaira-Spencer map acts as f → f (cid:0) dtdz (cid:1) dt, therefore the isomorphism Ω( S ) → M acts by sending dt for a function t to themodular form dtdz .Next we construct the canonical differential operator from the previous section.Let U be an open subset in S such that there exist modular forms f of weight n and g of weight n + 2 with non-zero values on U . Let D ( f, g ) be the operator φ → g (cid:18) ∂∂z (cid:19) n +1 φf . This is a differential operator which sends functions to functions, therefore D ( f, g ) ∈ D n +1 ( U ). Moreover D ( f, g ) f = 0, therefore B n ( U ) = g ⊗ D ( f, g ) ⊗ f M n +2 ⊗ R F n N . Its symbol is g f gdz n +1 f, which goes to 1 under the isomorphism M n +2 ⊗ R Der( R ) ⊗ ( n +1) ⊗ R M n ∼ = R .Let us consider the natural map H k − F k H k − + 2 π i H k − Z → H k − F H k − + 2 π i H k − Z ∼ = M − k / π i M − k Z , where H k − Z ⊂ H k − is the subsheaf of integral cohomology classes and M − j Z forany j ≥ M − j generated by 1 , z, . . . , z j . The image of the section AJ k, [ x ] under this map will be denoted by A x . More explicitly, one can obtain A x by choosing a modular form f of weight 2 k −
2, and then dividing by f the pairingof A x with the holomorphic differential 2 k − f . It isclear that the operator (cid:18) ∂∂z (cid:19) k − : M − k → M k vanishes on M − k Z . Therefore it is defined on M − k /M − k Z and it is clear thatΨ ′ an ( B k − ) = (cid:18) ∂∂z (cid:19) k − A x ∈ M k . Also we have the canonical (non-holomorphic) section of F k − H k − . This isdefined by the polynomial-valued function Q k − z = (cid:18) ( X − z )( X − z ) z − z (cid:19) k − ∈ Sym k − V = H k − . Theorem 3.2.9.
Suppose S is a smooth affine curve. Suppose we have a familyof elliptic curves { E s } s ∈ S and an algebraic family of higher cycles { x s } s ∈ S , x s ∈ Z k ( E k − s , . Suppose there is a map ϕ from S to H / Γ which lifts the canonicalmap from S to H /SL ( Z ) . Suppose the symmetric part of AJ k, [ x ] is constant alongthe fibers of φ . Take the corresponding modular form A x of weight − k on theimage of φ defined locally up to polynomials in z of degree not greater than k − as above. Suppose (cid:18) ∂∂z (cid:19) k − A x = ( − k − D k − αg H / Γ k,z ( z ) for a CM point z of discriminant D and a non-zero rational number α . Then forany point z in the image of φ , φ ( s ) = z , G H / Γ k ( z, z ) = 2 α − (2 k − k − ℜ (cid:18) D − k ( AJ k, [ x ] , Q k − z ) (cid:19) . ake N A , N B , N as in Corollary 1.5.6. Suppose z is a CM point of discriminant D and make N A larger if necessary to satisfy N A ( k − α − ∈ Z . Then the algebraicityconjecture is true and one has ( DD ) k − b G k, H / Γ ( z, z ) ≡ α − log( x s · Z z ) mod 2 π i N Z , where Z z is a subvariety of E k − s which intersects x s properly and has cohomologyclass cl Z z = (2 k − k − D k − Q k − z , and · denotes the intersection number as in Theorem 2.5.1, which is an algebraicnumber if x s is defined over Q .Proof. Note that the function A x is holomorphic of weight 2 − k . Hence it is (locally)of type F k, − k (see Section 1.2). Therefore the following function is of type F k, : f := ( − k − ( k − k − δ k − A x . We obtain the function e f with values in V k − . The function e f satisfies (by Theorem1.2.5) ( e f , ( X − z ) k − ) = ( − k − (2 k − k − e f , δ − k Q k − z ) = A x . There is another formula for e f (see Proposition 1.2.6): e f = k − X i =0 ( X − z ) i i ! (cid:18) ∂∂z (cid:19) i A x . This formula shows that when we add to A x a polynomial p ( z ), the function e f changes by p ( X ). Therefore e f is defined up to elements of 2 π i V Z k − . On the otherhand, AJ k, [ x ] is a function with values in V k − / ( F k V k − + 2 π i V Z k − )( F k corresponds to the polynomials divisible by ( X − z ) k ). So the difference satisfies AJ k, [ x ] − e f ∈ F V k − / ( F k V k − + 2 π i V Z k − ) . Consider the local sections of F k − N given by ξ i = ∂∂z ⊗ ( X − z ) i + i ( X − z ) i − ( i ≥ k ) . It is easy to check that Ψ ′ alg ξ i = 0 using the property (iii) of the function Ψ .Therefore Ψ ′ an ξ i = 0. This means ∂∂z ( AJ k, [ x ] , ( X − z ) i ) + i ( AJ k, [ x ] , ( X − z ) i − ) = 0 ( i ≥ k ) . e f satisfies similar property. Therefore their difference also does. Since( AJ k, [ x ] − e f , ( X − z ) k − ) = 0 , we prove by induction that( AJ k, [ x ] − e f , ( X − z ) i ) = 0 ( i ≥ k − . Next we note that (see Theorem 1.2.5) d e f = ( X − z ) k − (2 k − (cid:18) ∂∂z (cid:19) k − A x ! dz. So if the hypothesis is true, d e f = ( − k − αD k − (2 k − X − z ) k − g H / Γ k,z ( z ) dz. Therefore one can choose (see Theorem 1.3.4 and Corollary 1.5.6) I A, Γ N A ( X − z ) k − g H / Γ k,z ( z ) dz = ( − k − N A α − (2 k − D − k e f , recall that A = 2 π i (2 k − k − D − k V Z k − . This shows that b G H / Γ k ( z, z ) ≡ α − (2 k − k − D − k ( e f , Q k − z ) mod 2 π i( DD ) − k N Z , so the statement follows from Theorem 2.5.1. Remark . A cycle with cohomology class (2 k − k − D k − Q k − z was constructed inthe introduction by taking the graphs of complex multiplication by az and a ¯ z ( a isthe leading coefficient of the minimal quadratic equation of z ), denoted Y az and Y a ¯ z respectively, and adding the products of k − Y az − Y a ¯ z for all possiblesplittings of the 2 k − E k − z into pairs. Remark . As we will show later (Section 5.3) for the case Γ =
P SL ( Z ), k = 2Corollary 1.5.6 gives N B = 2, N A = 1. Since N = ( k − N A N B this gives N = 2.But if the numerator of α is greater then 1, we take N A to be the numerator of α to satisfy the conditions of the theorem, so the statement will hold for N = 2 N A .86 hapter 4 Cohomology of elliptic curves
This chapter studies the Weierstrass family of elliptic curves y = x + ax + b. We first study expansions at infinity of various functions. As a coordinate we use theformal integral of the holomorphic differential form dx y . Also we note that the basering C [ a, b ] is isomorphic to the ring of modular forms for SL ( Z ) and we choose anisomorphism µ .In Section 4.2 we state the precise relation between periods of differential formsof second kind and values of quasi-modular forms.In Section 4.3 we choose lifts of vector fields on the base to vector fields on thetotal space of the family. It happens that particularly nice formulae can be obtainedfor lifts of the Euler vector field and the Serre vector field. Therefore it is naturalto choose these vector fields as a basis. We represent cohomology of elliptic curvesby two differential forms of second kind dx y and xdx y .In Section 4.4 we choose representatives of two cohomology classes, correspond-ing to the forms of second kind dx y and xdx y , as hyperforms on the total space.This choice satisfies an important property. Whenever we apply the Gauss-Maninderivative (see Section 3.1.13) to these representatives, the result is again a linearcombination of