ITERATED INTEGRALS, DIAGONAL CYCLES AND RATIONAL POINTS ON ELLIPTIC CURVES

HENRI DARMON, VICTOR ROTGER AND IGNACIO SOLS

Abstract. The theme of this article is the connection between the pro-unipotent fundamental group π_{1}(X; o) of a pointed algebraic curve X, algebraic cycles, and special values of Lfunctions. The extension of mixed Hodge structures arising in the second stage in the lower central series of π_{1}(X; o) gives rise to a supply of complex points on the Jacobian Jac(X) of X indexed by Hodge cycles on X × X. The main results of this note relate these points to the Abel-Jacobi image of Gross-Kudla-Schoen’s modified diagonal in X^{3}, and express this Abel-Jacobi image in terms of iterated integrals. The resulting formula is the basis for the practical complex-analytic calculations of these points when X is a modular (or Shimura) curve, a setting where the recent work [YZZ] of X. Yuan, S. Zhang and W. Zhang relates their non-triviality to special values of certain L-series attached to modular forms.

Introduction

Let (Y, o) be a pointed algebraic variety over the complex numbers. Its pro-unipotent fundamental group π_{1}(Y; o) is a rich source of links between the topology, geometry and arithmetic of the variety: see for example the works of Bloch [Bl77], Chen [Chen], Deligne [De], Drinfeld [Dr], Hain [Hain3] and Kim [Ki] and the references therein.

Among these developments, the contributions of Carlson, Clemens and Morgan [CCM], Harris [Har] and Pulte [Pu] in the eighties exhibited examples where the values of Chen’s iterated integrals –which provide an explicit description of the underlying mixed Hodge structure of the de Rham realization of π_{1}(Y; o)– coincide with the Abel-Jacobi images of a suitable null-homologous cycle on Y or on some other variety that one may naturally construct out of Y. In geometric terms, these results can be phrased as the equality of two points in an intermediate jacobian of the variety: one, constructed as the class of the periods afforded by certain iterated integrals, and the other obtained as the image of a suitable cycle under the Abel-Jacobi map.

Let F ⊂ ℂ be a field embedded in the field of complex numbers, X a smooth projective curve over F and Y = X \ {∞} the complement of a single rational point ∞ ∈ X(F). Choose a base point o ∈ Y (F). In this article we focus on the construction of a supply of complex points

(1)  κ_{dR}(ξ) ∈ Jac(X)(ℂ)

on the jacobian of X, indexed by the set of Hodge classes

Hdg(H^{1}(X) ⊗ H^{1}(X)) := {ξ : ℤ(−1) → H^{1}(X) ⊗ H^{1}(X)}

1991 Mathematics Subject Classification. 11F67; 11G05; 11G40; 11G18; 14C15. The exchanges which led to this work were initiated in a series of visits by the first (H.D) and third (I.S)

author to the CRM in Barcelona on the occasion of its 2009-10 arithmetic special year which was organised in part by the second author (V.R). All three authors thank the CRM for its hospitality which greatly facilitated their collaboration. H.D. acknowledges the support of NSERC (Natural Sciences and Engineering Research Council of Canada). V.R. received financial support from DGICYT Grant MTM2009-13060-C02-01. I.S. also thanks the Spanish Ministerio de Educación for a Salvador Madariaga grant which funded his stay in the Institut de Recerca Matematica during the arithmetic special year.

in the Hodge structure underlying H^{1}(X) ⊗ H^{1}(X). This construction has also been extensively studied by Pulte [Pu] and Kaenders [Ka] and we review it in §1. These points arise from the pro-unipotent fundamental group π_{1}(Y (ℂ); o) and therefore depend on the choice of o, and their construction exploits the canonical isomorphism

(2)  Ext_{MHS}^{1}(ℤ(−1), H_{B}^{1}(X)) ≃ \frac{Ω^{1}(X)^{∨}}{H_{1}(X(ℂ), ℤ)} ≃ Jac(X)(ℂ)

between the module of extensions of ℤ(−1) by H_{B}^{1}(X) in the category of mixed Hodge structures, and the set of complex points of Jac(X), as described e.g. in [Ca].

Assume now that ξ = ξ_{Z} is the Hodge class associated by the cycle class map with an algebraic cycle Z ∈ Pic(X × X), as detailed in (44). Section 2 associates to each divisor Z in X × X an algebraic correspondence Π_{Z} : \xymatrix{X^{3} \ar@{.>}[r] & X} between X^{3} and X and our first main theorem is:

Theorem 1. For every divisor class Z ∈ Pic(X^{2}),

(3)  κ_{dR}(ξ_{Z}) = Π_{Z}(Δ_{GKS}) ∈ Jac(X)(ℂ),

where

• Π_{Z} : CH^{2}(X^{3})_{0} ⟶ CH^{1}(X)_{0} = Jac(X) is the transformation induced by Π_{Z} by functoriality, and

• Δ_{GKS} ∈ CH^{2}(X^{3})_{0} is the null-homologous cycle of codimension 2 in X^{3} with basepoint o studied by Gross, Kudla and Schoen [GrKu], [GrSc], cf. (47).

Since the right-hand side of (3) is purely algebraic, we derive the following corollary, which a priori does not follow directly from the construction of κ_{dR}(ξ_{Z}) stemming from π_{1}(Y (ℂ); o).

Corollary 2. If Z belongs to Pic(X × X)(F) then κ_{dR}(ξ_{Z}) is an F -rational point of Jac(X).

The proof of Theorem 1 is complex analytic: it proceeds by relating–in §1 and §2, respectively– the left and right-hand side of equality (3) to Chen’s iterated integrals. The resulting explicit formula (Theorem 2.5) was one of the main motivations for this note and underlies the numerical calculations of Chow-Heegner points that are carried out in [DDLR] when X is a modular curve. The authors also hope that Theorem 1 could form the basis for an eventual approach to the “Stark-Heegner points” of [DP] via iterated integrals.

Beyond the general expectation alluded to above that the pro-unipotent fundamental group π_{1}(Y; o) of a variety should encode the Abel-Jacobi images of non-trivial null-homologous algebraic cycles, it is also natural to expect, on the grounds of the conjectures of Bloch-Kato [BK] linking algebraic cycles to L-functions, that a good deal of the non-abelian arithmetic of π_{1}(Y; o) should be accounted for by the leading terms of the L-function of the motives appearing as Jordan-Hölder constituents of π_{1}(Y; o). There are however few examples in the literature in which this principle is illustrated^{1}. A celebrated instance is afforded by the curve Y = ℙ_{1} − {0, 1, ∞}, whose pro-unipotent fundamental group is an iterated extension of Tate motives. It is shown in [De] that the extension classes that arise in it encode special values of the Riemann zeta function at certain (non-critical) points. Theorem 3.7 in §3 describes analogous phenomena when Y is a punctured modular curve attached to a congruence subgroup of SL_{2}(ℤ), or a Shimura curve, exploiting [YZZ].

Let us close this introduction by relating Theorem 1 to existing results in the literature. In our notations and the normalizations adopted in definition (43) of the points κ_{dR}(ξ) –which are tailored so that there is no dependence on the choice of the point ∞–, the main theorem of Kaenders [Ka] says that when Z = X_{12} ∈ Pic(X ×X) is the class of the curve X diagonally embedded in X × X, we have

1while the connection between algebraic cycles and special values of L-functions as predicted by [BK] has already borne some fruit: cf. e.g. [GrZa], [Ne], [Zh], [YZZ].

(4)  κ_{dR}(ξ_{X_{12}}) = (2g − 2)o − K ∈ Jac(X),

where g stands for the genus of X and K is any canonical divisor on the curve. Note that, since K is an F -rational divisor, the conclusion of Corollary 2 follows directly from (4) for this particular choice of Z.

Kaenders proves (4) by showing that a suitable translate of κ_{dR}(ξ_{X_{12}}) is equal to twice Riemann’s constant; his result then follows by an immediate application of the Riemann-Roch theorem. It can also be recovered directly from Theorem 1: see Corollary 2.8.

It is also interesting to compare Theorem 1 with the work of B. Harris [Har], as recast by Pulte in [Pu]. For any n ∈ ℤ, let [n] ∈ End(Jac(X)) denote the endomorphism given by multiplication-by-n. Define

X_{o} := {[P − o] : P ∈ X}, X_{o}^{−} := [−1]_{∗}X_{o} = {[o − P] : P ∈ X} ⊂ Pic^{0}(X)

and let C_{o} := X_{o} − X_{o}^{−} ∈ CH^{g−1}(Jac(X))_{0} be Ceresa’s cycle, a null-homologous cycle of codimension g − 1 on the jacobian of X.

Every choice of Hodge class ξ ∈ Hdg(H_{B}^{1}(X)^{⊗2}) singles out a natural map^{2}

(5)  π_{ξ} : CH^{g−1}(Jac(X))_{0}(ℂ) ⟶ Jac(X)(ℂ),

and Theorem 4.9 of [Pu] asserts that

(6)  κ_{dR}(ξ) = 2π_{ξ}(C_{o}) ∈ Jac(X)(ℂ).

Notice that Corollary 2 does not follow automatically from (6), as it is not a priori obvious that the map π_{ξ} is Galois equivariant.

Our Theorem 1 is in the same spirit of Harris-Pulte’s, in that both relate the points κ_{dR}(ξ) arising from π_{1}(Y (ℂ); o) to algebraic cycles on some algebraic variety intimately related to X, namely X^{3} and Jac(X), respectively. These two varieties and their relevant Chow groups are related through their intermediate jacobians by the commutative diagram

(7)  \xymatrix{CH^{2}(X^{3})_{0}(ℂ) \ar[d]^{μ_{∗}} \ar[r] & J^{2}(X^{3}) = Fil^{2}H_{dR}^{3}(X^{3}, ℂ)/H_{3}(X^{3}, ℤ) \ar[d]^{AJ(μ_{∗})} \\ CH^{g−1}(Jac(X))_{0}(ℂ) \ar[r] & J_{1}(Jac(X)) = Fil^{2}H_{dR}^{3}(Jac(X), ℂ)/H_{3}(Jac(X), ℤ),}

where the horizontal arrows are the Abel-Jacobi maps AJ_{X^{3}} on X^{3} and AJ_{Jac(X)} on Jac(X), respectively, the left-most vertical arrow is the transformation induced by functoriality by the morphism μ : X^{3} → Jac(X) given by the rule (x_{1}, x_{2}, x_{3}) ↦ x_{1} +x_{2} +x_{3} −3o, and the vertical arrow AJ(μ) is the natural surjection given by the Künneth decomposition of H^{3}(X^{3}) and the classical isomorphism H^{3}(Jac(X)) = Λ^{3}H^{1}(X).

