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\begin{document}

\title[Algorithms for Chow-Heegner points]{Algorithms for  Chow-Heegner points \\ via iterated integrals}

\author[Henri Darmon, Michael Daub, Sam Lichtenstein and Victor Rotger]{Henri Darmon, Michael Daub, Sam Lichtenstein and Victor Rotger\\  \\ 
(with an appendix by William Stein)}


\begin{abstract}
Let $E_{/\mathbf Q}$ be an elliptic curve of conductor $N$ and let $f$ be
the weight $2$ newform on $\Gamma_0(N)$ associated to it by modularity. 
Building on an idea of S.~Zhang, the article \cite{DRS} describes 
 the construction of so-called  {\em Chow-Heegner points}
$P_{T,f}\in E({\bar{\mathbf Q}})$  
 indexed by algebraic correspondences $T\subset X_0(N)\times X_0(N)$. 
 It also gives an analytic formula, depending only on the image of 
 $T$ in cohomology under the complex cycle class map,  for calculating   $ P_{T,f}$ 
 numerically 
 via Chen's theory of iterated integrals. 
The present  work describes an  algorithm based on this formula
for computing the
Chow-Heegner points to arbitrarily high complex accuracy,  carries out the computation for all elliptic curves
 of rank $1$ and conductor $N< 100$ when the cycles $T$ arise from 
 Hecke correspondences, and discusses several important variants of the basic construction.
\end{abstract}

\address{H. D.: Montreal, Canada}
\email{darmon@math.mcgill.ca}
\address{M. D.: Berkeley, California}
\email{mwdaub@math.berkeley.edu}
\address{S. L.: Stanford, California}
\email{saml@math.stanford.edu}
\address{V. R.: UPC, Barcelona, Spain}
\email{victor.rotger@upc.edu}



\maketitle
\tableofcontents

\section{Introduction}

Let $E_{/\mathbf Q}$ be an elliptic curve of conductor $N$ 
and let 
$$f=\sum_{n\geq 1} a_n e^{2\pi i nz}\in S_2(\Gamma_0(N))$$ be the weight two 
newform associated to $E$ by modularity. 
The proof of the  Tate conjecture for curves implies the existence of a non-constant
{\em modular parametrisation} 
\begin{equation}
\label{eqn:mod-par-Q}
 \pi_E: X_0(N) \lra E
 \end{equation}
defined over $\Q$, together with a  regular differential  $\omega\in \Omega^1(E/\Q)$
satisfying 
\begin{equation}
\label{eqn:def-omega}
\pi_E^{*}(\omega) = \omega_f:= 2\pi i f(z) dz.
\end{equation}
The map $\pi_E$  plays a key role in the study of the 
  Birch and Swinnerton-Dyer conjecture (BSD).
  This conjecture
 predicts that, for any number field $H$, the rank of $E(H)$ 
  is equal to the order  of vanishing $r_{\mathrm{an}}(E_{/H})$ of the L-series $L(E_{/H},s)$ 
  at  $s=1$; 
  one is thus led to seek strategies for
  constructing a non-torsion  point $P\in E(H)$ when
 $L(E_{/H},1)=0$, or---better yet---a system of  $r_{\mathrm{an}}(E_{/H})$ independent
 such points.

When $H$ is  a ring class field  of an imaginary quadratic field $K$,
a fruitful approach to this question   is afforded by the theory of 
{\em Heegner points} attached to  $K$.  These  are the  images under $\pi_E$ 
of points in $X_0(N)(\bar K)$ attached 
to the moduli of elliptic curves with complex multiplication by some
order in $K$. 
For instance it is known, thanks to the work of Gross--Zagier  \cite{GZ} and its strengthening 
by Zhang \cite{Zh} combined with a mild generalisation
\cite{bd-crelle} of the work of Kolyvagin  that   Heegner points attached to $K$
can be made to generate all of $E(H)\otimes\Q$ 
when $r_{\an}(E_{/H})=[H:K]$, at least when the discriminant of $K$ is prime to $N$
 and   $L(E_{/K},s)$ has a simple zero at $s=1$.



Let 
$$\mathfrak H^*=\{ z\in \mathbf C: \mathrm{Im}(z)>0\} \cup \mathbf P^1(\mathbf Q) = 
\mathfrak H \cup \mathbf P^1(\mathbf Q)$$
 denote the extended upper-half plane, topologized in the usual manner.
 % (\textit{cf.}, for example, \cite[Ch. 2]{DiSh}). 
 The  Heegner points attached to $K$  correspond precisely to
  the quadratic
  irrationalities $z\in \mathfrak H\cap K$ under the usual identification 
  of $X_0(N)(\C)$ with
  $\Gamma_0(N)\backslash \cH^*$. 
  Furthermore, after identifying $E(\C)$ with the quotient $\C/\Lambda_E$ by 
  the period
  lattice  $\Lambda_E\subset\C$  attached to the regular differential $\omega$
  of \eqref{eqn:def-omega},
  the map $\pi_E$ can be computed 
  analytically  from the formula
  \begin{equation}
  \label{h}
\pi_E(z) =  \int_{i\infty}^z \omega_f =   \sum_{n=1}^\infty \frac{a_n}{n} e^{2\pi i n z} \in \C/\Lambda_E,
\end{equation} 
  where the integral is taken over the vertical line joining  the cusp  $i\infty\in \mathbf P^1(\mathbf Q)$ 
 to $z$ (or any path in $\Gamma_0(N)\backslash \cH^*$ homotopic to it).
 
Equation   \eqref{h} makes it possible to calculate Heegner points numerically on a computer\footnote{For the important role 
  of experiments in the early
 study of Heegner points, see for example 
 \cite{birch}, \cite{birch-gross}.},
and leads to a efficient algorithm for calculating algebraic points on elliptic curves 
 which is presumably better than
  the more general-purpose method of descent, at least  in situations where
  it can be made to yield non-torsion points.
   In addition, 
 a careful study of  \eqref{h} has guided  
  attempts to unearth more general analytic recipes for algebraic points on elliptic curves in settings 
  falling {\em outside} the scope of the theory of complex multiplication.
    To elaborate on this last point, let us mention that

 


\begin{itemize}
\item In \cite{BeDa} and \cite{Gr1}, the complex upper half-plane $\mathfrak H^*$ is replaced 
by its  $p$-adic counterpart $\cH_p:= \PP_1(\C_p)-\PP_1(\Q_p)$ regarded as a rigid analytic variety,
and the group   $\Gamma_0(N)$ by  the group of units in a 
$\Z[1/p]$-order of  a totally definite quaternion algebra.
 Replacing the complex integrals
of \eqref{h} by 
Coleman's $p$-adic path integrals leads to $p$-adic analytic  formulae for (and   explicit calculations of) 
  Heegner points arising from Shimura curve parametrisations, thanks to
     the Cerednik-Drinfeld theory of 
  $p$-adic uniformisation of these curves.
 \item By mimicking the  formulae for Shimura curve parametrisations
  involving $p$-adic path integrals, 
the article  \cite{Dar00} was able to propose  a  $p$-adic construction of  
points on elliptic curves--the so called {\em Stark-Heegner points}--conjecturally defined over
abelian extensions of {\em real } quadratic fields. 
In this setting,
 the global structure analogous to  \eqref{eqn:mod-par-Q}
 is not currently known to be available, and numerical 
verifications such as those carried out in  \cite{DaPo}  provide the best evidence at present
for the conjectures of \cite{Dar00}. 
\item Let  $E_{/F}$ be a modular elliptic curve over a totally real number field $F$. 
By calculations similar to those of \cite{Dar00}, but replacing
 \eqref{h} by a formula involving integrals of complex-analytic differential forms of {\em higher order} 
 associated to Hilbert modular forms, \cite[Ch.\,VIII]{Darbook} proposes 
 a purely complex  construction of Stark-Heegner points in $E(\mathbf C)$. These points are conjectured 
 to be defined over suitable abelian extensions   of quadratic extensions $K/F$ which have exactly one 
 non-real archimedean place. In this complex setting as well, numerical 
 experiments   such as those described in 
  \cite{DaLo}  are the best evidence so far for the algebraicity 
  of the  Stark-Heegner points.
  \item Complex Stark-Heegner points can be viewed as images of certain null-homologous topological cycles
  on Hilbert modular varieties under a kind of ``Abel-Jacobi map". In the hope of better understanding this 
  construction, the articles \cite{BDP2} and \cite{DRS} discuss ``generalised modular parametrisations" 
  $$ \Pi: \CH^{r+1}(V)_0 \lra E$$
  from the Chow-group
  $\CH^{r+1}(V)_0$ of null-homologous codimension $r+1$ algebraic
  cycles on  a $(2r+1)$-dimensional 
  variety $V$ to the Mordell-Weil group
  of an elliptic curve $E$, 
  arising from suitable algebraic correspondences $\Pi\in \CH^{r+1}(V\times E)$. 
  In the setting of 
 \cite{BDP2}, $E$ is an elliptic curve with complex multiplication, $V$  the product of
 an $(r+1)$-dimension Kuga-Sato variety with $E^r$, and
 the existence of $\Pi$ relies on an ostensibly difficult instance of the Hodge or Tate conjectures.
 Nonetheless, the global nature of the points in the image of $\Pi$, for which the term 
 ``Chow-Heegner point" was proposed in \cite{BDP2}, is better understood than in the setting 
 of Stark-Heegner points.
 \end{itemize}
 An even simpler instance of the Chow-Heegner point construction
 arises when $V= X_0(N)^3$ is the triple product of a modular curve
 and $\Pi\subset X_0(N)^3\times E$ is the product of a codimension $1$ cycle  $T\subset X_0(N)^2$ with the graph of
 the ``classical" modular parametrisation $\pi_E$ of equation
 \eqref{eqn:mod-par-Q}.
 The resulting map 
 $$ \Pi_T: \CH^2(X_0(N)^3)_0 \lra E, $$
 is a natural generalisation of \eqref{eqn:mod-par-Q}.
 In particular the image $P_{T,f} := \Pi_T(\Delta_{\rm GKS})$
 of the {\em Gross-Schoen modified diagonal cycle}
 $\Delta_{\rm GKS}$ (as defined in Theorem 1 of \cite{DRS} for example) 
  under $\Pi_T$ 
is an interesting  rational point on $E$ whose relations with special values of 
$L$-functions \cite{YZZ} and the pro-nilpotent
fundamental groups of modular 
curves \cite{DRS} are  well documented in the literature.
 The point $P_{T,f}\in E(\Q)$ shall be  called the {\em Chow-Heegner point} 
 attached to the cycle $T$ and the newform $f$.
An alternate, somewhat simpler description of $P_{T,f}$ 
proceeds by  essentially   intersecting $T$ with the diagonal in $X_0(N)\times X_0(N)$ to
obtain a class in $\Pic(X_0(N))$, and hence (after a suitable modification to make it of
degree $0$) a rational point $P_T$ of the Jacobian $J_0(N)$.
The point $P_{T,f}$ is  then just the image of $P_T$ under 
$\pi_E$.
The definition of the points $P_T$ and $P_{T,f}$ 
 is described more precisely in \S \ref{describe1} and \S \ref{describe} below. 
 
 
The article \cite{DRS} also gives a complex analytic formula for  the point
$P_{T,f}$ in terms of 
K.-T. Chen's iterated path integrals.
In order to provide some motivation for it,
note that  the homomorphism
$$
\pi_1(X_0(N)(\mathbf C);\infty)\lra  \mathbf C \quad \text{  defined by}  \qquad
 \gamma \mapsto \int_{\gamma} \omega_f
$$
factors through the maximal abelian quotient 
$H_1(X_0(N)(\mathbf C),\mathbf Z)$ of the fundamental group of  $X_0(N)(\C)$. 
Chen observed that it is also possible to obtain non-abelian information 
about this fundamental group through integration.
His theory of iterated integrals gives rise to functions
$$
J: \pi_1(X_0(N)(\mathbf C);\infty) \lra \mathbf C
$$
attached to an $r$-tuple  of differentials,
which need not factor through $H_1(X_0(N)(\mathbf C),\mathbf Z)$ but rather through
the quotient of $\pi_1(X_0(N)(\C),\Z)$ by the $r$-th term in its lower central series.
The  algorithm for the computation of 
the Chow-Heegner points $P_{T,f}$ described in this paper, in effect,
 replaces the integral appearing in \eqref{h} by 
a sum of  iterated integrals attached to  {\em pairs } of modular forms. 
These pairs depend only on the
so called {\em Hodge cycle} attached to $T$, namely, its
 image 
 $$\xi_T\in H^{1,1}_{\dR}(X_0(N)^2)\cap H^2_B(X_0(N)^2,\Q)$$
under the cycle class map.
While algebraic equations for the modular curve $X_0(N)$ (and, a fortiori, for the cycle
$T$ itself) can be hard to calculate in practice, the Hodge class $\xi_T$ 
is amenable to numerical calculation using the explicit description of the de Rham cohomology of
modular curves in terms of weakly holomorphic modular forms of weight two and their
$q$-expansions.
The algorithm for computing the Hodge class $\xi_T$ and the associated
iterated integrals has been  implemented  in a computer program 
using the free software package {\tt Sage} \cite{sage}, and makes the Chow-Heegner 
points $P_{T,f}$ 
computable in practice for elliptic curves $E$ of moderate conductor.

