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\begin{document}

\title[Iterated integrals, cycles and rational points]{Iterated integrals, diagonal cycles\\ and rational points on elliptic curves}



\author{Henri Darmon, Victor Rotger and Ignacio Sols}

\begin{abstract}
The theme of this article is the connection between the pro-unipotent fundamental group
 $\pi_1(X;o)$ of a pointed algebraic curve $X$, algebraic cycles, iterated integrals, 
 and special values of $L$- functions. The extension of mixed Hodge structures arising in the second stage in the lower central series of $\pi_1(X;o)$ gives rise to a supply of complex points on the Jacobian $\Jac(X)$ of $X$  indexed by 
 Hodge cycles on $X\times X$. 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 \cite{YZZ} 
   % for the height of the Gross-Kudla-Schoen cycle  obtained 
   of  X. Yuan, S. Zhang and W. Zhang  relates 
    their non-triviality  to  special
   values of certain $L$-series attached to modular forms.

    



\end{abstract}

\address{H. D.: Montreal, Department of Mathematics, McGill University, Montreal, Canada}
\email{henri.darmon@mcgill.ca}
\address{V. R.: Dep. Matem\`{a}tica Aplicada II, Universitat Polit\`{e}cnica de Catalunya, Barcelona, Spain}
\email{victor.rotger@upc.edu}
\address{I. S.:  Departamento de \'Algebra, Universidad Complutense de Madrid, Spain}
\email{isols@mat.ucm.es}

\subjclass{11F67; 11G05; 11G40; 11G18; 14C15}

\thanks{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 Educaci\'on for a
   Salvador Madariaga grant which funded his stay in the Institut de Recerca Matematica during the arithmetic special year.
}

\maketitle

%\tableofcontents

\section*{Introduction}

Let $(Y,o)$ be a pointed algebraic variety over the complex numbers. Its pro-unipotent fundamental group $\pi_1(Y;o)$ is a rich source of links between the topology, geometry and arithmetic of the variety: see for example the works of Bloch \cite{Bl77}, Chen \cite{Chen}, Deligne \cite{deligne}, Drinfeld \cite{Dr}, Hain \cite{Hain-survey} and Kim \cite{kim} and the references therein. 

Among these developments, the contributions of Carlson, Clemens and Morgan \cite{Carlson}, Harris  \cite{Har} and Pulte \cite{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 $\pi_1(Y;o)$-- 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 $F\subset \C$ be a field embedded in the field of complex numbers, $X$ a smooth projective curve over $F$ and $Y=X\setminus \{ \infty\}$ the complement of a single rational point $\infty\in X(F)$. Choose a base point $o\in Y(F)$. In this article we focus on the construction of a supply of complex points 
\begin{equation}\label{supply}
\kappa_{\dR}(\xi) \in \Jac(X)(\C)
\end{equation}
on the jacobian of $X$, indexed by the set of Hodge classes $$\mathrm{Hdg}(H^1(X)\otimes H^1(X)):= \{\xi: \Z(-1)\ra H^1(X)\otimes H^1(X)\}$$ in the Hodge structure underlying $H^1(X)\otimes H^1(X)$. This construction has also been extensively studied by Pulte \cite{Pu} and Kaenders \cite{Kae} and we review it in \S \ref{sec:mixed-motive}. These points arise from the pro-unipotent fundamental group $\pi_1(Y(\C);o)$ and therefore depend on the choice of $o$, and their construction exploits the canonical isomorphism
\begin{equation}\label{ext-mhs-jac}
\Ext^1_{\mhs}(\Z(-1),H_B^1(X)) \simeq \frac{\Omega^1(X)^\vee}{H_1(X(\C),\Z)} \simeq \Jac(X)(\C)
\end{equation}
between the module of extensions of $\Z(-1)$ by $H_{\Betti}^1(X)$ in the category of mixed Hodge structures, and the set of complex points of $\Jac(X)$, as described e.g. in \cite{Ca}.

Assume now that $\xi=\xi_Z$ is the Hodge class associated by the cycle class map with an algebraic cycle $Z\in \Pic(X\times X)$, as detailed in \eqref{xiZ}. Section \ref{sec:cycles} associates to each divisor $Z$ in $X \times X$ an algebraic correspondence $\Pi_Z: \xymatrix{ X^3\ar@{.>}[r] & X}$ between $X^3$ and $X$ and our first main theorem is:

\begin{mtheorem}
\label{thm:main}
For every divisor class $Z\in \Pic(X^2)$,
\begin{equation}\label{mainformula}
\kappa_{\dR}(\xi_Z) = \Pi_Z(\Delta_{\GKS})\in \Jac(X)(\C),
\end{equation}
where
\begin{itemize}

\item $\Pi_Z: \CH^2(X^3)_0 \lra \CH^1(X)_0=\Jac(X)$ is the transformation induced by $\Pi_Z$ by functoriality, and


\item $\Delta_{GKS}  \in \CH^2(X^3)_0$ is the null-homologous cycle of codimension $2$ in $X^3$ with base-point $o$ studied by Gross, Kudla and Schoen \cite{GrKu}, \cite{GrSc}, cf.\,\eqref{eqn:Delta-GKS}.
\end{itemize}
\end{mtheorem}

This theorem spells out  a relationship between
  diagonal cycles and the 
pro-unipotent fundamental group which is hinted at in the  penultimate sentence of
\S 5.1 of \cite{zhang-crm}; the reader is encouraged to consult 
\S 5 of loc.cit.~\!for further discussion of Gross-Kudla-Schoen cycles and their arithmetic
applications.
Since the right-hand side of \eqref{mainformula} is purely algebraic, we derive the following corollary:
% which a priori does not follow directly from the construction of $\kappa_{\dR}(\xi_Z)$ stemming from $\pi_1(Y(\C);o)$.

\begin{mcorollary}\label{cor1}
If $Z$ belongs to $\Pic(X\times X)(F)$ then $\kappa_{\dR}(\xi_Z)$ is an $F$-rational point of $\Jac(X)$.
\end{mcorollary}
\noindent
The proof of Theorem \ref{thm:main} is complex analytic: it proceeds by relating--in \S \ref{sec:mixed-motive} and \S \ref{sec:cycles}, respectively--the left and right-hand side of equality \eqref{mainformula} to Chen's iterated integrals. The resulting explicit formula (Theorem \ref{thm:ii-cycles}) was one of the main motivations for this note and underlies the numerical calculations of Chow-Heegner points that are carried out in \cite{DDLR} when $X$ is a modular curve. The authors also hope that Theorem \ref{thm:main} could 
form the basis for an eventual  approach to the ``Stark-Heegner points"
of \cite{darmon-pollack} via iterated integrals.

Beyond the general expectation alluded to above that the pro-unipotent fundamental group $\pi_1(Y;o)$ 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 \cite{BlKa} linking algebraic cycles to L-functions, that a good deal of the non-abelian arithmetic of $\pi_1(Y;o)$ should be accounted for by the leading terms of the $L$-function  of the motives   appearing as Jordan-H\"older constituents of $\pi_1(Y;o)$.
There are however few examples in the literature in which this principle is illustrated\footnote{while the connection between algebraic cycles and special values of L-functions as predicted by \cite{BlKa} has already borne some fruit: cf.\,e.g.\,\cite{GZ}, \cite{Ne}, \cite{Zh}, \cite{YZZ}.}. A celebrated instance is afforded by the curve $Y=\PP_1-\{0,1,\infty\}$, whose pro-unipotent fundamental group is an iterated extension of Tate motives. It is shown in 
\cite{deligne} that the extension classes that arise in it encode  special values of the Riemann zeta function at certain (non-critical) points. Theorem \ref{cor2} in \S \ref{secZh} describes analogous phenomena
 when $Y$ is a punctured modular curve attached to a congruence 
 subgroup of $\SL_2(\Z)$, or a Shimura curve, exploiting \cite{YZZ}. 


Let us close this introduction by relating Theorem \ref{thm:main} to existing results in the literature. In our notations and the normalizations adopted in definition \eqref{eqn:def-kappa-dr-xi} of the points $\kappa_{\dR}(\xi)$ --which are tailored so that there is no dependence on the choice of the point $\infty$--, the main theorem of Kaenders \cite{Kae} says that when $Z=X_{12}\in \Pic(X\times X)$ is the class of the curve $X$ diagonally embedded in $X\times X$, we have
\begin{equation}\label{kaenders}
\kappa_{\dR}(\xi_{X_{12}}) = (2g-2) o-K \in \Jac(X),
\end{equation}
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 \ref{cor1} follows directly from \eqref{kaenders} for this particular choice  of $Z$. 

Kaenders proves \eqref{kaenders} by showing that a suitable translate of $\kappa_{\dR}(\xi_{X_{12}})$ 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 \ref{thm:main}: see Corollary \ref{kaenders-by-us}.

It is also interesting to compare Theorem \ref{thm:main} with the work of B. Harris \cite{Har}, as recast by Pulte in \cite{Pu}. For any $n\in \Z$, let $[n]\in \End (\Jac(X))$ denote the endomorphism given by multiplication-by-$n$. Define
$$
X_o:= \{ [P-o]: P\in X\}, \quad X^-_o:= [-1]_* X_o =\{ [o-P]: P\in X\} \subset \Pic^0(X)
$$ 
and let $C_o:=X_o - X_o^-\in \CH^{g-1}(\Jac (X))_0$ be {\em Ceresa's cycle}, a null-homologous cycle of codimension $g-1$ on the jacobian of $X$.  

Every choice of Hodge class $\xi\in \Hdg(H_B^1(X)^{\otimes 2})$ singles out a natural map\footnote{In the notation of \S \ref{sec:hodge-classes}, $\pi_\xi$ is the composition of the Abel-Jacobi map $\CH^{g-1}(\Jac (X))_0(\C)\overset{\AJ}{\ra} J_1(\Jac(X))$, the monomorphism $J_1(\Jac(X))\hookrightarrow \Ext^1_{\mhs}(H_B^1(X)^{\otimes 2},H_B^1(X))$ considered by Pulte in \cite{Pu}, the projection $\Ext^1_{\mhs}(H_B^1(X)^{\otimes 2},H_B^1(X))\rightarrow  \Ext^1_{\mhs}(\Z(-1),H_B^1(X))$ given by $\xi$ as in \eqref{eqn:pullback-ext} and the isomorphism \eqref{ext-mhs-jac}.}
\begin{equation}\label{pultes-map}
\pi_\xi: \CH^{g-1}(\Jac (X))_0(\C) \lra \Jac(X)(\C),
\end{equation}
%which is $\Aut(\C/F)$-equivariant when $\xi$ arises from an algebraic cycle defined over $F$.
and Theorem 4.9 of \cite{Pu} asserts that 
\begin{equation}\label{pulte}
\kappa_{\dR}(\xi) = 2\pi_\xi(C_o)\in \Jac(X)(\C).
\end{equation} 
Notice that Corollary \ref{cor1} does not follow automatically from \eqref{pulte}, as it is not a priori obvious that the map $\pi_\xi$ is Galois equivariant.

Our Theorem \ref{thm:main} is in the same spirit of Harris-Pulte's, in that both relate the points $\kappa_{\dR}(\xi)$ arising from $\pi_1(Y(\C);o)$ 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
\begin{equation}
\label{X3JX}
\xymatrix{ 
\CH^2(X^3)_0(\C) \ar[d]^{\mu_*} \ar[r] & J^2(X^3)=\Fil^2 H_{\dR}^3(X^3,\C)/H_3(X^3,\Z) \ar[d]^{\AJ(\mu_*)}  \\
\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), }
\end{equation}
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 $\mu: X^3 \ra \Jac(X)$ given by the rule $(x_1,x_2,x_3)\mapsto x_1+x_2+x_3-3o$, and  the vertical arrow $\AJ(\mu)$ is the natural surjection given by the K\"unneth decomposition of $H^3(X^3)$ and the classical isomorphism $H^3(\Jac(X)) = \Lambda^3 H^1(X)$. 

It readily follows from the definitions that $$\mu_*(\Delta_{GKS}) = [3]_* X_o -3 [2]_* X_o + 3 X_o\in \CH^{g-1}(\Jac(X))_0$$ and Colombo and van Geemen proved in \cite[Proposition 2.9]{CG} that 
\begin{equation}\label{colombogeemen}
\AJ_{\Jac(X)}(\mu_*(\Delta_{GKS})) = 3\AJ_{\Jac(X)}(C_o).
\end{equation}

This result lies far from proving Theorem \ref{thm:main} via \eqref{pulte}, as the maps $\mu_*$ and $\AJ(\mu_*)$ have non-trivial kernel. In fact, the combination of Theorem \ref{thm:main} with Pulte's \eqref{pulte} yields the equality
\begin{equation}\label{us+pulte}
\Pi_Z(\Delta_{\GKS})=2\pi_{\xi_Z}(C_o)\in \Jac(X)(\C) \quad \mbox{ for all } Z\in \Pic(X\times X),
\end{equation}
which is yet another set of non-trivial relationships between Gross-Kudla-Schoen's cycle $\Delta_{GKS}\in \CH^2(X^3)_0$ and Ceresa's $C_o\in \CH^{g-1}(\Jac (X))_0$. 

%To conclude this introduction, let us discuss one corollary of our main result that follows when combined with the deep work of X. Yuan, S. Zhang and W. Zhang \cite{YZZ}, and
% whose study  was one of  the authors'   motivations for proving Theorem \ref{thm:main}. 


\section{Fundamental groups and mixed motives}
\label{sec:mixed-motive}

Let $F$ be a number field equipped with an embedding $F\hookrightarrow \C$ 
into the field of complex numbers, and write $\bar F$ for 
the Galois closure of $F$ in $\C$; as before, recall
that  $G_F:=\Gal(\bar F/F)$. 