these representatives. We express the hyperforms at infinity. Indeed,for computation of residues later it will be enough to know only these expressions,the result does not depend on the global information. Let R = k [ a, b ] be the ring of polynomials in two variables a , b . Denote by K thefield of fractions of R . Let G m be the multiplicative group. Let G m act on R by thelaw a → λ a, b → λ b ( λ ∈ G m ) . We consider the family over R given by the equation y = x + ax + b. E over R given by the homoge-neous equation in e x , e y , e z : e y e z = e x + a e x e z + b e z . The action of G m extends to the action on E in the following way: e x → λ e x, e y → λ e y, e z → e z ( λ ∈ G m ) . Therefore the affine chart e z = 1 is stable under the action. We denote this chart by U . In fact E is an elliptic curve outside the zero locus of the discriminant∆ = − a + 27 b ) . If a rational function φ on E transforms according to φ → λ k φ ( λ ∈ G m ) , then we say that φ is of weight k . Let us denote the space of rational functions ofweight k by F k . The action of G m gives rise to the vector field whose derivationis the Euler operator, δ e . This operator acts on homogeneous rational functions asfollows: δ e f = kf ( f ∈ F k ) . We have the zero section s : Spec R → E given by sending e x → , e y → , e z → . Let t = − x/y = − e x/ e y ∈ F − . This is a local parameter at s . We can express x and y as Laurent series in t : x = t − − at − bt − a t − abt + O ( t ) ,y = − t − x = − t − + at + bt + a t + 3 abt + O ( t ) . The invariant differential form ω = dx y has expansion ω = dx y = (1 + 2 at + 3 bt + 6 a t + 20 abt + O ( t )) dt. Consider the formal integral of ω : z = Z ω = t + 2 a t + 3 b t + 2 a t + 20 ab t + O ( t ) . In fact z is the logarithm for the formal group law of the elliptic curve. We can nowtake z as a new local parameter and express x and y in terms of z : x = z − − a z − b z + a z + 3 ab z + O ( z ) ,y = ∂ ∂z x = − z − − a z − b z + a z + 12 ab z + O ( z ) . R and the ring of modular forms in thefollowing way: µ ( a ) = − E , µ ( b ) = E . Then the integral of − xdz can be expressed as follows: v = − Z xdz = z − + a z + b z − a z − ab z = z − − E z + E z − E z + E E z + O ( z )= z − + X k ≥ B k E k (2 k )! z k − . In fact, this follows from the corresponding identity over the complex numbers whichcan be proved using the Taylor expansion of the Weierstrass ℘ -function. We define v = v + E z = z − + X k ≥ B k E k (2 k )! z k − ∈ R [ E ](( z )) . Note that for a = − , b = (this corresponds to E = 1, E = 1 and thecurve is degenerate) we can find expansions of v , x and y explicitly: v = 1 e z − − z , x = e z ( e z − + 112 , y = − e z + e z e z − . This corresponds to the fact that the Fourier expansion of E k starts with 1. Let us view E as an elliptic curve over K . We will consider odd differential formson U . Each such form has an expansion of the type X k ∈ Z a k z k dz ( a k ∈ K ) . In fact such a form is determined by its coefficients a , a − , a − , . . . . Moreover, givena polynomial P there is a unique form which has Laurent expansion starting with P ( z − ) dz . The space of odd differential forms on U has basis ω k = x k dxy ( k ≥ . An odd function on U is a function of the form Q ( x ) y . It has expansion X k ∈ Z a k − z k − ( a k − ∈ K ) . Such a function is determined by its coefficients a − , a − , . . . . Conversely, for eachpolynomial P there exists a form which has Laurent expansion starting with z − P ( z − ).89t follows that the space of odd forms modulo the space of exact odd forms is2-dimensional with basis ω = dx y , η = xdx y . This space is canonically isomorphic to the first de Rham cohomology group of E .We denote it by H K . Proposition 4.2.1.
Consider the map which sends an odd differential form κ tothe following element of K [ E ] : µ ( κ ) := res( κv ) . This map vanishes on exact forms. Therefore it defines a map from H K to K [ E ] .Proof. Indeed, if f = Q ( x ) y , thenres( vdf ) = − res( f dv ) = res( Q ( x ) y ( x − E
12 ) dz )= 12 res( Q ( x )( x − E
12 ) dx ) = 0 . One has µ ( ω ) = 1 , µ ( η ) = E . Therefore using µ one can build an isomorphism of algebras over K :Sym H K / ( ω − −→ ∼ K [ E ] . The symbol Sym H K denotes the algebra of symmetric tensors of H K .For any G m -module M which is G m -equivariant over some G m -field we denoteby M (1) the same module but with the twisted action. If m M is the action on M then the action on M (1) is defined as follows: m M (1) λ a = λm Mλ a ( a ∈ M, λ ∈ G m . )Using this notation the isomorphism constructed above can be made into a G m -equivariant isomorphism: µ : Sym H K (1) / ( ω − −→ ∼ K [ E ] . In the following two propositions the value of a quasi-modular form f of weight k on a pair of numbers ω , ω ∈ C with τ = ω ω ∈ H is defined as follows: f ( ω , ω ) := (2 π i) k f ( τ ) ω − k . Proposition 4.2.2.
Let k = C . Let a , b ∈ C and f ∈ K a rational function whichis defined at the point ( a , b ) . Suppose f has weight k . Choose a basis c , c of thefirst homology for the curve y = x + a x + b . Let ω i = R c i ω with τ = ω ω ∈ H .Then f ( a , b ) = µ ( f )( ω , ω ) . roposition 4.2.3. Let k = C . Let a , b ∈ C and [ κ ] ∈ H K represented by anodd differential form κ which is defined at the point ( a , b ) . Suppose κ has weight k . Choose a basis c , c of the first homology for the curve y = x + a x + b . Let ω i = R c i ω with τ = ω ω ∈ H . Then Z c κ = ω µ ( κ )( ω , ω ) . Consider the Gauss-Manin connection on the module H K . It is a map ∇ : H K → Ω ( K/k ) ⊗ H K . This map is equivariant with respect to the G m -action. Consider ∇ ω . This is anelement of Ω ( K/k ) ⊗ H K . Let us view this element as a map ∇ [ ω ] : Der( K/k ) → H K . Since both spaces have dimension 2 and one can check that the map is surjective, itis an isomorphism. For example, one has ∇ [ ω ]( δ e ) = − [ ω ] . We would like to compute ∇ [ ω ] for a general vector field ∂ . Let ∂ ∈ Der(
K/k )be a derivation of the field K . By ∂ ∗ we denote a lift of ∂ to the affine set U . Weassume that ∂ ∗ is even, so it is given by an even function ∂ ∗ x and an odd function ∂ ∗ y . There is no canonical choice of this lift. On the other hand we let ∂ act onformal Laurent series in z simply by setting ∂z = 0. We obtain2 y ( ∂y − ∂ ∗ y ) = (3 x + a )( ∂x − ∂ ∗ x ) . Therefore there exists a Laurent series α with the property ∂y = ∂ ∗ y + αy ′ , ∂x = ∂ ∗ x + αx ′ , where ′ denotes the derivative of a Laurent series with respect to z . Since ∂ commuteswith ′ , we have2 ∂y = ∂x ′ = ( ∂ ∗ x ) ′ + αx ′′ + α ′ x ′ = ( ∂ ∗ x ) ′ + 2 ∂y − ∂ ∗ y + α ′ x ′ . Therefore α ′ x ′ = 2 ∂ ∗ y − ( ∂ ∗ x ) ′ . It is easy to see that the right hand side is a regular odd function on U , hence it is aproduct of y and a polynomial of x . Since x ′ = 2 y , we obtain that α ′ is a polynomialof x . Consider the form α ′ dz.