 
As explained in the appendix,  there are other, somewhat more direct
approaches to computing $P_{T,f}$ for specific cycles  $T$ arising from
elliptic curve factors of $J_0(N)$.  
The approach based on iterated integrals appears to be somewhat more general and
 works for arbitrary Hodge
 cycles. In future work, we hope to extend our algorithm  to compute 
 Chow-Heegner points associated to ``exceptional" Hodge cycles on products of Kuga-Sato 
 varieties arising from CM forms. The rationality of Chow-Heegner points computed in this
  manner would provide some indirect  numerical evidence for certain open cases of the Hodge conjecture,
  in the spirit of the study carried out in \cite{BDP2}.

\subsection*{Plan of the paper} In \S  \ref{prelim} we recall necessary
 facts about iterated 
integrals and related ingredients for our main algorithm.
In \S  \ref{chpt} we specialize to the case of modular curves, define 
the points $P_{T,f}$ precisely, and write down an explicit analytic formula for them. 
In \S  \ref{alg} we describe in detail an algorithm for evaluating this formula numerically. 
The algorithm is illustrated with numerical examples in \S \ref{eg}, which also includes
 a table of triple Chow-Heegner points on rank one elliptic curves of small conductor.

% One interesting question is to determine the index $[A_f(\mathbf Q): \mathbf Z P_{g,f}]$ in terms of arithmetic invariants attached to the elliptic curve $A_f$. By BSD, this index should be related to the ``fudge factor'' connecting the derivative $L(f\otimes g\otimes g, 2)$ with the height of $P_{g,f}$; cf. \cite[Theorem 4.1]{DRS} and Yuan-Zhang-Zhang FIXME:add ref. (What is the fudge factor? Do our data match is?) The numerical data in our tables could be useful for investigating this matter.

\subsection*{Source code} Our {\tt Sage} implementation of the algorithm described
 in this paper can be found at \url{http://math.berkeley.edu/~mwdaub}.

\subsection*{Acknowledgments}
We warmly thank the organizers of the 2011 Arizona Winter School  for 
inviting the first and fourth author to deliver a series of lectures at the 
meeting, for creating such an inspiring atmosphere and giving  the four
 of us the opportunity of working together.

\section{Preliminaries}
\label{prelim}
\subsection{De Rham cohomology}
\label{hdr}
%Let $F$ be a number field (we take $F = \mathbf Q$ in the sequel)  and fix an embedding $\iota:F\hookrightarrow \mathbf C$.
% Let $\bar F$ denote the algebraic closure of $F$ in $\C$ and set  $G_F:=\Gal(\bar F/F)$. 
Let $X$ be a smooth, complete algebraic curve of genus $g\geq 2$ 
over $\mathbf Q$, and let $Y=X\setminus \{ \infty\}$ be the complement of a single point in $X(\mathbf Q)$. For a smooth variety $V$ defined over $\mathbf Q$ (such as $X$ or $Y$) we denote by $V^{\mathrm{an}}$ the complex manifold
$V(\mathbf C)$ with its analytic topology.

The {\em de Rham cohomology}  $H^1_{\dR}(X^{\mathrm{an}},\mathbf C)$ 
is the cohomology of the de Rham complex of smooth $\mathbf C$-valued
 differential forms on $X^{\mathrm{an}}$. 
 
Since the Riemann surface $X^\mathrm{an}$ is algebraic and defined over $\Q$, there is a canonical isomorphism
$$H^1_{\dR}(X^{\mathrm{an}},\mathbf C) = H^1_{\dR}(X/\Q)\otimes \mathbf C.$$
Here 
\begin{equation*}
\label{deRham-comparison}
H^1_{\dR}(X/\mathbf Q) := \mathbf H^1(0\rightarrow \cO_X \rightarrow \Omega^1_X \rightarrow 0)
\end{equation*}
is the {\em algebraic de Rham cohomology} of $X/\mathbf Q$, defined as 
the hypercohomology of the de Rham complex of sheaves of
regular differential forms on $X$.

Because $X$ is a curve, the group $H^1_{\dR}(X/\mathbf Q)$ has a particularly simple description in terms of the
space $\Omega^1_{\mathrm{II}}(X)$ of {\em differentials of the second kind} on $X$, defined as
$$ \Omega^1_{\mathrm{II}}(X) := \text{rational 1-forms on $X$ with vanishing residues at all points of $X$}.$$
Observe that $\Omega^1_{\mathrm{II}}(X)=\Omega^1_{\mathrm{II}}(Y)$, by the residue formula.
Thus there is a canonical isomorphism
\[H^1_{\mathrm{dR}}(X/\mathbf Q) = \Omega^1_{\mathrm{II}}(Y)/\mathrm{d}\mathbf Q(Y),\]
where $\mathbf Q(Y)=\mathbf Q(X)$ is the field of rational functions on $Y$ (or on $X$).
Using the  Riemann-Roch theorem, one sees that 
$$\Omega^1_{\mathrm{II}}(Y)/\mathrm{d}\mathbf Q(Y)
 \cong \Omega^1(Y)/\mathrm{d}\mathcal O(Y).$$
  So $H^1_{\mathrm{dR}}(X/\mathbf Q)$ can also be computed as the space of
\textit{regular} 1-forms on $Y$, modulo exact forms. For 
computational purposes, the latter description is the most useful: 
we will compute with classes in $H^1_{\mathrm{dR}}(Y)$ using rational 
1-forms on $X$ which  are regular away from the point $\infty$.
These are amenable to computation via their Laurent expansions about $\infty$.

Later on we shall make use of the symplectic {\em Poincar\'e pairing}
\[\langle,\rangle:H^1_{\dR}(X^\mathrm{an},\mathbf C)\times H^1_{\dR}(X^\mathrm{an}, \mathbf C)\to \mathbf C,\]
defined by 
  $$\langle\omega,\eta\rangle:=\frac{1}{2\pi i}\int_X \omega\wedge \eta$$
  for (the cohomology classes of) smooth $\mathbf C$-valued 1-forms $\omega,\eta$ on $X^\mathrm{an}$.
If $\omega$ and $\eta$ are differentials of the second kind on $X$, regular away from the cusp $\infty$, then the induced pairing on $H^1_{\mathrm{dR}}(X/\mathbf Q)$
can  be computed as
\[\langle \omega,\eta\rangle = \res_\infty (F_\omega \cdot \eta) =
 -\res_\infty (\omega \cdot F_\eta),\]
where $F_\nu$ denotes a local primitive of the differential $\nu$ at $\infty$.
\subsection{Iterated integrals}
We now turn to recalling the definition and basic properties of iterated integrals; see \cite{Chen}, \cite{Hain}, \cite{Hain2} for more details.

Fix a base point $o\in Y^{\mathrm{an}}$ and denote by $\Gamma := \pi_1(Y^{\mathrm{an}};o)$ the fundamental group
of the Riemann surface $Y^{\mathrm{an}}$.
We write $I\subset \mathbf Z[\Gamma]$  for augmentation ideal of the integral group ring of $\Gamma$.
Recall that $H_1(X^\mathrm{an},\mathbf Z) = H_1(Y^{\mathrm{an}},\mathbf Z) \cong \Gamma^{\mathrm{ab}}$,
as can be seen from the well-known presentation for the fundamental group of a Riemann surface, and that this abelian group 
is naturally identified with $I/I^2$.

\begin{definition}
The {\em path space} on $Y$ based at $o$, denoted  $\mathbf P(Y;o)$,  is  the set of
piecewise-smooth paths 
$$
\gamma:[0,1]\lra Y^{\mathrm{an}}, 
\quad  \mbox{ with  } \gamma(0)=o. $$ 
\end{definition}

Let $$\pi:{\tilde Y}\to Y^{\mathrm{an}},\qquad\text{resp.}\qquad \pi:\tilde X\to X^\mathrm{an},$$
 denote the universal covering space of  $Y^{\mathrm{an}}$ (resp. $X^\mathrm{an}$) corresponding to the basepoint $o$.
 The group $\Gamma $
 acts on $\tilde Y$ 
transitively and without fixed points, 
and the map $\gamma\mapsto \gamma(1)$  identifies
the quotient ${\tilde Y}/\Gamma$ with $Y^{\mathrm{an}}$. 

Recall that a closed, $\mathbf C$-valued smooth 1-form
(resp. a meromorphic 1-form of the second kind) $\eta$
 on $X^{\mathrm{an}}$ admits a smooth (resp. meromorphic) primitive 
 $F_\eta:{\tilde X}\to \mathbf C,$
 defined by the rule
$$F_\eta(\gamma) := \int_0^1 \gamma^\ast \eta.$$

\begin{definition}
The {\em  basic iterated integral }
attached to an ordered $n$-tuple  $(\omega_1,\ldots,\omega_n)$ of smooth 1-forms on $Y^{\mathrm{an}}$ is
the function $\mathbf P(Y;o)\to \mathbf C$, denoted $\int \omega_1\cdot \omega_2\cdot\ldots\cdot \omega_n$, defined by
\begin{equation*}
\label{eqn:def-ii}
\gamma\mapsto  \int_{\gamma} \omega_1\cdot\omega_2\cdot\ldots \cdot\omega_n := \int_{\Delta}
(\gamma^*\omega_1)(t_1) (\gamma^*\omega_2)(t_2) \cdots  (\gamma^*\omega_n)(t_n),
\end{equation*}
where $\Delta$ is the simplex in $[0,1]^n$ defined by $0\le t_n \le t_{n-1} \le \cdots \le t_1 \le 1$.
The integer $n$ is called the {\em length} of this basic iterated integral.
\end{definition}
\begin{example}
When $n=2$,
the basic iterated integral attached to
 $\omega$ and $\eta$ can be computed  by the formula
$$ \int_\gamma \omega \cdot \eta =  
\int_\gamma \omega F_{\eta} = 
\int_0^1  \gamma^\ast(\omega F_{\eta}).$$
In the expression in the middle, we abusively use the same notation $\omega$ for the differential $\pi^*\omega$ on  $\tilde Y$.
The 1-form $\omega F_\eta$  is to be integrated along a lift of $\gamma$ to $\tilde Y$, which is unique once a lift of $o$ to $\tilde Y$ is specified.
\end{example}
\begin{definition}
An {\em iterated integral} 
 is  a linear combination  of basic iterated integrals, viewed as a function on $\mathbf P(Y;o)$.
Its  {\em length} is defined to be the maximum of the lengths
of its constituent basic iterated integrals. It is said to be 
{\em homotopy invariant} if its value on any path $\gamma$  depends only on the homotopy class of
$\gamma$. 
\end{definition}
A homotopy-invariant iterated integral defines a $\mathbf C$-valued function on $\Gamma$, and
by extending linearly induces a homomorphism of abelian groups $\mathbf Z[\Gamma]\to \mathbf C$.
Observe that a homotopy invariant iterated integral of length $\leq n$ 
 vanishes on the $(n+1)^{\mathrm{st}}$ power $I^{n+1}$ of the augmentation ideal in $\mathbf Z[\Gamma]$, and hence gives rise to a well-defined  element of
$\operatorname{Hom}(I/I^{n+1},\mathbf C)$. 
The natural map
\begin{equation}\label{hiii}
\{\text{homotopy invariant iterated integrals of length $\leq n$}\} \lra \operatorname{Hom}(I/I^{n+1},\mathbf C)
\end{equation}
is an isomorphism; see, for example, \cite{Hain2}.