Let $X$ be a smooth, complete algebraic curve of genus $g\geq 2$ 
over $F$, and let $Y=X\setminus S$ for some finite, $G_F$-stable set $S$  of points on $X(\bar F)$. 
Fix a base point $o\in Y(F)$; the integral group ring 
$\Z[\Gamma]$ of the fundamental group $\Gamma := \pi_1(Y(\C);o)$ 
of the Riemann surface underlying 
$$Y_\C=Y\times_{\mathrm{Spec}(F)}\mathrm{Spec}(\C)$$ 
is equipped with  a decreasing
filtration 
$$\Z[\Gamma]\supset I \supset I^2 \supset \cdots \supset I^j \supset \cdots$$
by powers of the augmentation ideal.

The first few successive quotients in this filtration are given by
\begin{eqnarray}
\label{eqn:aug-triv}
\Z[\Gamma]/I = \Z; & &  \quad  \sum m_\sigma \sigma \mapsto \sum m_\sigma;\\
\label{eqn:ImodI2}
I/I^2 = \Gamma_{\ab} := \Gamma/[\Gamma,\Gamma]; & & \quad \sum m_\sigma \sigma \mapsto \prod \sigma^{m_\sigma}; \\
\label{eqn:I2modI3}
I^2/I^3 = (\Gamma_{\ab} \otimes \Gamma_{\ab})/\KK, & & 
\end{eqnarray}
where in the  last identification, $\KK$ is the kernel of the natural surjective homomorphism
\begin{equation}
\label{eqn:mapj}
j:{\Gamma_{\ab}\otimes \Gamma_{\ab}} {\lra} I^2/I^3, \qquad
\gamma_1 \otimes \gamma_2 \mapsto (\gamma_1-1)(\gamma_2-1).
\end{equation}
Let $M^\vee := \hom(M,\Z)$ be the $\Z$-linear dual of a $\Z$-module $M$.
Formula (\ref{eqn:ImodI2}) implies that
\begin{equation}
\label{eqn:ImodI2dual}
 (I/I^2)^\vee = \hom(\Gamma,\Z) = H^1_{\Betti}(Y),
\end{equation}
where $H^1_{\Betti}(Y)=H^1(Y(\C),\Z)$ refers to the singular Betti cohomology of the
smooth real manifold underlying $Y(\C)$.
Furthermore, it can be shown
that the   map 
$$j^\vee: (I^2/I^3)^\vee \hookrightarrow 
 H^1_{\Betti}(Y)^{\otimes 2} $$
identifies $(I^2/I^3)^\vee$ with the kernel of the map
$$ H^1_{\Betti}(Y)^{\otimes 2} \lra H^2_{\Betti}(Y) $$
given by the cup product.
(Cf.\,\cite[Cor. 8.2]{Hain-survey} for more details.)
Letting $H^1_{\Betti}(Y)^{\otimes 2}_{\cup=0}$ denote this kernel, the
map $j^\vee$ induces an identification
\begin{equation}
\label{eqn:I2modI3dual}
(I^2/I^3)^\vee =  H^1_{\Betti}(Y)^{\otimes 2}_{\cup=0}.
\end{equation}
By dualising  the exact sequence
$$ 0 \lra I^2/I^3 \lra I/I^3 \lra I/I^2 \lra 0,$$
invoking (\ref{eqn:ImodI2dual}) and  (\ref{eqn:I2modI3dual}),
and setting
$$ M_{\Betti} := (I/I^3)^\vee,$$
we obtain the fundamental  exact sequence of $\Z$-modules
\begin{equation}
\label{eqn:fundamental-exact-sequence}
 0 \lra H^1_{\Betti}(Y) \lra M_{\Betti} \lra H^1_{\Betti}(Y)^{\otimes 2}_{\cup=0} \lra 0.
\end{equation} 

Note that $H^1_{\Betti}(Y)^{\otimes 2}_{\cup=0}=H^1_{\Betti}(Y)^{\otimes 2}$ as soon as $S\ne \emptyset$, because in this case $H^2_{\Betti}(Y)=\{0\}$. The abelian groups that appear on the left and right of 
\eqref{eqn:fundamental-exact-sequence} are
not just ``unadorned"  $\Z$-modules, but the {\em Betti realisations} 
of certain 
motives over $F$. In
 the terminology of \cite{deligne},   the exact sequence
\eqref{eqn:fundamental-exact-sequence} suggests that $M_B$ ought to be interpreted as the
Betti realization of a putative mixed motive $M^?$ (in the sense of \cite{deligne}, say) which would arise as an extension in that
 category of the motive $H^1(Y)^{\otimes 2}_{\cup=0}$ 
(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:
\begin{enumerate}
\item The existence of a decreasing, separated and exhaustive {\em Hodge filtration}
on $$M_{\dR,\C}:= M_{\Betti}\otimes\C,$$
for which the complexification
\begin{equation}
\label{eqn:fes-C}
 0 \lra H^1_{\dR}(Y/\C) \stackrel{i}{\lra} M_{\dR,\C} \stackrel{\mathrm{p}}{\lra} H^1_{\dR}(Y/\C)^{\otimes 2}_{\cup=0} \lra 0
\end{equation}
of \eqref{eqn:fundamental-exact-sequence}
becomes an exact sequence of filtered complex vector spaces. 
\item The existence, for each rational  prime $p$, of the
 $p$-adic \'etale realisation 
$M_{\et,p}$,   fitting into an exact sequence of  continuous
$\Q_p[G_F]$-modules
\begin{equation}
\label{eqn:fundamental-exact-sequence-p}
 0 \lra H^1_{\et}(X_{\bar F},\Q_p) \lra M_{\et,p} \lra H^1_{\et}(X_{\bar F},\Q_p)^{\otimes 2}_{\cup=0} \lra 0
\end{equation}
which is the $p$-adic \'etale counterpart of \eqref{eqn:fundamental-exact-sequence}.
\end{enumerate}
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 \eqref{eqn:fes-C}. 

Let us briefly recall the well-known mixed Hodge structures underlying the Betti cohomology group
$H^1_{\Betti}(Y)$ and its tensor square. Its weight filtration 
arises from the exact sequence 
\begin{equation}\label{es-YX}
0\lra H^1_{\Betti}(X) \lra H^1_{\Betti}(Y) \lra \oplus_{P\in S} \Z(-1) \lra \Z(-1)\lra 0,
\end{equation}
and is given by
\begin{equation}
\label{weightH1}
 W_j H^1_{\Betti}(X) = \left\{
\begin{array}{ll}
0 & \mbox{ if } j \le 0,\\
H^1_{\Betti}(X) & \mbox{ if } j=1 \\
H^1_{\Betti}(Y) & \mbox{ if } j\ge 2.
\end{array} \right.
\end{equation}

The {\em de Rham cohomology}  $H^1_{\dR}(Y(\C))$ 
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
$H^1_{\Betti}(Y) \otimes \C = H^1_{\dR}(Y(\C))$
arising from integration of smooth differential forms against smooth chains, and its Hodge filtration is given by
\begin{equation}
\label{eqn:hodge-filtration-C}
\ \Fil^j H^1_{\dR}(Y(\C)) = \left\{
\begin{array}{ll}
H^1_{\dR}(Y(\C)) & \mbox{ if } j \le 0,\\
\Omega^1_{\log}(Y(\C)) & \mbox{ if } j = 1,\\
0  & \mbox{ if } j \ge 2,
\end{array} \right.
\end{equation}
where 
$$\Omega^1_{\log}(Y(\C)) = \{ \omega \in \Omega^1(Y(\C)) 
\quad \mbox{ with } \quad  \mathrm{ord}_P(\omega)\geq -1, \ \  \,\,\forall P\in S\} $$ 
and $\mathrm{ord_P}(\omega)$ denotes  the valuation of the differential
 $\omega$ at the  point $P\in X(\C)$. 
%Recall that a differential is said to have {\em logarithmic poles} at a point if its expansion in terms of a local parameter $q$ at this point is of the form $\sum_{n=0}^\infty a_n q^n \frac{dq}{q}$.

Because the Riemann surface $Y(\C)$ arises from an algebraic curve over $F$, we can 
identify $H^1_{\dR}(Y(\C))$ with $H^1_{\dR}(Y/F)\otimes \C$, where 
\begin{equation}\label{deRham-comparison}
H^1_{\dR}(Y/F) := \bbH^1(0\rightarrow \cO_Y \rightarrow \Omega^1(Y) \rightarrow 0)
\end{equation}
is the {\em 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 $H^1_{\dR}(Y/F)$ may also be described concretely by
\begin{equation}
\label{filtrationH1}
\Fil^j H^1_{\dR}(Y/F) = \left\{ 
\begin{array}{ll}
H^1_{\dR}(Y/F) \simeq \Omega^{II}(Y)/dF(Y) & \mbox{ if } j\le 0, \\
\Omega^1_{\log}(Y/F) & \mbox{ if } j = 1 \\
0 & \mbox{ if } j \ge 2,
\end{array} \right.
\end{equation}
where 
\begin{enumerate}
\item
$\Omega^{II}(Y)$ is the space 
of  rational differential forms on $Y$ over $F$
 with vanishing residues at all points of $Y$; 
these are also called {\em differentials
of the second kind} on $Y$;
\item 
$F(Y)$ is the field of rational functions on $Y$ over $F$;
\item  $\Omega^1_{\log}(Y/F)$ is the subspace of
$\Omega^1(Y/F)$ consisting of differentials $\omega$ satisfying
$\mathrm{ord}_P(\omega)\geq -1$ for all $P\in S$. 
\end{enumerate}

%The datum of the  weight filtration 
%\eqref{weightH1} together with the Hodge filtration 
%\eqref{eqn:hodge-filtration-C}, \eqref{filtrationH1} 
%makes the $\Z$-module $H^1_{\Betti}(Y)$ into a 
%{\em mixed Hodge structure}. 
The tensor product
$H^1_{\Betti}(Y)^{\otimes 2}$ is again a mixed Hodge structure, and the Hodge filtration on
  $H^1_{\dR}(Y/F)^{\otimes 2}$ is given by
$$ \Fil^j H^1_{\dR}(Y/F)^{\otimes 2} = \left\{
\begin{array}{ll}
H^1_{\dR}(Y/F)^{\otimes 2} & \mbox{ if } j\le 0;\\
\Omega^1_{\log}(Y/F)\otimes H^1_{\dR}(Y/F) + H^1_{\dR}(Y/F)\otimes \Omega^1_{\log}(Y/F) & \mbox{ if } j=1;\\
\Omega^1_{\log}(Y/F)^{\otimes 2} & \mbox{ if } j= 2 \\
0 & \mbox{ if } j \ge 3.
\end{array}
\right.
$$


\subsection{Iterated integrals and the Hodge filtration on $M_{\dR,\C}$}

The extension $M_B$ of $H^1_{\Betti}(Y)^{\otimes 2}$ by $H^1_{\Betti}(Y)$
arising in \eqref{eqn:fundamental-exact-sequence}
is equipped with a natural mixed Hodge structure.
The definition of the   Hodge filtration on $M_{\dR,\C}$ 
was given 
by  R. Hain in \cite{Hain-filtration}, 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  \cite{Chen}, \cite{Hain-survey} 
and the references therein for further details.

The {\em path space} on $Y$ based at $o$,
denoted $\PP(Y;o)$,  is  the set of
piecewise smooth paths 
$$
\gamma:[0,1]\lra Y(\C), 
\quad  \mbox{ with  } \gamma(0)=o. $$ 
The
universal covering space of  $Y(\C)$,  denoted
  ${\tilde Y}$, is 
the space of homotopy classes in 
$\PP(Y;o)$. It
 is conformally equivalent to the open unit disc in $\C$. 
The group $\Gamma = \pi_1(Y(\C);o) $
 acts on it 
transitively and without fixed points, 
and the map $\gamma\mapsto \gamma(1)$  identifies
the quotient ${\tilde Y}/\Gamma$ with $Y(\C)$. 
If $\eta$ 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 $F_\eta$ on
${\tilde Y}$, defined by the rule
$$F_\eta(\gamma) := \int_0^1 \gamma^\ast \eta.$$


The {\em  basic iterated integral }
attached to smooth  differential forms $\omega_1,\ldots,\omega_n$ along a path 
$\gamma\in \PP(Y;o)$
is defined to be
\begin{equation}
\label{eqn:def-ii}
 \int_{\gamma} \omega_1\cdot\omega_2\cdots \omega_n := \int_{0\le t_n \le t_{n-1} \le \cdots \le t_1 \le 1}
\gamma^*(\omega_1)(t_1) \cdots \gamma^*(\omega_n)(t_n).
\end{equation}
The integer $n$ is called the {\em length} of this basic iterated integral.
In particular,   when $n=2$,
the basic iterated integral attached to
 $\eta_1$ and $\eta_2$ is  given  by the formula
$$ \int_\gamma \eta_1 \cdot \eta_2 :=  
\int_\gamma \eta_1 F_{\eta_2} = 
\int_0^1  \gamma^\ast(\eta_1 F_{\eta_2}).$$
It can also be expressed in terms of  $\eta_1\wedge \eta_2$,
{\em viewed as a two-form  on 
$X\times X$}, as the expression
\begin{equation}
\label{eqn:T-gamma}
\int_{\gamma} \eta_1\cdot \eta_2 := \int_{T_\gamma} \eta_1\wedge\eta_2, \qquad \mbox{ where } T_\gamma =\{ (\gamma(r),\gamma(s)), \  \ \ 0\le s\le r \le 1 \}\subset X\times X.
\end{equation}