91t is a linear combination over K of an exact form, ω , and η . Therefore α is a linearcombination of a regular odd function on U , z , and v . We obtain ∂x = P ( x ) + ( A ∂ z + B ∂ v ) x ′ ( P ∈ K [ x ] , A ∂ ∈ K, B ∂ ∈ K ) . Note that ∂x = − ∂a z − ∂b z + O ( z ) . Therefore deg P ≤
2. Moreover, looking at the expansions x = z − − a − b z + a z + O ( z ) ,x = z − − a z − b z + O ( z ) ,v x ′ = − z − − a − b z + a z + O ( z ) ,zx ′ = − z − − a z − b z + O ( z ) , we see that ∂x = A ∂ ( zx ′ + 2 x ) + B ∂ ( v x ′ + 2 x + 4 a . The elements A ∂ , B ∂ can be found from the equations: − ∂a − a A ∂ − b B ∂ , − ∂b − b A ∂ + 4 a B ∂ , or ∂a = 4 aA ∂ + 6 bB ∂ , ∂b = 6 bA ∂ − a B ∂ . We see that it is convenient to choose the following vector fields as a basis: δ e : δ e a = 4 a, δ e b = 6 b,δ s : δ s a = 6 b, δ s b = − a . The first one, δ e is the Euler derivative, which was already defined. The secondone is the Serre derivative. One can check that our definition coincides with thestandard one, i.e. δ s E = − E , δ s E = − E . In particular, we see that δ e and δ s define vector fields on Spec R . Choosing α e = z , α s = v we also obtain liftings of these vector fields to regular vector fields on U : δ ∗ e = δ e − α e ddz , δ ∗ s = δ s − α s ddz . For reference we summarise the values of the derivations: δ ∗ e z = − z, δ ∗ e v = v , δ ∗ e x = 2 x, δ ∗ e y = 3 y,δ ∗ s z = − v , δ ∗ s v = − y − a z, δ ∗ s x = 2 x + 4 a , δ ∗ s y = 3 xy,z ′ = 1 , v ′ = − x, x ′ = 2 y, y ′ = 3 x + a.
92t is also convenient to introduce differential forms d e , d s on the base as the dualbasis to the basis δ e , δ s . Then we have dx = 2 y ( dz + zd e + v d s ) + 2 xd e + (2 x + 4 a d s , (4.1) dy = (3 x + a )( dz + zd e + v d s ) + 3 yd e + 3 xyd s . (4.2)It is now easy to compute ∇ ∂ [ ω ]: ∇ ∂ [ ω ] = [ d∂ ∗ z ] = − [ α ′ ∂ dz ] , so we obtain: ∇ δ e [ ω ] = − [ ω ] , ∇ δ s [ ω ] = [ η ] . The derivatives of [ η ] are given by ∇ δ e [ η ] = [ η ] , ∇ δ s [ η ] = a ω ] . In this section we consider in details how to represent cohomology classes on ellipticcurves using the language of hypercovers (Section 3.1). We would like to representcohomology classes for families of elliptic curves by differential forms on the totalspace. For an elliptic curve E we consider two affine open sets U and U . The set U was mentioned before, it is the complement of the neutral element [ ∞ ] of E . Theequation of U is y = x + ax + b . The set U can be chosen to be any affine openset which contains [ ∞ ]. We will not fix U since sometimes we need U to be “smallenough”. The intersection is denoted U int = U ∩ U . The triple U , U , U int definesa hypercover of E which is a particular case of a ˇCech hypercover, the correspondingabstract chain complex is the segment, ∆ . Next, 0-hyperforms are triples ( f , f , f int ) where f and f are functions on U , U correspondingly, and f int is forced to be 0. 1-hyperforms are triples ( θ , θ , θ int )where θ and θ are differential 1-forms on U , U correspondingly, and θ int is afunction on U int . 2-hyperforms are triples ( ι , ι , ι int ) where ι and ι are differential2-forms on U , U correspondingly, and ι int is a differential 1-form on U int . Notethat in the case of a single elliptic curve or relative forms for a family of ellipticcurves ι and ι must be zero. However if we consider absolute differential forms ona family of elliptic curves this is not the case.According to the formulae of Section 3.1 the differentials are given as follows: d ( f , f ,
0) = ( df , df , f − f ) ,d ( θ , θ , θ int ) = ( dθ , dθ , dθ int − θ + θ ) . .4.2 The class [ ω ] . We are going to write down some representatives for H ( E, C ). Consider the dif-ferential form dx y . We want to write it down as a regular form on the Weierstrassfamily. For this we recall that we lifted two vector fields δ e and δ s on the base toregular vector fields δ ∗ e , δ ∗ e on U . Consider an arbitrary vector field ∂ on the base.Expressing it as a linear combination of δ e and δ s and using the lifts δ ∗ e and δ ∗ s weconstruct a lift of ∂ , denoted ∂ ∗ . Then we have2 y∂ ∗ y = (3 x + a ) ∂ ∗ x + x∂a + ∂b. Therefore i ∂ ∗ ( dx ∧ dy ) = ∂ ∗ xdy − ∂ ∗ ydx = ( − x∂a − ∂b ) dx y mod da, db. We see that if we choose ∂ = − ∂∂b we obtain a differential form which is regular on U and represents the same relative form as dx y .More explicitly, one can check that ∂∂b = 12∆ − (4 aδ s − bδ e ) , ∂∂a = − − (6 bδ s + 4 a δ e ) . Then we compute i − ( ∂∂b ) ∗ ( dx ∧ dy ) = 8∆ − (( − ax + 18 bx − a ) dy + (18 axy − by ) dx ) . Let us compute the Laurent expansion of the form above. More generally, forany ∂ we have ∂ ∗ xdy − ∂ ∗ ydx = ( ∂ ∗ xy ′ − ∂ ∗ yx ′ )( dz + zd e + v d s ) + ( ∂ ∗ xδ ∗ e y − ∂ ∗ yδ ∗ e x ) d e + ( ∂ ∗ xδ ∗ s y − ∂ ∗ yδ ∗ s x ) d s . We have already seen that ∂ ∗ xy ′ − ∂ ∗ yx ′ = − x∂a − ∂b. For the remaining part it is enough to compute δ ∗ s xδ ∗ e y − δ ∗ e xδ ∗ s y = 4 ay. Hence i − ( ∂∂b ) ∗ ( dx ∧ dy ) = dz + zd e + v d s − − (16 a yd e + 24 abyd s ) . Noting that da = 4 ad e + 6 bd s we obtain i − ( ∂∂b ) ∗ ( dx ∧ dy ) = dz + zd e + v d s − − ayda. We would like to construct forms which have as small order of pole at infinity aspossible. Therefore we take as ω the form ω = i − ( ∂∂b ) ∗ ( dx ∧ dy ) + 48∆ − ayda = dz + zd e + v d s .