It readily follows from the definitions that

μ_{∗}(Δ_{GKS}) = [3]_{∗}X_{o} − 3[2]_{∗}X_{o} + 3X_{o} ∈ CH^{g−1}(Jac(X))_{0}

and Colombo and van Geemen proved in [CG, Proposition 2.9] that

(8)  AJ_{Jac(X)}(μ_{∗}(Δ_{GKS})) = 3AJ_{Jac(X)}(C_{o}).

This result lies far from proving Theorem 1 via (6), as the maps μ_{∗} and AJ(μ_{∗}) have non-trivial kernel. In fact, the combination of Theorem 1 with Pulte’s (6) yields the equality

(9)  Π_{Z}(Δ_{GKS}) = 2π_{ξ_{Z}}(C_{o}) ∈ Jac(X)(ℂ) for all Z ∈ Pic(X × X),

2In the notation of §1.2, π_{ξ} is the composition of the Abel-Jacobi map CH^{g−1}(Jac(X))_{0}(ℂ) →^{AJ} J_{1}(Jac(X)), the monomorphism J_{1}(Jac(X)) ↪ Ext_{MHS}^{1}(H_{B}^{1}(X)^{⊗2}, H_{B}^{1}(X)) considered by Pulte in [Pu], the projection Ext_{MHS}^{1}(H_{B}^{1}(X)^{⊗2}, H_{B}^{1}(X)) → Ext_{MHS}^{1}(ℤ(−1), H_{B}^{1}(X)) given by ξ as in (36) and the isomorphism (2).

which is yet another set of non-trivial relationships between Gross-Kudla-Schoen’s cycle Δ_{GKS} ∈ CH^{2}(X^{3})_{0} and Ceresa’s C_{o} ∈ CH^{g−1}(Jac(X))_{0}.

1. Fundamental groups and mixed motives

Let F be a number field equipped with an embedding F ↪ ℂ into the field of complex numbers, and write F̄ for the Galois closure of F in ℂ; as before, recall that G_{F} := Gal (F̄/F).

Let X be a smooth, complete algebraic curve of genus g ≥ 2 over F, and let Y = X \ S for some finite, G_{F} -stable set S of points on X(F̄). Fix a base point o ∈ Y (F); the integral group ring ℤ[Γ] of the fundamental group Γ := π_{1}(Y (ℂ); o) of the Riemann surface underlying

Y_{ℂ} = Y ×_{Spec(F)} Spec(ℂ)

is equipped with a decreasing filtration

ℤ[Γ] ⊃ I ⊃ I^{2} ⊃ ⋯ ⊃ I^{j} ⊃ ⋯

by powers of the augmentation ideal.

The first few successive quotients in this filtration are given by

(10)  ℤ[Γ]/I = ℤ;  ∑ m_{σ}σ ↦ ∑ m_{σ};
(11)  I/I^{2} = Γ_{ab} := Γ/[Γ, Γ];  ∑ m_{σ}σ ↦ ∏ σ^{m_{σ}};

I^{2}/I^{3} = (Γ_{ab} ⊗ Γ_{ab})/K, (12)

where in the last identification, K is the kernel of the natural surjective homomorphism

(13)  j : Γ_{ab} ⊗ Γ_{ab}⟶I^{2}/I^{3},  γ_{1} ⊗ γ_{2} ↦ (γ_{1} − 1)(γ_{2} − 1).

Let M^{∨} := hom(M, ℤ) be the ℤ-linear dual of a ℤ-module M. Formula (11) implies that

(14)  (I/I^{2})^{∨} = hom(Γ, ℤ) = H_{B}^{1}(Y),

where H_{B}^{1}(Y) = H^{1}(Y (ℂ), ℤ) refers to the singular Betti cohomology of the smooth real manifold underlying Y (ℂ). Furthermore, it can be shown that the map

j^{∨} : (I^{2}/I^{3})^{∨} ↪ H_{B}^{1}(Y)^{⊗2}

identifies (I^{2}/I^{3})^{∨} with the kernel of the map

H_{B}^{1}(Y)^{⊗2} ⟶ H_{B}^{2}(Y)

given by the cup product. (Cf. [Hain3, Cor. 8.2] for more details.) Letting H_{B}^{1}(Y)_{∪=0}^{⊗2} denote this kernel, the map j^{∨} induces an identification

(15)  (I^{2}/I^{3})^{∨} = H_{B}^{1}(Y)_{∪=0}^{⊗2}.

By dualising the exact sequence

0 ⟶ I^{2}/I^{3} ⟶ I/I^{3} ⟶ I/I^{2} ⟶ 0,

invoking (14) and (15), and setting

M_{B} := (I/I^{3})^{∨},

we obtain the fundamental exact sequence of ℤ-modules

(16)  0 ⟶ H_{B}^{1}(Y) ⟶ M_{B} ⟶ H_{B}^{1}(Y)_{∪=0}^{⊗2} ⟶ 0.

Note that H_{B}^{1}(Y)_{∪=0}^{⊗2} = H_{B}^{1}(Y)^{⊗2} as soon as S ≠ ∅, because in this case H_{B}^{2}(Y) = {0}. The abelian groups that appear on the left and right of (16) are not just “unadorned” ℤ-modules, but the Betti realisations of certain motives over F. In the terminology of [De], the exact sequence (16) suggests that M_{B} ought to be interpreted as the Betti realization of a putative mixed motive M^{?} (in the sense of [De], say) which would arise as an extension in that category of the motive H^{1}(Y)_{∪=0}^{⊗2} (which is pure of weight 2 when Y = X) by the motive H^{1}(Y) (which is pure of weight 1 when Y = X). Implicit in the existence of M^{?}, which we do not address in this paper, are the following concrete consequences:

(1) The existence of a decreasing, separated and exhaustive Hodge filtration on

M_{dR,ℂ} := M_{B} ⊗ ℂ,

for which the complexification

(17)  0 ⟶ H_{dR}^{1}(Y/ℂ) ⟶^{i} M_{dR,ℂ} ⟶^{p} H_{dR}^{1}(Y/ℂ)_{∪=0}^{⊗2} ⟶ 0

of (16) becomes an exact sequence of filtered complex vector spaces.

(2) The existence, for each rational prime p, of the p-adic étale realisation M_{et,p}, fitting into an exact sequence of continuous ℚ_{p}[G_{F}]-modules

(18)  0 ⟶ H_{et}^{1}(XF̄, ℚ_{p}) ⟶ M_{et,p} ⟶ H_{et}^{1}(XF̄, ℚ_{p})_{∪=0}^{⊗2} ⟶ 0

which is the p-adic étale counterpart of (16). There are other aspects as well (notably, the cristalline realisations attached to the fundamental groups of the reduction of Y modulo p, equipped with its semi-linear frobenius endomorphism at p and a comparison isomorphism with the de Rham cohomology over p-adic fields) which shall not be touched upon at all here. In this note we focus almost exclusively on (17).

Let us briefly recall the well-known mixed Hodge structures underlying the Betti cohomology group H_{B}^{1}(Y) and its tensor square. Its weight filtration arises from the exact sequence

(19)  0 ⟶ H_{B}^{1}(X) ⟶ H_{B}^{1}(Y) ⟶ ⊕_{P ∈S}ℤ(−1) ⟶ ℤ(−1) ⟶ 0,

and is given by

(20)  W_{j}H_{B}^{1}(X) = \{ \begin{matrix}0 & if j ≤ 0, \\ H_{B}^{1}(X) & if j = 1 \\ H_{B}^{1}(Y) & if j ≥ 2. \end{matrix}

The de Rham cohomology H_{dR}^{1}(Y (ℂ)) is the cohomology of the de Rham complex of smooth ℂ-valued differential forms on Y (ℂ) with at worst logarithmic poles at the points in S. There is a canonical identification H_{B}^{1}(Y) ⊗ ℂ = H_{dR}^{1}(Y (ℂ)) arising from integration of smooth differential forms against smooth chains, and its Hodge filtration is given by

(21)  Fil^{j}H_{dR}^{1}(Y (ℂ)) = \{ \begin{matrix}H_{dR}^{1}(Y (ℂ)) & if j ≤ 0, \\ Ω_{log}^{1}(Y (ℂ)) & if j = 1, \\ 0 & if j ≥ 2, \end{matrix}

where

Ω_{log}^{1}(Y (ℂ)) = {ω ∈ Ω^{1}(Y (ℂ)) with ord_{P} (ω) ≥ −1, ∀P ∈ S}

and ord_{P}(ω) denotes the valuation of the differential ω at the point P ∈ X(ℂ).

Because the Riemann surface Y (ℂ) arises from an algebraic curve over F, we can identify H_{dR}^{1}(Y (ℂ)) with H_{dR}^{1}(Y/F) ⊗ ℂ, where

(22)  H_{dR}^{1}(Y/F) := ℍ^{1}(0 → O_{Y} → Ω^{1}(Y) → 0)

is the algebraic de Rham cohomology of Y/F, defined as the hypercohomology of the de Rham complex of sheaves of regular differential forms on Y. The Hodge filtration on H_{dR}^{1}(Y/F) may also be described concretely by

(23)  Fil^{j}H_{dR}^{1}(Y/F) = \{ \begin{matrix}H_{dR}^{1}(Y/F) ≃ Ω^{II}(Y)/dF (Y) & if j ≤ 0, \\ Ω_{log}^{1}(Y/F) & if j = 1 \\ 0 & if j ≥ 2, \end{matrix}

(1) Ω^{II}(Y) is the space of rational differential forms on Y over F with vanishing residues at all points of Y; these are also called differentials of the second kind on Y;

(2) F (Y) is the field of rational functions on Y over F;

(3) Ω_{log}^{1}(Y/F) is the subspace of Ω^{1}(Y/F) consisting of differentials ω satisfying ord_{P} (ω) ≥ −1 for all P ∈ S.

The tensor product H_{B}^{1}(Y)^{⊗2} is again a mixed Hodge structure, and the Hodge filtration on H_{dR}^{1}(Y/F)^{⊗2} is given by

Fil^{j}H_{dR}^{1}(Y/F)^{⊗2} = \{ \begin{matrix}H_{dR}^{1}(Y/F)^{⊗2} & if j ≤ 0; \\ Ω_{log}^{1}(Y/F) ⊗ H_{dR}^{1}(Y/F) + H_{dR}^{1}(Y/F) ⊗ Ω_{log}^{1}(Y/F) & if j = 1; \\ Ω_{log}^{1}(Y/F)^{⊗2} & if j = 2 \\ 0 & if j ≥ 3. \end{matrix}

1.1. Iterated integrals and the Hodge filtration on M_{dR,ℂ}. The extension M_{B} of H_{B}^{1}(Y)^{⊗2} by H_{B}^{1}(Y) arising in (16) is equipped with a natural mixed Hodge structure. The definition of the Hodge filtration on M_{dR,ℂ} was given by R. Hain in [Hain2], and rests on the explicit description of M_{dR,ℂ} in terms of K.-T. Chen’s iterated integrals of smooth differential forms on path spaces. We now briefly recall this description, referring the reader to [Chen],

[Hain3] and the references therein for further details.