An {\em iterated integral}
 is  any linear combination  of basic iterated integrals,
of possibly differing lengths, viewed as a function on $\PP(Y;o)$.
 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 
{\em homotopy invariant} if its value on a path $\gamma$  
depends only on the homotopy class of
$\gamma$. The space $\II(Y)$ of homotopy invariant iterated integrals will
 be viewed as a subspace
of the space of $\C$-valued functions on $\Gamma$, in the obvious way.
By extending   $J\in \II(Y)$ to 
the group ring $\C[\Gamma]$ by
$\C$-linearity,  
we will exploit the resulting natural inclusion
$$  \II(Y) \subset \C[\Gamma]^\vee$$
to realise $\II(Y)$ as a  space of complex functionals on $\C[\Gamma]$. 
For each $n$, let
 $\II^{\le n}(Y)$ 
denote the subspace of  homotopy invariant
iterated integrals of length $\le n$. This space plays an important
role in our description of $M_{\dR,\C}$, because of  the following facts:
\begin{enumerate}
\item  Any element $J\in \II^{\le n}(Y)\subset \C[\Gamma]^\vee$ 
 vanishes on $I^{n+1}$, and hence gives rise to a well-defined  element of
$(I/I^{n+1})^\vee\otimes_\Z \C$. 
\item
The natural map
\begin{equation}
\label{eqn:II-fg}
 \II^{\le n}\lra (I/I^{n+1})^\vee \otimes_\Z \C
\end{equation}
is an isomorphism. In particular,
\begin{equation}
\label{eqn:MdrII}
 M_{\dR,\C} = \II^{\le 2}(Y).
\end{equation}
\end{enumerate}

The following lemma is helpful in describing $\II^{\le 2}(Y)$ concretely.
\begin{lemma}
Let $\eta_1$ and $\eta_2$ be smooth closed one-forms  on $Y(\C)$
 whose classes in de Rham cohomology are 
orthogonal with respect to the Poincar\'e pairing, 
  and let 
 $\alpha$
 be a smooth one-form on $Y(\C)$ 
 satisfying the differential equation 
\begin{equation}
\label{eqn:diff-hard}
 d \alpha = \eta_1 \wedge \eta_2.
\end{equation}
\begin{enumerate}
\item The  iterated integral
$$ J_{\eta_1\otimes\eta_2,\alpha}(\gamma) := \int_{\gamma} \left(\eta_1 \cdot \eta_2 -
 \alpha\right)$$
is homotopy invariant. 
\item  The image of $J_{\eta_1\otimes\eta_2,\alpha}\in M_{\dR,\C}$ under the projection 
$\mathrm{p}$ of \eqref{eqn:fes-C} is the element $[\eta_1]\otimes[\eta_2]$. 
\item The space $\II^{\le 2}(Y)$ is spanned by
 iterated integrals of the form $J_{\eta_1\otimes\eta_2,\alpha}$. 
\end{enumerate}
\end{lemma}
\begin{proof}
The first claim follows directly from the fact that the differential 
one-form
on ${\tilde Y}$ 
appearing in the expression
$$ J_{\eta_1\otimes \eta_2,\alpha}(\gamma) = \int_\gamma \eta_1 F_{\eta_2} - 
\alpha$$
is closed.
The second statement follows from a direct calculation showing that
$$J_{\eta_1\otimes \eta_2,\alpha}((\gamma_1-1)(\gamma_2-1)) = 
\int_{\gamma_1} \eta_1 \int_{\gamma_2} \eta_2.$$
Finally, given any $J\in M_{\dR,\C}$, after  writing $\mathrm{p}(J) = \sum \omega_i \otimes \eta_i$,
it can be checked that
$$ J = \left(\sum J_{\omega_i\otimes \eta_i,\alpha_i}\right)  - J_{0,0,\alpha}$$
for suitable smooth differentials $\alpha_i$ on $Y(\C)$ 
  and $\alpha \in H^1_{\dR}(Y(\C))$. 
\end{proof}

In the special case where $Y(\C) = X(\C) - \{\infty\}$ with $X$ projective,
the smooth differential $\alpha$ on $Y(\C)$ can be expressed in terms of the Green's function 
$g_{\eta_1\wedge\eta_2, \infty}$ attached to the smooth $(1,1)$-form $\eta_1\wedge\eta_2$ and the point $\infty$. 
Recall (cf.~ \cite{Lang-book}, \S II.1 for example) that $g_{\eta_1\wedge\eta_2,\infty}$ is a smooth $\C$-valued function on
$Y(\C)$ satisfying the following two conditions:
\begin{enumerate}
\item
Letting $\partial$ and $\bar\partial: \cO_{\smooth}(Y) \lra \Omega^1_{\smooth}(Y)$ be the usual differential operators 
defined by 
$$ \partial f = \frac{\partial f}{\partial z} dz = \frac{1}{2}\left(\frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y}\right)(dx+idy),
\quad
 \bar \partial f = \frac{\partial f}{\partial \bar z} d\bar z = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y}\right)(dx-idy),
$$
the function $g_{\eta_1\wedge\eta_2,\infty}$ 
is a solution of the differential equation 
$$ \frac{-1}{2\pi i} \partial \bar\partial g_{\eta_1\wedge\eta_2,\infty} = \eta_1\wedge \eta_2.$$
\item The function $g_{\eta_1\wedge\eta_2}$ has at worst logarithmic
growth at $\infty$. More precisely, letting 
$$m_{\eta_1\wedge\eta_2}:= \int_{X(\C)} \eta_1\wedge \eta_2,$$
the function $g_{\eta_1\wedge\eta_2,\infty}$ can be written as
$$ g_{\eta_1\wedge\eta_2,\infty}(x) = - m_{\eta_1\wedge \eta_2} \log|f_\infty(x)|^2 + h(x),$$
where $f_\infty$ is a rational function on $X$ with 
a simple zero at $\infty$ and $h(x)$ is smooth in a neighbourhood
of $\infty$.
\end{enumerate}
The differential 
$$\alpha_\infty := \frac{1}{2\pi i}  \partial g_{\eta_1\wedge\eta_2,\infty}$$
is a smooth one-form of type $(1,0)$ on $Y(\C)$ satisfying
$$  d \alpha_\infty = (\partial + \bar\partial ) \alpha_\infty = \bar\partial\alpha_\infty = \eta_1\wedge\eta_2.$$
Its Laurent development around $\infty$ has a principal part of the form
$$ \alpha_\infty = \frac{-m_{\eta_1\wedge\eta_2}}{2\pi i} \frac{dz}{z}.$$
This implies that, for any smooth region $R\subset X(\C)$ whose boundary does not contain $\infty$,
\begin{equation}
\int_R \eta_1\wedge \eta_2 \ \ \  = \ \ \ \int_{\partial R} \alpha_\infty \ \ \ + \ \ \ m_{\eta_1\wedge\eta_2}\int_R \delta_\infty,
\end{equation}
where $\delta_\infty$ is the Dirac delta current, satisfying
$$ \int_R \delta_\infty = \left\{ \begin{array}{ll}
1 & \mbox{ if } \infty\in R;\\
0 & \mbox{ if } \infty\notin R. \end{array} \right.
$$
In other words, we have the equality of currents:
\begin{equation}
\label{eqn:delta-alpha}
\eta_1\wedge \eta_2 = d \alpha_\infty + m_{\eta_1\wedge\eta_2} \delta_\infty.
\end{equation}
The following lemma concerning
 the differential $\alpha_\infty$  shall  be used in 
the proofs of  Section 
\ref{sec:hodge-classes}.
\begin{lemma} 
\label{lemma:dirac-horizontal}
After setting $X=X_1=X_2$, viewing $\eta_1\wedge \eta_2$ as a
 $(1,1)$-form on $X_1$, and 
letting $\rho$ be a  closed one-form on $X_2$,
consider the
three-form $(\eta_1\wedge \eta_2)\wedge \rho$ on $X_1\times X_2$.
Let 
 $W$ be any smooth three-chain on $X_1\times X_2$ 
which  intersects $\{\infty\}\times X_2$ in
a smooth one-chain, so that 
$$ (\partial W)':= \partial W- \partial(W\cap (\{\infty\}\times X_2))$$
 is the complement of a finite set in $\partial W$. 
Then the integral
$$ \int_{\partial W} \alpha_\infty\wedge \rho := \int_{(\partial W)'}\alpha_\infty\wedge\rho$$
converges, and satisfies
$$ \int_W (\eta_1\wedge\eta_2)\wedge \rho \ \ 
= \ \ \int_{\partial W} \alpha_\infty \wedge\rho \ \  
+ \ \  m_{\eta_1\wedge\eta_2} \int_{W\cap (\{\infty\}\times X_2)} \rho.$$
\end{lemma}
\begin{proof}
Since $\rho$ is closed, equation \eqref{eqn:delta-alpha} 
implies that $$ (\eta_1\wedge\eta_2)\wedge\rho 
= d(\alpha_\infty\wedge\rho) + m_{\eta_1\wedge\eta_2} \cdot 
\delta_\infty \wedge \rho.$$
Integrating over $W$ and applying Stokes' theorem, we obtain
$$ \int_W (\eta_1\wedge\eta_2)\wedge \rho \ \ =  \ \ 
\int_{\partial W} \alpha_\infty\wedge\rho \ \  + \ \ 
m_{\eta_1\wedge\eta_2}\int_W \delta_\infty\wedge\rho,$$ 
and the lemma follows from the fact that $\int_W \delta_\infty \wedge \rho = \int_{W\cap(\{\infty\}\times X_2)} \rho.$
\end{proof}


\sk\noindent
Following  Hain
(cf. \cite[\S 13.1]{Hain-survey}),
 the Hodge filtration on 
$(I/I^{n+1})^\vee \otimes_\Z \C$  is given  in terms of 
 (\ref{eqn:II-fg}) by defining $\Fil^j((I/I^{n+1})^\vee \otimes_\Z \C)$ to be the set of
$J\in \II^{\le n}$ of the form
$$ J = \left(\int \omega_1^{(1)}\cdot\omega_2^{(1)}\cdots \omega^{(1)}_{n_1}\right) + \cdots + 
\left(\int \omega_1^{(t)}\cdot \omega_2^{(t)} \cdots \omega_{n_t}^{(t)}\right)$$
in which each $n_i$-tuple $(\omega_1^{(i)},\ldots,\omega_{n_i}^{(i)})$ involves {\em 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
\begin{enumerate}
\item[(i)]
$\Fil^j M_{\dR,\C} =
M_{\dR,\C}$
for all $j\le 0$;
\item[(ii)]
The space $\Fil^1 M_{\dR,\C}$
is spanned by iterated integrals of the form $J_{\eta_1,\eta_2,\alpha}$ in which $\alpha$ and at least
one of $\eta_1$ and $\eta_2$ lie in $\Omega^1_{\log}(Y(\C))$.
\item[(iii)] 
The space $\Fil^2 M_{\dR,\C}$ is spanned by iterated integrals of the form
$J_{\omega_1,\omega_2,0}$ in which $\omega_1$ and   $\omega_2$  both 
  belong to $\Omega^1_{\log}(Y(\C))$. 
\item[(iv)] 
For all $j\ge 3$,  $\Fil^j M_{\dR,\C} = 0$.
\end{enumerate}
With this    description
  of the Hodge filtration on $M_{\dR,\C}$, it is
 routine to verify that the exact sequence 
\eqref{eqn:fes-C} is an exact sequence of filtered vector spaces.
The $\Z$-module $M_{\Betti}$
is  in fact a prototypical example of an  integral mixed Hodge structure.
(See \cite[Cor. 9.3]{Hain-survey} and 
\cite{Hain-proof} for a discussion in a more general setting.)
Let
\begin{equation}
\label{eqn:kappadr}
\kappa_{\dR} \in  \Ext^1_{\mhs}(H^1_{\Betti}(Y)^{\otimes 2}_{\cup=0}, H^1_{\Betti}(Y)) 
\end{equation}
denote the class of the  extension \eqref{eqn:fundamental-exact-sequence} 
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.


\sk\noindent
{\bf An alternate description of the Hodge filtration}

The analytic description of the Hodge filtration on $M_{\dR,\C}$ which we have just given is 
well-suited for the theoretical calculations of this note. However, the difficulties inherent in 
solving  equation \eqref{eqn:diff-hard} seem to make it ill-suited for the practical numerical
calculations of Abel-Jacobi maps that are carried out in \cite{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  $\omega\in \Omega^1_{\log}(Y(\C))$ and a
smooth closed one-form $\eta$ on $Y(\C)$,  the  two-form
$\omega\wedge\eta + \eta\wedge\omega$ vanishes identically on 
$Y$. In particular, the class of  $\omega\otimes\eta + \eta\otimes \omega$
always belongs to $H^1(Y)^{\otimes 2}_{\cup=0}$, even when $Y$ is projective,
and the elements of $M_{\dR,\C}$ which map to this class
are those of the form
\begin{eqnarray*}\label{Joe}
J_{\omega\otimes \eta + \eta\otimes \omega,\alpha}, & \mbox{ with } d \alpha =0; \\
J_{\omega\otimes \eta+\eta\otimes\omega,\alpha}(\gamma) :=  & \int_\gamma (\omega \cdot \eta+\eta\cdot \omega) - \int_\gamma \alpha.
\end{eqnarray*}

A direct calculation reveals that
\begin{equation*}
\label{square}
J_{\omega\otimes \eta+\eta\otimes\omega,\alpha}(\gamma)  = 
\left(\int_\gamma \omega \right) 
\left(\int_\gamma\eta\right) - \int_\gamma \alpha,
\end{equation*}
so that the values of $J_{\omega\otimes \eta+\eta\otimes \omega,\alpha}$ are expressed
in terms of products of periods of $\gamma$ relative to the de Rham cohomology
 classes  of $\omega$, $\eta$ and $\alpha$. 
The function $J_{\omega\otimes \eta + \eta\otimes \omega,\alpha}$ 
 can therefore be readily calculated in practice using meromorphic representatives
of the classes of $\omega$, $\eta$, and $\alpha$. It is then clear from the description of the Hodge filtration on $M_{\dR,\C}$ given in (ii)  above that 
\begin{equation}
\label{eqn:hodge-fil-symm}
J_{\omega\otimes\eta+\eta\otimes\omega,\alpha} \,\mbox{ belongs to }\, \Fil^1 M_{\dR,\C} 
\mbox{ if and only if } \alpha\in \Omega^1_{\log}(Y/\C).
\end{equation}

Secondly, let us represent a cohomology class $\xi\in H^1_{\dR}(Y/\C)$ by a 
 meromorphic differential form $\eta$ of the second kind on 
$Y(\C)$ with at worst logarithmic poles along $S$,
making use of the algebraic description for $H^1_{\dR}(Y/F)$ given in 
\eqref{filtrationH1}. 
The principal part of the meromorphic differential $1$-form $\omega  F_{\eta}$ 
at a point $x\in \tilde Y$, denoted 
$\pp_x(\omega F_\eta)$,    depends only
 on the image of $x$ in $Y(\C)$. Indeed, let
 $\gamma x\in \tilde Y$, for $\gamma\in \Gamma$, be any other point in 
${\tilde Y}$ lying above the same point of $Y(\C)$. 
Comparing   principal parts, we find: 
$$\pp_{\gamma x}(\omega F_\eta) = \pp_x(\gamma^*(\omega F_\eta)) 
= \pp_x(\omega F_\eta + c\omega)$$ 
for some constant $c$, and the claim follows because 
$\omega$ is regular. In particular, it makes sense to talk about 
the principal part
$\pp_P(\omega F_\eta)$ at  a point $P\in Y(\C)$, even though
the one-form $\omega F_\eta$ is only defined on ${\tilde Y}$. 
A direct application of  the  Riemann-Roch theorem shows that
  there exists a meromorphic 
differential form $\alpha$ on $Y(\C)$ with at worst logarithmic poles along $S$,
satisfying
\begin{equation}
\label{eqn:alphapp}
\pp_P(\alpha) = \pp_P(\omega\cdot F_{\eta}), \qquad \mbox{ for all } P\in Y(\C). 
\end{equation}
The one-form $ \omega\cdot F_{\eta}-\alpha$ 
is then holomorphic on $\tilde Y$. 
The differentials $\alpha$ satisfying (\ref{eqn:alphapp}) all differ by
 elements of $\Omega^1_{\log}(Y(\C)) = 
\Fil^1 H^1_{\dR}(Y/\C)$, and we again have that
\begin{equation}
\label{iv}
J_{\omega\otimes \eta,\alpha} \mbox{ belongs to }
\Fil^1 M_{\dR,\C}.
\end{equation} 

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 \eqref{eqn:hodge-fil-symm}
 and \eqref{iv}.