94s the form ω , since it is always possible to approximate power series in z byrational functions, one can choose any form such that ω = dz + zd e + O ( z N ) d e + O ( z N ) d s for some N >> t = x/y . This is a function on a neighborhood of infinity.Then express dt = α dx y mod da, db , where α is a rational function which equals to − α − dt = dz mod da, db . One can add correction terms ofthe form f da , f db to construct ω as needed.Next since we want ( ω , ω , ω int ) to be in the F of the Hodge filtration we put ω int = 0.In this way we have constructed a hyperform which represents the cohomologyclass [ ω ]: ω = ( ω , ω ,
0) = ( dz + zd e + v d s , dz + zd e + O ( z N ) d e + O ( z N ) d s , . [ η ] To construct a representative of [ η ] we consider xω . Its Laurent expansion is xω = xdz + xzd e + xv d s . Here it is possible to make the pole of the coefficient at d s smaller by adding functionsregular on U . Indeed, it is easy to see that the Laurent series xv + y has no pole.Therefore we put η = xω + yd s = xdz + xzd e + ( xv + y ) d s . Next we choose η int . Since we need dη int + η − η = 0 mod da, db it is naturalto take as η int some kind of a formal integral of − η . Therefore we choose η int asany function which satisfies η int = v + O ( z N ) . We finally choose η as follows: η = − a zd s + O ( z N ) d z + O ( z N ) d e + O ( z N ) d s . Then the hyperform η is defined as η = ( η , η , η int ) = ( xdz + xzd e + ( xv + y ) d s , − a zd s + O ( z N ) dz + O ( z N ) d e + O ( z N ) d s , v + O ( z N )) . .4.4 Gauss-Manin derivatives Let us now compute the Gauss-Manin derivatives for the hyperforms constructedabove. First we compute dω : dω = − d e ∧ dz + xd s ∧ dz = − d e ∧ ω + d s ∧ η mod d e ∧ d s ,dω = − d e ∧ dz + O ( z N − ) d e ∧ dz + O ( z N − ) d s ∧ dz = − d e ∧ ω + d s ∧ η + O ( z N − ) d e ∧ dz + O ( z N − ) d s ∧ dz mod d e ∧ d s .dω int − ω + ω = v d s + O ( z N ) d e + O ( z N ) d s . This shows that dω = − d e ∧ ω + d s ∧ η + (0 , O ( z N ) d e ∧ dz + O ( z N ) d s ∧ dz, O ( z N ) d e + O ( z N ) d s ) mod d e ∧ d s . Similarly dη = xd e ∧ dz + a d s ∧ dz = d e ∧ η + a d s ∧ ω mod d e ∧ d s .dη = a d s ∧ dz + O ( z N − ) d e ∧ dz + O ( z N − ) d s ∧ dz mod d e ∧ d s .dη int − η + η = v d e + O ( z N − ) d e + O ( z N − ) d s . Therefore dη = d e ∧ η + a d s ∧ η + (0 , O ( z N − ) d e ∧ dz + O ( z N − ) d s ∧ dz, O ( z N − ) d e + O ( z N − ) d s ) mod d e ∧ d s . Thus we have proved:
Proposition 4.4.1.
For hyperforms ω , η defined above the following identities holdup to hyperforms of type (0 , O ( z N − ) , O ( z N − ) d e + O ( z N − ) d s ) and hyperforms fromthe piece G of the filtration G • : ∇ δ e ω = − ω, ∇ δ s ω = η, ∇ δ e η = η, ∇ δ s η = a ω. We would like to compute the Poincar´e pairing of ω and η to illustrate Corollary3.1.23. By the definition h ω, η i = Z E [ ω ∧ η ] = Z ∆ E ⊂ E × E [ ω × η ] . ω × η has components indexed by pairs α, β where α, β ∈ { , , int } . According toSection 3.1.7) ω × η = ω × η ω × η ω × η int ω × η ω × η ω × η int − ω int × η − ω int × η ω int × η int . Recall that ω int = 0. Therefore the last row is zero.We have Z ∆ E [ ω × η ] = Tr int∆ E ω × η = 2 π i Tr ∆ E ω × η, where Tr ∆ E ω × η = X L ⊂{ , } , | L | =1 X Z • ∈ Fl L (∆ E ) res L,Z • ( ω × η ) a L ( Z • ) . Clearly, we have only one flag in Fl L (∆ E ) for each L ⊂ { , } , | L | = 1. The flagis ∆ E , [ ∞ × ∞ ]. Let us denote this flag fl. We find a { } (fl) = [0 , × [1] and a { } (fl) = [0] × [0 , ∆ E ω × η = 2 π i( − res { } , fl ω int η + res { } , fl ω η int ) . The first residue is zero since ω int is zero. The second one equals tores [ ∞ ] × [ ∞ ] η int ω . The residue in the last expression can be computed using Laurent series expansion:res [ ∞ ] × [ ∞ ] η int ω = res v dz = 1 . Therefore we have proved that Tr ∆ E ω × η = 1 and h ω, η i = 2 π i . hapter 5 Examples
In the first section of this chapter we give examples of higher cycles on the squareof the Weierstrass family. We construct two higher cycles cycle , cycle and thenshow that their difference is annihilated by 12 in the Chow group. The goal here issimply to give example of showing that two cycles are equivalent. The constructionof these cycles was inspired by other constructions of higher cycles on products ofelliptic curves in [GL99], [GL00]. The author came up first with the cycle cycle and the main theorem was proved using the cycle cycle . This cycle may seembetter than cycle since it is composed of smooth varieties. Later the author foundthe more simple and natural construction of cycle , but since it is composed of asingular variety (and using the singularity is crucial since we have a function whichhas at the same time zero and pole there), we needed to develop somewhat moreinvolved machinery to deal with general varieties in the third chapter, though thecomputations with cycle given here became simpler than it was before with cycle .In the second section we compute the corresponding D -module and the invariantΨ ′ alg for the second cycle (we could use the first one, but due to the first sectionwe know that the result would be the same up to torsion). We verify that theconstructed cycle satisfies requirements for Theorem 3.2.9 for Γ = P SL ( Z ), k = 2and z = i. This proves the algebraicity conjecture in this case for any second CMpoint z which is not equivalent to i (Theorem 5.2.1). Then we give example ofcomputing the value of the Green function which proves that √ b G H /P SL ( Z )2 (cid:18) − √− , i (cid:19) ≡ − √
7) mod π i Z and G H /P SL ( Z )2 (cid:18) − √− , i (cid:19) = 8 √ − √ . In the last section we obtain values of the constants N A = 1, N B = 2 and N = 2for the case Γ = P SL ( Z ), k = 2. In this section we construct some higher cycles in Z ( E × E,