We will be interested in numerically evaluating certain homotopy invariant iterated integrals
on $Y$ of length $\leq 2$.
Suppose $\omega$ and $\eta$ are two  differentials of the second kind on $X$, regular on $Y$, representing cohomology classes $\omega,\eta\in H^1_{\dR}(X/\mathbf Q)$ in the manner of \S \ref{hdr}.
The basic iterated integral  $\int\omega\cdot\eta$ of length 2 is not generally homotopy invariant.
But when either $\omega$ or $\eta$ is holomorphic on $X$ --- i.e., has no pole at $\infty$ ---
a suitable modification of $\int \omega \cdot \eta$ will be homotopy invariant, as we now explain.

  Recall that a differential on a Riemann surface is said to have a
 {\em logarithmic pole} at a point if its expansion in terms of a local parameter $q$
  at this point is of the form $\sum_{n=0}^\infty a_n q^n \frac{dq}{q}$.
When $\omega$ is holomorphic at $\infty$, we let $\alpha_{\omega,\eta}$ be a meromorphic 1-form on $X$ that is regular on $Y$ 
and is such that the induced differential $\omega F_{\eta}-\alpha_{\omega,\eta}$ on $\tilde X$ has at worst a logarithmic pole at (any point lying over) $\infty$. This condition is well-posed because the principal part of $\omega F_\eta$ at $\tilde x\in \tilde X$
depends only on the image $x$ of $\tilde x$; see \cite[\S 2]{DRS}.
The form $\alpha_{\omega,\eta}$ exists --- and in fact can even be taken to be algebraic and defined over $\mathbf Q$ ---  by Riemann-Roch.
If $\omega$ is not holomorphic at $\infty$ but $\eta$ is, then we define $\alpha_{\omega,\eta} := -\alpha_{\eta,\omega}$.

\begin{lemma}
\label{homotopyinv}
Let $\omega$ and $\eta$ be as above, and assume that either $\omega$ or $\eta$ is holomorphic at $\infty$. Then
the iterated integral $J_{\omega,\eta}:= \int \omega\cdot \eta - \alpha_{\omega,\eta}$ is homotopy-invariant.
\end{lemma}

\begin{proof}
The homotopy invariance of $J_{\omega,\eta}$ follows from the 
fact that $J_{\omega,\eta}(\gamma) = \int_\gamma \omega F_\eta - \alpha_{\omega,\eta}$,
and the 1-form on $\tilde X $ in the integrand is holomorphic when restricted to $\tilde Y$.

\end{proof}

\begin{remarkwr}
\label{alpha=0} 
Note that if $\omega$ and $\eta$ are both 
holomorphic at $\infty$, then we can take $\alpha_{\omega,\eta} = 0$.
\end{remarkwr}

Now consider an integral Hodge class $\xi \in H^1(X^\mathrm{an},\mathbf Z)\otimes H^1(X^\mathrm{an},\mathbf Z)$.
The Hodge condition implies that we can choose a basis $\{\omega_i\}$ such that when we write $\xi = \sum c_{i,j} \omega_i \otimes \omega_j$, then either $\omega_i$ or $\omega_j$ is holomorphic at $\infty$ whenever $c_{i,j} \not=0$.
By the previous lemma, the iterated integral $J_\xi = \sum J_{\omega_i,\omega_j}$
is homotopy invariant
\begin{lemma}
\label{lemma:homotopyinv-bis}
Suppose that $\xi$ is an integral Hodge class on $X\times X$ as
above. Using \eqref{hiii}, identify $J_{\xi}$ with a homomorphism of abelian groups $I/I^3\to \mathbf C$.
Then the restriction of $J_{\xi}$
to $I^2/I^3$ is $\mathbf Z$-valued and agrees   with $\xi$
viewed as an element of
\begin{eqnarray*}
H^1(X^\mathrm{an},\Z)\otimes H^1(X^\mathrm{an},\Z) & \cong & (H_1(X^\mathrm{an},\Z)\otimes H_1(X^\mathrm{an},\Z))^\vee \\
 &=&  (I/I^2 \otimes I/I^2)^\vee =  (I^2/I^3)^\vee.
 \end{eqnarray*}
(Here $A^\vee$ denotes $\operatorname{Hom}(A,\Z)$, for any abelian group $A$.)
 \end{lemma}
 \begin{proof}
See the discussion at the beginning of \S 2 of \cite{DRS}, and  \textit{loc. cit.}, Lemma 1.1(2).
\end{proof}
By Lemma \ref{lemma:homotopyinv-bis}, the map $J_\xi$ 
 induces a homomorphism
\[J_\xi:H_1(X^\mathrm{an},\mathbf Z) = I/I^2 \to \mathbf C/\mathbf Z.\]

The following observation, which is extended in Theorem 
\ref{iterformula} below to the entire Jacobian of $X$, is key in our
approach to calculating Chow-Heegner points. 
Fix any holomorphic 1-form $\rho \in  H^{1,0}(X_{\mathbf C})\subset H^1(X^\mathrm{an},\mathbf C)$
 corresponding to an {\em elliptic 
curve}  factor $E$ of the Jacobian of $X$, and 
denote by $\Lambda$ the period lattice
 $$\Lambda := \left\{ \int_\gamma \rho,  \quad \gamma\in H_1(X^\mathrm{an},\Z)\right\}\subset \mathbf C$$
 attached to $\rho$. 
  The class $\gamma_\rho \in H_1(X^\mathrm{an},\mathbf C)$ that is Poincar\'e dual
to $\rho$ actually belongs to $H_1(X^\mathrm{an},\mathbf Z)\otimes \Lambda$.
Consequently $J_\xi(\gamma_\rho)$ can be viewed as  a well-defined element of $\mathbf C/\Lambda$, and hence
of $E(\C)$. 


\subsection{Chow-Heegner points}
\label{describe1}
Let $X_1,X_2$ denote copies of $X$, and $X_{12}$ the diagonal copy of $X$
in $X_1\times X_2$.  We  fix a choice of base
point $\infty\in X$ and use it to 
identify $X_1$ and $X_2$ with the curves $X_1\times\{\infty\}$ and $\{\infty\}\times X_2$ respectively in 
$X_1\times X_2$. 
 To a codimension-one algebraic cycle
$Z\subset X\times X= X_1\times X_2$ (defined over $\mathbf Q$) we associate the divisor class
$$D_Z= (Z\cap X_{12})-(Z\cap X_1) - (Z\cap X_2).$$
Define a corresponding degree-zero divisor class
\[P_Z = D_Z - \deg(D_Z)o\in \operatorname{Pic}^0(X).\]
(Recall that $o\in X(\mathbf Q)$ is a fixed base point  which
 is different from the previously chosen base point
$\infty$.)

We now state the iterated integral formula from \cite{DRS}
for the image of $P_Z$ under the Abel-Jacobi map
\[\mathrm{AJ}_X: \operatorname{Pic}^0(X) \to  \Omega^1(X^{\mathrm{an}})^\vee/H_1(X^{\mathrm{an}},\mathbf Z).\]
Let $\epsilon_o$ be the projector on $\operatorname{Pic}(X\times X)$
defined by
\[\epsilon_o(Z) = Z - i_{1*}\pi_{1*}-i_{2*}\pi_{2*}\]
where $\pi_{1},\pi_2:X\times X\rightrightarrows X$ are the projections
and $i_1,i_2:X\rightrightarrows  X\times X$ are the inclusions of ``vertical and horizontal'' copies of $X$ over the basepoint $o$.

Let $$\mathrm{cl}(\epsilon_o-):\operatorname{Pic}(X\times X)\to  H^1(X^{\mathrm{an}},\mathbf Z)\otimes H^1(X^{\mathrm{an}}, \mathbf Z)$$
denote the composition of the cycle class map and the projector $\epsilon_o$.
(The effect of $\epsilon_0$ is to annihilate the $H^2\otimes H^0$ and $H^0\otimes H^2$
factors in the K\"unneth decomposition of the second Betti cohomology group of $X\times X$.)
Suppose $\mathrm{cl}(\epsilon_o Z)$ is represented by $\sum_{i,j} c_{i,j} \omega_i\otimes \omega_j$,
where $\omega_i$ or $\omega_j$ is holomorphic at $\infty$ whenever $c_{i,j} \not=0$. 
\begin{theorem}
[\cite{DRS}, Corollary 3.6]
\label{iterformula}
The image $\mathrm{AJ}_X(P_Z)\in 
\Omega^1(X^{\mathrm{an}})^\vee/H_1(X^{\mathrm{an}},\mathbf Z)$
is represented by the linear functional
that maps  $\rho\in \Omega^1(X^{\mathrm{an}})$
to 
\[  \sum_{i,j} c_{i,j} \int_{\gamma_\rho} (\omega_i\cdot\omega_j - \alpha_{\omega_i,\omega_j})+\deg(D_Z) \int_o^\infty \rho  \in \mathbf C,\]
where
$\gamma_\rho\in H_1(X^{\mathrm{an}},\mathbf C)$ is Poincar\'e dual
to $\rho\in H^{1,0}(X^{\mathrm{an}})\subset 
H^1_{\mathrm{dR}}(X^{\mathrm{an}},\mathbf C)$.
\qed
\end{theorem}


\section{Chow-Heegner points on modular curves}
\label{chpt}
We now specialize the discussion of the preceding section to the case of classical 
modular curves $X$. We shall define certain rational points on an arbitrary 
elliptic curve $E_{/\mathbf Q}$ called \textit{Chow-Heegner points}, 
such that the corresponding points of $E(\mathbf C)\cong \mathbf C/\Lambda_E$ 
can be computed using iterated path integrals via Theorem \ref{iterformula}.

\subsection{Definitions}
\label{describe}
Let $N>1$ be an integer and $X=X_0(N)$ denote the canonical model over $\mathbf Q$ of the classical modular curve of level $N$; write $J_0(N)$ for the Jacobian of $X_0(N)$.
With this choice of $X$ we place ourselves in the setup of \S  1,
taking the ground field $F$ to be $\mathbf Q$
and the point $\infty\in X(\mathbf Q)$ to be the usual cusp at infinity.
Thus $Y:= X_0(N)-\{\infty\}$. (Note that $Y$ differs from the usual open modular curve $Y_0(N)$.) 
For the moment we do not specify the basepoint $o\in Y^{\mathrm{an}}$ used for topological constructions.

We write $S_2(\Gamma_0(N))$ for the space of cuspidal weight 2 modular forms for $\Gamma_0(N)$.
This space is canonically isomorphic to the vector space $\Omega^1(X^\mathrm{an})$ of holomorphic 1-forms
on $X$, via the map which associates to a modular form $f:\mathfrak H^*\to \mathbf C$
the 1-form $\omega_f = 2\pi i f(z)\operatorname{d}z$ on $X$.
We recall that if $f$ has Fourier expansion $\sum_{n\geq 1} a_n q^n$
then a Laurent expansion for $\omega_f$ near $\infty$ is given by $\sum_{n\geq 1} a_n q^n \frac{\operatorname{d}q}{q}$.


Let $E$ be an elliptic curve over $\mathbf Q$ of conductor $N_E$ dividing $N$. Its isogeny class corresponds to a newform $f\in S_2(\Gamma_0(N_E))$ with rational Fourier coefficients, which gives rise to a \textit{modular parametrization} $\pi_E: J_0(N_E) \to E$, a morphism of abelian varieties defined over $\mathbf Q$.

For the ease of exposition, we will assume throughout that $\ker \pi_E$ is connected. In this case, assuming the Manin constant $c=1$, the N\'eron lattice of $E$ coincides with the period lattice $\Lambda_f$ of the differential $\omega_f = 2\pi if(z)\mathrm{d}z\in \Omega^1(X_0(N_E)^{\mathrm{an}})$ corresponding to $f$.