The path space on Y based at o, denoted ℙ(Y; o), is the set of piecewise smooth paths

γ : [0, 1] ⟶ Y (ℂ), with γ(0) = o.

The universal covering space of Y (ℂ), denoted Ỹ, is the space of homotopy classes in ℙ(Y; o). It is conformally equivalent to the open unit disc in ℂ. The group Γ = π_{1}(Y (ℂ); o) acts on it transitively and without fixed points, and the map γ ↦ γ(1) identifies the quotient Ỹ /Γ with Y (ℂ). If η is a smooth closed one-form (resp. a meromorphic one-form of the second kind) on Y (ℂ), then it admits a smooth (resp. meromorphic) primitive F_{η} on Ỹ, defined by the rule

F_{η}(γ) := ∫_{0}^{1} γ^{∗}η.

The basic iterated integral attached to smooth differential forms ω_{1}, …, ω_{n} along a path γ ∈ ℙ(Y; o) is defined to be

(24)  ∫_{γ} ω_{1} · ω_{2} ⋯ ω_{n} := ∫_{0≤t_{n}≤t_{n−1}≤···≤t_{1}≤1} γ^{∗}(ω_{1})(t_{1}) ⋯ γ^{∗}(ω_{n})(t_{n}).

The integer n is called the length of this basic iterated integral. In particular, when n = 2, the basic iterated integral attached to η_{1} and η_{2} is given by the formula

∫_{γ} η_{1} · η_{2} := ∫_{γ} η_{1}F_{η_{2}} = ∫_{0}^{1} γ^{∗}(η_{1}F_{η_{2}}).

It can also be expressed in terms of η_{1} ∧ η_{2}, viewed as a two-form on X × X, as the expression

(25) ∫_{γ} η_{1} · η_{2} := ∫_{T_{γ}} η_{1} ∧ η_{2},  where T_{γ} = {(γ(r), γ(s)), 0 ≤ s ≤ r ≤ 1} ⊂ X × X.

An iterated integral is any linear combination of basic iterated integrals, of possibly differing lengths, viewed as a function on ℙ(Y; o). The length of an iterated integral is then defined to be the maximal length arising in its expression as a linear combination of basic iterated integrals.

An iterated integral is said to be homotopy invariant if its value on a path γ depends only on the homotopy class of γ. The space II(Y) of homotopy invariant iterated integrals will be viewed as a subspace of the space of ℂ-valued functions on Γ, in the obvious way. By extending J ∈ II(Y) to the group ring ℂ[Γ] by ℂ-linearity, we will exploit the resulting natural inclusion

II(Y) ⊂ ℂ[Γ]^{∨}

to realise II(Y) as a space of complex functionals on ℂ[Γ]. For each n, let II^{≤n}(Y) denote the subspace of homotopy invariant iterated integrals of length ≤ n. This space plays an important role in our description of M_{dR,ℂ}, because of the following facts:

(1) Any element J ∈ II^{≤n}(Y) ⊂ ℂ[Γ]^{∨} vanishes on I^{n+1}, and hence gives rise to a welldefined element of (I/I^{n+1})^{∨} ⊗_{ℤ} ℂ.

(2) The natural map

(26)  II^{≤n} ⟶ (I/I^{n+1})^{∨} ⊗_{ℤ} ℂ

is an isomorphism. In particular,

(27)  M_{dR,ℂ} = II^{≤2}(Y).

The following lemma is helpful in describing II^{≤2}(Y) concretely.

Lemma 1.1. Let η_{1} and η_{2} be smooth closed one-forms on Y (ℂ) whose classes in de Rham cohomology are orthogonal with respect to the Poincaré pairing, and let α be a smooth one-form on Y (ℂ) satisfying the differential equation

(28)  dα = η_{1} ∧ η_{2}.

(1) The iterated integral

J_{η_{1}⊗η_{2},α}(γ) := ∫_{γ} (η_{1} · η_{2} − α)

is homotopy invariant.

(2) The image of J_{η_{1}⊗η_{2},α} ∈ M_{dR,ℂ} under the projection p of (17) is the element [η_{1}]⊗[η_{2}].

(3) The space II^{≤2}(Y) is spanned by iterated integrals of the form J_{η_{1}⊗η_{2},α}.

Proof. The first claim follows directly from the fact that the differential one-form on Ỹ appearing in the expression

J_{η_{1}⊗η_{2},α}(γ) = ∫_{γ} η_{1}F_{η_{2}} − α

is closed. The second statement follows from a direct calculation showing that

J_{η_{1}⊗η_{2},α}((γ_{1} − 1)(γ_{2} − 1)) = ∫_{γ_{1}} η_{1} ∫_{γ_{2}} η_{2}.

Finally, given any J ∈ M_{dR,ℂ}, after writing p(J) = ∑ ω_{i} ⊗ η_{i}, it can be checked that

J = (∑ J_{ω_{i}⊗η_{i},α_{i}}) − J_{0,0,α}

for suitable smooth differentials α_{i} on Y (ℂ) and α ∈ H_{dR}^{1}(Y (ℂ)). □

In the special case where Y (ℂ) = X(ℂ) − {∞} with X projective, the smooth differential α on Y (ℂ) can be expressed in terms of the Green’s function g_{η_{1}∧η_{2},∞} attached to the smooth (1, 1)-form η_{1} ∧ η_{2} and the point ∞. Recall (cf. [La], §II.1 for example) that g_{η_{1}∧η_{2},∞} is a smooth ℂ-valued function on Y (ℂ) satisfying the following two conditions:

(1) Letting ∂ and ¯∂ : O_{smooth}(Y) ⟶ Ω_{smooth}^{1}(Y) be the usual differential operators defined by

∂f = \frac{∂f}{∂z} dz = \frac{1}{2} (\frac{∂f}{∂x} − i\frac{∂f}{∂y} ) (dx + idy), ¯∂f = \frac{∂f}{∂z̄}dz̄ = \frac{1}{2} (\frac{∂f}{∂x} + i\frac{∂f}{∂y} ) (dx − idy),

the function g_{η_{1}∧η_{2},∞} is a solution of the differential equation

\frac{−1}{2πi}∂ ¯∂g_{η_{1}∧η_{2},∞} = η_{1} ∧ η_{2}.

(2) The function g_{η_{1}∧η_{2}} has at worst logarithmic growth at ∞. More precisely, letting

m_{η_{1}∧η_{2}} := ∫_{X(ℂ)} η_{1} ∧ η_{2},

the function g_{η_{1}∧η_{2},∞} can be written as

g_{η_{1}∧η_{2},∞}(x) = −m_{η_{1}∧η_{2}} log |f_{∞}(x)|^{2} + h(x),

where f_{∞} is a rational function on X with a simple zero at ∞ and h(x) is smooth in a neighbourhood of ∞.

The differential

α_{∞} := \frac{1}{2πi}∂g_{η_{1}∧η_{2},∞}

is a smooth one-form of type (1, 0) on Y (ℂ) satisfying

dα_{∞} = (∂ + ¯∂)α_{∞} = ¯∂α_{∞} = η_{1} ∧ η_{2}.

Its Laurent development around ∞ has a principal part of the form

α_{∞} = \frac{−m_{η_{1}∧η_{2}}}{2πi} \frac{dz}{z} .

This implies that, for any smooth region R ⊂ X(ℂ) whose boundary does not contain ∞,

(29)  ∫_{R} η_{1} ∧ η_{2} = ∫_{∂R} α_{∞} + m_{η_{1}∧η_{2}} ∫_{R} δ_{∞},

where δ_{∞} is the Dirac delta current, satisfying

∫_{R} δ_{∞} = \{ \begin{matrix}1 & if ∞ ∈ R; \\ 0 & if ∞ ∉ R. \end{matrix}

In other words, we have the equality of currents:

(30)  η_{1} ∧ η_{2} = dα_{∞} + m_{η_{1}∧η_{2}}δ_{∞}.

The following lemma concerning the differential α_{∞} shall be used in the proofs of Section 1.2.

Lemma 1.2. After setting X = X_{1} = X_{2}, viewing η_{1} ∧ η_{2} as a (1, 1)-form on X_{1}, and letting ρ be a closed one-form on X_{2}, consider the three-form (η_{1} ∧ η_{2}) ∧ ρ on X_{1} × X_{2}. Let W be any smooth three-chain on X_{1} × X_{2} which intersects {∞} × X_{2} in a smooth one-chain, so that

(∂W)′ := ∂W − ∂(W ∩ ({∞} × X_{2}))

is the complement of a finite set in ∂W. Then the integral

∫_{∂W} α_{∞} ∧ ρ := ∫_{(∂W)′} α_{∞} ∧ ρ

converges, and satisfies

∫_{W} (η_{1} ∧ η_{2}) ∧ ρ = ∫_{∂W} α_{∞} ∧ ρ + m_{η_{1}∧η_{2}} ∫_{W ∩({∞}×X_{2})} ρ.

Proof. Since ρ is closed, equation (30) implies that

(η_{1} ∧ η_{2}) ∧ ρ = d(α_{∞} ∧ ρ) + m_{η_{1}∧η_{2}} · δ_{∞} ∧ ρ.

Integrating over W and applying Stokes’ theorem, we obtain

∫_{W} (η_{1} ∧ η_{2}) ∧ ρ = ∫_{∂W} α_{∞} ∧ ρ + m_{η_{1}∧η_{2}} ∫_{W} δ_{∞} ∧ ρ,

and the lemma follows from the fact that ∫_{W} δ_{∞} ∧ ρ = ∫_{W ∩({∞}×X_{2})} ρ. □

Following Hain (cf. [Hain3, §13.1]), the Hodge filtration on (I/I^{n+1})^{∨} ⊗_{ℤ} ℂ is given in terms of (26) by defining Fil^{j}((I/I^{n+1})^{∨} ⊗_{ℤ} ℂ) to be the set of J ∈ II^{≤n} of the form

J = (∫ ω_{1}^{(1)} · ω_{2}^{(1)} ⋯ ω_{n_{1}}^{(1)}) + ⋯ + (∫ ω_{1}^{(t)} · ω_{2}^{(t)} ⋯ ω_{n_{t}}^{(t)})

in which each n_{i}-tuple (ω_{1}^{(i)}, …, ω_{n_{i}}^{(i)}) involves at least j smooth differential forms of type (1, 0) on Y (ℂ) with logarithmic growth along S. In particular, for n = 2 we have

(i) Fil^{j}M_{dR,ℂ} = M_{dR,ℂ} for all j ≤ 0; (ii) The space Fil^{1}M_{dR,ℂ} is spanned by iterated integrals of the form J_{η_{1},η_{2},α} in which α

and at least one of η_{1} and η_{2} lie in Ω_{log}^{1}(Y (ℂ)). (iii) The space Fil^{2}M_{dR,ℂ} is spanned by iterated integrals of the form J_{ω_{1},ω_{2},0} in which ω_{1}

and ω_{2} both belong to Ω_{log}^{1}(Y (ℂ)). (iv) For all j ≥ 3, Fil^{j}M_{dR,ℂ} = 0.