1) where E is an ellipticcurve. 98 .1.1 Notations We consider the Weierstrass family y = x + ax + b . Denote the total space by E .A fiber, i.e., a single elliptic curve, is denoted by E . We cover E by two charts. Thefirst one is U = E \ { [ ∞ ] } , which was mentioned before. The second one is U = E \ { ( y = 0) } . The coordinates on U are t, u and the equation is u = t + atu + bu . The gluing maps are given as follows: U → U : ( x, y ) → ( xy − , y − ) , U → U : ( t, u ) → ( tu − , u − ) . The first cycle will be on E × E for b = 0. To denote the coordinates on the first E in E × E we will use index 1, for the second one we use 2. So U × U is given bytwo equations in 4 variables: y = x + ax + b, y = x + ax + b. Consider the subvariety W which is the closure in E × E of the subvariety givenby the equation x y + i x y = 0. To compute the closure first consider U × U .We have equations: y = x + ax + b, u = t + at u + bu , t u − y + i x u − = 0 . This gives x = i y t , y = − i t y + i at y + b. when ( t , u ) = 0 we obtain ( x , y ) = (0 , ±√ b ). So we see that W contains twomore points on U × U . In fact t is a local parameter on W at these points, u hasorder 3 and x has order 1.Analogously W contains two more points on U × U , namely ( x , y ) = ( ±√ b, U × U . The equations are u = t + at u + bu , u = t + at u + bu , t u − u − + i t u − u − = 0 . Therefore we have t = i t . One can check that the point [ ∞ ] × [ ∞ ] also belongs to W and the local parameterthere can be chosen as t or t .Let f be the rational function on W given as y − i y . We first compute thedivisor of f . Since y has a triple pole at [ ∞ ] and the projections W → E are99nramified at [ ∞ ] we see that f has triple pole at the points [ ∞ ] × (0 , ±√ b ) and(0 , ±√ b ) × [ ∞ ]. To study the behaviour of f at [ ∞ ] × [ ∞ ] we write f as f = u − − i u − = ( u − i u ) u − u − . Using the fact that u = t + at + bt + · · · on U at [ ∞ ] we see that f ∼ b i t at [ ∞ ] × [ ∞ ],so f has a triple zero.Finally we look for zeroes of f on the set U × U . We need to find all commonsolutions of the equations y = x + ax + b, y = x + ax + b, x y + i x y = 0 , y − i y = 0 . We get y = i y , ( x + x ) y y = 0 . In the case y = y = 0 we see that for λ , λ , λ — the distinct roots of thepolynomial x + ax + b the 9 points ( λ i , × ( λ j ,
0) are solutions. The case x + x = 0does not give any solution unless b = 0. We will not check that the 9 zeroes areindeed simple since we already know that f has total multiplicity of poles 12 andtriple zero was already found. Thereforediv f = X i,j ( λ i , × ( λ j ,
0) + 3[ ∞ ] × [ ∞ ] − ∞ ] × (0 , √ b ) − ∞ ] × (0 , −√ b ) − , √ b ) × [ ∞ ] − , −√ b ) × [ ∞ ] . It is not difficult to correct (
W, f ) and obtain a cycle. The following combinationis a cycle in Z ( E × E, W, f ) − X i (( λ i , × E, y ) − E × [ ∞ ] , y ) + 3([ ∞ ] × E, x ) + 3( E × [ ∞ ] , x ) . The second cycle is also on E × E , but it is given by a single summand. We take W as the closure of the subvariety defined by equation x + x = 0. It is clear fromthe equation that when x is infinite, x must be infinite as well and vice versa.Therefore the closure contains only one additional point [ ∞ ] × [ ∞ ].The problem with this W is that it is singular. Namely on U × U the equationsare u = t + at u + bu , u = t + at u + bu , t u − + t u − = 0 . So we obtain the following equations for W inside E × E : t u = − u t , t + t = 2 bt u . One can see that the function t t − belongs to the integral closure of the structurering, but does not belong to the structure ring itself. It is easy to check that if we100ass to the normalization of W we obtain 2 points over [ ∞ ] × [ ∞ ], namely the onewith t t − = i and the one with t t − = − I .The function f remains the same, f = y − i y . It is clear that f does not have zeroes or poles on W ∩ U × U . The onlyremaining point is [ ∞ ] × [ ∞ ]. We use expansion u = t + at + bt + · · · . The secondequation for W gives t + t = 2 bt ( t + 2 at + 2 bt + · · · ) . This implies t = − t − bt − abt − b t + · · · . There are 2 solutions of this equation: t = ± i( t + bt + 2 abt + 72 b t + · · · ) . The corresponding value of u can be computed: u = ∓ i( t + at + 2 bt + · · · ) . We can now compute the expansion of f . Along the branch t ∼ i t we have f = u − i u u u ∼ t − . Along the branch t ∼ − i t we have f = u − i u u u ∼ bt . Therefore div f = 0 and ( W, f ) ∈ Z ( E × E, Denote f := x y + i x y , f := x + x , f := y − i y . Denote some algebraic sets D := Z ( f ) , D := Z ( f ) , D := Z ( f ) , Y := Z ( y ) , Y := Z ( y ) ,X := Z ( x ) , X := Z ( x ) , Z := [ ∞ ] × E, Z := E × [ ∞ ] , where Z ( f ) denotes the closure of the zero locus of a function f .Then we havediv f = D − Z − Z , div f = D − Z − Z , div f = D − Z − Z , div y = Y − Z , div y = Y − Z . = ( D , f ) − ( Y , y ) + 3( Z , x ) + 3( Z , x y − ) . The second cycle is cycle = ( D , f ) . Let us write { f, g } := (div f, g | div f ) − (div g, f | div g ) for functions f , g .Note that all { f, g } are trivial in the Chow group by definition.We consider the following element:Ψ := { f f − y − y − , f } . It can be expressed as follows:Ψ = 3( D , f ) − D , f ) − ( Y , f ) − ( Y , f ) − ( D , f f − y − y − )+ 3( Z , f f − y − y − ) + 3( Z , f f − y − y − ) . Now each term will be treated separately. The second and the third ones give − ( Y , y − i y ) − ( Y , y − i y ) = − ( Y , y ) − ( Y , y ) + ( Y , i) . Using the fact that y = i y on D the fourth term equals to − (cid:18) D , ( x y + i x y ) y y ( x + x ) (cid:19) = − ( D , − y ) = − ( D , y ) + ( D , − . Computing asymptotics at infinity we get (cid:18) Z , ( x y + i x y ) y y ( x + x ) (cid:19) = (cid:18) Z , x y (cid:19) , (cid:18) Z , ( x y + i x y ) y y ( x + x ) (cid:19) = (cid:18) Z , − i x y (cid:19) . ThereforeΨ = 3cycle − − ( D , y ) + 2( Y , y ) − ( Y , y ) − Z , y ) + 6( Z , y )+ ( Y + 2 D + Z , i) . The last term is torsion so we could neglect it but for curiosity we will track itfurther.Next put Ψ ′ = { ( y − i y ) x − , y } . We haveΨ ′ = 2( D , y ) − Z , y ) − X , y ) − Y , y ) + 3( Y , x ) + 3( Z , − . ′ = 6(cycle − cycle ) + 4( Y , y ) − Y , y ) − Z , y ) + 12( Z , y )+ 3( Y , x ) − X , y ) + ( Y , − . Next we putΨ ′′ = { y , y } = ( Y , y ) − Z , y ) − ( Y , y ) + 3( Z , y ) . This gives2Ψ + Ψ ′ − ′′ = 6(cycle − cycle ) + 3( Y , x ) − X , y ) + ( Y , − . To kill the remaining terms (which are already decomposable ) we take X i =1 { x − λ i , λ i } = 2( Y , x ) − Z , λ λ λ ) = 2( Y , x ) − Z , − b ) , { x , b } = 2( X , y ) − Z , b ) . Concluding, 12(cycle − cycle ) = 4Ψ + 2Ψ ′ − ′′ . Therefore
Proposition 5.1.1.