Given a  regular differential $\omega$ 
and  a meromorphic differential
$\eta$ of the second kind on $Y$, the task of
calculating the meromorphic one-form $\alpha$
satisfying \eqref{eqn:alphapp} 
can be carried out explicitly in the case of modular curves,
and seems easier  in practice than
solving the equation
\eqref{eqn:diff-hard} associated to  a  smooth  representative of the class
of $\eta$. See the forthcoming article 
\cite{DDLR} for a more detailed discussion of these computational
aspects.






% \sk\noindent
% {\bf Remark on the $p$-adic \'etale realisation}

%Fix a prime number $p$. The $p$-adic \'etale cohomology of the curves $Y_{\bar F}$ and $X_{\bar F}$ over ${\bar F}$  give rise to $p$-adic avatars of the Betti cohomology, denoted  $H^1_{\et}(Y_{\bar F},\Q_p)$ and  $H^1_{\et}(X_{\bar F},\Q_p)$, respectively. These are finite dimensional $\Q_p$-vector spaces equipped with a continuous action of $G_F$. 

% A $p$-adic representation of $G_F$ is said to be pure of weight $j$ if the eigenvalues of the geometric frobenius at all unramified prime ideals $\ell$ of $F$ have absolute value $N(\ell)^{j/2}$. This is for instance the case of $H^1_{\et}(X_{\bar F},\Q_p)$ and thus also of $H^1_{\et}(X_{\bar F},\Q_p)^{\otimes 2}$, which thanks to the Weil conjectures (proved by Weil in the case of curves) is pure of weight $1$ (resp. weight $2$). 

% Just as  \eqref{eqn:fundamental-exact-sequence} need not split in the category of mixed Hodge structures, it is also true that \eqref{eqn:fundamental-exact-sequence-p} need not  split in the category $\Rep_{F,p}$ of continuous $p$-adic representations of $G_F$. Let \begin{eqnarray*}   \kappa_{F,\et} &\in&  \Ext^1_{\Rep_{F,p}}(H^1_{\et}(Y_{\bar F},\Q_p)^{\otimes 2}, 
% H^1_{\et}(Y_{\bar F},\Q_p))  \\
% &=& H^1(F, \mathrm{Hom}(H^1_{\et}(Y_{\bar F},\Q_p)^{\otimes 2}, H^1_{\et}(Y_{\bar F},\Q_p))
% \end{eqnarray*}
% be the corresponding extension class in $\Rep_{F,p}$. It is known that this extension class is unramified at the primes 
% $\ell\ne p$ at which $X$ has good reduction,
% and that it is {\em cristalline} at $p$ if $X$ has good reduction at $p$. 
% In other words, it corresponds to an element in the pro-$p$ Selmer group attached to the 
% $p$-adic Galois representation
% $\mathrm{Hom}(H^1_{\et}(Y_{\bar F},\Q_p)^{\otimes 2}, H^1_{\et}(Y_{\bar F},\Q_p))$. 





\subsection{Points arising from Hodge classes and the fundamental group}
\label{sec:hodge-classes}
The goal of this section is to recall a construction which goes back at least to \cite{Kae} and \cite{Pu} whereby the extension class
$\kappa_{\dR}$ of \eqref{eqn:kappadr} arising from the  pro-unipotent fundamental group are    used to 
  construct  rational points in the Jacobian of a (smooth, projective)  curve\footnote{A tantalising ongoing program (cf.\,\cite{kim}) also exploits
the pro-unipotent fundamental group of a (hyperbolic, affine) curve $Y$ to  {\em 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 {\em Selmer varieties} which parametrise them.}. We retain the same notations as in the previous section and assume that $Y=X\setminus \{ \infty\}$ is the complement of a point $\infty\in X(F)$. 
%A {\em Hodge class of weight $2j$} in   a mixed hodge structure $N$
%is  a morphism
%of Hodge structures
%\begin{equation}
%\label{eqn:xi-hodge}
%\xi: \Z(-j) \lra N,
%\end{equation}
%where
%$\Z(-j)$ denoted the usual pure Hodge structure of weight $2j$
%with Hodge filtration given by 
%$$
%\Fil^i \Z(-1)= \left\{
%\begin{array}{ll} 
%\Z & \mbox{ if }  i \le j, \\
%0 & \mbox{ if } i \ge j+1.
%\end{array}
%\right.
%$$
%Such a class is equivalent to the datum of 
%the vector 
%\begin{equation}
%\label{identification}
%\xi(1)\in\Fil^j(N\otimes \C) \cap N. 
%\end{equation}
%The space of all Hodge classes in $N$ 
%is denoted $\Hdg(N)$. 

A {\em Hodge class} in a mixed Hodge structure $N$ of weight two  is a morphism  
\begin{equation}
\label{eqn:cycle-map}
 \xi: \Z(-1) \lra N
\end{equation}
of mixed Hodge structures, where
$\Z(-1)$ denotes the  weight $2$ Hodge structure
attached to the Tate motive $H^2(\PP_1)$.
The $\Z$-module of all such Hodge classes is denoted 
$\Hdg(N)$. 

The extension class
$$
\kappa_{\dR}(\infty) \in \Ext^1_{\mhs}(H_B^1(X)^{\otimes 2},H_B^1(X)),
$$
of $M_{\Betti}$ in the category of mixed Hodge structures arising from the exact sequence \eqref{eqn:fundamental-exact-sequence} is a natural object of 
study. We do so by  considering, for each class $\xi\in \Hdg(H_B^1(X)^{\otimes 2})$, the element
$$\kappa_{\dR}(\xi;\infty)\in \Ext^1_{\mhs}(\Z(-1),H_B^1(X))$$ 
associated to 
the   class 
of the extension
$M(\xi)=M(\xi;\infty)$ 
 appearing in the top row of the following 
commutative diagram 
with exact rows and cartesian squares:
\begin{equation}
\label{eqn:pullback-ext}
\xymatrix{ 
0 \ar[r] & H_B^1(X) \ar@{=}[d] \ar[r]  & M_B(\xi) \ar[d] \ar[r] & \Z(-1) \ar[d]^{\xi} 
\ar[r] & 0 \\
0 \ar[r] & H_B^1(X)  \ar[r]  & M_B  \ar[r] & H^1_B(X)^{\otimes 2}  \ar[r] & 0, }
\end{equation}
where the lower exact sequence is  \eqref{eqn:fundamental-exact-sequence} applied to $Y$, once we identify $H^1_B(Y)=H^1_B(X)$ thanks to \eqref{es-YX}.

There is a canonical isomorphism (cf.\,e.g.\,\cite{Ca}, \cite[\S 10.2]{Hain-survey})
\begin{equation}
\label{isos}
\Ext^1_{\mhs}(\Z(-1), H^1_{\Betti}(X)) =  \frac{H^1_{\dR}(X/\C)}{\Omega^1(X(\C)) + 
H^1_{\Betti}(X)}.
\end{equation}

Recalling the exact sequence \eqref{eqn:fes-C}, this identification is obtained by choosing elements 
$v_{\Hodge} \in \Fil^1(M_B(\xi)\otimes\C)$ and
$v_\Z\in M_B(\xi)$ 
 satisfying
\begin{equation}
\label{eqn:lift-of-1}
\mathrm{p}(v_{\Hodge}) = \mathrm{p}(v_\Z) = 1,
\end{equation}
and assigning to the extension class $M_B(\xi)$ the element
\begin{equation}
\label{eqn:vect-ext-class}
 v_\xi := i^{-1}(v_{\Hodge}-v_\Z) \in H^1_{\dR}(X/\C).
\end{equation}
Since \eqref{eqn:lift-of-1} determines  $v_{\Hodge}$ and
$v_\Z$  up to elements of $\Fil^1 H^1_{\dR}(X/\C)= \Omega^1(X/\C)$ and
$H^1_B(X(\C),\Z)$ respectively, the image of 
$v_\xi$ in the quotient  on the right-hand side 
of \eqref{isos} depends only on the 
isomorphism class of the extension $M_B(\xi)$. 

 
The Poincar\'e pairing
\begin{equation}
\label{pairing}
H^1_{\dR}(X/F)\times H^1_{\dR}(X/F) \lra F
\end{equation}
with $F=\C$ 
 induces an isomorphism
\begin{equation}
\label{duality}
\frac{H^1_{\dR}(X/\C)}{\Omega^1(X(\C)) + H^1_{\Betti}(X)}  \lra \frac{\Omega^1(X(\C))^\vee}{H^1_{\Betti}(X)} = \Jac(X)(\C).
\end{equation}

For all $\xi \in \Hdg(H^1_B(X)^{\otimes 2})$, 
we continue to use $\kappa_{\dR}(\xi;\infty)\in \Jac(X)(\C)$ to describe the  canonical
point in the jacobian of $X$ arising from the identifications \eqref{isos}
and \eqref{duality}.

The assignment $\xi\mapsto \kappa_{\dR}(\xi;\infty)$  
therefore  sets up a map
\begin{equation}
\label{eqn:projective-case}
\Hdg(H^1_B(X)^{\otimes 2}) \lra  \Jac(X)(\C)
\end{equation}
sending Hodge classes in $H^1_{\dR}(X)^{\otimes 2}$ to complex points in the Jacobian of $X$. 



%This is the extension appearing at the top row in the following 
%commutative diagram 
%with exact rows and cartesian squares in the category of mixed Hodge structures:
%\begin{equation}
%\label{eqn:pullback-xi}
%\xymatrix{ 
%0 \ar[r] &  H^1_{\Betti}(Y) \ar@{=}[d] \ar[r]^i  & M_{B}(\xi) \ar[d] \ar[r]^p & \Z(-1) \ar[d]^{\xi}
% \ar[r] & 0 \\
%0 \ar[r] &  H^1_{\Betti}(Y)  \ar[r]  & M_{B}  \ar[r] & H^1_{\Betti}(Y)_{\cup=0}^{\otimes 2}  \ar[r] & 0. }
%\end{equation}
%Assume now that the Hodge structure $H^1_{\Betti}(Y)$ is pure of weight one.
%This condition is satisfied if and only if
%$Y=X$ is projective, or  $Y=X-\{\infty\}$ is the complement of a 
%single  point $\infty\in X(\C)-\{o\}$ in a projective curve $X$.  
%In either case,  $H^1_{\dR}(Y/F) =  H^1_{\dR}(X/F)$, and


%If $\xi$ is merely a Hodge class in $H^1_{\dR}(X/\C)^{\otimes 2}$
%and does not lie in the kernel of the cup product on $H^1_{\dR}(X/\C)^{\otimes 2}$, 
%we then write
%$\kappa_{\dR}(\xi;\infty) $ for the  point obtained  via the exact
%sequence  (\ref{eqn:pullback-xi}) with 
 %$Y=X-\{\infty\}$.

\begin{lemma}
\label{lemma:dependence-on-infty}
Given $\infty_1,\infty_2\in X(\C) - \{o\}$,
we have
$$ \kappa_{\dR}(\xi;\infty_1) = \kappa_{\dR}(\xi;\infty_2) + m_\xi ((\infty_2)-(\infty_1)),$$
where we recall that $m_\xi = \int_{X(\C)} \xi$ is the integral of any smooth two-form representing  $\xi$ over the curve $X(\C)$ diagonally embedded in $X(\C)\times X(\C)$.
\end{lemma}
\begin{proof}
Let 
$$v_\xi(\infty_1) \in H^1_{\dR}(X-\{\infty_1\}/\C), \qquad
v_\xi(\infty_2) \in H^1_{\dR}(X-\{\infty_2\}/\C) $$
be the be the elements 
 attached to the extension classes 
$M_B(\xi)$ with $\infty=\infty_1$ and $\infty_2$ respectively,
following  equation
\eqref{eqn:vect-ext-class}. 
The difference $v_\xi(\infty_1) - v_\xi(\infty_2)$  corresponds, modulo elements of 
$\Omega^1(X)$ and $H^1_B(X(\C),\Z)$, to the class  in
$H^1_{\dR}(X-\{\infty_1,\infty_2\})$  of a regular differential  $\omega\in\Omega^1_{\log}(X-\{\infty_1,\infty_2\})$
whose residual divisor is $m_\xi((\infty_2)-(\infty_1))$. 
But such an element describes the image of the degree zero
divisor  $m_\xi((\infty_2)-(\infty_1))$ in $\Ext^1(\Z(-1),H^1_{\dR}(X/\C))$, and the result follows.
\end{proof}

It is occasionally convenient to attach an
 invariant $\kappa_{\dR}(\xi)\in \Ext^1(\Z(-1),H^1_{\dR}(X/\C))$ 
to an arbitrary element of $\Hdg(H^1_B(X)^{\otimes 2})$ 
depending only on the pointed curve $(X;o)$ and its fundamental group, and not on the choice of an auxiliary base point 
$\infty\in X-\{o\}$. We can do this by setting
\begin{equation}
\label{eqn:def-kappa-dr-xi}
\kappa_{\dR}(\xi) := \kappa_{\dR}(\xi;\infty) + m_\xi ((\infty)-(o)).
\end{equation}
By Lemma \ref{lemma:dependence-on-infty}, the right hand side 
does not depend on the point $\infty$. It  depends only on the Hodge class $\xi$ and the 
exact sequence \eqref{eqn:fundamental-exact-sequence} arising from the fundamental group
$\pi_1(X;o)$. 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 \cite{deligne}.