With this description of the Hodge filtration on M_{dR,ℂ}, it is routine to verify that the exact sequence (17) is an exact sequence of filtered vector spaces. The ℤ-module M_{B} is in fact a prototypical example of an integral mixed Hodge structure. (See [Hain3, Cor. 9.3] and [Hain1] for a discussion in a more general setting.) Let

(31)  κ_{dR} ∈ Ext_{MHS}^{1}(H_{B}^{1}(Y)_{∪=0}^{⊗2}, H_{B}^{1}(Y))

denote the class of the extension (16) in the category MHS of mixed Hodge structures. We will primarily be concerned with understanding this extension class and finding criteria for it to be non-trivial.

An alternate description of the Hodge filtration

The analytic description of the Hodge filtration on M_{dR,ℂ} which we have just given is wellsuited for the theoretical calculations of this note. However, the difficulties inherent in solving equation (28) seem to make it ill-suited for the practical numerical calculations of Abel-Jacobi maps that are carried out in [DDLR]. For the latter, it is convenient to exploit an algebraic description of M_{dR,ℂ} and its Hodge filtration involving meromorphic differentials of the second kind rather than smooth differentials.

Note that the definitions of Fil^{j}M_{dR,ℂ} given above apply just as well to the algebraic setting, with the exception of Fil^{1}M_{dR,ℂ} which is described in terms of smooth forms. We now proceed to give a description of this space involving meromorphic differentials on Y of the second kind.

Firstly, given ω ∈ Ω_{log}^{1}(Y (ℂ)) and a smooth closed one-form η on Y (ℂ), the two-form ω ∧ η + η ∧ ω vanishes identically on Y. In particular, the class of ω ⊗ η + η ⊗ ω always belongs to H^{1}(Y)_{∪=0}^{⊗2}, even when Y is projective, and the elements of M_{dR,ℂ} which map to this class are those of the form

J_{ω⊗η+η⊗ω,α},  with dα = 0;
J_{ω⊗η+η⊗ω,α}(γ) := ∫_{γ}(ω · η + η · ω) − ∫_{γ} α.

A direct calculation reveals that

J_{ω⊗η+η⊗ω,α}(γ) = (∫_{γ} ω) (∫_{γ} η) − ∫_{γ} α,

so that the values of J_{ω⊗η+η⊗ω,α} are expressed in terms of products of periods of γ relative to the de Rham cohomology classes of ω, η and α. The function J_{ω⊗η+η⊗ω,α} can therefore be readily calculated in practice using meromorphic representatives of the classes of ω, η, and α. It is then clear from the description of the Hodge filtration on M_{dR,ℂ} given in (ii) above that

(32)  J_{ω⊗η+η⊗ω,α} belongs to Fil^{1}M_{dR,ℂ} if and only if α ∈ Ω_{log}^{1}(Y/ℂ).

Secondly, let us represent a cohomology class ξ ∈ H_{dR}^{1}(Y/ℂ) by a meromorphic differential form η of the second kind on Y (ℂ) with at worst logarithmic poles along S, making use of the algebraic description for H_{dR}^{1}(Y/F) given in (23). The principal part of the meromorphic differential 1-form ωF_{η} at a point x ∈ Ỹ, denoted \underline{pp}_{x}(ωF_{η}), depends only on the image of x in Y (ℂ). Indeed, let γx ∈ Ỹ, for γ ∈ Γ, be any other point in Ỹ lying above the same point of Y (ℂ). Comparing principal parts, we find:

\underline{pp}_{γx}(ωF_{η}) = \underline{pp}_{x}(γ^{∗}(ωF_{η})) = \underline{pp}_{x}(ωF_{η} + cω)

for some constant c, and the claim follows because ω is regular. In particular, it makes sense to talk about the principal part \underline{pp}_{P} (ωF_{η}) at a point P ∈ Y (ℂ), even though the one-form ωF_{η} is only defined on Ỹ. A direct application of the Riemann-Roch theorem shows that there exists a meromorphic differential form α on Y (ℂ) with at worst logarithmic poles along S, satisfying

(33)  \underline{pp}_{P} (α) = \underline{pp}_{P} (ω · F_{η}),  for all P ∈ Y (ℂ).

The one-form ω · F_{η} − α is then holomorphic on Ỹ. The differentials α satisfying (33) all differ by elements of Ω_{log}^{1}(Y (ℂ)) = Fil^{1}H_{dR}^{1}(Y/ℂ), and we again have that

(34)  J_{ω⊗η,α} belongs to Fil^{1}M_{dR,ℂ}.

It is then obvious from the definition of Fil^{1}M_{dR,ℂ} given in (ii) above that this space can be recovered in terms of meromorphic differentials as the vector space generated by the iterated integrals appearing in (32) and (34).

Given a regular differential ω and a meromorphic differential η of the second kind on Y, the task of calculating the meromorphic one-form α satisfying (33) can be carried out explicitly in the case of modular curves, and seems easier in practice than solving the equation (28) associated to a smooth representative of the class of η. See the forthcoming article [DDLR] for a more detailed discussion of these computational aspects.

1.2. Points arising from Hodge classes and the fundamental group. The goal of this section is to recall a construction which goes back at least to [Ka] and [Pu] whereby the extension class κ_{dR} of (31) arising from the pro-unipotent fundamental group are used to construct rational points in the Jacobian of a (smooth, projective) curve^{3}. We retain the same notations as in the previous section and assume that Y = X \ {∞} is the complement of a point ∞ ∈ X(F).

A Hodge class in a mixed Hodge structure N of weight two is a morphism

(35)  ξ : ℤ(−1) ⟶ N

of mixed Hodge structures, where ℤ(−1) denotes the weight 2 Hodge structure attached to the Tate motive H^{2}(ℙ_{1}). The ℤ-module of all such Hodge classes is denoted Hdg(N).

The extension class

κ_{dR}(∞) ∈ Ext_{MHS}^{1}(H_{B}^{1}(X)^{⊗2}, H_{B}^{1}(X)),

3A tantalising ongoing program (cf. [Ki]) also exploits the pro-unipotent fundamental group of a (hyperbolic, affine) curve Y to bound its set of integral points, the rough idea being to associate to each point in Y (ℤ) a system of non-abelian torsors, and then to control the Selmer varieties which parametrise them.

of M_{B} in the category of mixed Hodge structures arising from the exact sequence (16) is a natural object of study. We do so by considering, for each class ξ ∈ Hdg(H_{B}^{1}(X)^{⊗2}), the element

κ_{dR}(ξ; ∞) ∈ Ext_{MHS}^{1}(ℤ(−1), H_{B}^{1}(X))

associated to the class of the extension M(ξ) = M(ξ; ∞) appearing in the top row of the following commutative diagram with exact rows and cartesian squares:

(36)  [figure: 0; H_{B}^{1}(X); M_{B}(ξ); ℤ(−1); ξ; 0; 0; H_{B}^{1}(X); M_{B}; H_{B}^{1}(X)^{⊗2}; 0,]

where the lower exact sequence is (16) applied to Y, once we identify H_{B}^{1}(Y) = H_{B}^{1}(X) thanks to (19).

There is a canonical isomorphism (cf. e.g. [Ca], [Hain3, §10.2])

(37)  Ext_{MHS}^{1}(ℤ(−1), H_{B}^{1}(X)) = \frac{H_{dR}^{1}(X/ℂ)}{Ω^{1}(X(ℂ)) + H_{B}^{1}(X)}.

Recalling the exact sequence (17), this identification is obtained by choosing elements v_{Hodge} ∈ Fil^{1}(M_{B}(ξ) ⊗ ℂ) and v_{ℤ} ∈ M_{B}(ξ) satisfying

(38)  p(v_{Hodge}) = p(v_{ℤ}) = 1,

and assigning to the extension class M_{B}(ξ) the element

(39)  v_{ξ} := i^{−1}(v_{Hodge} − v_{ℤ}) ∈ H_{dR}^{1}(X/ℂ).

Since (38) determines v_{Hodge} and v_{ℤ} up to elements of Fil^{1}H_{dR}^{1}(X/ℂ) = Ω^{1}(X/ℂ) and H_{B}^{1}(X(ℂ), ℤ) respectively, the image of v_{ξ} in the quotient on the right-hand side of (37) depends only on the isomorphism class of the extension M_{B}(ξ).

The Poincaré pairing

(40)  H_{dR}^{1}(X/F) × H_{dR}^{1}(X/F) ⟶ F

with F = ℂ induces an isomorphism

(41)  \frac{H_{dR}^{1}(X/ℂ)}{Ω^{1}(X(ℂ)) + H_{B}^{1}(X)} ⟶ \frac{Ω^{1}(X(ℂ))^{∨}}{H_{B}^{1}(X)} = Jac(X)(ℂ).

For all ξ ∈ Hdg(H_{B}^{1}(X)^{⊗2}), we continue to use κ_{dR}(ξ; ∞) ∈ Jac(X)(ℂ) to describe the canonical point in the jacobian of X arising from the identifications (37) and (41).

The assignment ξ ↦ κ_{dR}(ξ; ∞) therefore sets up a map

(42)  Hdg(H_{B}^{1}(X)^{⊗2}) ⟶ Jac(X)(ℂ)

sending Hodge classes in H_{dR}^{1}(X)^{⊗2} to complex points in the Jacobian of X.

Lemma 1.3. Given ∞_{1}, ∞_{2} ∈ X(ℂ) − {o}, we have

κ_{dR}(ξ; ∞_{1}) = κ_{dR}(ξ; ∞_{2}) + m_{ξ}((∞_{2}) − (∞_{1})),

where we recall that m_{ξ} = ∫_{X(ℂ)} ξ is the integral of any smooth two-form representing ξ over the curve X(ℂ) diagonally embedded in X(ℂ) × X(ℂ).