The difference between the two cycles constructed above is tor-sion in the higher Chow group CH ( E × E, . More precisely, it is annihilated by . Let us compute the associated invariants of the Abel-Jacobi map for the cyclesconstructed above. We will use the second cycle (
W, f ). We need to compute Ψ forcertain hyperforms (see Section 3.2.4). The map Ψ is defined as an integral over W . The situation is similar to the one in Section 4.4.5.Since the only intersection of W and Z or Z is at [ ∞ ] × [ ∞ ] we have only oneflag fl = ( W, [ ∞ ] × [ ∞ ]) to consider. For a hyperform θ with components θ , , θ , ,. . . , θ int , int we obtain Z W θ = 2 π i(res [ ∞ ] × [ ∞ ] θ int , + res [ ∞ ] × [ ∞ ] θ , int ) . Let us compute Ψ for the following hyperforms: θ := ω × ω, θ := η × ω + ω × η, θ := η × η. .2.2 Some Laurent series expansions Let us use as the coordinate z = z , which is the formal integral of dx y . The equationof W becomes z − − a z − b z + a z + 3 ab z + · · · = − z − + a z + b z − a z − ab z + · · · We can solve this equation inverting both sides and considering the right hand sideas a power series in z : z = − z − b z + 4 ab z + · · · . Then we find z by taking the square root: z = ± i( z + b z − ab z + · · · ) . Let us call the case with plus “case 1” and the case with minus “case 2”. For thefunction f we obtain f = − z − − a z + 3 b z + 2 a z − ab z + · · · in case 1, (5.1) f = − bz + ab z + · · · in case 2. (5.2)In fact we know that the product of the two series above must be 2 b since ( y − i y )( y + i y ) = 2 b on W . Therefore we can get a better approximation for case 2using case 1: f = − bz + ab z − b z − a b z + 17 ab z + · · · in case 2.For computation of φ f we need the logarithmic derivative of f . dff = ( − z − + 4 a z − b z − a z + 86 ab z + · · · ) dz + ( 4 a z − b z − a z + 86 ab z + · · · ) d e + ( 6 b z + 2 a z − ab z + − a + 774 b z + · · · ) d s in case 1, dff = (3 z − − a z + 9 b z + 12 a z − ab z + · · · ) dz + (6 − a z + 9 b z + 12 a z − ab z + · · · ) d e + ( − a b − b z + 2 a z + 18 ab z − a + 774 b z + · · · ) d s in case 2.It is clear that Ψ ( ω × ω ) = 0. We turn to the hyperform θ := η × ω + ω × η. .2.3 The hyperform θ There are two components which we need, namely θ , int and θ , . Denote by ω ( i ) , η ( i ) , v ( i )0 the corresponding objects coming from the i -th curve ( i = 1 , O ( z N )): θ , int = η (2)int ω (1)0 = v (2)0 ( dz + z d e + v (1)0 d s ) ,θ , = − η (1)int ω (2)1 = − v (1)0 ( dz + z d e ) . Since in the end we will take the sum of residues on a curve at the same point,we can restrict our considerations to the sum θ s := θ , int + θ , . We compute the corresponding expansion in case 1. θ s = i( − z − − a z − b z + 2 a z + 62 ab z + · · · ) dz + i( − − a z − b z + 2 a z + 62 ab z + · · · ) d e + i( − z − − a z + b z + 299 a z − ab z + · · · ) d s . It is easy to see that for case 2 the expansion is precisely the negative of this one.Finally we take dff ∧ θ s = ( − z − + · · · ) d e ∧ dz + ( − z − + 2i a z + · · · ) d s ∧ dz + ( · · · ) d s ∧ d e in case 1, dff ∧ θ s = (6i z − + · · · ) d e ∧ dz + ( − z − − a b z − + 2i a z + · · · ) d s ∧ dz + ( · · · ) d s ∧ d e in case 2.This means that φ f θ s = d e ⊗ ( − z − + · · · ) dz + d s ⊗ ( − z − + 2i a z + · · · ) dz in case 1, φ f θ s = d e ⊗ (6i z − + · · · ) dz + d s ⊗ ( − z − − a b z − + · · · ) dz in case 2.Hence, taking residues and adding,Ψ ( θ ) = − a b d s . .2.4 The hyperform θ Analogously to the previous case we first write down θ , int = η (2)int η (1)0 = v (2)0 ( x dz + x z d e + ( x v (1)0 + y ) d s ) ,θ , = − η (1)int η (2)1 = v (1)0 ( a z d s ) . Adding we obtain the following expansion (in case 1): θ s = i( − z − + 2 a z + 11 b z + · · · ) dz + i( − z − + 2 a z + 11 b z + · · · ) d e + i( 2 a b z + 2 a z + · · · ) d s . This gives φ f θ s = d s ⊗ (2i az − + · · · ) dz + d e ⊗ ( − z − + 2i a z + · · · ) dz in case 1, φ f θ s = d s ⊗ ( − a b z − + 2i az − + · · · ) dz + d e ⊗ (3i z − − a z + · · · ) dz in case 2.Again we add residues to see thatΨ ( θ ) = 4i ad s . D -modules Using the computation of Ψ above and since the basis of hyperforms was chosenso that their Gauss-Manin derivatives can be again expressed in this basis, we canexplicitly describe the extension of D -modules from Section 3.2.4. The D -module M is generated over O S by 1, θ , θ , θ . The action of the derivations δ s and δ e (denoted δ ′ s , δ ′ e ) is defined as follows: δ ′ e , δ ′ e θ = − θ , δ ′ e θ = 0 , δ ′ e θ = 2 θ ,δ ′ s , δ ′ s θ = θ , δ ′ s θ = 2 θ + 2 a θ + 8i a b , δ ′ s θ = a θ − a. Consider a curve of the following type:∆ = − a + 27 b ) = const . Note that δ s is a derivation on such curve. For this curve the element B ∈ M ⊗ D ⊗ M must be of the form m ⊗ ( δ s − a δ s − b ) ⊗ [ ω ] , m ∈ M . ω ], the fiberwise cohomology class of ω , is a modular form ofweight 1. The canonical pairing identifies the image of [ η ] in H /F H with 2 π i[ ω ] − .Therefore the Kodaira-Spencer map acts as KS : [ ω ] → d s ⊗ π i[ ω ] − . Therefore the symbol of δ s is 2 π i[ ω ] − . Hence B = (2 π i) − [ ω ] ⊗ ( δ s − a δ s − b ) ⊗ [ ω ] . We evaluate Ψ ′ alg on B . We have[ ω ] = θ , δ ′ s [ ω ] = θ , δ ′ s [ ω ] = 2 θ + 2 a θ + 8i a b ,δ ′ s [ ω ] = 4 a θ + 4 bθ + 24i a + 32i a b . Therefore Ψ ′ alg ( B ) = (2 π i) − [ ω ] (cid:18) a + 32i a b (cid:19) . The modular form as a function on H which computes the periods of the coho-mology class above can be found using Proposition 4.2.2. It is the function2 π i µ (cid:18) a + 32i a b (cid:19) = − πE ( E − E ) E = − π E ( τ ) j ( τ ) − j (i) . The Abel-Jacobi map for the family of cycles we study is a function of a and b . As