The map $\pi_E$ can be computed on complex points explicitly, 
using the Abel-Jacobi  isomorphism 
$$\mathrm{AJ}_X:J_0(N_E)(\mathbf C)\cong \Omega^1(X_0(N_E)^{\mathrm{an}})^\vee/H_1(X_0(N_E)^{\mathrm{an}},\mathbf Z),$$
 the Weierstrass uniformization $W:\mathbf C/\Lambda_f\cong E(\mathbf C)$, and
  the analytic parametrization
 $$\pi_E^\mathrm{an}:\Omega^1(X_0(N_E)^\mathrm{an})^\vee/H_1(X_0(N_E)^\mathrm{an},\mathbf Z) \to \mathbf C/\Lambda_f.$$
The map $\pi_E^\mathrm{an}$ sends the coset of a functional 
on $\Omega^1(X_0(N_E)^\mathrm{an})$ to the evaluation of that functional at $\omega_f$.
Thus for $P_{\mathbf C}\in J_0(N_E)(\mathbf C)$ we have
\begin{equation}
\label{modularparam}
\pi_{E,\mathbf C}(P_{\mathbf C}) =
 W(\pi_E^\mathrm{an}(\mathrm{AJ}_{X_0(N_E)}(P_{\mathbf C}))) = 
 W(\mathrm{AJ}_{X_0(N_E)}(P_{\mathbf C})(\omega_f)).
 \end{equation}

For each divisor $d$ of $N/N_E$ there is a degeneracy map $\pi_d: X_0(N) \to X_0(N_E)$ that induces by the Albanese universal property a morphism of abelian varieties $\pi_d: J_0(N) \to J_0(N_E)$, which we denote with the same symbol. Put $\pi^d_{E}=\pi_E \cdot \pi_d: J_0(N) \to E$ for the composition of $\pi_d$ with $\pi_E$. When $d=1$, the map $\pi_1$ is the projection arising from the natural inclusion $\Gamma_0(N)\subseteq \Gamma_0(N_E)$ and we shall simply write $\pi^1_{E}=\pi_E$; it will be clear from the context whether the source of this map is $J_0(N)$ or $J_0(N_E)$. 

Let $\mathbf T = \mathbf Q[\ldots,T_n,\ldots]$ be the Hecke algebra of level $N$, and let $\mathbf T_0$ denote the subalgebra generated by Hecke operators prime to the level $N$. Then 
$$
\mathbf T_0  \simeq \prod_{h} K_{h} \subseteq \mathbf T  \simeq \prod_{h} L_{h} 
$$ 
where $h$ runs over Galois conjugacy classes of newforms of all
 levels $M$ dividing $N$, 
 $$K_{h} = \mathbf Q(\{a_n(h)\}_{n\geq 1})$$
  is the number field generated by the Hecke eigenvalues of $h$, and, if 
  $N_h\mid N$ stands for the level of $h$, the ring  $L_h$ is a commutative Artinian 
  $K_h$-algebra of dimension
  $[L_h:K_h] =\sigma(N/N_h)$, the number of factors of $N/N_h$ (cf.\,\cite{Ka}, \cite[Ch.\,14]{RiSt}). 
  Let $\End(\Jac(X))$ denote the ring of endomorphisms of $\Jac(X)$ which are defined over $\Q$. 
  The endomorphism algebra $\End^0(\Jac(X)):=\mathbf Q\otimes \End (\Jac(X))$  
  contains the Hecke algebra $\mathbf T_0$ as its center and $\mathbf T$ 
  as a maximal commutative subalgebra (and $\End^0(\Jac(X))\simeq \mathbf T \simeq \mathbf T_0$ 
  when $N$ is prime), as 
\begin{equation*}
\End^0(\Jac(X)) = \langle \mathbf T, \{ \delta_d\}_{d\mid \sigma(N/N_h)}\rangle\simeq \prod_h M_{\sigma(N/N_h)}(K_h),
\end{equation*}
where $\delta_d$ are the degeneracy operators as defined e.g. in \cite{Ka}.

\begin{definition}
For a divisor $M$ of $N$ and a newform $g \in S_2(\Gamma_0(M))$, denote by 
$$
T_g \in \mathbf T_0  \cong \prod_{h} K_{h}
$$
the idempotent with 1 in the $K_g$ component and 0 elsewhere. 

We shall also view $T_g$ as an idempotent of $\mathbf T $ and of $\End^0(\Jac(X))$ by the natural inclusion, so that $\mathbf T[g]:=T_g \cdot \mathbf T = L_g$ and $\End^0(\Jac(X))[g]:=T_g\cdot \End^0(\Jac(X)) = M_{\sigma(N/M)}(K_g)$. 


\end{definition}

There are natural isomorphisms 
\begin{equation}\label{corr-iso}
\End(\Jac(X))\otimes \mathbf Q \simeq \mathrm{Corr}(X):= \Pic(X\times X)\otimes \mathbf Q /( \pi_1^*\Pic(X)\otimes \mathbf Q \oplus \pi_2^* \Pic(X)\otimes \mathbf Q ).
\end{equation}

Under these identifications, an endomorphism $T\in \End^0(\Jac(X))$ is associated to a divisor class in $\operatorname{Pic}(X\times X) \otimes\mathbf Q$, which we also denote by the same symbol, and is well-defined only up to horizontal and vertical divisors.


In \S \ref{describe1} a point $P_T\in \operatorname{Pic}^0(X) = J_0(N)(\mathbf Q)$ was
attached to a divisor $T$ on $X\times X$; this point only depends on the class of $T$ in $ \Pic(X\times X)\otimes \mathbf Q /( \pi_1^*\Pic(X)\otimes \mathbf Q \oplus \pi_2^* \Pic(X)\otimes \mathbf Q )$.
This construction gives rise to a map
\begin{eqnarray*}
\operatorname{Pic}(X\times X)\otimes \mathbf Q&\to& J_0(N)(\mathbf Q)\otimes \mathbf Q\\
T&\mapsto& P_T.\end{eqnarray*}

Note that both these divisor classes and the projector $\epsilon_o$ depend on the basepoint $o$,
so we must now specify a particular choice of $o$.
In the following definition, \textit{we fix the choice of basepoint $o$
to be the cusp $0\in X$}, which is distinct from $\infty$ because $N>1$.

\begin{definition}\label{gf}
For an operator $T\in \End^0(\Jac(X))[g]$ and a divisor $d$ of $N/N_E$, define the Chow-Heegner point
$$
P_{T,f^{(d)}} = \pi^d_E(P_T) \in E(\mathbf Q)\otimes \mathbf Q
$$

When we take $d=1$ we shall just write $P_{T,f}$ for $P_{T,f^{(d)}}$; in addition, the Chow-Heegner point associated with $T=T_g$ and $d=1$ shall be denoted $P_{g,f}$.

For any positive integer $n$, define $T_{g,n} := T_g \cdot T_n \in \mathbf T $ as the product of $T_g$ and $T_n$, and define 
$$
P_{g,f,n} := P_{T_{g,n},f}.
$$
\end{definition}

Note that $P_{g,f}=P_{g,f,1}$ and that $P_{g,f,n}$ is a multiplie of $P_{g,f}$ if the 
eigenvalue $a_n(g)$ of $T_n$ acting on the $g$-isotypic component 
is rational. Thus, the points $P_{g,f,n}$ for $n>1$ are only of interest
 when $g$ has irrational Fourier coefficients or is an oldform.
For a general discussion of the notion of Chow-Heegner point, of 
which the construction above is a special case, see the introduction 
of \cite{BDP2} and \cite{DRS}. 


\begin{remarkwr}
\label{pointrelations}
To avoid the tensor products with $\mathbf Q$ in the definition
 of the above Chow-Heegner points $P_{T,f^{(d)}}$,
one must work with an actual algebraic cycle on $X\times X$, 
as opposed to $T$ which is merely a $\mathbf Q$-linear combination of such cycles.
Define the ``denominator'' $d_T$ of $T\in \mathbf T$ to be the smallest positive integer such that $d_{T} T$ lies in the integral Hecke algebra $\mathbf T_{\mathbf Z} := \mathbf Z[\ldots,T_n,\ldots]$. Then $d_T P_{T,f^{(d)}}$ belongs to $E(\mathbf Q)$; see also \S \ref{denom}.

However, we regard $d_{g,n} P_{g,f,n}$ as well-defined only modulo the torsion subgroup $E(\mathbf Q)^{\mathrm{tor}}$.
One  reason for this is that to obtain a computable formula for $P_{g,f,n}$
one can take the basepoint $o$ to be \textit{any} cusp other than $\infty$. There being no distinguished choice, 
one is left with an ambiguity valued in the image in $E(\mathbf Q)$ of the (torsion) cuspidal subgroup of $J_0(N)$;
see \eqref{pgf} below and the subsequent remarks.

\end{remarkwr}


\begin{theorem}[\cite{DRS}, Corollary 1.4]\label{DRS1.4}
Assume that the local signs $\varepsilon_p(g,g,f)$ of Garrett's triple product L-function $L(g,g,f,s)$ are $+1$ at the primes $p\mid N$.
Then the module of points 
$$
\underline{P}_{g,f} := \langle P_{T,f^{(d)}} : T\in \End^0(\Jac(X))[g], d\mid \tfrac{N}{N_E}\rangle \subseteq E(\mathbf Q)\otimes \mathbf Q
$$ 
is nonzero
(equivalently, there is a point $\sum_{d\mid N/N_E} n_d \pi^d_E(d_T P_{T})\in E(\mathbf Q)$ which is non-torsion, for some $T$ and integers $n_d$) if and only if the following conditions hold:
\begin{itemize}
\item[i.] $L(f,1)=0$,
\item[ii.] $L'(f,1)\neq 0$, and
\item[iii.] $L(f\otimes \operatorname{Sym}^2({}^{\sigma}g),2)\neq 0$ for all $\sigma \in \Gal(\bar{\mathbf Q}/\mathbf Q)$.
\end{itemize}
\end{theorem}


\subsection{Algorithms}

The aim of this section is to explain how to explicitly compute the slightly smaller submodule
$$
\underline{P}_{\mathbf T[g],f} := \{ P_{T,f} : T\in \mathbf T[g] \} = \langle P_{g,f,n}: n\geq 1\rangle \subseteq  \underline{P}_{g,f}
$$
of the Mordell-Weil group $E(\mathbf Q)\otimes \mathbf Q$ of $E$. Although the methods generalize to the computation of the full module of points $\underline{P}_{g,f}$, the computation of $\underline{P}_{\mathbf T[g],f}$ is simpler, and often sufficient for the purpose of finding a non-torsion point when one exists by Theorem \ref{DRS1.4}. Indeed, the authors have not come across an example where $\underline{P}_{\mathbf T[g],f}$ is torsion but the a priori larger $\underline{P}_{g,f}$ is non-torsion.
 The interested reader may consult \cite{Ka} for details relevant to the 
 eventual   adaptations of our methods to  calculating  $\underline{P}_{g,f}$ in its entirety.

Theorem \ref{iterformula} gives rise to an explicit formula for a Chow-Heegner point $P_{g,f,n}$ in terms of iterated integrals. This formula is stated in terms of  the components of the cohomology class $\mathrm{cl}(\epsilon_o T_{g,n})\in H^1_{\dR}(X/\mathbf Q)^{\otimes2}$ expressed as a sum of pure tensors. In this section we find an explicit formula for $\mathrm{cl}(\epsilon_o T_{g,n})$.

The action of the Hecke algebra $\mathbf T_0$ on modular forms extends to an action on the de Rham cohomology of $X$. Under this action, we have
$$H^1_{\mathrm{dR}}(X/\mathbf Q) \cong H^1_{\mathrm{dR}}(X/\mathbf Q)[g_1] \oplus \cdots \oplus H^1_{\mathrm{dR}}(X/\mathbf Q)[g_n],$$
indexed by Galois conjugacy classes of newforms of all levels $M$ dividing $N$. Let $g\in S_2(\Gamma_0(M))$ be a newform and suppose $\{\omega_{g,1},\ldots,\omega_{g,2k}\}$ is a collection of differentials of the second kind on $X$ representing a basis for $H^1_{\dR}(X/\mathbf Q)[g]$. Write $T_n\omega_{g,i} = \sum_j a_{ij}^n\omega_{g,j}$, and denote by $A_n$ and $B$ the matrices $(a_{ij}^n)_{1\leq i,j\leq 2k}$ and $(\langle \omega_{g,i}, \omega_{g,j} \rangle)_{1\leq i,j\leq 2k}$, respectively, where $\langle~,~\rangle$ denotes the Poincar\'e pairing. Then we have the following:


\begin{lemma}\label{hodgeclass}
$\mathrm{cl}(\epsilon_o T_{g,n}) = \sum_{i,j} c_{ij}^n \omega_{g,i} \otimes \omega_{g,j}$, where $(c_{ij}^n)_{1\leq i,j\leq 2k} = -B^{-1}A_n$.
\end{lemma}
\begin{proof}
The projector $\epsilon_o$ acts on $H^2_{\dR}(X \times X)$ by annihilating
 the $H^0_{\dR}(X) \otimes H^2_{\dR}(X)$ and $H^2_{\dR}(X) \otimes H^0_{\dR}(X)$ 
 components of the K\"unneth decomposition, so we have 
 $$\mathrm{cl}(\epsilon_o T_{g,n}) \in H^1_{\dR}(X) \otimes H^1_{\dR}(X).$$
  Note from the definition that $T_{g,n}$ acts on $H^1_{\dR}(X)[h]$ as $T_n$ if $h=g$ and 0 otherwise, so $\mathrm{cl}(\epsilon_o T_{g,n})$ is equal to the image of $T_n$ under the identification:
$$\End(H^1_{\dR}(X)[g]) \simeq H^1_{\dR}(X)[g]^{\vee}\otimes H^1_{\dR}(X)[g] \simeq H^1_{\dR}(X)[g] \otimes H^1_{\dR}(X)[g]$$
The first map is the canonical isomorphism of finite dimensional vector spaces, and the second is induced from the inverse of the identification $H^1_{\dR}(X)[g] \simeq H^1_{\dR}(X)[g]^{\vee}$ via the map $v \mapsto (w \mapsto \langle v,w\rangle)$. The remainder of the proof is a straightforward exercise in linear algebra, and is left to the reader.
\end{proof}