Proof. Let

v_{ξ}(∞_{1}) ∈ H_{dR}^{1}(X − {∞_{1}}/ℂ), v_{ξ}(∞_{2}) ∈ H_{dR}^{1}(X − {∞_{2}}/ℂ)

be the be the elements attached to the extension classes M_{B}(ξ) with ∞ = ∞_{1} and ∞_{2} respectively, following equation (39). The difference v_{ξ}(∞_{1})−v_{ξ}(∞_{2}) corresponds, modulo elements of Ω^{1}(X) and H_{B}^{1}(X(ℂ), ℤ), to the class in H_{dR}^{1}(X − {∞_{1}, ∞_{2}}) of a regular differential ω ∈ Ω_{log}^{1}(X − {∞_{1}, ∞_{2}}) whose residual divisor is m_{ξ}((∞_{2}) − (∞_{1})). But such an element describes the image of the degree zero divisor m_{ξ}((∞_{2}) − (∞_{1})) in Ext^{1}(ℤ(−1), H_{dR}^{1}(X/ℂ)), and the result follows. □

It is occasionally convenient to attach an invariant κ_{dR}(ξ) ∈ Ext^{1}(ℤ(−1), H_{dR}^{1}(X/ℂ)) to an arbitrary element of Hdg(H_{B}^{1}(X)^{⊗2}) depending only on the pointed curve (X; o) and its fundamental group, and not on the choice of an auxiliary base point ∞ ∈ X − {o}. We can do this by setting

(43)  κ_{dR}(ξ) := κ_{dR}(ξ; ∞) + m_{ξ}((∞) − (o)).

By Lemma 1.3, the right hand side does not depend on the point ∞. It depends only on the Hodge class ξ and the exact sequence (16) arising from the fundamental group π_{1}(X; o). It can be interpreted as arising from the pro-nilpotent fundamental group of X − {o} with “tangential base point” at o, in the sense of the article [De].

The next two propositions express the points κ_{dR}(ξ; ∞) and κ_{dR}(ξ) in terms of iterated integrals. Let us first introduce a few notations.

Given a Hodge class ξ ∈ Hdg(H_{B}^{1}(X)^{⊗2}), choose a representative

ξ̃ = ∑_{j=1}^{t} ω_{j} ⊗ η_{j},

for its de Rham cohomology class, in such a way that ω_{j} and η_{j} are smooth closed one-forms on X for each j, at least one of which is of type (1, 0) (and hence, holomorphic on X). If ρ ∈ Ω^{1}(X) is any holomorphic differential, let γ_{ρ} ∈ ℂ[Γ_{ab}] be a path whose homology class is Poincaré dual to ρ, i.e., such that

∫_{γ_{ρ}} η̃ = ⟨ρ, η⟩, for all η ∈ H_{dR}^{1}(X/ℂ).

Proposition 1.4. With notations as above, let α_{∞} be a smooth differential on Y (ℂ) of type (1, 0), with at worst logarithmic poles at ∞, satisfying

dα_{∞} = ∑_{j=1}^{t} ω_{j} ∧ η_{j} on Y (ℂ).

Then, for all ρ ∈ Ω^{1}(X/ℂ),

κ_{dR}(ξ; ∞)(ρ) = ∫_{γ_{ρ}} (∑_{j=1}^{t} ω_{j} · η_{j} − α_{∞}) and

κ_{dR}(ξ)(ρ) = ∫_{γ_{ρ}} (∑_{j=1}^{t} ω_{j} · η_{j} − α_{∞}) + m_{ξ} ∫_{o}^{∞} ρ.

Proof. The first equality is a direct consequence of the definitions, using the fact that the class of the extension M_{B}(ξ; ∞) attached to the pair (ξ, ∞) is described by the iterated integral J_{ξ̃,α_{∞}}. The second equality then follows from equation (43) defining κ_{dR}(ξ). □

We conclude this section with a formula analogous to the one in Proposition 1.4, expressing κ_{dR}(ξ; ∞) and κ_{dR}(ξ) in terms of differentials of the second kind. This formula turns out to be particularly useful in computing the points κ_{dR}(ξ) numerically, as it is done in [DDLR] for Hodge classes arising from Hecke correspondences on a classical elliptic modular curve.

Choose a representative

ξ̃ = ∑_{j=1}^{s} ω_{j}^{+} ⊗ η_{j}^{+} + ∑_{j=1}^{t} η_{j}^{−} ⊗ ω_{j}^{−} ∈ Ω^{1}(X/ℂ) ⊗ Ω_{mer}^{1}(X/ℂ) + Ω_{mer}^{1}(X/ℂ) ⊗ Ω^{1}(X/ℂ)

of the class ξ, in such a way that the elements ω_{j}^{±} belong to Ω^{1}(X) = Fil^{1}H_{dR}^{1}(X/ℂ) and the differentials η_{j}^{±} are of the second kind and regular at ∞.

Proposition 1.5. Which notations as above, let α_{∞} be a meromorphic differential on X, with at worst logarithmic poles at ∞, satisfying

\underline{pp}_{x}(α_{∞}) = ∑_{j=1}^{s} \underline{pp}_{x}(ω_{j}^{+}F_{η_{j}^{+}}) − ∑_{j=1}^{t} \underline{pp}_{x}(ω_{j}^{−}F_{η_{j}^{−}}), for all x ∈ Y (ℂ).

Then, for all ρ ∈ Ω^{1}(X/ℂ),

κ_{dR}(ξ; ∞)(ρ) = ∫_{γ_{ρ}} (∑_{j=1}^{s} ω_{j}^{+} · η_{j}^{+} + ∑_{j=1}^{t} η_{j}^{−} · ω_{j}^{−} − α_{∞}) .

Proof. The proof is the same as for Proposition 1.4, using this time the algebraic description of the Hodge filtration on M_{dR,ℂ} in terms of differentials of the second kind. □

2. Points arising from algebraic cycles

Hodge classes in H_{B}^{1}(X)^{⊗2} can be constructed naturally from geometry, by exploiting the cycle class maps

cl : Pic(X × X) ⟶ Hdg(H_{B}^{2}(X × X)).

More precisely, let ε be the projector on Pic(X × X) given by the rule

ε(Z) = Z − i_{1,∗}π_{1,∗}(Z) − i_{2,∗}π_{2,∗}(Z),

where

π_{1}, π_{2} : X × X ⟶ X, π_{1}(P_{1}, P_{2}) = P_{1}, π_{2}(P_{1}, P_{2}) = P_{2}

are the natural projections onto the first and second factors, and

i_{1}, i_{2} : X ↪ X × X, i_{1}(P) = (P, o), i_{2}(P) = (o, P)

are the natural inclusions of X into the horizontal and vertical copies of X in X × X over the point o.

The idempotent ε annihilates the factors H^{2}(X) ⊗ H^{0}(X) and H^{0}(X) ⊗ H^{2}(X) in the Künneth decomposition of H^{2}(X^{2}) (for any of the cohomology theories, Betti, de Rham or étale). For any Z ∈ Pic(X × X)(F) we can thus define

(44)  ξ_{Z} := cl(εZ) ∈ Hdg(H_{B}^{1}(X, ℤ)^{⊗2}).

In this way, we can assign to the cycle Z ⊂ X × X a point

κ_{dR}(ξ_{Z}) ∈ Jac(X)(ℂ),

The goal of this section is to associate to any Z ∈ Pic(X × X)(F) a point

P_{Z} ∈ Jac(X)(F),

in such a way that for all complex embeddings ι : F ⟶ ℂ,

κ_{dR}(ξ_{Z}) = P_{Z} ×_{ι} Spec(ℂ).

Let us describe a geometric recipe for constructing the desired point P_{Z}. Given a smooth projective variety V_{/F} and a positive integer c ≥ 1, let

CH^{c}(V) = Z^{c}(V)/ ∼_{rat}

denote the abelian group of rational equivalence classes of cycles of codimension c in V. We regard it as a functor from the category of field extensions of F in ℂ to the category of abelian groups, given by the rule

K/F ↦ CH^{c}(V)(K) := {[Z] : Z ∈ Z^{c}(V), such that ^{σ}Z ∼_{rat} Z for all σ ∈ Aut_{K}(ℂ)}.

For any integer i ≥ 1, let X_{i} = X be a copy of the curve X and for any subset {i_{1}, …, i_{m}} ⊆ {1, …, n} with m ≤ n of indices in a given finite set, let X_{i_{1},...,i_{m}} denote a copy of the curve X embedded in X_{1} × ⋯ × X_{n} via the map

x ↦ (x_{i})_{i=1}^{n}, where x_{i} = \{ \begin{matrix}x & if i ∈ \{i_{1}, ..., i_{m}\}, \\ o & otherwise. \end{matrix}

The one-dimensional cycle Z ⊂ X_{1} × X_{2} gives rise to the following divisor classes on X:

D_{12} = Z · X_{12}, D_{1} = Z · X_{1}, D_{2} = Z · X_{2}.

Set

(45)  P_{Z} = D_{12} − deg(D_{12}) · o ∈ Pic^{0}(X) = Jac(X).

We now give an alternate geometric description of P_{Z} which will be used to relate this point to iterated integrals. Let

Π_{Z} := Z_{12} × X_{34} ⊂ X_{1} × X_{2} × X_{3} × X_{4},

where Z_{12} denotes a copy of the cycle Z embedded in X_{1} × X_{2}. The class of the cycle Π_{Z} belongs to CH^{2}(X_{1} × X_{2} × X_{3} × X_{4}), and induces a morphism

Π_{Z} : CH^{2}(X_{1} × X_{2} × X_{3}) ⟶ CH^{1}(X_{4})

by the usual rule

(46)  Π_{Z}(Δ) = π_{4∗}(Π_{Z} · π_{123}^{∗}Δ),

where

π_{4} : X_{1} × X_{2} × X_{3} × X_{4} ⟶ X_{4}, π_{123} : X_{1} × X_{2} × X_{3} × X_{4} ⟶ X_{1} × X_{2} × X_{3}

denote the natural projections onto the fourth and first three factors respectively. The Gross-Kudla-Schoen modified diagonal cycle in X^{3} = X_{1} × X_{2} × X_{3} is defined to be

(47)  Δ_{GKS} = X_{123} − X_{23} − X_{13} − X_{12} + X_{1} + X_{2} + X_{3} ∈ CH^{2}(X_{1} × X_{2} × X_{3}).

Lemma 2.1. For all Z ∈ Pic(X × X),

P_{Z} = Π_{Z}(Δ_{GKS}).

Proof. A direct calculation using only the definition (46) shows that

Π_{Z}(X_{123}) = D_{12},
Π_{Z}(X_{23}) = D_{2}, Π_{Z}(X_{13}) = D_{1}, Π_{Z}(X_{12}) = deg(D_{12}) · o,
Π_{Z}(X_{1}) = deg(D_{1}) · o, Π_{Z}(X_{2}) = deg(D_{2}) · o, Π_{Z}(X_{3}) = 0.