wehave proved, the derivative of AJ , s [cycle ] (the symmetric part) is 0 along the vectorfield δ e . Therefore AJ , [cycle ] is invariant under the action of the multiplicativegroup G m , which sends a, b to λ a, λ b . Restricted to the curve ∆ = const thismeans that AJ , s [cycle ]( a, b ) = AJ , s [cycle ]( εa, b ) = AJ , s [cycle ] ( ε + ε + 1 = 0) . In particular AJ , [cycle ] depends only on the j -invariant AJ , s [cycle ]( a, b ) = AJ , s [cycle ]( j ) , j = − a ∆ . Therefore AJ , s [cycle ] is constant along the fibers of j . We have proved that (seeSection 3.2.6, keep in mind that k = 2, n = 2)Ψ ′ an ( B ) = 2 π i Ψ ′ alg ( B ) = − π i E ( E − E ) E . We aim to prove that g := Ψ ′ an ( B ) is proportional to g H /P SL ( Z )2 , i . Since g has onlypoles at the orbit of i and is zero at the cusp, it is enough to study the expansion107f g at I . We have (note that E (i) = 0 and E (i) = π and use the Ramanujan’sformulae for the derivatives of E , E , E ) E ( τ ) = E (i) (1 + 2i( τ − i) + · · · ) ,E ( τ ) = − π i E (i) ( τ − i) (cid:18) τ − i)2 + · · · (cid:19) . Therefore E ( E − E ) E = − π − (( τ − i) − + i( τ − i) − + O (1)) , whereas Q i ( τ ) − = − τ − i) ( τ + i) = ( τ − i) − + i( τ − i) − + O (1) . This gives g = 2i Q i ( τ ) − + O (1) , thereby proving that (see Theorem 1.5.3 and note that m = 2 for the point i)Ψ ′ an ( B ) = − i g H /P SL ( Z )2 , i ( τ ) . Therefore one may apply Theorem 3.2.9 ( α = ). This implies our first maintheorem: Theorem 5.2.1.
For any z ∈ H which is not equivalent to i let a , b be such thatthe j -invariant of the curve E a ,b (given by equation y = x + a x + b ) is j ( z ) .Then G H /P SL ( Z )2 ( z, i) = 8 ℜ ( √− AJ , [cycle ( a , b )] , Q z ) , where cycle ( a , b ) is the higher cycle on E a ,b × E a ,b constructed in Section 5.1.3and Q z is the polynomial ( X − z )( X − ¯ z ) z − ¯ z ∈ Sym H ( E a ,b , C ) ⊂ H ( E a ,b × E a ,b , C ) (recall that we identify H ( E a ,b , C ) with the space of polynomials of degree ofvariable X ). For any CM point z not equivalent to i the algebraicity conjecture istrue and √− D b G H /P SL ( Z )2 ( z, i) ≡ ( a , b ) · Z z ) mod π i Z for any algebraic cycle Z z on E a ,b × E a ,b whose cohomology class is √ DQ z . .2.7 Special values of the Green function The identity proved in the last section allows us to compute the values of the Green’sfunction G H /P SL ( Z ) k ( τ, I ) in points of complex multiplication. We consider in thissection an example with τ = τ = − √− (in this section we denote the variableon the upper half plane by τ ).The following elliptic curve corresponds to the point τ : y = x − x − . Let us denote µ = τ . It is a root of the polynomial x + x + 2. Note that we havea decomposition x − x −
98 = ( x − x + µ + 4)( x − µ + 3) . The curve has the following endomorphism (we denote µ = τ , µ + µ + 2 = 0): φ ( x, y ) = (cid:18) µ − (cid:18) x + (3 µ + 5) x + µ + 4 (cid:19) , µ − y (cid:18) − (3 µ + 5) ( x + µ + 4) (cid:19)(cid:19) . It’s graph Γ φ has a class in H ( E × E, Z ). This class can be represented as[Γ φ ] = c [ Z ] + c [ Z ] + c [∆ E ] + c ( X − τ )( X − τ ) τ − τ c i ∈ C . Here recall that we identify Sym H ( E, C ) with the space of polynomials of degree2 in variable X , in fact (see Section 3.2.6)( X − τ )( X − τ ) τ − τ = [ ω ] × [ ω ] + [ ω ] × [ ω ]4ivol ω E .
To find some of the constants above we compute intersections[Γ φ ] · [ Z ] = 1 , [Γ φ ] · [ Z ] = 2 , [Γ φ ] · [∆ E ] = 4 . Since we know that[ Z ] · [ Z ] = 1 , [ Z i ] · [ Z i ] = 0 , [∆ E ] · [ Z i ] = 1 , [∆ E ] · [∆ E ] = 0 , we obtain c = 52 , c = 32 , c = − . To find the last coefficient consider the following integral: Z Γ φ ( ω × ω + ω × ω ) = Z E ( ω ∧ φ ∗ ω + ω ∧ φ ∗ ω ) . Since φ ∗ ω = µω , we obtain that the integral in question equals to the following one:( µ − µ ) Z E ω ∧ ω = 2i( µ − µ )vol ω E. φ ] · [Γ φ ] = c [ Z ] · [Γ φ ] + c [ Z ] · [Γ φ ] + c [∆ E ] · [Γ φ ] + c µ − µ , which implies c = √− X − τ )( X − τ ) τ − τ = 1 √− (cid:18) [Γ φ ] −
52 [ Z ] −
32 [ Z ] + 12 [∆ E ] (cid:19) . Therefore we can choose Z τ = 2[Γ φ ] − Z ] − Z ] + [∆ E ] , or Z τ = [Γ φ ] − [Γ ¯ φ ], as in the introduction. We will use the first expression.Our next step is to employ Theorem 5.2.1. So we need to compute Z τ · cycle .For any curve Z ⊂ E × E which does not pass through [ ∞ ] × [ ∞ ] we have definedthe intersection number cycle · Z as the following element in C × :cycle · Z = Y p ∈ W ∩ Z f ( p ) ord p ( W · Z ) . Since this number depends only on the cohomology class of Z , the definition canbe extended to curves which pass through [ ∞ ] × [ ∞ ]. In particular, Z τ · cycle alsomakes sense.To compute cycle · Z we cannot directly take the value of f at the intersection W · Z since the only intersection of W · is at [ ∞ ] × [ ∞ ] and f is not defined as thispoint. Instead we consider the deformation Z ( z ) = { z } × E , where z is a formalparameter and by the point z we mean the point with z = z . Then there are twopoints of intersection with values of f given by the series (5.1), (5.2). Thereforecycle · Z = 2 b = − . For the computation of cycle · Z we take deformation Z ( z ) = E × { z } . Theexpansions of the branches of f begin with 2i z − and b i z , so we obtain − b as theproduct, cycle · Z = − b = 14 . Let us turn to the computation of cycle · ∆ E . There are now 3 points of in-tersection, namely [ ∞ ] × [ ∞ ], (0 , √ b ) × (0 , √ b ), and (0 , −√ b ) × (0 , −√ b ). The lasttwo points are simple. To find the value at infinity we consider the deformation∆ E ( z ) = ∆ E + ( z, z ∼ i z , z − z ∼ z . In this case z ∼ z , hence f ∼ (4 + 4i) z − .The second point satisfies z ∼ − i z , z − z ∼ z . In this case z ∼ − i2 z and f ∼ bz . Multiplication of all the principal parts givescycle · ∆ E = (4 + 4i) 1 + i4 b ( √ b − i √ b ) ( −√ b + i √ b ) = − b = − . · Γ φ is more difficult. Consider the equation µ − (cid:18) x + (3 µ + 5) x + µ + 4 (cid:19) = − x . It is equivalent to x being a root of the quadratic equation t + t ( µ + 4) − µ −