Combining the previous results, we obtain the following formula for $P_{g,f,n}$. Let $\gamma_f$ be the Poincar\'e dual of $\omega_f$
and let $\omega_{g,1},\ldots, \omega_{g,2k}$
be differentials of the second kind that give rise to a symplectic basis
 for the $g$-isotypic $\mathbf Q$-subspace 
$H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$. That is, for $1 \leq i \leq k$
 we have $\langle \omega_{g,i}, \omega_{g,i+k} \rangle = 1$ and $\langle \omega_{g,i},\omega_{g,j}\rangle = 0$
  for $j\not= i+k$. Assume moreover that this basis is adapted to the Hodge
 filtration, in the sense that the 1-forms $\omega_{g,1},\ldots,\omega_{g,k}$ 
 are  regular at $\infty$. 
 Since $H^0(X,\Omega^1(X/\mathbf Q))[g]\subset H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$ 
 is a maximal isotropic subspace with respect to the Poincar\'e pairing, we can simply
  take any basis $\omega_{g,1},\ldots, \omega_{g,k}$ for 
  $H^0(X,\Omega^1(X/\mathbf Q))[g] = S_2(\Gamma_0(N))[g]$ and
   extend it to a symplectic basis of the sort desired. Write
\begin{equation}\label{Tgn}
\mathrm{cl}(\epsilon_o T_{g,n}) = \sum_{i,j}^{2k} c^n_{ij} \omega_{g,i} \otimes \omega_{g,j}
\end{equation}
where the coefficients can be computed as in Lemma \ref{hodgeclass}. 
Since we chose the basis to be symplectic, the matrix $B$ is the standard symplectic matrix.
 Additionally, because $T_n$ preserves the holomorphic subspace of $H^1_{\dR}(X/\mathbf Q)$, 
 then the computation of $-B^{-1}A_n$ shows that $c^n_{ij}=0$ when $k+1 \leq i,j \leq 2k$. 
 Thus, the Hodge class in \eqref{Tgn} is of the form discussed after Remark \ref{alpha=0}.

Combining \eqref{Tgn} with \eqref{modularparam} and Theorem \ref{iterformula}, we obtain the following formula for the point $P_{g,f,n}$:
\begin{equation}\label{pgf}
P_{g,f,n} = W\left(\sum_{i,j}^{2k} c^n_{ij}\left(\int_{\gamma_f} \omega_{g,i}\cdot \omega_{g,j} - \alpha_{\omega_{g,i},\omega_{g,j}}\right)\right).
\end{equation}

Note that \eqref{pgf} omits the term $\deg(D_{T_{g,n}})\int_o^\infty \omega_f$
from the formula in Theorem \ref{iterformula}. This is justified by our choice of basepoint $o$,
for it is well-known (cf.\,e.g.\,\cite{Man}) that the difference $[\infty]-[0]$ is a torsion point of $J_0(N)(\mathbf Q)$.
Since $P_{g,f,n}$ is defined as an element of $E(\mathbf Q)\otimes \mathbf Q$, torsion can be disregarded.

We emphasize that by Lemmas \ref{homotopyinv}  and \ref{lemma:homotopyinv-bis}, 
the right-hand side of \eqref{pgf} depends only on the \textit{homology} class
$$\gamma_f\in  H_1(Y^\mathrm{an}, \mathbf Z) = H_1(X^\mathrm{an},\mathbf Z) $$ 
Poincar\'e dual to $\omega_f$.
It can therefore be evaluated by lifting $\gamma_f$ arbitrarily to an element $\tilde \gamma_f\in \pi_1(Y^\mathrm{an};o)$
and evaluating the length $\leq 2$ homotopy invariant iterated integral \eqref{pgf} on any loop in the homotopy class $\tilde\gamma_f$.

\begin{remarkwr}
\label{Tg} 
When $n=1$, formula \eqref{pgf} greatly simplifies. In the notation of Lemma \ref{hodgeclass}, $A_1$ is the identity matrix and $B$ is the standard symplectic matrix by our choice of basis, and so if we write $\eta_{g,i}$ for $\omega_{g,i+k}$, then
$$P_{g,f} = W\left(\sum_{i=1}^k \left(\int_{\gamma_f} \omega_{g,i} \cdot \eta_{g,i} - \eta_{g,i} \cdot \omega_{g,i} - 2\alpha_{\omega_{g,i},\eta_{g,i}}\right)\right)$$
\end{remarkwr}
%  \begin{example} 
% When $g$ has \textit{rational} Fourier coefficients the $g$-isotypic component is 2-dimensional, and the above formula reads
% $$
% P_{g,f} := \int_{\gamma_f} (\omega_g\cdot \eta_g - \eta_g \cdot \omega_g) -2\int_{\gamma_f} \alpha_{\omega_g,\eta_g} =$$ $$=2(\int_{\gamma_f} \omega_g\cdot \eta_g -\int_{\gamma_f}\alpha_{\omega_g,\eta_g})-\int_{\gamma_f} (\omega_g\cdot \eta_g + \eta_g \cdot \omega_g) \in \C/\Lambda_f \sim E(\C).
% $$
% \end{example}


Although $P_{g,f,n}$ is defined relative to the choice of base point $o$ as the cusp $0$ on $Y$,
it will be convenient to \textit{compute} it relative to a different choice of basepoint.
\begin{lemma}\label{basepointindependence}
The right hand side of \eqref{pgf} is independent of the choice of $o\in Y(\mathbf Q)$.
\end{lemma}
% FIXME: Should this lemma be in the preceeding section? Is the theorem from DRS valid (i.e. is the RHS of the formula for $P^{\mathrm{an}}_{Z,\omega}$ even well-defined) when $\mathrm{cl}(Z)$ is not orthogonal to $\omega$?
\begin{proof}
Changing the basepoint from $o$ to $o'$ amounts to conjugating the representative path $\gamma_f$ for the homology class Poincar\'e dual to $\omega_f$
by a path $\beta$ from $o$ to $o'$. This manifestly does not affect
the value of the integral of the meromorphic 1-form $\alpha_{\omega_{g,i}, \omega_{g,j}}$. Thus the issue is whether we have an identity
\begin{equation}\label{basepoint}
  \int_{\gamma_f} \omega_{g,i} \cdot \omega_{g,j} \stackrel{?}{=}\int_{\beta\gamma_f\beta^{-1}} \omega_{g,i}\cdot \omega_{g,j}.
\end{equation}
But by \cite[Exer. 8]{Hain}, for any 1-forms $\omega,\eta$, loop $\gamma$, and path $\beta$,  we have
\begin{equation}\label{cobp}
\int_{\beta\gamma\beta^{-1}} \omega\cdot\eta = \int_\gamma \omega\cdot \eta + \left|\begin{smallmatrix}\int_\gamma \omega&\int_\gamma \eta\\
\int_\beta \omega&\int_\beta \eta\end{smallmatrix}\right|.  
\end{equation}
In our situation, the determinants expressing the difference between the two sides of \eqref{basepoint} vanish. Indeed, $\int_{\gamma_f} \omega_{g,i} = \langle \omega_f, \omega_{g,i}\rangle = 0$,
since the decomposition into isotypic components for the action of the Hecke algebra is orthogonal with respect to the Poincar\'e pairing.
\end{proof}

% However, the point $P_{g,f}$ {\em does depend} on the choice of the cusp $\infty$ we chose in order to define the open curve $Y\subset X= X_0(N)$. (Note that in general $X_0(N)$ has several cusps.)
% We should keep this in mind, and if necessary we may actually write it as $P_{\infty, g;f}$. 

\section{Details of the algorithm}
\label{alg}
We now turn to the question of numerically evaluating formula \eqref{pgf} for a Chow-Heegner point $P_{g,f,n}\in E(\mathbf Q)\otimes\mathbf Q$ for an  elliptic curve $E = E_f$. We retain all the notation from \S \S \ref{prelim}--\ref{chpt}.
% \begin{remarkwr}\label{}
% Some of the previous assumptions could be relaxed: there are rational points $P_{g_1;g_2;f}\in J_0(N)(\mathbf Q)$ associated to any 3-tuple of modular forms
% (whose respective weights satisfy a certain combinatorial condition).
% Relaxing the hypothesis that $f$ has {\em rational} fourier coefficients 
% amounts to investigating the images of these rational points in an abelian variety quotient $A_f$ of $J_0(N)$ which has dimension $>1$.
%  Relaxing the  assumption that $E$ has rank 1 is only of moderate interest: 
% by \cite{DRS1}, when $E$ has (analytic?) rank $\neq 1$, the resulting points $P_{g,f}$ are all torsion.  
%  The hypothesis on the sign of the functional equation of $g$ is made so that the sign of the functional equation of $L(\mathrm{Sym}^2(g)\otimes f,s)$ is $-1$ and thus $L'(\mathrm{Sym}^2(g)\otimes f,2)$ can be (and is expected often to be) non-zero.
% \end{remarkwr}
The following ingredients occur in \eqref{pgf}:
\begin{itemize}
\item[1.] The Poincar\'e dual $\gamma_f\in H_1(X,\mathbf C)$
of $\omega_f\in H^1_{\dR}(X^{\mathrm{an}},\mathbf  C)$. 
\item[2.] A collection of rational differentials of the second kind $\omega_{g,1},\ldots, \omega_{g,2k}$ on $X$, regular away from $\infty$, whose images in $H^1_{\dR}(X/\mathbf Q)$ are a symplectic basis for the $g$-isotypic component $H^1_{\dR}(X/\mathbf Q)[g]$.
This basis should be adapated to the Hodge filtration on $H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$
in the sense that $\omega_{g,1},\ldots, \omega_{g, k}$ are holomorphic everywhere on $X$, including the point $\infty$.
\item[3.] The coefficients $c^n_{ij}$ arising from the action of $T_n$ on $H^1_{\dR}(X)$ with respect to the basis $\omega_{g,1},\ldots,\omega_{g,2k}$.
\item[4.] Meromorphic differentials $\alpha_{\omega_{g,i},\omega_{g,j}}$ on $X$, regular on $Y$, such that $\omega_{g,i} F_{\eta_{g,j}}-\alpha_{\omega_{g,i},\eta_{g,j}}$ has at worst a logarithmic pole at (any point lying over) $\infty$ for $1 \leq i \leq k$.
\end{itemize}
These data must be ``known'' in a sufficiently concrete form
to evaluate the iterated integrals occuring in \eqref{pgf}.
It is also desirable to know 
\begin{itemize}
\item[5.] the denominator $d_{g,n}$ of the projector
onto the $g$-isotypic component of the cohomology of $X$.
\end{itemize}
This last item will allow for the computation of a point in $E(\mathbf Q)$, as opposed to one in $E(\mathbf Q) \otimes \mathbf Q$. This section is devoted to methods of computing these five ingredients.

\subsection{Evaluating iterated integrals}\label{integration}
Let $J = \int \omega\cdot \eta -\alpha_{\omega,\eta}$ be a homotopy-invariant iterated integral of length $\leq 2$ on $Y$, expressed in terms of differentials of the second kind on $X$, regular on $Y$. We seek to compute the right-hand side of formula \eqref{pgf}, which is a $\mathbf{Q}$-linear combination of $J(\gamma)$ for various choices of $J$ and (the homotopy class of) a path $\gamma\in \pi_1(Y;o)$. As remarked earlier, \eqref{pgf} actually depends only on the homology class $\gamma_0$ of $\gamma$.
This homology class belongs to  $H_1(Y^\mathrm{an},\mathbf Z) = H_1(X^{\mathrm{an}},\mathbf Z)$, which
is the abelianization of the quotient $\pi_1(X^{\mathrm{an}},o)=\bar \Gamma_0(N)$
of $\Gamma_0(N)$ by the smallest normal subgroup containing the elliptic and parabolic elements. 