The next two propositions express the points $\kappa_{\dR}(\xi;\infty)$  and
$\kappa_{\dR}(\xi)$  in terms
of iterated integrals. Let us first introduce a few notations.


Given a Hodge class
 $\xi \in
 \Hdg(H_B^1(X)^{\otimes 2})$, choose a representative 
$$ \tilde\xi  = \sum_{j=1}^t \omega_j\otimes \eta_j,$$  
for its de Rham cohomology class, in such a way that
 $\omega_j$ and $\eta_j$  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 $\rho \in \Omega^1(X)$ is any holomorphic differential,
let  $\gamma_\rho\in \C[\Gamma_{\ab}]$ be 
a path whose homology class is Poincar\'e dual to $\rho$, i.e., such that
$$ \int_{\gamma_\rho} \tilde\eta = \langle \rho,\eta\rangle, \quad
\mbox{ for all } \eta \in H^1_{\dR}(X/\C).$$

\begin{proposition}
\label{thm:ii-hodge-smooth}
With notations as above,
let $\alpha_\infty$ be a smooth differential on $Y(\C)$ of
type $(1,0)$, with at worst
logarithmic poles at $\infty$, satisfying 
$$ d\alpha_\infty = \sum_{j=1}^t \omega_j \wedge\eta_j \,\mbox{ on } Y(\C).$$
Then, for all $\rho\in \Omega^1(X/\C),$
\begin{eqnarray*}
 \kappa_{\dR}(\xi;\infty)(\rho)  &=& 
\int_{\gamma_\rho} \left(
\sum_{j=1}^t  \omega_j \cdot \eta_j
  - \alpha_\infty \right)  \,\mbox{ and}\\
 \kappa_{\dR}(\xi)(\rho)  &=& 
\int_{\gamma_\rho} \left(
\sum_{j=1}^t  \omega_j \cdot \eta_j
  - \alpha_\infty \right) +  m_\xi\int_o^\infty \rho.
\end{eqnarray*}
\end{proposition}

\begin{proof}
The first equality is a direct consequence of the definitions,
using the fact that the class of the extension $M_B(\xi;\infty)$ attached to the pair
$(\xi,\infty)$  is described by the 
iterated integral $J_{\tilde \xi,\alpha_\infty}$.
The second equality then follows from equation
\eqref{eqn:def-kappa-dr-xi} defining $\kappa_{\dR}(\xi)$. 
\end{proof}



We conclude this section with a formula analogous to the one in Proposition \ref{thm:ii-hodge-smooth},
expressing $\kappa_{\dR}(\xi;\infty)$ and $\kappa_{\dR}(\xi)$ in terms of differentials of the 
second kind. This formula turns out to be particularly useful in computing   the points $\kappa_{\dR}(\xi)$ numerically, as it is done in \cite{DDLR} for Hodge classes arising from Hecke correspondences on a classical elliptic modular curve. Choose a representative
$$ \tilde \xi  = \sum_{j=1}^s \omega_j^+ \otimes \eta_j^+  
+ \sum_{j=1}^t \eta_j^- \otimes  \omega_j^- \in 
\Omega^1(X/\C)\otimes \Omega^1_{\mer}(X/\C) + 
\Omega^1_{\mer}(X/\C)\otimes \Omega^1(X/\C)
$$
of the class $\xi$, in such a way that
 the elements  $\omega_j^\pm$  belong to 
$\Omega^1(X) = \Fil^1 H^1_{\dR}(X/\C)$ and the differentials $\eta_j^\pm$ are of
the second kind and regular at $\infty$. 

\begin{proposition}
\label{thm:ii-hodge-algebraic}
Which notations as above,
let $\alpha_\infty$ be a meromorphic differential on $X$, with at worst
logarithmic poles at $\infty$, satisfying
$$ \pp_x(\alpha_\infty) =   
\sum_{j=1}^s \pp_x(\omega_j^+ F_{\eta_j^+})    
-  \sum_{j=1}^t \pp_x(\omega_j^- F_{\eta_j^-}), \qquad \mbox{ for all } x\in Y(\C).$$
Then, for all $\rho\in \Omega^1(X/\C),$
$$ \kappa_{\dR}(\xi;\infty)(\rho) = 
\int_{\gamma_\rho} \left(
\sum_{j=1}^s  \omega_j^+ \cdot \eta_j^+
+  \sum_{j=1}^t \eta_j^-\cdot \omega_j^-   - \alpha_\infty \right).$$
\end{proposition}

\begin{proof}
The proof is the same as for Proposition \ref{thm:ii-hodge-smooth}, using this time the algebraic description of the Hodge filtration on
$M_{\dR,\C}$ in terms of differentials of the second kind.
\end{proof}





%\sk
%\noindent
%{\bf The $p$-adic \'etale picture}. 
%Fix, as in the previous section, a prime number $p$. Letting $\Q_p(j)$ denote the $p$-adic representation of $G_F$ attached to the $j$-th power
%of the cyclotomic character, a {\em Tate class} of weight two
%attached to the $p$-adic representation
%$H^1_{\et}(X_{\bar F},\Q_p)^{\otimes 2}$ is a $G_F$-equivariant homomorphism
%\begin{equation}
%\label{eqn:inceta-bis}
%\xi_p: \Q_p(-1) \lra 
%H^1_{\et}(X_{\bar F},\Q_p)^{\otimes 2} \subset H^2_{\et}(X_{\bar F}\times X_{\bar F},\Q_p).
%\end{equation}

%Tate classes play the same role, in the context of 
%$p$-adic representations  of $G_F$, as Hodge classes in the context
%of mixed Hodge structures. Given a Tate class $\xi_p$ as in \eqref{eqn:inceta-bis}, 
%the extension class $\kappa_{F,\et}$ induces by functoriality a class
%\begin{equation}\label{isos-et}
% \kappa_{F,\et}(\xi_p)  \in   \Ext^1_{\Rep_{F,p}}(\Q_p(-1),
%H^1_{\et}(X_{\bar F},\Q_p)) = H^1(F, H^1_{\et}(X_{\bar F},\Q_p)(1))\simeq H^1(F, V_p(X))
%\end{equation}
%where 
%$$V_p(X) := \Q_p\otimes (\lim_{\leftarrow} \Jac(X)[p^n])$$
%is the $p$-adic Tate module of the Jacobian $\Jac(X)$ of $X$. 
%One would like to understand when the global cohomology classes
%$\kappa_{F,\et}(\xi_p)$ arise as the image of a global point
%$P\in \Jac(X)(F)$.


\section{Points arising from algebraic cycles}
\label{sec:cycles} 

Hodge classes in $H^1_{\Betti}(X)^{\otimes 2}$
%and Tate classes in $(H^1_{\et}(X_{\bar F},\Q_p)^{\otimes 2}(1))^{G_F}$,
can be constructed naturally from geometry, by exploiting the 
{\em cycle class maps}
\begin{equation*}
\cl:  \Pic(X\times X)  \lra  \Hdg(H^2_B(X\times X)).
%,\\\cl_p: \Pic(X\times X) & \lra &  H^2_{\et}((X\times X)_{\bar F},\Q_p)(1)^{G_F}.
\end{equation*}
More precisely, let $\varepsilon$ be the projector on $\Pic(X\times X)$ given 
by the rule 
$$\varepsilon(Z)=Z-i_{1,*}\pi_{1,*}(Z)-i_{2,*}\pi_{2,*}(Z),$$
where 
$$\pi_1, \pi_2: X\times X\lra X,  \qquad
\pi_1(P_1,P_2)=P_1, \quad \pi_2(P_1,P_2)=P_2$$
are the natural projections onto the first and second factors, and
$$i_1,i_2: X\hookrightarrow X\times X, \qquad i_1(P)=(P,o), 
\quad i_2(P)=(o,P)$$
are the natural inclusions of $X$ into the horizontal and vertical
copies of $X$ in $X\times X$ over the point $o$.

The idempotent $\varepsilon$ annihilates the  factors
 $H^2(X)\otimes H^0(X)$ and $H^0(X)\otimes H^2(X)$ in the K\"unneth 
decomposition of  $H^2(X^2)$ (for any of the cohomology theories,
Betti, de Rham or \'etale). 
For any $Z\in \Pic(X\times X)(F)$ we can thus define
\begin{equation}\label{xiZ}
\xi_{Z}:=\cl(\varepsilon Z) \in \Hdg(H^1_B(X,\Z)^{\otimes 2}).
%, \\\xi_{Z,p} &:=&\cl_p(\varepsilon Z) \in (H^1_{\et}(\bar X,\Q_p)^{\otimes 2}(1))^{G_F}.
\end{equation}

In this way,  we can assign to the cycle  $Z\subset X\times X$ a 
point  
$$ \kappa_{\dR}(\xi_Z)\in \Jac(X)(\C),$$
%and a global Galois cohomology class 
%\begin{equation}
%\label{defPZ}
%\kappa_{F,\et}(\xi_{Z,p})\in H^1(F, V_p(X))
%\end{equation}
%with values in the $p$-adic  Tate module 
%$$V_p(X) = (\lim_{\leftarrow} \Jac(X)(\bar F)[p^n])\otimes\Q_p$$  of $X$. 
The goal of this  section is to 
associate to any $Z\in \Pic(X\times X)(F)$ a point
$$P_Z\in \Jac(X)(F),$$ in such a way that
%\begin{enumerate}
for all complex embeddings $\iota:F\lra \C$,
$$\kappa_{\dR}(\xi_Z) = P_{Z}\times_{\iota}\mathrm{Spec}(\C).$$
%\item For all rational primes $p$,
%$$ \kappa_{F,\et}(\xi_{Z,p}) = \delta_p(P_Z),$$
%\end{enumerate}
%where $$ \delta_p: \Jac(X)(F)\otimes \Q_p \lra H^1(F,V_p(X))$$ is
 %the connecting homomorphism arising from
 %the $p$-power descent exact sequences of Kummer theory.


Let us describe a geometric recipe for constructing the desired point $P_Z$. Given a  smooth projective variety $V_{/F}$ and a positive integer
 $c\geq 1$, let 
$$\CH^c(V)=\mathcal Z^c(V)/\sim_{\rat} $$ 
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 
$$
K/F \,\,\mapsto \,\,\CH^c(V)(K):=\{ [Z]:\, Z\in \mathcal Z^c(V), \quad \mbox{ such that } ^{\sigma }Z\sim_{\rat} Z \, \mbox{ for all } \sigma \in \Aut_K(\C)\} .
$$



For any integer $i\geq 1$, let $X_i=X$ be a copy of the curve $X$ and for any subset 
$\{ i_1,\ldots,i_m \}\subseteq \{ 1,\ldots,n\}$ with  $m\leq n$ of indices in a given finite set,
 let $X_{i_1,\ldots,i_m}$ denote a copy of the  curve $X$    embedded in 
$X_{1}\times \cdots \times X_{n}$ via
 the map 
$$x\mapsto (x_i)_{i=1}^n, \qquad \mbox{ where  } x_i=
\left\{\begin{array}{cl}
x  &  \mbox{ if } i\in \{ i_1,...,i_m \}, \\
 o & \mbox{  otherwise.} 
\end{array} \right. $$

The one-dimensional cycle $Z\subset X_1\times X_2$ gives rise to the following  divisor classes on $X$:
$$
D_{12} = Z\cdot X_{12}, \quad D_1 = Z\cdot X_1, \quad D_2 =   Z\cdot X_2. 
$$
Set
\begin{equation}
\label{eqn:first-def-PZ}
 P_Z = D_{12} - \deg(D_{12}) \cdot o \ \ \  \in \ \ \   \Pic^0(X)  = \Jac(X).
\end{equation}

We now give an alternate  geometric description of $P_Z$ 
which will be used to 
relate this point to iterated integrals. 
Let 
$$ \Pi_Z := Z_{12} \times X_{34} \subset X_1\times X_2 \times X_3\times X_4,$$
where $Z_{12}$ denotes a copy of the cycle $Z$ embedded in $X_1\times X_2$. 
The class of the cycle $\Pi_Z$ belongs to $\CH^2(X_1\times X_2\times X_3\times X_4)$, and induces a morphism
$$ \Pi_Z: \CH^2(X_1\times X_2\times X_3) \lra \CH^1(X_4)$$ 
by the usual rule
\begin{equation}
\label{eqn:usual-rule}
 \Pi_Z(\Delta)  = \pi_{4\ast}(\Pi_Z \cdot \pi_{123}^\ast\Delta),
\end{equation}
where 
$$\pi_4: X_1\times X_2\times X_3\times X_4\lra X_4,\qquad
\pi_{123}: X_1\times X_2\times X_3\times X_4\lra 
X_1\times X_2\times X_3$$
denote the natural projections  onto the fourth and first three factors 
respectively.