The lemma follows. □

Remark 2.2. Let ∞ be any point in X (possibly equal to o). Viewing X_{1} ×{∞} and {∞}×X_{2} as divisors in X_{1} × X_{2}, a direct calculation shows that

P_{X_{1}×{∞}} = P_{{∞}×X_{2}} = 0.

Hence P_{Z} = P_{εZ} and the point P_{Z} depends only on the Hodge class ξ_{Z} ∈ Hdg(H_{B}^{1}(X)^{⊗2}).

We will use Lemma 2.1 to derive an analytic formula for the point P_{Z}, or rather, for its image under the Abel-Jacobi map

AJ_{X} : CH^{1}(X)_{0} ⟶ \frac{Ω^{1}(X/ℂ)^{∨}}{H_{1}(X(ℂ), ℤ)}.

This Abel-Jacobi map is defined on degree 0 divisors D by the familiar rule:

(48)  AJ_{X}(D)(ω) = ∫_{∂^{−1}(D)} ω,

where ∂^{−1}(D) is any smooth one-chain on X(ℂ) whose boundary is D. It admits a higher dimensional generalisation

(49)  AJ_{X^{3}} : CH^{2}(X^{3})_{0}(ℂ) ⟶ J^{2}(X^{3}),

where CH^{2}(X^{3})_{0} is the Chow group of null-homologous cycles on X^{3}, and

J^{2}(X^{3}) = \frac{Fil^{2}H_{dR}^{3}(X^{3}/ℂ)^{∨}}{H_{3}(X^{3}(ℂ), ℤ)}

is the intermediate Jacobian associated to H^{3}(X^{3}). This Abel-Jacobi map is defined by the rule generalising (48),

(50)  AJ_{X^{3}}(Δ)(ω) = ∫_{∂^{−1}(Δ)} ω,  ω ∈ Fil^{2}H_{dR}^{3}(X^{3}/ℂ).

The following lemma verifies that the Gross-Kudla-Schoen cyle Δ_{GKS} lies in the domain of this Abel-Jacobi map.

Lemma 2.3. The cycle Δ_{GKS} is null-homologous.

Proof. The obvious method to prove this lemma is to exploit the Künneth decomposition of the cohomology of X_{1} × X_{2} × X_{3}, as it is done in [GrSc] for example. We give a more detailed, direct argument, which yields a differentiable 3-chain on X_{1} × X_{2} × X_{3} having Δ_{GKS} as boundary. While not strictly necessary to prove Lemma 2.3, such an explicit 3-chain will be used in subsequent calculations.

Let g be the genus of the projective curve X, and let γ_{1}, …, γ_{2g} be a collection of paths in ℙ(X; o) whose associated homology classes, denoted [γ_{1}], …, [γ_{2g}], form an integral basis for H_{1}(X(ℂ), ℤ). Let γ_{1}^{∗}, …, γ_{2g}^{∗} be another collection of paths, whose homology classes give rise to the dual basis in H_{1}(X(ℂ), ℤ) relative to the intersection product. In other words, we require that

([γ_{i}] · [γ_{j}^{∗}]) = δ_{ij}, for all 1 ≤ i, j ≤ 2g,

where δ_{ij} is the Kronecker delta. Let Δ denote the diagonal in X ×X. Its class in H_{2}(X ×X) is given by

cl(Δ) = cl(X × {o}) + cl({o} × X) + ∑_{i=1}^{2g}[γ_{i}] ⊗ [γ_{i}^{∗}],

and therefore there is a smooth 3-chain W ⊂ X × X such that

(51)  Δ = (X × {o}) + ({o} × X) + ∑_{i=1}^{2g} γ_{i} × γ_{i}^{∗} + ∂(W).

Applying this relation to X_{123}, viewed as the diagonal in X_{12} × X_{3}, gives

(52)  X_{123} = X_{12} + X_{3} + ∑_{i=1}^{2g} γ_{i}(12) × γ_{i}^{∗}(3) + ∂(W_{12,3}),

(1) For {r, s} ∈ {1, 2, 3}, the regions γ_{i}(rs) or γ_{i}(r) are the paths γ_{i} diagonally embedded in X_{rs} ⊂ X_{1} × X_{2} × X_{3} and in X_{r} respectively, and similar conventions are adopted for the paths γ_{j}^{∗}.

(2) The three-chain W_{12,3} is a copy of the region W of (51), embedded in X_{12} × X_{3} in the

natural way. By an analogous reasoning, we find

(53)  X_{13} = X_{1} + X_{3} + ∑_{i=1}^{2g} γ_{i}(1) × γ_{i}^{∗}(3) + ∂(W_{13}),

and

(54)  X_{23} = X_{2} + X_{3} + ∑_{i=1}^{2g} γ_{i}(2) × γ_{i}^{∗}(3) + ∂(W_{23}),

with the obvious meaning given to W_{13} ⊂ X_{1} × {o} × X_{3} and to W_{23} ⊂ {o} × X_{2} × X_{3}. Subtracting (53) and (54) from (52), we obtain

Δ_{GKS} = ∑_{i=1}^{2g}(γ_{i}(12) − γ_{i}(1) − γ_{i}(2)) × γ_{i}^{∗}(3) + ∂(W_{12,3} − W_{13} − W_{23}).

But

γ_{i}(12) − γ_{i}(1) − γ_{i}(2) = ∂(T_{γ_{i}}(12)),

where

T_{γ}(12) = {(γ(r), γ(s)), 0 ≤ s ≤ r ≤ 1} ⊂ X_{1} × X_{2}

is the two-dimensional region introduced in (25). Therefore,

(55)  Δ_{GKS} = ∂ (∑_{i=1}^{2g} T_{γ_{i}}(12) × γ_{i}^{∗}(3) + W_{12,3} − W_{13} − W_{23}) ,

and the result follows. □

Lemma 2.4. For all Z ∈ CH^{2}(X × X), and all ρ ∈ Ω^{1}(X),

AJ_{X}(P_{Z})(ρ) = AJ_{X^{3}}(Δ_{GKS})(ξ_{Z}(12) ∧ ρ(3)),

where the classes

ξ_{Z}(12) ∈ Fil^{1}H_{dR}^{2}(X_{1} × X_{2}/ℂ), ρ(3) ∈ Ω^{1}(X_{3}) = Fil^{1}H_{dR}^{1}(X_{3}/ℂ)

are obtained in the natural way from the classes ξ and ρ respectively (so that, in particular, ξ_{Z}(12) ∧ ρ(3) belongs to Fil^{2}H_{dR}^{3}(X^{3}/ℂ)).

Proof. By Lemma 2.1 and the functoriality properties of the Abel-Jacobi map under correspondences,

AJ_{X}(P_{Z})(ρ) = AJ_{X}(Π_{Z}Δ_{GKS})(ρ) = AJ_{X^{3}}(Δ_{GKS})(Π_{Z}^{∗}ρ),

where

Π_{Z}^{∗} : H_{dR}^{1}(X_{4}/ℂ) ⟶ H_{dR}^{3}(X_{1} × X_{2} × X_{3}/ℂ)

is the morphism induced on de Rham cohomology by the correspondence Π_{Z} = Z_{12} × X_{34}. A direct calculation reveals that

Π_{Z}^{∗}(ρ) = cl(Z) ∧ ρ,

and therefore

AJ_{X}(P_{Z})(ρ) = AJ_{X^{3}}(Δ_{GKS})(cl(Z) ∧ ρ).

By a similar reasoning,

AJ_{X}(P_{εZ})(ρ) = AJ_{X^{3}}(Δ_{GKS})(ξ_{Z} ∧ ρ),

and the result follows from Remark 2.2. □

For the next proposition, we adopt the same notations that were introduced in the statement of Proposition 1.4 and the discussion immediately preceding it.

Theorem 2.5. For all ξ ∈ Hdg(H_{B}^{1}(X)^{⊗2}) and all ρ ∈ Ω^{1}(X/ℂ),

AJ_{X^{3}}(Δ_{GKS})(ξ(12) ∧ ρ(3)) = ∫_{γ_{ρ}} (∑_{j=1}^{t} ω_{j} · η_{j} − α_{∞}) + m_{ξ} ∫_{o}^{∞} ρ.

Proof. By (55), and the definition of the Abel-Jacobi map,

(56)  AJ(Δ_{GKS})(ξ ∧ ρ) = ∑_{i=1}^{2g} ∫_{T_{γ_{i}}(12)×γ_{i}^{∗}(3)} ξ̃∧ ρ + ∫_{W_{12,3}} ξ̃∧ ρ,

where we have used the fact that ξ̃∧ ρ vanishes identically on W_{13} ⊂ X_{1} × {o} × X_{3} and on W_{23} ⊂ {o} × X_{2} × X_{3} to dispense with the integrals over these two regions. The first term appearing in the right hand side of (56) is equal to

∑_{i=1}^{2g} ∫_{T_{γ_{i}}(12)×γ_{i}^{∗}(3)} ξ̃∧ ρ = ∑_{i=1}^{2g} ∫_{T_{γ_{i}}(12)} ξ̃∫_{γ_{i}^{∗}(3)} ρ.

This last expression can also be viewed as the integral of the two-form ξ̃ over the 2-chain

∑_{i=1}^{2g} (∫_{γ_{i}^{∗}} ρ) T_{γ_{i}} = T_{γ_{ρ}},

where

T_{γ_{ρ}} = ∑_{i=1}^{2g} λ_{i}T_{γ_{i}}, with [γ_{ρ}] = ∑ λ_{i}[γ_{i}].

By the formula for the iterated integral that is given in (25), it follows that

(57)  ∑_{i=1}^{2g} ∫_{T_{γ_{i}}(12)×γ_{i}^{∗}(3)} ξ̃∧ ρ = ∫_{γ_{ρ}} ∑_{j=1}^{t} ω_{j} · η_{j}.

To deal with the second term in the right-hand side of (56), we choose an auxiliary point ∞ which is not in the image of any of the paths γ_{i} and γ_{i}^{∗}, and such that W_{12,3} ∩ ((∞, ∞) × X_{3}) is a smooth one-chain. By Lemma 1.2,

(58)  ∫_{W_{12,3}} ξ̃∧ ρ = ∫_{∂W_{12,3}} α_{∞} ∧ ρ + m_{ξ} ∫_{W_{12,3}∩((∞,∞)×X_{3})} ρ.