To evaluate $J(\gamma_0)$ for $\gamma_0\in H_1(Y^{\mathrm{an}},\mathbf Z)$, choose the basepoint $o$ away from the set $S$ of elliptic points on $Y_0(N)^{\mathrm{an}}\subset Y^{\mathrm{an}}$ and lift $\gamma_0$ arbitrarily to a path $\tilde \gamma \in \pi_1(Y_0(N)^{\mathrm{an}}\setminus S,o)$. For each elliptic point $x\in S$, let $e_x=|\mathrm{Stab}_{\Gamma_0(N)}(x)/\{\pm 1\}|$ denote the index of $x$ (which is either $2$ or $3$) and $\gamma_x$ be a sufficiently small counterclockwise loop around $x$. Writing $H$ for the normal subgroup of $\pi_1(Y_0(N)^{\mathrm{an}}\setminus S,o)$ generated by $\{\gamma_x^{e_x}, x \in S\}$, there is a natural isomorphism $\Gamma_0(N) \simeq \pi_1(Y_0(N)^{\mathrm{an}}-S;o)/H$. 

We may regard then $\tilde \gamma$ as an element of $\Gamma_0(N)$; this causes no ambiguity because $H$ lies in the kernel of the natural projection $H_1(Y_0(N)^{\mathrm{an}}-S,\mathbf Z)\ra H_1(Y_0(N)^{\mathrm{an}},\mathbf Z) \ra H_1(Y^{\mathrm{an}},\mathbf Z)$. The path $\tilde \gamma$ can then also be viewed as a path in $\mathfrak H$ from $\tau_0$ to $\gamma\tau_0$, where $\gamma\in \Gamma_0(N)$ is a lift
of $\gamma_0$. 

\begin{lemma}\label{l1}
Suppose $\gamma_0$ is Poincar\'e-dual to $\rho$.
As an element of $\mathbf C/\Lambda_\rho$, we have
  $$J(\gamma_0) = \int_{\tau_0}^{\gamma \tau_0} \omega F_{\eta} - \alpha_{\omega,\eta}$$
where we conflate 1-forms on $X$ with their pullbacks to $\mathfrak H^* = \mathfrak H\cup\{\infty\}$.
Moreover, $F_{\eta}$ has Laurent expansion about $\infty\in \mathfrak h^*$
given by formally integrating the Laurent expansion of $\eta$
about the cusp $\infty\in X$.
\end{lemma}
\begin{proof}
This follows from the preceding discussion, using the definition of iterated integrals and the homotopy invariance of $J$.
\end{proof}

Given any differential form $\lambda$ of the second kind on $X$, and any $\gamma\in\Gamma_0(N)$, let
$$ I(\lambda;\gamma) := \int_{\tau_0}^{\gamma\tau_0} \lambda.$$
(As above, in the righthand side of this expression $\lambda$ is conflated with its pullback to $\mathfrak H^*$.)
By the residue formula,
this expression is independent of the choice of path on the upper half-plane $\mathfrak H$  from $\tau_0$ to $\gamma\tau_0$. The $\Gamma_0(N)$-invariance of
$\lambda$ also shows that this expression is independent of the choice of  base point 
$\tau_0\in \mathfrak H$, which justifies suppressing $\tau_0$ from the notation. 

If $\lambda$ instead denotes a differential of the second kind on  $\tilde X$ then the integral above still makes sense
but depends on both the basepoint $o$ and the chosen lift of $o$ to $\tau_0\in \mathfrak H$. We will primarily be interested in evaluating such integrals in the context of \eqref{pgf}, for which the choice of basepoint is ultimately irrelevant. (This is because the Poincar\'e dual of the homology class of $\gamma$, is orthogonal
to the 1-forms in the iterated integral giving rise to the path integral we seek to evaluate; \textit{cf.} Lemma \ref{basepointindependence}.)
However, as we are about to see, for the purposes of algorithmic efficiency it is necessary to break up the path of integration into pieces that can be computed relatively quickly. The integrals over these pieces may no longer be basepoint-independent: when we express $\gamma$ as a product of computationally-amenable elements $\gamma^{(j)}\in \Gamma_0(N)$, the corresponding 
homology classes may no longer lie in (the Poincar\'e dual of) the orthogonal complement of $H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$.
Thus for a general meromorphic 1-form $\lambda$ on $\tilde X$ and a general $\gamma\in\Gamma_0(N)$, we adopt the notation
\[I_{\tau_0}(\lambda;\gamma) = \int_{\tau_0}^{\gamma\tau_0}\lambda\]
to emphase the dependence on the choice of basepoint.

By meromorphicity, for $\lambda$ as above (defined on either $X$ or $\tilde X$) the integral $I(\lambda;\gamma)$ or $I_{\tau_0}(\lambda;\gamma)$ can be computed by integrating term-wise a Laurent expansion for $\lambda$
using the fundamental theorem of calculus.
Thus, in practice, one computes the Laurent expansion for the primitive 
$F_\lambda$ about $\infty \in X$ (or a choice of $\tilde \infty\in \tilde X$ lying over $\infty$), regarded as function given by a convergent power series
in $q = e^{2\pi i \tau}$ on $\mathfrak h$, and evaluates it at $\tau_0$ and $\tau_0'=\gamma \tau_0$. The larger the imaginary parts of $\tau_0$ and $\tau'_0$ are, the faster this series converges and the fewer coefficients of the Laurent series of $\lambda$ are necessary to approximate $I(\lambda;\gamma)$ or $I_{\tau_0}(\lambda;\gamma)$ to a give degree of accuracy.
Writing $\gamma = \begin{smallmat} abcd\end{smallmat}$, it is well-known that the best compromise between $\mathrm{Im}(\tau_0)$ and $\mathrm{Im}(\tau'_0)$ is achieved when we choose $\tau_0=-\frac{d}{c} + \frac{1}{|c|} i$ (\textit{cf.}, for example, \cite[p. 35]{cremona}).
This {\em optimal basepoint for $\gamma$} will be denoted $\tau^*_\gamma$.

With this remark in mind, we take the following approach to computing $J(\gamma_0)$ as in the lemma above. First compute Laurent expansions for the differentials $\omega,\eta, \alpha_{\omega,\eta}$. Then find a ``good'' expression for the homology class $\gamma_0\in H_1(X^{\mathrm{an}},\mathbf C)$, writing it as a $\mathbf C$-linear combination of classes $\gamma_0^{(j)}\in H_1(X^{\mathrm{an}},\mathbf Z)$ that lift to elements $\gamma^{(j)}\in \Gamma_0(N)$ with small lower-left entries $cN$. Finally, calculate approximations to the integrals $I_{\tau_0}(\omega F_{\eta}; \gamma^{(j)})$ and $I(\alpha_{\omega,\eta};\gamma^{(j)})$. The appropriate linear combination of these integrals is an (approximate) representative for the coset $J(\gamma_0)\in \mathbf C/\Lambda_\rho$.)

To calculate $I(\alpha_{\omega,\eta};\gamma^{(j)})$, one is free to change the basepoint from $\tau_0$
to the optimal basepoint $\tau_j^*:=\tau_{\gamma^{(j)}}^*$ for $\gamma^{(j)}$, since $\alpha_{\omega,\eta}$ is defined on $X$
and not only on $\tilde X$. The same is \textit{not} true for $\omega F_{\eta}$.
To evaluate $I_{\tau_0}(\omega F_{\eta};\gamma^{(j)})$ we appeal to the following lemma.
\begin{lemma}\label{randomlemma}
$I_{\tau_0}(\omega F_{\eta};\gamma^{(j)}) = I_{\tau_j^*}(\omega F_{\eta}; \gamma^{(j)}) -
I(\eta;\gamma^{(j)}) \int_{\tau_0}^{\tau_j^*}\omega$.
\end{lemma}
Observe that every term on the righthand side can be computed using the fundamental theorem of calculus, evaluating power series only at the points $\tau_0$ and $\tau_j^*$. In particular, taking $\tau_0 = i/N$,
each such evaluation converges at least as fast as an evaluation at $\tau_j^*$, so this formula for the integral is ``optimally efficient''. 
\begin{proof}[Proof of Lemma \ref{randomlemma}]
Since $\lambda = \omega F_{\eta}$ is a holomorphic 1-form on $\mathfrak H$, its integral along a closed contour vanishes. Thus 
\[I_{\tau_0}(\lambda;\gamma^{(j)}) = I_{\tau_j^*}(\lambda;\gamma^{(j)}) + \int_{\tau_0}^{\tau_j^*}\lambda 
- \int_{\gamma^{(j)} \tau_0}^{\gamma^{(j)} \tau_j^*}\lambda.\]
To evaluate the second term on the righthand side, we observe
that $\omega$ comes from a 1-form on $X$, so it is $\Gamma_0(N)$-invariant; it thus pulls back to itself along the fractional linear transformation defined by
$\gamma^{(j)}$.
On the other hand,
\[I(\eta;\gamma^{(j)})=\int_{\gamma^{(j)}}\eta = F_{\eta}(\gamma^{(j)}\tau) - F_{\eta}(\tau),\qquad \text{for all }\tau\in \mathfrak H.\]
Hence $(\gamma^{(j)})^* F_{\eta} = F_{\eta} + I(\eta;\gamma^{(j)})$.
So $(\gamma^{(j)})^* \lambda = \lambda + I(\eta;\gamma^{(j)})\omega$, and we find
\[\int_{\gamma^{(j)} \tau_0}^{\gamma^{(j)} \tau_j^*}\lambda = \int_{\tau_0}^{\tau_j^*} (\gamma^{(j)})^*\lambda
= \int_{\tau_0}^{\tau_j^*}\lambda + I(\eta;\gamma^{(j)}) \int_{\tau_0}^{\tau_j^*}\omega,\]
which yields the lemma.
\end{proof}

\begin{remarkwr}\label{r1}
We warn the reader  that possibly $I_{\tau_0}(\omega F_{\eta};\gamma^{(j)}) \neq \int_{\gamma^{(j)}}\omega\cdot \eta$ (regarding $\gamma^{(j)}$ as an element of $\pi_1(Y^\mathrm{an};o)$).
Indeed, the iterated integral attached to
$\omega\cdot \eta$ need not even be homotopy invariant (!) 
so $\int_{\gamma^{(j)}}\omega\cdot \eta$ is ill-defined.
 In particular, one \textit{cannot}
relate $I_{\tau_0}(\omega F_{\eta};\gamma^{(j)})$ to $I_{\tau_j^*}(\omega F_{\eta};\gamma^{(j)})$
using the change-of-basepoint formula \eqref{cobp} for iterated integrals.
\end{remarkwr}

To efficiently evaluate the integrals in \eqref{pgf} using the method just explained, it is therefore necessary to know:
\begin{itemize}{}
\item[a.] Laurent expansions about $\infty$ for a symplectic basis $\omega_{g,1},\ldots,\omega_{g,2k}$
of $H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$ and the forms $\alpha_{\omega_{g,i},\omega_{g,j}}$; and,
\item[b.]  the homology class $\gamma_f$ as a $\mathbf C$-linear combinationation
of class $\gamma_0^{(j)}$ whose lifts to $\Gamma_0(N)$ have small lower-left entries $cN$.
\end{itemize}
In the rest of this section we turn to the task of computing these data.




\subsection{Calculating a symplectic basis for $H^1_{\dR}(X/\mathbf Q)[g]$}
\label{symplecticbasis}
The calculation of a basis for the de Rham cohomology can be carried out
by first writing down a modular function $u$ --- that is, a rational function on $X=X_0(N)$ --- 
which is regular away from $\infty$. Such a function exists by Riemann-Roch 
and a $q$-expansion for one such function can sometimes be computed 
explicitly using the Dedekind eta-function, as explained in the next subsection.

Using a modular symbol algorithm, one can compute $q$-expansions for a  basis
of $S_2(\Gamma_0(N))$ consisting of cusp forms with rational Fourier coefficients; 
\textit{cf.} for example \cite{Stn}. 
Write $\omega_1,\ldots, \omega_{t}$ for the corresponding holomorphic 1-forms on $X$,
where for convenience we denote by $t=\dim_{\mathbf C} S_2(\Gamma_0(N))$ the genus of $X$.