21 = 0 . Let us denote the solutions of the quadratic equation above by t , t . For each t = t i we put x = t and get an equation on y as follows: y = t − t − . In this way we obtain 4 points of intersection W ∩ Γ φ . Using the equation on t wetransform t − t −
98 = 14 µ ( t + 6 µ + 3) . Also it is easy to see that 3 µ + 5 t + µ + 4 = t − µ − µ + 5 = t − µ . Therefore for each point of intersection we have y = y µ − (cid:18) − t (3 − µ ) (cid:19) = y µ − (cid:18) t ( µ + 4) − µ + 3)7 − µ (cid:19) = − y ( t − µ − µ + 4 . It is not hard to see that at these four points the intersection is transversal. Theproduct of values of f over these four points is equal to( t − t − t − t − (cid:18) µ + 4 ( t − µ − (cid:19) × (cid:18) µ + 4 ( t − µ − (cid:19) The product is in fact a product of norms from Q ( µ, t, i) to Q ( µ, i) (we denote thisnorm by N ): = µ N ( t + 6 µ + 3) N ( t − µ − − i(8 µ + 4)) . We have N ( t + 6 µ + 3) = (6 µ + 3) − (6 µ + 3)( µ + 4) − µ −
21 = − µ + 3) ,N ( t − µ − − i(8 µ + 4)) = (3 µ + 5 + i(8 µ + 4)) + (3 µ + 5 + i(8 µ + 4))( µ + 4) − µ −
21 = − µ (2 µ + 1)(i µ + 1) . µ ( Iµ + 1) = 14 ( − µ − − i µ ) = 14 u, where u = − µ − − i µ ( u is a unit.)It remains to consider the intersection point at infinity. This is done as in theprevious cases. We consider the deformation Γ φ ( z ) = Γ φ + ( z, z ∼ µ ( z − z ) and z ∼ ± i z . The expansions of f start with f = − (cid:18) µ − i µ (cid:19) z − + · · · , f = − b (cid:18) µµ + i (cid:19) z + · · · . Multiplication of the principal terms gives2 b (cid:18) µ − i µ + i (cid:19) = − bu . Therefore we have proved thatcycle · Γ φ = − b u = 14 u . Finally we obtaincycle · Z τ = cycle · (2Γ φ − Z − Z + ∆ E ) = u . Theorem 5.2.1 says in this case (note that u = i(8 − √ Proposition 5.2.2. √ b G H /P SL ( Z )2 (cid:18) − √− , i (cid:19) ≡ − √
7) mod π i Z . Therefore G H /P SL ( Z )2 (cid:18) − √− , i (cid:19) = 8 √ − √ . We compute the constants N A and N B for the group P SL ( Z ), k = 2. This cor-responds to the representation V . To compute N B we need to find the group H ( P SL ( Z ) , V Z ). The group P SL ( Z ) is generated by two elements: S = (cid:18) −
11 0 (cid:19) , T = (cid:18) (cid:19) . For v ∈ V Z , v = v X + v X + v we have Sv − v = ( v − v )( X − − v X, T v − v = − v X + v − v . This shows that H ( P SL ( Z ) , V Z ) ∼ = Z / Z , N B = 2.To find N A we need to compute the parabolic cohomology group H ( P SL ( Z ) , V /V Z ) . So let c be a parabolic cocycle with values in V /V Z , which is given by its valueson S and T . Since we know that the cocycle is parabolic, we can modify it by acoboundary to ensure that c T = 0. Let v = c ( S ) = v X + v X + v ( v , v , v ∈ C / Z ) . Then we have c ( ST ) = c ( S ) and v + Sv = ( v + v )( X + 1) , v + ST v + ( ST ) v = (2 v − v + 2 v )(1 + X + X ) . This shows that v ≡ − v mod Z , v ≡ Z . Moreover, we are free to add to c a coboundary which is zero on T . Therefore wecan add S ( v ) − v = ( X − v , which makes the cocycle zero. Therefore H ( P SL ( Z ) , V /V Z ) = 0 , and we can choose N A = 1. This gives N = 2.113 ibliography [Blo72] Spencer Bloch. Semi-regularity and deRham cohomology. Invent. Math. ,17:51–66, 1972.[Blo86] Spencer Bloch. Algebraic cycles and the Be˘ılinson conjectures. In
TheLefschetz centennial conference, Part I (Mexico City, 1984) , volume 58 of
Contemp. Math. , pages 65–79. Amer. Math. Soc., Providence, RI, 1986.[Dol80] Albrecht Dold.
Lectures on algebraic topology , volume 200 of
Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathe-matical Sciences] . Springer-Verlag, Berlin, second edition, 1980.[GH78] Phillip Griffiths and Joseph Harris.
Principles of algebraic geometry .Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure andApplied Mathematics.[GKZ87] B. Gross, W. Kohnen, and D. Zagier. Heegner points and derivatives of L -series. II. Math. Ann. , 278(1-4):497–562, 1987.[GL99] B. Brent Gordon and James D. Lewis. Indecomposable higher Chowcycles on products of elliptic curves.
J. Algebraic Geom. , 8(3):543–567,1999.[GL00] B. Brent Gordon and James D. Lewis. Indecomposable higher Chowcycles. In
The arithmetic and geometry of algebraic cycles (Banff, AB,1998) , volume 548 of
NATO Sci. Ser. C Math. Phys. Sci. , pages 193–224.Kluwer Acad. Publ., Dordrecht, 2000.[Gro62] Alexander Grothendieck.
Fondements de la g´eom´etrie alg´ebrique. [Ex-traits du S´eminaire Bourbaki, 1957–1962.] . Secr´etariat math´ematique,Paris, 1962.[GZ86] Benedict H. Gross and Don B. Zagier. Heegner points and derivatives of L -series. Invent. Math. , 84(2):225–320, 1986.[Har66] Robin Hartshorne.
Residues and duality . Lecture notes of a seminaron the work of A. Grothendieck, given at Harvard 1963/64. With anappendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin, 1966. 114Hir75] Heisuke Hironaka. Triangulations of algebraic sets. In
Algebraic geome-try (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata,Calif., 1974) , pages 165–185. Amer. Math. Soc., Providence, R.I., 1975.[KLMS06] Matt Kerr, James D. Lewis, and Stefan M¨uller-Stach. The Abel-Jacobimap for higher Chow groups.
Compos. Math. , 142(2):374–396, 2006.[KZ01] Maxim Kontsevich and Don Zagier. Periods. In
Mathematics unlimited—2001 and beyond , pages 771–808. Springer, Berlin, 2001.[WW62] E. T. Whittaker and G. N. Watson.