Recall that

∂W_{12,3} = X_{123} − X_{12} − X_{3} − ∑_{i=1}^{2g} γ_{i}(12) × γ_{i}^{∗}(3),

and note that the differential α_{∞} ∧ ρ vanishes identically on X_{123}, X_{12} and X_{3} since it is of type (2, 0). Therefore, by a similar argument as above,

(59)  ∫_{∂W_{12,3}} α_{∞} ∧ ρ = − ∑_{i=1}^{2g} ∫_{γ_{i}} α_{∞} ∫_{γ_{i}^{∗}} ρ = − ∫_{γ_{ρ}} α_{∞}.

Furthermore, the region of integration that appears in the second term of (58) has boundary

∂(W_{12,3} ∩ ((∞, ∞) × X_{3})) = (X_{123} − X_{12} − X_{3} − ∑_{i=1}^{2g} γ_{i} × γ_{i}^{∗}) ∩ ((∞, ∞) × X_{3})
= (∞, ∞, ∞) − (∞, ∞, o).

It follows that

(60)  m_{ξ} ∫_{W_{12,3}∩((∞,∞)×X_{3})} ρ = m_{ξ} ∫_{o}^{∞} ρ.

Substituting (59) and (60) into (58) yields

(61)  ∫_{W_{12,3}} ξ̃∧ ρ = − ∫_{γ_{ρ}} α_{∞} + m_{ξ} ∫_{o}^{∞} ρ.

Finally, using (57) and (61) to rewrite the right hand side of (56), we obtain

(62)  AJ(Δ_{GKS})(ξ ∧ ρ) = ∫_{γ_{ρ}} ∑_{j=1}^{t} ω_{j} · η_{j} − ∫_{γ_{ρ}} α_{∞} + m_{ξ} ∫_{o}^{∞} ρ,

as was to be shown. □

Corollary 2.6. For all Z ∈ Pic(X × X), and all ρ ∈ Ω^{1}(X),

AJ_{X}(P_{Z})(ρ) = ∫_{γ_{ρ}} (∑_{j=1}^{t} ω_{j} · η_{j} − α_{∞}) + m_{ξ_{Z}} ∫_{o}^{∞} ρ,

where ξ̃_{Z} = ∑_{i=1}^{t} ω_{i} ⊗ η_{i} is a representative of ξ_{Z} := cl(εZ) and α_{∞} and γ_{ρ} are as in the statement of Theorem 2.5.

Proof. This follows directly from Lemma 2.4 and Theorem 2.5. □

By combining Proposition 1.4 and Corollary 2.6 we obtain the sought-for relation between the extension classes κ_{dR}(ξ) arising from π_{1}(X; o) and the points P_{Z}.

Theorem 2.7. For all Z ∈ Pic(X × X), κ_{dR}(ξ_{Z}) = P_{Z} in Jac(X)(ℂ).

In light of Lemma 2.1, this is Theorem 1. As anticipated in the introduction, we can recover from it a result that is equivalent to the one obtained by Kaenders in [Ka]:

Corollary 2.8. Let X_{12} ∈ Pic(X × X) be the class of the diagonal divisor in X × X up to rational equivalence. Then κ_{dR}(ξ_{X_{12}}) = (2g − 2)o − K in Jac(X)(ℂ), where K ∈ Pic(X) is the canonical divisor on X.

Proof. By Theorem 2.7 and (45), κ_{dR}(ξ_{X_{12}}) = P_{X_{12}} = X_{12} · X_{12} − deg(X_{12} · X_{12})o. An easy application of the Riemann-Roch theorem and the adjunction formula shows that the self-intersection of the diagonal in the square of a curve is −K. The corollary follows. □

3. Modular and Shimura curves

In this last section we specialize to the setting where X is a Shimura curve over a totally real field.

Let F be a totally real number field of degree d, and R_{F} its ring of integers. Fix a classical Hilbert modular newform g of parallel weight 2, level N_{g} and trivial nebentypus. We refer the reader to [Zh] for the precise definitions of Shimura curves, Hecke correspondences and L-functions that we use throughout this section.

Choose a square-free ideal D = ℘_{1} ·...·℘_{r} || N_{g} of R_{F} dividing it exactly (by which we mean that D and N_{g}/D are relatively prime). Assume that r + d is odd. (If d is odd, a possible choice for D is thus D = R_{F}).

Choose an ideal N of R_{F} such that N_{g} | N and D || N, and put M = N/D. Let X_{0}^{D}(M) denote the Shimura curve associated with the Eichler order of level M in a quaternion algebra ramified precisely at D and at all archimedean places of F but one, and write J_{0}^{D}(M) for its jacobian.

Let 𝕋 := ⟨T_{℘}, ℘ ∤ N⟩_{ℚ} be the Hecke algebra generated by the set of Hecke operators at primes ℘ not dividing N, acting on J_{0}^{D}(M). This ℚ-algebra is isomorphic to a free product of number fields, namely

(63)  𝕋 ≃ ∏_{D||n|N} ∏_{h} K_{h}

where for each ideal n such that D || n and n | N, h runs through a set of representatives for the orbits of Gal (ℚ̄/ℚ) acting on the set of Hilbert newforms of level n, and K_{h} is the number field generated by the eigenvalues of 𝕋 acting on h. In particular 𝕋 contains a copy of K_{g}; write T_{g} ∈ 𝕋 for the corresponding idempotent in the decomposition (63).

If M is prime then in fact 𝕋 ≃ EndJ_{0}^{D}(M). But as soon as M is a composite ideal, the endomorphism algebra of J_{0}^{D}(M) is larger, namely a product of matrix algebras:

(64)  End^{0}(J_{0}^{D}(M)) := End(J_{0}^{D}(M)) ⊗ ℚ ≃ ∏_{D||n|N} ∏_{h} M_{σ(N/n)}(K_{h}),

where σ(N/n) is the number of ideal dividing N/n. Its center is thus 𝕋.

Note that End^{0}(J_{0}^{D}(M))[g] := T_{g} · End^{0}(J_{0}^{D}(M)) ≃ M_{σ(N/N_{g})}(K_{g}), the center of which is 𝕋[g] := T_{g} · 𝕋.

Since

(65)  End(J_{0}^{D}(M)) ≃ \frac{Pic(X_{0}^{D}(M) × X_{0}^{D}(M))}{π_{1}^{∗}(Pic(X_{0}^{D}(M)) + π_{2}^{∗}(Pic(X_{0}^{D}(M))}

by [LaBi, Theorem 11.5.1], we can define

(66)  H_{g} := {ξ_{T} , T ∈ 𝕋[g]} ⊆ Hdg(H^{1}(X_{0}^{D}(M))^{⊗2} ⊗ ℚ)

to be the module of Hodge classes associated with g.

Definition 3.1. \underline{P}_{g} := κ_{dR}(H_{g}) ⊆ ℚ ⊗ J_{D}^{0}(M)(F).

The Hodge classes in H_{g} can be written down explicitly. For instance, a representative for ξ_{g} := ξ_{T_{g}} ∈ H_{g} is given as follows: let {ω_{g,i}} be a basis of the g-isotypical component of the module of regular differential forms on X_{0}^{D}(M) and complete it to a symplectic basis {ω_{g,i}, η_{g,i}} of H_{dR}^{1}(X_{0}^{D}(M))[g]. Then

(67)  ξ̃_{g} = ∑_{i} ω_{g,i} ⊗ η_{g,i} − η_{g,i} ⊗ ω_{g,i}

is a differential form on X_{0}^{D}(M) × X_{0}^{D}(M), rational over F, which represents ξ_{g}. The point P_{g} := κ_{dR}(ξ_{g}) ∈ \underline{P}_{g} can then be computed as the tuple of periods of the iterated integral associated with (67); similar formulas can be obtained of course for a complete basis of \underline{P}_{g}: see [DDLR] for explicit calculations.

Recall that κ_{dR}(ξ) depends on the choice of the base point o. If we are dealing with the classical elliptic modular curve associated to the split algebra M_{2}(ℚ), we can take o to be the cusp at infinity, which is a rational point on X_{0}(M).

If X_{0}^{D}(M) is the Shimura curve attached to an order in a division quaternion algebra over an arbitrary totally real number field F, then X_{0}^{D}(M)(F) may be empty: see e.g. [Sh75, Theorem 5.1], which establishes criteria for the existence of real points on these curves and shows in particular that X_{0}^{D}(M)(ℝ) = ∅ when F = ℚ. Instead of fixing a single point o rational over F, one proceeds as in [GrKu, §13], [YZZ] and [Zh] by rather considering a canonical Gal (F̄/F)-invariant divisor class o = ∑ a_{i}o_{i} of degree 1 with rational coefficients on X_{0}^{D}(M), which in [Zh] is called the Hodge class of X_{0}^{D}(M). In this case, by κ_{dR}(ξ) we mean ∑ a_{i}κ_{dR}(ξ; o_{i}).

Choose now another normalised Hilbert newform f of weight 2, level N_{f} | N and trivial nebentypus, different from g. The triple product L-function L(g ⊗ g ⊗ f, s), which was first studied by Garrett in [Gar], is the L-function associated with the tensor product V_{g} ⊗V_{g} ⊗V_{f}, where V_{g} and V_{f} denote the two-dimensional ℓ-adic representations of G_{F} := Gal (F̄/F) attached to g and f respectively. It is explicitly given by an Euler product with Euler factors of degree 8 at the primes ℘ ∤ N, and factors as

(68)  L(g ⊗ g ⊗ f, s) = L(f, s − 1)L(Sym^{2} g ⊗ f, s),

mirroring the decomposition

V_{g} ⊗ V_{g} = ℚ_{ℓ}(−1) ⊕ Sym^{2} V_{g}

of ℓ-adic representations of G_{F}.

Thanks to the work of Garrett, Piatetski-Shapiro–Rallis, Harris–Kudla, Prasad and others (cf. [Gar],[Pr] and the references therein), it is known that L(g ⊗ g ⊗ f, s) admits analytic continuation to ℂ and satisfies a functional equation relating its values at s and 4 − s. The sign ϵ in this functional equation is expressed as a product

(69)  ϵ = (−1)^{d} ∏_{℘|gcd(N_{g},N_{f})} ϵ_{℘}

of local signs ϵ_{℘} ∈ {±1}; for primes ℘ dividing exactly both N_{g} and N_{f},

ϵ_{℘}(g ⊗ g ⊗ f) = −a_{℘}(f)

is the negative of the eigenvalue of the Hecke operator T_{℘} acting on f (cf. [GrKu], [Pr]), and in particular does not depend on the choice of g. When ℘^{2} | N_{g} or N_{f}, the value of ϵ_{℘}(g ⊗ g ⊗ f) depends on the local behavior of the automorphic representations associated with g and f; see [Pr] for more details. Assume that

(70)  ϵ(f) = −1,  ϵ(Sym^{2}(g) ⊗ f) = +1.