Define $\eta_i=u\omega_i$, which is a differential of the second kind by the residue theorem, and let $\mathcal B = \{ \omega_1,...,\omega_{t}, \eta_1,...,\eta_{t}\}\subset H^1_{\mathrm{dR}}(X/\mathbf Q)$ be the corresponding set of cohomology classes.  A simple application of Riemann-Roch shows the following.
\begin{lemma}\label{h1basis}
  The set $\mathcal B$ is basis for $H^1_{\mathrm{dR}}(X/\mathbf Q)$ whenever $\infty$ is not a Weierstrass point on $X$ and $u$ has a pole of order $t+1$ (i.e., the smallest possible) at $\infty$. 
\end{lemma}
\begin{proof}
Since $\infty$ is not a Weierstrass point on $X$, we may assume that $\ord_\infty (\omega_i) = i-1$, and thus $\ord_\infty (\eta_i) = i-t-2$. For any differential of the second kind $\omega'$, we can find a linear combination of $\eta_1,\ldots,\eta_t$ and $dh$ for an appropriate rational function $h$ having the same principal part as $\omega'$. Thus the difference is holomorphic, and lies in the span of $\{\omega_1,\ldots,\omega_t\}$.
\end{proof}
\begin{remarkwr}\label{oggremark}
By a result of Ogg \cite{Ogg}, the cusp $\infty$ is not a Weierstrass point when the level $N$ is prime, or more generally when $N = pM$ for prime $p$ and
an integer $M\geq 1$ such that $X_0(M)$ has genus zero and $p \nmid M$. 

When $\infty$ is a Weierstrass point, there is a rational function with a single pole at $\infty$ of order $\leq g(X)$. When $u$ is taken to be such a function, then the set $\mathcal B$ will never be a basis. Indeed, since $\infty$ is a Weierstrass point, there exists a holomorphic differential form $\omega$ with order of vanishing $\geq g(X)$ at $\infty$. Then $u\omega$ is still holomorphic, and thus lies in the span of $\{\omega_1,\ldots,\omega_t\}$. But $u\omega$ also is in the span of $\{\eta_1,\ldots,\eta_t\}$ by definition of the $\eta_i$, giving rise to a linear dependence relation. Hence, in order for $\mathcal B$ to be a basis, it is necessary for $u$ to have a pole at $\infty$ of order greater than the order of vanishing at $\infty$ of any holomorphic differential.

This lemma is not strictly necessary for the computation, but rather serves to guarantee its success in certain cases. Even if the hypotheses of the lemma do not hold (for example, if $\infty$ is not a Weierstrass point but $u$ has a pole of order $> t+1$), the set $\mathcal B$ may still be a basis of $ H^1_{\mathrm{dR}}(X/\mathbf Q)$, and almost always is at levels $< 200$. Moreover this can be checked easily in any particular example by computing the matrix for the Poincar\'e pairing.
\end{remarkwr}

Given one basis $\mathcal B$ for $ H^1_{\mathrm{dR}}(X/\mathbf Q)$ --- for example, one computed as above --- it is then a matter of linear algebra to produce a better basis that is adapted to the action of the Hecke algebra. Note that the usual formula for the action of the Hecke algebra $\mathbf T_0$ on holomorphic modular forms in terms of $q$-expansions extends to \textit{weakly holomorphic modular forms} (i.e., meromorphic 1-forms on $X$
with possible poles only at the cusps), and preserves the subspace of differentials regular on $Y$.
In particular one can compute the action of $\mathbf T_0$ on any 1-form representing an element of $H^1_{\mathrm{dR}}(X/\mathbf Q)$.
Using $q$-series
for the elements of the basis $\mathcal B$, we can thus write down the
matrix $[T_p]\in\mathrm{Mat}_{2t\times 2t}(\mathbf Q)$ that describes the action of any $T_p\in \mathbf T_0$.
After identifying $H^1_{\mathrm{dR}}(X/\mathbf Q)\approx \mathbf Q^{2t}$ via the basis $\mathcal B$, by finding the eigenspaces of finitely many such matrices one can write down $\mathbf Q$-bases for each isotypic component of $H^1_{\mathrm{dR}}(X/\mathbf Q)$. As is shown in \cite{Stn}, the Hecke algebra $\mathbf T_0$ is generated as a $\mathbf Z$-module by $T_i$ for $1 \leq i \leq \frac{m}{6} - \frac{m-1}{N}$, where $m = [\Gamma(1) : \Gamma_0(N)]$. This gives an upper bound on the number of matrices needed, although in practice considerably fewer are necessary.
Using these it is simple linear algebra to produce the desired symplectic basis $\omega_{g,1},\ldots,\omega_{g,2k}$ for each isotypic component $H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$.

%\begin{remarkwr}
%FIXME: Add a remark about the alternative method which avoids computing Hecke actions on differentials explicitly.
%(M-symbols, periods, blah blah) Note from Mike: I have been unable to implement this method without ending up with complex matrices, rather than ones with rational entries. Sage cannot compute eigenspaces of complex matrices, so I'm not sure if this method can produce explicit bases of isotypic components. The computation can be done without explicitly finding isotypic bases, but it is far less efficient.
%\end{remarkwr}

\subsection{Modular units and $\eta$-products}\label{eta} 
The preceding discussion raises the question of how to compute the $q$-expansion about $\infty$ of a rational function $u$ used to write down an initial choice of basis $\mathcal B$ for $H^1_{\mathrm{dR}}(X/\mathbf Q)$.


Recall that a \textit{modular unit} (for $\Gamma_0(N)$) is a modular function $u\in \mathbf Q(X)^\times$ whose associated divisor is supported on the cusps of $X=X_0(N)$. Denote by $U$ the multiplicative group of modular units. 
\begin{definition}\label{etadef}
The {\em eta group} $U_\eta$ is the subgroup of $ \mathbf Q(X)^\times$ of rational functions of the form $$u(q) =
\lambda \prod_{0<d\mid N} \eta(q^d)^{r_d},$$ where $\lambda\in \mathbf Q^\times$, $\eta(q)=q^{1/24} \prod_{n>0} (1-q^n)$ is the classical eta function
 and $\{ r_d\}_{d\mid N}$ is a collection of integers satisfying the following conditions.
\begin{itemize}
\item[i.]  $\sum_{d\mid N} r_d =0$,
\item[ii.] $\prod_{d\mid N} d^{r_d}\in \mathbf Q^\times$ is a square,
\item[iii.] $(n_d):=A_N\cdot (r_d)$ is a vector (indexed by divisors $d$ of $N$) of integers divisible by 24, where $A_N$ is the $\sigma(N)\times \sigma(N)$-matrix whose entry indexed by $(d,d')$ is $\frac{N\cdot (d,d')^2}{d d' (d',N/d')}$.
\end{itemize}
\end{definition}
Work of Newman and Ligozat shows that such functions are indeed modular units on $X$; that is, $U_\eta\subset U$. In fact more is true:

\begin{proposition}\label{QUeta}
$\mathbf{Q}\otimes_\mathbf Z U_\eta = \mathbf{Q}\otimes_\mathbf Z U$.
\end{proposition}
\begin{proof}
It easy to see that the set $\{ \frac{a}{d}:\, d\mid N, a\in (\mathbf Z/(d,N/d)\mathbf Z)^\times\}\subset \mathbf P^1(\mathbf Q)$ is a complete set of representatives of the cusps of $X$. The subspace $\mathbf Q \otimes U_\eta \subset \mathbf Q\otimes U$ coincides with $\mathbf Q\otimes U'$,
where $U'\subset U$ consists of modular units that have the same valuation at any two cusps $a/d$, $a'/d$ with the same denominator; \textit{cf.} \cite[Prop. 2]{Gonzalez}.
This implies the proposition in light of the next lemma, since an element  $u\in U\subset \mathbf Q(X)$ has the same valuation at any two Galois-conjugate cusps. 
\end{proof}
\begin{lemma}\label{l2}
  Let $d|N$. Then the cusp $1/d$ is rational if and only if $(d,N/d)=1$.
More generally, the Galois orbit of the cusp $1/d$ is $\Gal(\bar {\mathbf Q}/\mathbf Q)\cdot \frac{1}{d} = \{ \frac{a}{d}:\, a\in (\mathbf Z/(d,N/d)\mathbf Z)^\times\}$.
\end{lemma}
\begin{proof}[Proof of the lemma]
We prove the first statement using the results of \cite[\S 1.3]{Stevens}.
Namely, it is known that the cusps of $X$ are rational over $\mathbf Q(\zeta_N)$, and the Galois action of $\Gal(\mathbf Q(\zeta_N))/\mathbf Q)\simeq (\mathbf Z/N\mathbf Z)^\times$ can be described explicitly as follows \cite[Thm. 1.3.1]{Stevens}: given $b\in  (\mathbf Z/N\mathbf Z)^\times$, let $\tau_b$ be the automorphism of $\mathbf Q(\zeta_N)$ that sends $\zeta_N\mapsto \zeta_N^b$. If $a\in \mathbf Z$ is chosen so that $a b\equiv 1\pmod N$ then $\tau_b$ sends the cusp $\frac{1}{d}$ to $\frac{1}{a d}=\frac{a}{d}$. Hence the Galois orbit of $\frac{1}{d}$ is $$\{ \tfrac{a}{d}: a\in (\mathbf Z/N\mathbf Z)^\times\},$$ and it can be shown by an elementary argument that this set of cusps is equal to the image of $$\{ \tfrac{a}{d} \mid a\in (\mathbf Z/(d,N/d)\mathbf  Z)^\times \}$$ in $\Gamma_0(N)\backslash \mathbf P^1(\mathbf Q)$. 
\end{proof}


By the Riemann-Roch theorem, there exist nonconstant rational functions on $X$ that are regular away from $\infty$. The proposition implies that an integer power of such a function belongs to the subgroup $U_\eta\subset U$, which yields the following.
\begin{corollary}\label{uexists}
There exists an eta product $u\in U_\eta$
that is regular away from $\infty$.\qed
\end{corollary}
It is thus possible to compute the rational function $u$ required in the compation of a basis for $H^1_{\mathrm{dR}}(X)$ as an eta product.

 A practical approach to finding the vector $(r_d)_{d|N}$ giving rise to the $u$ we seek is to apply a linear programming algorithm: one minimizes the pole order $-n_N$ of $u$ at $\infty$ subject to the criteria of Newman-Ligozat in Definition \ref{etadef} and the condition that the orders $n_d$ of $u$ at other cusps are non-negative. 

\begin{remarkwr}\label{ueff}
To determine the complexity of the algorithm described in this paper (see \S \ref{complexity}), it is necessary to bound effectively (as a function of $N$) the order of the pole at $\infty$ of the eta quotient $u$ in Corollary \ref{uexists}. This can be done by examining the proof of Corollary \ref{uexists}.
By the Riemann-Roch theorem, there is a nonconstant rational function $w$ on $X$ that is regular on $Y$
and has a pole of order $\leq t = \text{genus}(X)$ at $\infty$.
From the formula for $\text{genus}(X)$ as a function on $N$, one can thus extract the bound $-\ord_\infty(w)= O(N\log\log N)$; \textit{cf.} for example \cite{genera}. (We adopt the convention that unless decorated with a subscript, an expression $O(-)$ denotes
a bound with an absolute implied constant.)
An examination of the proof of \cite[Prop. 2]{Gonzalez}, which was invoked to show Proposition \ref{QUeta}, 
actually gives the more precise result that $w^\mu$ belongs to $U_\eta$ for an integer $\mu = O(\det A_N)$.
Combining this with the explicit formula \cite[Prop. 1]{Gonzalez} for $\det A_N$,
one deduces  the estimate $-\ord_\infty(u) = O(N^{2\sigma_0(N)+2})$,
where $\sigma_0(N)$ denotes the number of positive divisors of $N$.
\end{remarkwr}


 \subsection{Computing the Poincar\'e dual $\gamma_f$ of $\omega_f$}\label{computepoincaredual}

Assume that $\{\gamma^{(j)}\}$ is a collection of elements of $\Gamma_0(N)$
with small lower-left entries $cN$, whose homology classes $\gamma_0^{(j)}$ generate $H_1(X^{\mathrm{an}},\mathbf Z)$. By a brute-force search it is straightforward to find such elements $\gamma^{(j)}$ in practice. (For small $N$, often one need take $c$ no greater than 2 or 3.)
% Expressing each $m_i,m_i'$ above as a $\mathbf Z$-linear combination of the $s_j$'s, we can thus efficiently compute the periods $I(\gamma_i,\omega_j)$
% and $I(\gamma_i',\omega_j)$ as linear combinations of periods $I(\sigma_i,\omega_j)$ which can be calculated using relatively few Fourier coefficients for $\omega_j$.