The Gross-Kudla-Schoen {\em modified diagonal cycle} in $X^3 = X_1\times X_2\times X_3$ is defined to be
\begin{equation}
\label{eqn:Delta-GKS}
\Delta_{\GKS} =X_{123}-X_{23}-X_{13}-X_{12}+X_{1}+X_{2}+X_3 \in \CH^2(X_1\times X_2\times X_3).
\end{equation}

\begin{lemma}
\label{lemma:PZ-GKS}
For all $Z
\in \Pic(X\times X)$, 
$$ P_Z = \Pi_Z(\Delta_{\GKS}).$$
\end{lemma}
\begin{proof}
A direct calculation using only the definition \eqref{eqn:usual-rule}
shows that
$$\Pi_Z(X_{123}) = D_{12}, $$
$$ \Pi_Z(X_{23}) = D_2, 
\quad \Pi_Z(X_{13}) = D_1, 
\quad \Pi_Z(X_{12}) = \deg(D_{12})\cdot o,$$
$$ \Pi_Z(X_1) = \deg(D_1) \cdot o, \quad
 \Pi_Z(X_2) = \deg(D_2)\cdot o, \quad
 \Pi_Z(X_3) = 0.$$ 
The lemma follows.
\end{proof}

\begin{remark}
\label{rem:Z-or-epsilonZ}
 Let $\infty$ be any point in $X$ (possibly equal to $o$). 
Viewing $X_1\times\{\infty\}$ and $\{\infty\}\times X_2$ as divisors in $X_1\times X_2$, 
a direct calculation shows that  
$$  P_{X_1\times \{\infty\}} = P_{\{\infty\}\times X_2} = 0.$$
Hence $P_Z = P_{\varepsilon Z} $  and the point $P_Z$ depends only on the Hodge class
$\xi_Z\in \Hdg(H^1_B(X)^{\otimes 2}).$ 
\end{remark}

%\begin{remark}
%There are many other ways of viewing the divisor classes
% $D_{12}$, $D_1$, $D_2$, 
%$D_Z$, and $P_Z$. For instance, one might  prefer to work with a 
%fixed $\Pi \in \CH^2(X^4)$ not depending on $Z$, such as the cycle
%$$ \Pi = X_{12}\times X_{34}\in \CH^2(X_1\times X_2\times X_3\times X_4), $$
%and to evaluate  the resulting morphism 
%$$ \Pi: \CH^2(X^3) \lra \CH^1(X)$$
%on a varying collection of codimension two cycles on $X^3$,
 %depending eventually on one-dimensional 
%cycles  $Z\subset X^2$.
%It is not hard to see, in this vein, that
%\begin{equation}
%\label{eqn:useless}
% \Pi(Z\times_{\pi_1} X_3) = D_{12}, \quad \Pi(Z\times \{o\}) = \deg(D_{12}) \cdot o,
%\quad \Pi(\{o\}\times Z) = D_2,
%\end{equation}
%and so on and so forth.
%While this alternate  description is 
%more consistent with the  point of view of ``Chow-Heegner points"
%developped in \cite{BDP2} and \cite{Dar}, 
%the approach taken in Lemma
%\ref{lemma:PZ-GKS} turns out to be more convenient for making an efficient  
%connection between the point
%$P_Z$ and iterated integrals. Hence   no use  shall be made of the formulae  in 
%\eqref{eqn:useless}.
%\end{remark}







  

We will  use Lemma \ref{lemma:PZ-GKS} to derive an analytic formula
for the point $P_Z$, or rather, for its image under the Abel-Jacobi map
$$ \AJ_X: \CH^1(X)_0 \lra \frac{\Omega^1(X/\C)^\vee}{H_1(X(\C),\Z)}.$$
This Abel-Jacobi map is defined on degree $0$ divisors $D$ by the
familiar rule:
\begin{equation}
\label{eqn:aj-on-X}
 \AJ_X(D)(\omega) = \int_{\partial^{-1}(D)} \omega,
\end{equation}
where $\partial^{-1}(D)$ is any smooth one-chain on $X(\C)$ whose 
boundary is $D$.
It admits a higher dimensional generalisation
\begin{equation}\label{AJX3}
\AJ_{X^3}: \CH^2(X^3)_0(\C) \lra J^2(X^3),
\end{equation}
where $\CH^2(X^3)_0$ is the Chow group of {\em null-homologous} cycles on 
$X^3$, and 
$$J^2(X^3)= \frac{\Fil^2 H^3_{\dR}(X^3/\C)^{\vee}}{H_3(X^3(\C),\Z)}$$ 
is the intermediate Jacobian associated to $H^3(X^3)$. 
This Abel-Jacobi map is defined by the rule generalising 
\eqref{eqn:aj-on-X}, 
\begin{equation}
\label{eqn:aj-on-X3}
 \AJ_{X^3}(\Delta)(\omega) = \int_{\partial^{-1}(\Delta)} \omega,
\qquad \quad \omega \in \Fil^2 H^3_{\dR}(X^3/\C).
\end{equation}

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


\begin{lemma}
\label{formulai}
The cycle $\Delta_{\GKS}$ is null-homologous. 
\end{lemma}
\begin{proof}
The obvious  method to prove this lemma is to
 exploit the K\"unneth decomposition of the cohomology of
 $X_1\times X_2\times X_3$, as it is done in \cite{GrSc} for example.
We give a more detailed, direct  argument, which yields 
  a differentiable $3$-chain on $X_1\times X_2\times X_3$ 
having $\Delta_{\GKS}$ as boundary.
While not strictly necessary to prove
Lemma \ref{formulai},
such an explicit $3$-chain will be used in subsequent calculations.

Let $g$ be the genus of the projective curve $X$, and let
 $\gamma_1,\ldots,\gamma_{2g}$  be a collection of paths in 
$\PP(X;o)$ whose associated homology classes, denoted
$[\gamma_1],\ldots,[\gamma_{2g}]$,   form an integral
basis for $H_1(X(\C),\Z)$. 
Let $\gamma_1^\ast,\ldots,\gamma_{2g}^\ast$ 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
$$ ([\gamma_i]\cdot [\gamma_j^*]) = \delta_{ij}, \qquad \mbox{ for all } 1\le i,j\le 2g,$$
where $\delta_{ij}$ is the Kronecker delta.
Let $\Delta$ denote the diagonal in $X\times X$. Its class in 
 $H_2(X\times X)$ is given by
$$ \cl(\Delta) = \cl(X\times \{o\}) + \cl(\{o\}\times X) + \sum_{i=1}^{2g} [\gamma_i]\otimes [\gamma_i^*],$$
and therefore there is 
a smooth $3$-chain $W\subset X\times X$ such that
\begin{equation}
\label{eqn:univ-relation}
\Delta = (X\times \{o\}) + (\{o\}\times X) + \sum_{i=1}^{2g} \gamma_i \times \gamma_i^*   + \partial(W).
\end{equation}
Applying this relation to $X_{123}$, viewed as the diagonal in $X_{12}\times X_3$, 
gives
\begin{equation}
\label{eqn:diag123}
X_{123} = X_{12} + X_3 + \sum_{i=1}^{2g} \gamma_i(12) \times \gamma_i^*(3)  + \partial(W_{12,3}),
\end{equation}
where
\begin{enumerate}
\item For $\{r,s\}\in \{1,2,3\}$, the regions
 $\gamma_i(rs)$ or $\gamma_i(r)$ 
are the paths $\gamma_i$ diagonally embedded in $X_{rs}\subset X_1\times X_2\times X_3$ and
in $X_r$ respectively, and similar conventions are adopted for the paths
$\gamma_j^*$.
\item The three-chain $W_{12,3}$  is a copy of the region $W$ of \eqref{eqn:univ-relation}, 
embedded in $X_{12}\times X_3$ in the natural way.
\end{enumerate}
By an analogous reasoning, we find
\begin{equation}
\label{eqn:diag13}
X_{13} = X_{1} + X_3 + \sum_{i=1}^{2g} \gamma_i(1) \times \gamma_i^*(3)  + \partial(W_{13}),
\end{equation}
and 
\begin{equation}
\label{eqn:diag23}
X_{23} = X_{2} + X_3 + \sum_{i=1}^{2g} \gamma_i(2) \times \gamma_i^*(3)  + \partial(W_{23}),
\end{equation}
with the obvious meaning given to $W_{13}\subset X_1\times \{o\} \times X_3$ and to 
$W_{23}\subset \{o\}\times X_2\times X_3$. 
Subtracting (\ref{eqn:diag13}) and (\ref{eqn:diag23}) from (\ref{eqn:diag123}),
we obtain
$$ \Delta_{\GKS} = \sum_{i=1}^{2g} (\gamma_i(12) - \gamma_i(1) - \gamma_i(2))\times \gamma_i^*(3)  + \partial(W_{12,3}-W_{13}-W_{23}).$$
But 
$$ \gamma_{i}(12) - \gamma_i(1) - \gamma_i(2) = \partial(T_{\gamma_i}(12)),$$
where 
$$T_\gamma(12) = \{ (\gamma(r),\gamma(s)), \  \ \ 0\le s\le r \le 1 \}\subset X_1\times X_2 $$
is the two-dimensional region introduced in \eqref{eqn:T-gamma}.
Therefore,
\begin{equation}
\label{eqn:Delta-GKS-boundary}
\Delta_{\GKS} = \partial\left( \sum_{i=1}^{2g} T_{\gamma_i}(12)\times \gamma_i^*(3)  + W_{12,3} - W_{13}-W_{23}\right),
\end{equation}
and the result follows.
\end{proof}



\begin{lemma}
\label{lemma:AJ-PZ}
For all $Z\in \CH^1(X\times X)$,  and all $\rho\in \Omega^1(X)$, 
$$ \AJ_X(P_Z)(\rho) = \AJ_{X^3}(\Delta_{\GKS})(\xi_Z(12)\wedge \rho(3)),$$
where the classes
$$\xi_Z(12) \in \Fil^1 H^2_{\dR}(X_1\times X_2/\C), \qquad
\rho(3) \in \Omega^1(X_3) = \Fil^1 H^1_{\dR}(X_3/\C)$$
are obtained in the natural way from the classes $\xi$ and $\rho$
respectively (so that, in particular,
$\xi_Z(12)\wedge\rho(3)$ belongs to $\Fil^2 H^3_{\dR}(X^3/\C)$).
\end{lemma}
\begin{proof}
By Lemma \ref{lemma:PZ-GKS} and the functoriality properties of the Abel-Jacobi map 
under correspondences,
$$ \AJ_X(P_Z)(\rho) = \AJ_X(\Pi_Z \Delta_{\GKS})(\rho) 
= \AJ_{X^3}(\Delta_{\GKS})(\Pi_Z^\ast \rho),$$
where
$$\Pi_Z^\ast: H^1_{\dR}(X_4/\C) \lra H^3_{\dR}(X_1\times X_2\times X_3/\C)$$
is the morphism induced on de Rham cohomology by the correspondence
$\Pi_Z= Z_{12}\times X_{34}$. 
A direct calculation reveals that
$$ \Pi_Z^\ast (\rho) = \cl(Z) \wedge \rho,$$
and therefore
$$ \AJ_X(P_Z)(\rho) = \AJ_{X^3}(\Delta_{\GKS})(\cl(Z)\wedge\rho).$$
By a similar reasoning,
$$ \AJ_X(P_{\varepsilon Z})(\rho) =
 \AJ_{X^3}(\Delta_{\GKS})(\xi_Z\wedge\rho),$$
and the result follows from Remark \ref{rem:Z-or-epsilonZ}.
\end{proof}