Then ϵ = −1 and we can set D = D_{g;f} = ∏_{ϵ_{℘}=−1} ℘; note that D is the product of r = ♯{ϵ_{℘} = −1} distinct prime ideals, and that r + d is odd by (69). Assume that D || N and put as above M = M_{g;f} = N/D.

The quotient of J_{0}^{D}(M) associated to f by the theory of Eichler-Shimura is an abelian variety which is well-defined only up to F -isogenies, and is isogenous to the power A_{f}^{σ(N/N_{f})} of an F -simple abelian variety A_{f}. This means that there are σ(N/N_{f}) F -linear independent modular parametrizations

π_{f}^{d} : J_{0}^{D}(M) ⟶ A_{f}

of A_{f}, one for each divisor d of N/N_{f}.

Definition 3.2. Let

(71)  \underline{P}_{g,f} := Σ_{d| \frac{N}{N_{f}}} π_{f}^{d}(\underline{P}_{g}) ⊆ A_{f}(F) ⊗ ℚ

be the subspace of the Mordell-Weil group (tensored with ℚ) of A_{f} associated to the pair of modular forms (g, f).

In particular, if we let π_{f} : J_{D}^{0}(M) → J_{0}^{D}(N_{f}/D) → A_{f} denote the map which factors through the natural projection of J_{0}^{D}(M) onto J_{0}^{D}(N/N_{f}), the point

(72)  P_{g,f} = π_{f}(P_{g})

belongs to this space, and can be explictly computed by means of (67).

Formula (72), combined with Theorem 2.5, provides a suggestive construction of a point on a modular abelian variety by means of iterated integrals. Theorem 1 can be used to show that, in the particular case that g has rational fourier coefficients, P_{g,f} is essentially equal to a point that was first introduced by S. Zhang by a different method, whose definition we recall here for the convenience of the reader:

Definition 3.3. [S. Zhang] Assume g has fourier coefficients in ℤ. Let E_{g}/F be the elliptic curve corresponding to g by the Eichler-Shimura construction and π_{g} : X_{0}^{D}(M) ⟶ E_{g} be a modular parametrization of curves. Letting O ∈ E_{g}(F) denote the identity element, define

P_{g,f}^{Zh} := π_{f,∗}π_{g}^{∗}(O).

We refer the reader to the forthcoming Ph. D thesis [Daub] of M. Daub for more details concerning the connection between (72) and Definition 3.3. See also W. Stein’s appendix to [DDLR] for the effective numerical computation of the points P_{g,f}^{Zh} in the classical setting of elliptic modular curves over F = ℚ. Note that Definition 3.3 does not readily generalize to the case in which the fourier coefficients of g are not integral, as the resulting modular abelian variety has dimension greater than 1. On the other hand, formula (72) is available in that broader setting: see again [DDLR] for explicit examples.

The algebra 𝕋 of Hecke operators acts by correspondences on CH^{1}(X_{0}^{D}(M)) ⊗ ℚ and gives rise via (65) to a linear action of 𝕋 ⊗ 𝕋 ⊗ 𝕋 on CH^{2}(X^{3}) ⊗ ℚ and on its subspace CH^{2}(X_{0}^{D}(M)^{3})_{0} ⊗ ℚ of null-homologous cycles.

Definition 3.4. Set

MW(g, f) := Hom (𝕋[g] ⊗ 𝕋[g] ⊗ 𝕋[f], CH^{2}(X_{0}^{D}(M)^{3})_{0} ⊗ ℚ)

and let y_{g,f} ∈ MW(g, f) be the homomorphism defined by

y_{g,f}(T_{1} ⊗ T_{2} ⊗ T_{3}) := (T_{1} ⊗ T_{2} ⊗ T_{3})(Δ_{GKS}).

By working with a suitable integral model of X^{3}, B. Gross and C. Schoen have shown in [GrSc] that Beilinson-Bloch’s [Bl84] height pairing ⟨ , ⟩^{BB} on CH^{2}(X_{0}^{D}(M)^{3})_{0} gives rise to a well-defined pairing

⟨ , ⟩^{BB} : MW(g, f) × MW(g, f) ⟶ Hom((𝕋[g] ⊗ 𝕋[g] ⊗ 𝕋[f])^{⊗2}, ℂ),

independently of any choices; see also [YZZ, §1.3].

X. Yuan, S. Zhang and W. Zhang have recently proved in [YZZ] the following formula, which was conjectured by B. Gross and S. Kudla in [GrKu, Conjecture 13.2]:

Theorem 3.5. [YZZ, Theorem 1.3.1, Corollary 1.4.1 (1)] Assume as in (70) that ϵ(f) = −1 and ϵ(Sym^{2}(g) ⊗ f) = +1. Then

⟨y_{g,f}, y_{g,f}⟩^{BB} = L′(f, 1) · ∏_{σ:K_{g}→ℂ}L(g^{σ} ⊗ g^{σ} ⊗ f, 2) · λ

for some non-zero homomorphism λ ∈ Hom((𝕋[g] ⊗ 𝕋[g] ⊗ 𝕋[f])^{⊗2}, ℂ).

Remark 3.6. Note the different language in which the results are presented in [YZZ] in contrast to [GrSc, §8] and here: while the former exploits the language of automorphic representations, the latter adopt the more classical parlance of modular forms and Hecke operators, working with Shimura curves of fixed finite level.

The combination of Theorem 1 together with Theorem 3.5 yields the following result.

Theorem 3.7. Assume as in (70) that ϵ(f) = −1 and ϵ(Sym^{2}(g) ⊗ f) = +1. Then the space \underline{P}_{g,f} ⊆ A_{f}(F) ⊗ ℚ has positive rank if and only if

(73)  L′(f, 1) ≠ 0, and L(Sym^{2}(g^{σ}) ⊗ f, 2) ≠ 0 for all σ : K_{g} → ℂ.

Proof. The automorphism τ : X^{3} → X^{3}, (x_{1}, x_{2}, x_{3}) ↦ (x_{2}, x_{1}, x_{3}) gives rise to an involution τ^{∗} on CH^{2}(X^{3})_{0} ⊗ ℚ and on J^{2}(X^{3}) which preserves the (g, g, f)-isotypical components. By [YZZ, Corollary 1.4.1 (1)], the homomorphism y_{g,f} is invariant under the action of τ.

Hence, combined with Theorem 3.5, condition (73) holds if and only if there exist correspondences T_{g,i} ∈ 𝕋[g], T_{f,i} ∈ 𝕋[f] such that,

⟨Δ[ğ, ğ, f̆], Δ[ğ, ğ, f̆]⟩^{BB} ≠ 0,

where we set

Δ[ğ, ğ, f̆] := ∑_{i} T_{g,i} ⊗ T_{g,i} ⊗ T_{f,i}(Δ_{GKS}).

Lemma 2.1 combined with Lemma 3.8 below show that

∑_{i} Π_{X_{12}} · (T_{g,i} ⊗ T_{g,i} ⊗ 1)(Δ_{GKS}) = ∑_{i} Π_{T_{g,i}}(Δ_{GKS}).

Since T ∘ Π_{X_{12}} = Π_{X_{12}}(1 ⊗ 1 ⊗ T) for any T ∈ CH^{1}(X^{2}), we obtain that

(74)  ∑ T_{f,i}Π_{T_{g,i}}(Δ_{GKS}) = Π_{X_{12}}(Δ[ğ, ğ, f̆]).

For each i, T_{f,i} is a self-correspondence on X that gives rise to an endomorphism π_{f,i} of Jac(X) which is a linear combination of the projections π_{f}^{d}, d | N/N_{f}. Set

P [ğ, ğ, f̆] := ∑_{i} π_{f,i}Π_{T_{g,i}}(Δ_{GKS})] ∈ A_{f}

and write ⟨ , ⟩^{NT} for the Néron-Tate canonical height pairing on an abelian variety. By [Bl84, p. 120-121, A.2], the pairing ⟨ , ⟩^{BB} is functorial with respect to correspondences, and from (74) it follows that

⟨Δ[ğ, ğ, f̆], Δ[ğ, ğ, f̆]⟩^{BB} =^{·} ⟨P [ğ, ğ, f̆], P [ğ, ğ, f̆]⟩^{NT} ,

up to a positive constant.

Note that, by (71), P [ğ, ğ, f̆] is a point in \underline{P}_{g,f}. By Theorem 1 we have P [ğ, ğ, f̆] = ∑_{i} π_{f,i}κ_{dR}(ξ_{T_{g,i}}). The corollary follows. □

Lemma 3.8. Given a self-correspondence Z ∈ CH^{1}(X^{2}) of X, it holds that

Π_{X_{12}} · (Z ⊗ Z ⊗ 1)(Δ_{GKS}) = Z · X_{12} − deg(Z · X_{12})o.

Proof. By applying (46), it directly follows that Π_{X_{12}} · (Z ⊗ Z ⊗ 1)(X_{123}) = Z · X_{12} ∈ Jac(X). Similarly, when one applies (46) to the six other components X_{j}, X_{jk} of Δ_{GKS} one obtains that, for all j, k = 1, 2, 3, j ≠ k, the divisors Π_{X_{12}} ·(Z ⊗Z ⊗1)(X_{j}) and Π_{X_{12}} ·(Z ⊗Z ⊗1)(X_{jk}) are all linearly equivalent to some multiple of the degree 1 divisor o.

Since Δ_{GKS} is null-homologous, Π_{X_{12}} · (Z ⊗ Z ⊗ 1)(Δ_{GKS}) is a divisor class on X of degree 0. The lemma follows. □

In light of Theorem 1, Theorem 3.7 can be viewed to some extent as an analogue for modular and Shimura curves of the Deligne-Wojtkowiak calculations [De] in which values of the Riemann zeta-function ζ(s) at odd positive integers emerge from the study of the mixed Tate motives arising in the pro-unipotent fundamental group of ℙ_{1} − {0, 1, ∞}. It was motivated by the authors’ study of the Chow-Heegner points attached to diagonal cycles in triple products of Kuga-Sato-varieties, which is detailed more systematically in [DR1]. Note however that while [De] obtains formulae for the values of ζ(s) at certain non-critical points, Theorem 3.7 applies only to the critical value of the L-function L(g ⊗ g ⊗ f, s) at s = 2.

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H. D.: Montreal, Department of Mathematics, McGill University, Montreal, Canada

   E-mail address: henri.darmon@mcgill.ca

V. R.: Dep. Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

E-mail address: victor.rotger@upc.edu

I. S.: Departamento de Álgebra, Universidad Complutense de Madrid, Spain

E-mail address: isols@mat.ucm.es