For the next proposition, we adopt the same notations that
 were introduced in the statement of Proposition \ref{thm:ii-hodge-smooth}  and the discussion immediately preceding it.
\begin{theorem}
\label{thm:ii-cycles}
For all $\xi \in \Hdg(H^1_B(X)^{\otimes 2})$ and all $\rho\in\Omega^1(X/\C)$, 
$$ \AJ_{X^3}(\Delta_{\GKS})(\xi(12)\wedge \rho(3)) = 
\int_{\gamma_\rho} \left(\sum_{j=1}^t \omega_j \cdot \eta_j - \alpha_\infty\right) + 
m_\xi \int_o^\infty \rho.$$
\end{theorem}
\begin{proof}
By (\ref{eqn:Delta-GKS-boundary}), and the definition of the Abel-Jacobi map,
\begin{equation}
\label{eqn:two-terms}
 \AJ(\Delta_{\GKS})(\xi\wedge \rho)  =
\sum_{i=1}^{2g} \int_{T_{\gamma_i}(12)\times \gamma_i^*(3)} \!\!\!\!\!\!\! \tilde\xi\wedge\rho  \ \  + 
\int_{W_{12,3}} \tilde\xi\wedge\rho,
\end{equation}
where we have used the fact that 
$\tilde\xi\wedge\rho$ vanishes identically on 
$W_{13}\subset X_1\times\{o\}\times X_3$ and on $W_{23}\subset \{o\}\times X_2\times X_3$ to dispense with the
integrals over these two regions.
The first term appearing in the right hand side of \eqref{eqn:two-terms} is 
equal to 
$$
\sum_{i=1}^{2g} \int_{T_{\gamma_i}(12)\times \gamma_i^*(3)} \!\!\!\!\!\!\! \tilde\xi\wedge\rho  = 
\sum_{i=1}^{2g} \int_{T_{\gamma_i}(12)} \tilde\xi \int_{\gamma_i^*(3)} \rho.$$
This last expression can also be viewed as the integral of the two-form $\tilde\xi$ over the $2$-chain
$$ \sum_{i=1}^{2g} \left(\int_{\gamma_i^*}\rho\right) T_{\gamma_i} = T_{\gamma_\rho},$$
where 
$$ T_{\gamma_\rho} = \sum_{i=1}^{2g} \lambda_i T_{\gamma_i}, \qquad \mbox{ with } [\gamma_\rho] = \sum \lambda_i [\gamma_i].$$
By the formula for the iterated integral that is given in 
\eqref{eqn:T-gamma}, it follows that
\begin{equation}
\label{eqn:first-term}
\sum_{i=1}^{2g} \int_{T_{\gamma_i}(12)\times \gamma_i^*(3)} \!\!\!\!\!\!\! \tilde\xi\wedge\rho  = 
 \int_{\gamma_\rho } \sum_{j=1}^t \omega_j \cdot\eta_j.
\end{equation}
To deal with the second term in the right-hand side of
(\ref{eqn:two-terms}),
we choose an auxiliary point $\infty$ which is 
not in the image of any of the paths $\gamma_i$ and $\gamma_i^*$, and
such that $W_{12,3}\cap ((\infty,\infty)\times X_3) $ is a smooth
one-chain. 
By Lemma \ref{lemma:dirac-horizontal}, 
\begin{equation}
\label{eqn:intonW1}
\int_{W_{12,3}} \tilde\xi\wedge \rho = 
\int_{\partial W_{12,3}} \alpha_\infty\wedge \rho +
 m_{\xi} \int_{W_{12,3}\cap((\infty,\infty)\times X_3)} \rho.
\end{equation}
Recall that 
$$\partial W_{12,3} = X_{123} - X_{12}- X_3 -\sum_{i=1}^{2g} \gamma_i(12)\times \gamma_i^*(3),$$
and note that the differential $\alpha_\infty \wedge\rho$ vanishes identically on $X_{123}$, $X_{12}$ and $X_3$ since it is of
type $(2,0)$. Therefore, by a similar argument as above,
\begin{equation}
\label{eqn:intonW1-bis}
\int_{\partial W_{12,3}} \alpha_\infty\wedge \rho  
= -\sum_{i=1}^{2g} \int_{\gamma_i} \alpha_\infty \int_{\gamma_i^*}\rho  
= - \int_{\gamma_\rho} \alpha_\infty.
\end{equation}
Furthermore, the region of integration that appears in the second term of \eqref{eqn:intonW1} has boundary
\begin{eqnarray*}
 \partial( W_{12,3}\cap ((\infty,\infty)\times X_3)) 
&=&  
\left(X_{123}-X_{12}-X_3 -\sum_{i=1}^{2g} \gamma_i\times \gamma_i^*\right)
\cap 
( (\infty,\infty)\times X_3) 
 \\
&=&  (\infty,\infty,\infty) - (\infty, \infty,o).
\end{eqnarray*}
It follows that 
\begin{equation}
\label{eqn:intonW1-bisbis}
  m_{\xi} \int_{W_{12,3} \cap ((\infty,\infty)\times X_3) } \rho 
= m_\xi \int_o^\infty \rho.
\end{equation}
Substituting \eqref{eqn:intonW1-bis} and 
\eqref{eqn:intonW1-bisbis} into
\eqref{eqn:intonW1} yields
\begin{equation}
\label{eqn:second-term}
\int_{W_{12,3}} \tilde\xi\wedge \rho =  - \int_{\gamma_\rho} \alpha_\infty 
+ m_{\xi} \int_o^\infty \rho.
\end{equation}
Finally, using \eqref{eqn:first-term} and \eqref{eqn:second-term} to rewrite 
the right hand side of \eqref{eqn:two-terms}, we obtain
\begin{equation}
 \AJ(\Delta_{\GKS})(\xi\wedge \rho)  =
\int_{\gamma_\rho } \sum_{j=1}^t \omega_j\cdot\eta_j  -
\int_{\gamma_\rho} \alpha_\infty + m_{\xi} \int_o^\infty \rho, 
\end{equation}
as was to be shown.
\end{proof}

\begin{corollary}
\label{cor:ii-PZ}
For all $Z\in \Pic(X\times X)$,  and all $\rho\in \Omega^1(X)$, 
$$ \AJ_X(P_Z)(\rho) = 
\int_{\gamma_\rho} \left(\sum_{j=1}^t \omega_j \cdot \eta_j - \alpha_\infty\right) + 
m_{\xi_Z} \int_o^\infty \rho,$$
where $\tilde\xi_Z = \sum_{i=1}^t \omega_i \otimes \eta_i$ is a representative
of $\xi_Z:=\cl(\varepsilon Z)$ and $\alpha_\infty$  and $\gamma_\rho$ are as in the statement of Theorem \ref{thm:ii-cycles}. 
\end{corollary}
\begin{proof} This follows directly from Lemma 
\ref{lemma:AJ-PZ}
 and  Theorem
\ref{thm:ii-cycles}.
\end{proof}



By combining Proposition \ref{thm:ii-hodge-smooth} and 
Corollary \ref{cor:ii-PZ}  we obtain the sought-for
relation between the extension classes $\kappa_{\dR}(\xi)$ arising from
$\pi_1(X;o)$  and the points $P_Z$. 

\begin{theorem}
\label{thm:main-bis}
For all $Z\in \Pic(X\times X)$, $\kappa_{\dR}(\xi_Z) = P_Z$ in $\Jac(X)(\C).$
\end{theorem}

In light of Lemma \ref{lemma:PZ-GKS}, this is Theorem \ref{thm:main}. As anticipated in the introduction, we can recover from it a result that is equivalent to the one obtained by Kaenders in \cite{Kae}:

\begin{corollary}\label{kaenders-by-us}
Let $X_{12}\in \Pic(X\times X)$ be the class of the diagonal divisor in $X\times X$ up to rational equivalence. Then $\kappa_{\dR}(\xi_{X_{12}}) = (2g-2) o -K$ in $\Jac(X)(\C)$, where $K\in \Pic(X)$ is the canonical divisor on $X$.
\end{corollary}

\begin{proof}
By Theorem \ref{thm:main-bis} and \eqref{eqn:first-def-PZ}, $\kappa_{\dR}(\xi_{X_{12}}) = P_{X_{12}}=X_{12}\cdot X_{12}-\deg(X_{12}\cdot X_{12}) o$. 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.
\end{proof}

\section{Modular and Shimura curves}
\label{secZh}

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 \cite{Zh} for the precise definitions of Shimura curves, Hecke correspondences and L-functions that we use throughout this section.


Choose a square-free ideal $D=\wp_1\cdot...\cdot\wp_r\mid \mid N_g$ of $R_F$ dividing it exactly (by which we mean that $D$ and $N_g/D$ are relatively prime). Assume that $r+d$ is {\em odd}. (If $d$ is odd, a possible choice for $D$ is thus $D=R_F$). 

Choose an ideal $N$ of $R_F$ such that $N_g\mid N$ and $D\mid \mid N$, 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 $J^D_0(M)$ for its jacobian. 

Let $\T:=\langle T_\wp, \wp\nmid N\rangle_\Q$ be the Hecke algebra generated by the set of Hecke operators at primes $\wp$ not dividing $N$, acting on $J^D_0(M)$. This $\Q$-algebra is isomorphic to a free product of number fields, namely 
\begin{equation}\label{decomT0}
\T \simeq \prod_{D\mid \mid n \mid N} \prod_h K_h
\end{equation}
where for each ideal $n$ such that $D\mid \mid n$ and $n \mid N$,  $h$ runs through a set of representatives for the orbits of $\Gal(\bar \Q/\Q)$ 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 $T_g\in \T$ for the corresponding idempotent in the decomposition \eqref{decomT0}.  

If $M$ is prime then in fact $\T\simeq \End J_0^D(M)$. 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:
\begin{equation}\label{decom-end}
\End^0 (J_0^D(M)) :=\End (J_0^D(M))\otimes \Q \simeq \prod_{D\mid \mid n \mid N} \prod_h M_{\sigma(N/n)}(K_h),
\end{equation}
where $\sigma(N/n)$ is the number of ideal dividing $N/n$. Its center is thus $\T$. 

Note that $\End^0 (J_0^D(M))[g]:=T_g\cdot \End^0 (J_0^D(M)) \simeq M_{\sigma(N/N_g)}(K_g)$, the center of which is $\T[g]:=T_g\cdot \T$.

Since 
\begin{equation}
\label{labi}
\End(J^D_0(M)) \, \simeq \,\frac{\Pic(X^D_0(M)\times X^D_0(M))}{\pi_1^*(\Pic(X^D_0(M)) + \pi_2^*(\Pic(X^D_0(M))}
\end{equation}
by \cite[Theorem 11.5.1]{LaBi}, we can define 
\begin{equation}
%H_g:=\{ \xi_{T}, T \in \End^0 (J_0^D(M))[g] \}\subseteq \Q\otimes \Hdg(H^1(X_0^D(M))^{\otimes 2})
H_g:=\{ \xi_{T}, T \in \T[g] \}\subseteq \Hdg(H^1(X_0^D(M))^{\otimes 2}\otimes \Q)
\end{equation}
to be the module of Hodge classes associated with $g$. 

\begin{definition}\label{Pg}
$\underline{P}_{g}:=\kappa_{\dR}(H_g)\subseteq \Q \otimes J^0_D(M)(F)$.
\end{definition}

The Hodge classes in $H_g$ can be written down explicitly. For instance, a representative for $\xi_g:=\xi_{T_g}\in H_g$ is given as follows: let $\{\omega_{g,i}\}$ 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 $\{ \omega_{g,i}, \eta_{g,i}\}$ of $H^1_{\dR}(X_0^D(M))[g]$. Then
\begin{equation}\label{tildexig}
\tilde{\xi}_g= \sum_{i} \omega_{g,i}\otimes \eta_{g,i} -\eta_{g,i}\otimes \omega_{g,i}
\end{equation}
is a differential form on $X_0^D(M)\times X_0^D(M)$, rational over $F$, which represents $\xi_g$. The point $P_g:=\kappa_{\dR}(\xi_g)\in \underline{P}_{g}$ can then be computed as the tuple of periods of the iterated integral associated with \eqref{tildexig}; similar formulas can be obtained of course for a complete basis of $\underline{P}_{g}$: see \cite{DDLR} for explicit calculations.


Recall that $\kappa_{\dR}(\xi)$ 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 $X^D_0(M)$ is the Shimura curve attached to an order in a {\em division} quaternion algebra over an arbitrary totally real number field $F$, then $X^D_0(M)(F)$ may be empty: see e.g. \cite[Theorem 5.1]{Shimura}, which establishes criteria for the existence of real points on these curves and shows in particular that $X^D_0(M)(\R)=\emptyset$ when $F=\Q$. Instead of fixing a single point $o$ rational over $F$, one proceeds as in \cite[\S 13]{GrKu}, \cite{YZZ} and \cite{Zh} by rather considering a canonical $\Gal(\bar F/F)$-invariant divisor class $o=\sum a_i o_i$ of degree $1$ with rational coefficients on $X^D_0(M)$, which in \cite{Zh} is called the {\em Hodge class} of $X^D_0(M)$. In this case, by $\kappa_{\dR}(\xi)$ we mean $\sum a_i \kappa_{\dR}(\xi;o_i)$.


Choose now another normalised Hilbert newform $f$ of weight $2$, level $N_f\mid N$ and trivial nebentypus, different from $g$. The triple product L-function 
$L(g\otimes g\otimes f,s)$, which was first studied by Garrett in \cite{Gar}, is 
the L-function associated with the tensor product $V_g\otimes V_g\otimes V_f$, where $V_g$ and $V_f$ 
denote the two-dimensional $\ell$-adic representations of $G_F:=\Gal(\bar F/F)$ 
attached to
$g$ and $f$ respectively. It is explicitly given by an Euler product with
Euler factors of degree $8$ at the primes $\wp\nmid N$, 
and factors as 
\begin{equation}\label{sym-factors}
L(g\otimes g\otimes f,s) = L(f,s-1) L(\sym^2g\otimes f,s),
\end{equation} 
mirroring the decomposition $$V_g \otimes V_g = \Q_\ell(-1) \oplus \sym^2 V_g$$ 
of $\ell$-adic representations of $G_F$. 

Thanks to the work of Garrett, Piatetski-Shapiro--Rallis, Harris--Kudla, Prasad and others (cf.\,\cite{Gar},\cite{Pr} and the references therein), it is known
that $L(g\otimes g\otimes f,s)$ admits  analytic continuation to $\C$ and satisfies a functional equation relating its
values at $s$ and $4-s$. The  sign $\epsilon$ in this functional equation is 
expressed as a product 
\begin{equation}\label{global-eps}
\epsilon = (-1)^d\prod_{\wp \mid \mathrm{gcd}(N_g,N_f)} \epsilon_\wp
\end{equation} 
of local 
signs $\epsilon_\wp\in \{\pm 1\}$; for primes $\wp$ dividing exactly both $N_g$ and $N_f$, $$\epsilon_\wp = -a_\wp(f)$$ is the negative of the eigenvalue of the Hecke operator $T_\wp$ acting on $f$ (cf.\,\cite{GrKu},\,\cite{Pr}), and in particular does not depend on the choice of $g$. When $\wp^2\mid N_g$ or $N_f$, the value of $\epsilon_\wp$ depends on the local behavior of the automorphic representations associated with $g$ and $f$; see \cite{Pr} for more details. Assume that 
\begin{equation}\label{signs}
\epsilon(f)=-1, \qquad \epsilon(\sym^2(g)\otimes f)=+1.
\end{equation}

Then $\epsilon = -1$ and we can set $D=D_{g;f} = \prod_{\epsilon_\wp=-1} \wp$; note that $D$ is the product of $r=\sharp \{\epsilon_\wp=-1\}$ distinct prime ideals, and that $r+d$ is odd by \eqref{global-eps}. Assume that $D\mid \mid N$ and put as above 
$M=M_{g;f}=N/D$. 

The quotient of $J^D_0(M)$ 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 $A_f^{\sigma(N/N_f)}$ of an $F$-simple abelian variety $A_f$. This means that there are $\sigma(N/N_f)$ $F$-linear independent modular parametrizations
$$
\pi^d_f:J^D_0(M) \lra A_f
$$ 
of $A_f$, one for each divisor $d$ of $N/N_f$. 

\begin{definition} Let
\begin{equation}\label{MWgf}
\underline{P}_{g,f} := \sum_{d\mid \frac{N}{N_f}}\,\, \pi^d_f(\underline{P}_g) \subseteq A_f(F)\otimes \Q
\end{equation} 
be the subspace of the Mordell-Weil group (tensored with $\Q$) of $A_f$ associated to the pair of modular forms $(g,f)$. 
\end{definition}

In particular, if we let $\pi_f: J_D^0(M) \ra J^D_0(N_f/D) \ra A_f$ denote the map which factors through the natural projection of $J_0^D(M)$ onto $J_0^D(N/N_f)$, the point 
\begin{equation}\label{Pgf}
P_{g,f}=\pi_f(P_g)
\end{equation}
 belongs to this space, and can be explictly computed by means of \eqref{tildexig}.


Formula \eqref{Pgf}, combined with Theorem \ref{thm:ii-cycles}, provides a suggestive construction of a point on a modular abelian variety by means of iterated integrals. Theorem \ref{thm:main} can be used to show that, in the particular case that $g$ has {\em 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:

\begin{definition}\label{def-zh}[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  $\pi_g: X^D_0(M)\lra E_g$ be a modular parametrization {\em of curves}. Letting $O\in E_g(F)$ denote the identity element, define
$$
P^{Zh}_{g,f}:=\pi_{f,*} \pi_g^*(O).
$$
\end{definition}

We refer the reader to the forthcoming Ph.\,D thesis \cite{Daub} of M. Daub for more details concerning the connection between \eqref{Pgf} and Definition \ref{def-zh}.
See also W. Stein's appendix to \cite{DDLR} for the effective numerical computation of the points $P^{Zh}_{g,f}$ in the classical setting of elliptic modular curves over $F=\Q$. Note that Definition \ref{def-zh} 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 \eqref{Pgf} is available in that broader setting: see again \cite{DDLR} for explicit examples.

\vspace{0.2cm}



The algebra $\T$ of Hecke operators acts by correspondences on $\CH^1(X^D_0(M))\otimes \Q$ and gives rise via \eqref{labi} to a linear action of $\T\otimes \T\otimes \T$ on $\CH^2(X^D_0(M)^3)\otimes \Q$ and on its subspace $\CH^2(X^D_0(M)^3)_0\otimes\Q$ of null-homologous cycles. 

\begin{definition}\label{ygf}
Set $$\mathrm{MW}(g,f):= \Hom\big(\T[g]\otimes \T[g]\otimes \T[f], \CH^2(X^D_0(M)^3)_0\otimes \Q\big)$$ and let $y_{g,f}\in \mathrm{MW}(g,f)$
be the homomorphism defined by
$$
y_{g,f}(T_1\otimes T_2\otimes T_3) := (T_1\otimes T_2\otimes T_3)(\Delta_{GKS}).
$$
\end{definition}

%Define $\CH^2(X^D_0(M)^3)_0[g,g,f]=(T_g\otimes T_g\otimes T_f)\Q\otimes \CH^2(X^D_0(M)^3)_0$ to be the $(g,g,f)$-isotypical component of %$\Q\otimes \CH^2(X^D_0(M)^3)_0$ and $$\Delta_{g,f}:=(T_g\otimes T_g\otimes T_f)(\Delta_{GKS})\in \CH^2(X^D_0(M)^3)_0[g,g,f].$$ 

By working with a suitable integral model of $X^D_0(M)^3$, B. Gross and C. Schoen have shown in \cite{GrSc} that Beilinson-Bloch's \cite{Bl} height pairing $\langle \,,\, \rangle^{\mathrm{BB}}$ on $\CH^2(X^D_0(M)^3)_0$ 
%and the height $h^{BB}(\Delta_{g,f}) := \langle \Delta_{g,f},\Delta_{g,f}\rangle^{BB}$ 
gives rise to a well-defined pairing
$$
\langle\,,\rangle^{\mathrm{BB}}: \mathrm{MW}(g,f)\times \mathrm{MW}(g,f) \lra \Hom((\T[g]\otimes \T[g]\otimes \T[f])^{\otimes 2},\C),
$$
independently of any choices; see also \cite[\S 1.3]{YZZ}.


% and the classes $\xi_g \otimes \varrho$ for any $\varrho \in \Omega^1(X)$. 
%On the other hand, for $\xi \otimes \varrho$ lying in the eigenspace $\Fil^2 H^3_{\dR}(X^3)^-$ on which $\tau^*$ acts as $-\Id$, we have
%\begin{equation}\label{vanishes}
%\AJ_{X^3}(\Delta_{g,f})(\xi\otimes \varrho)=0.
%\end{equation}
%as the left-hand side is equal to $$\AJ_{X^3}(\tau^*\Delta_{g,f})(\xi\otimes \varrho)=\AJ_{X^3}(\Delta_{g,f})(\tau^*\xi\otimes \varrho) = %-\AJ_{X^3}(\Delta_{g,f})(\xi\otimes \varrho).$$

%This computation reveals that the cycle $\Delta_{g,f}$ is only expected to contribute in a non-trivial way to the first factor of the decomposition %$M_g\otimes M_g\otimes M_f\simeq M_f(-1)\oplus M_{\sym^2(g)}\otimes M_f$ of the tensor product of the motives associated to $g$ and $f$; %note that this is in consonance with the scenario \eqref{signs} encountered on the analytic side.


X. Yuan, S. Zhang and W. Zhang have recently proved in \cite{YZZ} the following formula, which was conjectured by B. Gross and S. Kudla in \cite[Conjecture 13.2]{GrKu}:
%and is reminiscent of the classical Gross-Zagier formula:

\begin{theorem}\label{yuzh}\cite[Theorem 1.3.1, Corollary 1.4.1 (1)]{YZZ} Assume as in \eqref{signs} that
$\epsilon(f)=-1$ and $\epsilon(\sym^2(g)\otimes f)=+1$. Then
$$
\langle y_{g,f}, y_{g,f}\rangle^{\mathrm{BB}} = L'(f,1)\cdot \prod_{\sigma: K_g \ra \C} L(g^\sigma \otimes g^\sigma \otimes f,2)\cdot \lambda
%}{\Omega(g^\sigma ,g^\sigma,f)}
$$
for some non-zero homomorphism $\lambda \in  \Hom((\T[g]\otimes \T[g]\otimes \T[f])^{\otimes 2},\C)$.
%$\Omega(g^\sigma,g^\sigma,f)$ is a transcendental period, defined as the Petersson inner product of the form $g^\sigma\otimes %g^\sigma\otimes f$ on $X^3$, and 
%the symbol $\,\overset{\cdot}{=} \,$ stands for equality up to non-zero fudge factors. 
\end{theorem}

\begin{remark}
Note the different language in which the results are presented in \cite{YZZ} in contrast to \cite[\S 8]{GrSc} 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.
\end{remark}

The combination of Theorem \ref{thm:main} together with Theorem \ref{yuzh} yields the following result.

\begin{theorem}
\label{cor2} 
Assume as in \eqref{signs} that
$\epsilon(f)=-1$ and $\epsilon(\sym^2(g)\otimes f)=+1$. Then
the space $\underline{P}_{g,f}\subseteq  A_f(F)\otimes\Q $ has positive rank if and only if 
\begin{equation}\label{Lvalues}
L'(f,1)\ne 0, \quad \mbox{ and } \quad L(\sym^2(g^{\sigma})\otimes f,2)\ne 0 \, \mbox{ for all } \sigma: K_g \ra \C.
\end{equation}
\end{theorem}

\begin{proof}  The automorphism $\tau: X^D_0(M)^3 \ra X^D_0(M)^3$, $(x_1,x_2,x_3)\mapsto (x_2,x_1,x_3)$ gives rise to an involution $\tau^*$ on $\CH^2(X^D_0(M)^3)_0\otimes\Q$ and on $J^2(X^D_0(M)^3)$  which preserves the $(g,g,f)$-isotypical components. By \cite[Corollary 1.4.1 (1)]{YZZ}, the homomorphism $y_{g,f}$ is invariant under the action of $\tau$.

%By Theorem \ref{yuzh}, condition \eqref{Lvalues} holds if and only if there exists some linear combination $T=\sum T_i\otimes T_j\otimes T_k$ of Hecke operators in $\T[g]\otimes \T[g]\otimes \T[f]$ such that $$\langle %T(\Delta_{GKS}), T(\Delta_{GKS})\rangle^{\mathrm{BB}}\ne 0.$$

Hence, combined with Theorem \ref{yuzh}, condition \eqref{Lvalues} holds if and only if there exist correspondences $T_{g,i}\in \T[g]$, $T_{f,i}\in \T[f]$ such that, 
$$\langle \Delta[\breve{g},\breve{g},\breve{f}],  \Delta[\breve{g},\breve{g},\breve{f}]\rangle^{\mathrm{BB}}\ne 0,$$
where we set $$\Delta[\breve{g},\breve{g},\breve{f}]:=\sum_i T_{g,i}\otimes T_{g,i}\otimes T_{f,i}(\Delta_{GKS}).$$

%Given a self-correspondence $T\in \CH^1(X^2)$ of $X$, define
%$$
%D_T:=\pi_{34,*}(\pi^*_{12}(X_{12})\cdot  (T_{13}\times T_{24}))= \{ (T\cdot x,T\cdot x), x\in X  \}\in \CH^1(X^2),
%$$
%where $T_{jk}$ denotes a copy of $T$ in $X_j\times X_k\simeq X^2$.

%An exercise in intersection theory, using \cite[precise references?]{Ful}, shows that
%\begin{equation}\label{intersection}
%T\cdot X_{12} = D_T\cdot X_{12}
%\end{equation}
%as divisors on $X=X_{12}$.

Lemma \ref{lemma:PZ-GKS} combined with Lemma \ref{below} below show that
%$$
%\Pi_{T_{g,i}} (\Delta_{GKS}) =  
%T_{g,i}\cdot X_{12} - \deg(T_{g,i}\cdot X_{12}) o.
%$$
%while 
$$
\sum_i \Pi_{X_{12}}\cdot (T_{g,i}\otimes T_{g,i}\otimes 1)(\Delta_{GKS}) = \sum_i \Pi_{T_{g,i}} (\Delta_{GKS}).
$$
Since $T\circ \Pi_{X_{12}} = \Pi_{X_{12}}(1\otimes 1\otimes T)$ for any $T\in \CH^1(X^D_0(M)^2)$, we obtain that 
\begin{equation}\label{link}
\sum T_{f,i} \Pi_{T_{g,i}} (\Delta_{GKS}) =  \Pi_{X_{12}}(\Delta[\breve{g},\breve{g},\breve{f}]).
\end{equation}


For each $i$, $T_{f,i}$ is a self-correspondence on $X^D_0(M)$ that gives rise to an endomorphism $\pi_{f,i}$ of $\Jac(X^D_0(M))$ which is a linear combination of the projections $\pi_f^d$, $d\mid N/N_f$. Set $$P[\breve{g}, \breve{g}, \breve{f}] := \sum_i \pi_{f,i} \Pi_{T_{g,i}} (\Delta_{GKS})]\in A_f$$ and write $\langle \,,  \rangle^{NT}$ for the N\'eron-Tate canonical height pairing on an abelian variety.
By \cite[p.\,120-121, A.2]{Bl}, the pairing $\langle \,,\rangle^{\mathrm{BB}}$  is functorial with respect to correspondences, and from \eqref{link} it follows that
$$
\langle \Delta[\breve{g},\breve{g},\breve{f}],  \Delta[\breve{g},\breve{g},\breve{f}]\rangle^{\mathrm{BB}} \overset{\cdot}= \langle P[\breve{g}, \breve{g}, \breve{f}], P[\breve{g}, \breve{g}, \breve{f}] \rangle^{NT},
$$
up to a positive constant.

Note that, by \eqref{MWgf}, $P[\breve{g}, \breve{g}, \breve{f}]$ is a point in $\underline{P}_{g,f}$. By Theorem \ref{thm:main} we have  $P[\breve{g}, \breve{g}, \breve{f}]=\sum_i \pi_{f,i} \kappa_{\dR}(\xi_{T_{g,i}})$. The corollary follows.
\end{proof}


\begin{lemma}\label{below}
Given a self-correspondence $Z\in \CH^1(X^D_0(M)^2)$ of $X^D_0(M)$, it holds that
$$
\Pi_{X_{12}}\cdot (Z\otimes Z\otimes 1)(\Delta_{GKS}) = Z\cdot X_{12} - \deg(Z \cdot X_{12}) o.
$$
\end{lemma}

\begin{proof}
By applying \eqref{eqn:usual-rule}, it directly follows that $\Pi_{X_{12}}\cdot (Z\otimes Z\otimes 1)(X_{123}) = Z\cdot X_{12}\in \Jac(X^D_0(M))$. Similarly, when one applies \eqref{eqn:usual-rule} to the six other components $X_j$, $X_{jk}$ of $\Delta_{GKS}$ one obtains that, for all $j,k=1,2,3$, $j\ne k$, the divisors $\Pi_{X_{12}}\cdot (Z\otimes Z\otimes 1)(X_{j})$ and $\Pi_{X_{12}}\cdot (Z\otimes Z\otimes 1)(X_{jk})$ are all linearly equivalent to some multiple of the degree $1$ divisor $o$. 

Since $\Delta_{GKS}$ is null-homologous, $\Pi_{X_{12}}\cdot (Z\otimes Z\otimes 1)(\Delta_{GKS})$ is a divisor class on $X^D_0(M)$ of degree $0$. The lemma follows.
\end{proof}


In light of Theorem \ref{thm:main}, Theorem \ref{cor2} can be viewed to some extent as an analogue for modular and Shimura
curves  of the
Deligne-Wojtkowiak calculations \cite{deligne}
 in which values of the Riemann zeta-function $\zeta(s)$
at odd positive integers  emerge from the
study of the mixed Tate motives arising in the pro-unipotent fundamental group
of $\PP_1-\{0,1,\infty\}$.
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 
\cite{DR1}. Note however that  while \cite{deligne} obtains formulae for  the values of $\zeta(s)$ at certain {\em non-critical points},   Theorem \ref{cor2} applies  only to  the {\em critical value}   of the L-function $L(g\otimes g\otimes f,s)$ at $s=2$.




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\end{document}