For any $m\in H_1(X^{\mathrm{an}},\mathbf C)$, write $\eta_m$ for the Poincar\'e dual of $c$; conversely, for any differential $\eta$ of the second kind on $X$, let $m_\eta\in H_1(X^{\mathrm{an}},\mathbf C)$ denote the Poincar\'e dual
of its cohomology class. We normalize the Poincar\'e duality isomorphism so that it is characterized by the property
\begin{equation}\label{pd}
\langle \eta_m,\eta\rangle  = \int_m \eta.
\end{equation} 

% (In \cite[Ch. III, (1.1.1)]{FK} they rather agree to say that "Poincar\'e dual" means $\int_m \eta = \int_{X_0(N)} \eta_m\wedge \eta$; we use a different convention, which differs from \cite{FK} by the factor $2\pi i$. If we stick to this, we should at least be consistent all the time. Check throughout that we didn't put a wrong factor $2\pi i$ anywhere.)

The vector space $H_1(X^{\mathrm{an}},\mathbf C)$ is also equipped an intersection product, which is related to the Poincar\'e pairing by the formula %(cf.\, \cite[Ch. III, (1.1.2)]{FK})
\begin{equation}\label{intersect}
m\cdot m_\eta = \frac{1}{2 \pi i}\langle \eta_m,\eta\rangle.
\end{equation}
The homology of $X$ also admits a natural action of the Hecke algebra, compatible with the action on cohomology via Poincar\'e duality. For any normalized eigenform $f\in S_2(\Gamma_0(N))$ and any $m\in H_1(X^{\mathrm{an}},\mathbf C)$, write $m^f\in H_1(X^{\mathrm{an}},\mathbf C)[f]$ for the projection of $m$ onto the $f$-isotypic component of homology.
Similarly, for $\eta\in H^1_{\mathrm{dR}}(X/\mathbf Q)$ write
$\eta^f$ for its projection onto the $f$-isotypic component.

We can assume that via the method described above a symplectic basis
$$\mathcal S = \{\omega_{f,1},\ldots, \omega_{f, \ell}, \eta_{f,1},\ldots, \eta_{f, \ell}\}$$ for $H^1_{\mathrm{dR}}(X/\mathbf Q)[f]$ has already been computed, where $\langle \omega_{f,i}, \eta_{f,j}\rangle = \delta_{i,j}$.

\begin{lemma}\label{fact} Fix $\gamma_1$, $\gamma_2\in \Gamma_0(N)$ and let $m_1$, $m_2\in H_1(X^{\mathrm{an}},\mathbf Z)$ denote the corresponding homology classes on $X$.
For any normalized eigenform $f\in S_2(\Gamma_0(N))$, we have
$$m_1^f\cdot m^f_2=\frac{1}{2\pi i}\sum_{i=1}^{\ell} I(\omega_{f,i};m_1)I(\eta_{f,i};m_2) - I(\omega_{f,i};m_2)I(\eta_{f,i};m_1),$$
where $\omega_f  = 2\pi i f(z)\mathrm{d}z$ is the 1-form corresponding to $f$.
\end{lemma}

\begin{proof}
Let $\eta_k=\eta_{m_k}$ and write $\eta^{f}_k=\sum c^{(k)}_i\omega_{f,i} + \sum d^{(k)}_i\eta_{f,i}$. Then we compute $m^f_1\cdot m^f_2 =  \frac{1}{2 \pi i} \langle \eta^f_1,\eta^f_2\rangle = \sum_i \frac{1}{2 \pi i} (c^{(1)}_i d^{(2)}_i-c^{(2)}_i d^{(1)}_i )= \frac{1}{2 \pi i} \sum_i (I(\omega_{f,i};\gamma_1)I(\eta_{f,i},\gamma_2)-I(\eta_{f,i};\gamma_1)I(\omega_{f,i};\gamma_2) )$. 
\end{proof}

Using \eqref{pd}, \eqref{intersect}, and Lemma \ref{fact}, we can compute the Poincar\'e dual $\gamma_f$ of $\omega_f$. Let $m_1,\ldots,m_{2\ell}$ be modular symbols giving rise to a basis of $H_1(X^{\mathrm{an}},\mathbf Z)[f]$, which can be computed using a modular symbols algorithm (\textit{cf}. \cite{Stn}). In particular, if $f$ is new, then $\ell=1$. Write $A$ for the matrix $(m_i \cdot m_j)_{1\leq i,j \leq 2\ell}$, which can be computed using Lemma \ref{fact}, and let $v$ be the column vector $(m_i \cdot m_{\omega_f})_{i=1}^{2\ell}$, which can be computed using \eqref{intersect} in conjunction with \eqref{pd}. Then the vector $A^{-1}v$ gives the coefficients expressing $m_{\omega_f}$ as a linear combination of $m_1,\ldots,m_{2\ell}$. These coefficients can then easily be converted to an expression for $\gamma_f$ as a linear combination of $\{\gamma^{(j)}_0\}$.





\subsection{Computing the adjustments $\int_{\gamma_f}\alpha$}\label{computealpha}
Write the homology class $\gamma_f$ Poincar\'e dual to $\omega_f$ as
\[\gamma_f = \sum \beta_j \gamma_0^{(j)}\]
for $\beta_j\in \mathbf C$
and homology classes $\gamma_0^{(j)}$ whose lifts $\gamma^{(j)}$ to $\Gamma_0(N)$ have small lower-left entries $cN$.
Let $\omega$ and $\eta$ be differentials of the second kind, at least one of which is regular at $\infty$. Using the methods described so far, we are already able to compute
\[z_{\omega,\eta} := \sum_j \beta_j \int_{\tau_0}^{\gamma^{(j)}\tau_0} \omega F_{\eta}.\]
We stress that the value of $z_{\omega,\eta}$ depends on $\tau_0$ and the choices we made in representing $\gamma_f$. It is simply one part of the iterated integral $J_{\omega,\eta}(\gamma_f) = \int_{\gamma_f} \omega \cdot \eta - \alpha_{\omega,\eta}$, which is independent of these choices. In this section, we describe a method for computing $J_{\omega,\eta}(\gamma_f) - z_{\omega,\eta}$. This amounts to computing the $q$-expansion of $\alpha_{\omega,\eta}$, for then we have
\[J_{\omega,\eta}(\gamma_f) - z_{\omega,\eta} = -\sum_j \beta_j \int_{\tau_0}^{\gamma^{(j)}\tau_0} \alpha_{\omega,\eta}.\]

Recall that the defining property of $\alpha_{\omega,\eta}$ is that its principal part at $\infty$ agrees with that of  $\omega F_{\eta}$ on $\tilde X$, modulo $\mathrm{d}q/q$; i.e., their difference has at worst logarithmic poles. However, note that since $\int_{\gamma_f} \lambda = 0$ for exact 1-forms $\lambda$, we may replace $\alpha_{\omega,\eta}$ by any cohomologous 1-form. The cohomology class of $\alpha_{\omega,\eta}$ is determined by the data $\langle \lambda_i, \alpha_{\omega,\eta} \rangle$, where $\lambda_1,\ldots,\lambda_{2t}$ (for $t$ the genus of $X$) form a basis of $H^1_{\dR}(X/\mathbf Q)$, so it suffices to compute these values of the Poincar\'e pairing.

We can choose $\lambda_1,\ldots,\lambda_t$ to be holomorphic. In this case, we can compute
$$\langle \lambda_i, \alpha_{\omega,\eta} \rangle = \res_\infty(F_{\lambda_i} \cdot \alpha_{\omega,\eta}) = \res_{\tilde \infty}(F_{\lambda_i}\cdot F_\eta\cdot\omega),$$
where the second equality holds because $\res_{\infty}(F_{\lambda_i}\cdot \alpha_{\omega,\eta})$
depends only on $$\mathrm{pp}_\infty(\alpha_{\omega,\eta})\mod \frac{\operatorname{d}q}{q} = \mathrm{pp}_{\tilde\infty}(\omega F_\eta).$$
\begin{lemma}\label{l3} Let $\lambda_1,\ldots,\lambda_t \in H^0(X,\Omega^1_{X/\mathbf Q})$ be a basis of regular 1-forms on $X$. Then $\alpha \in H^1_{\dR}(X/\mathbf Q)$ lies in the subspace $H^0(X,\Omega^1_{X/\mathbf Q})$ if and only if $\langle \lambda_i, \alpha\rangle = 0$ for all $1\leq i \leq t$.
\end{lemma}
\begin{proof} The subspace $H^0(X,\Omega^1_{X/\mathbf Q}) \subseteq H^1_{\dR}(X)$ is isotropic for the Poincar\'e pairing because the pairing can be computed using residues. For dimension reasons, it is maximal isotropic, and the lemma follows.
\end{proof}
By the lemma, if $\langle \lambda_i,\alpha\rangle = \langle \lambda_i, \alpha'\rangle$ for $i = 1,\ldots,t$, then $\alpha - \alpha'$ is cohomologous to a regular 1-form. Since $\alpha_{\omega,\eta}$ is only well-defined modulo $H^0(X,\Omega^1_{X/\mathbf Q})$, it follows that we can choose $\langle \lambda_i, \alpha_{\omega,\eta}\rangle$ for $i=t+1,\ldots,2t$ arbitrarily. For convenience, we choose $\langle \lambda_i,\alpha_{\omega,\eta}\rangle=0$ for $i=t+1,\ldots,2t$. Define the matrix $B = (\langle \lambda_i,\lambda_j\rangle)_{i,j=1}^{2t}$ and the vector 
$$w=(\langle \lambda_i, \alpha_{\omega,\eta}\rangle)_{i = 1}^{2t} = (\res_{\tilde\infty}(F_{\lambda_1}F_\eta\omega), \cdots, \res_{\tilde\infty}(F_{\lambda_t}F_\eta\omega),\overbrace{0,\ldots, 0}^t).$$ It then follows by elementary linear algebra that the vector $B^{-1}w$ yields the coefficients of an expression for $\alpha_{\omega,\eta}$ as a linear combination of $\lambda_1,\ldots,\lambda_{2t}$.

\subsection{Computing the coefficients $c_{ij}$}\label{Tnmatrix}
Using the methods described so far, we may compute all the integrals $\int_{\gamma_f} (\omega \cdot \eta - \alpha_{\omega,\eta})$ occuring in \eqref{pgf}. The last input needed to evaluate \eqref{pgf} is the set of coefficients $c^n_{ij}$ appearing in that formula. Using Lemma \ref{hodgeclass}, these coefficients can be computed using the matrix $A_n$ which expresses the right action of $T_n$ on $H^1_{\mathrm{dR}}(X/\mathbf Q)[g]$ with respect to the basis $\omega_{g,1},\ldots,\omega_{g,2k}$, and the matrix $B = (\langle \omega_{g,i},\omega_{g,j}\rangle)_{1\leq i,j\leq 2k}$, which by our choice of basis is the standard symplectic matrix. In section \ref{symplecticbasis}, we gave a method for computing $A_n$ using the action of $T_n$ on $q$-expansions. However, this only works if $\gcd(n,N)=1$, so we must resort to other methods. We exploit the fact that the action of $T_n$ on $H_1(X^\mathrm{an},\mathbf C)[g]$ is readily computable using modular symbols; see \cite{Stn} for de!
 tails. Recall that we have a Hecke-equivariant duality
$$H^1_{\dR}(X^{\mathrm{an}}, \mathbf C)[g] \times H_1(X^{\mathrm{an}},\mathbf C)[g] \to \mathbf C$$
given by the integration pairing $\langle \omega, \lambda \rangle = \int_{\gamma} \lambda$;
here the Hecke-equivariance means that $\langle T_n\omega, \alpha \rangle = \langle \omega, T_n \alpha \rangle$. 
Using modular symbols and the techniques of \cite{Stn} one can compute the matrix $B_n$ of $T_n$
acting on $H_1(X^\mathrm{an},\mathbf C)[g]$ on the left, with respect to a basis $m_1,\ldots, m_{2k}$.
 Write $Q = (\langle \omega_i, m_j\rangle )_{i,j=1}^{2k}$, which can be computed efficiently via the method explained in \S \ref{integration} (using an appropriate basis $\{m_j\}$ derived from the generators $\gamma^{(j)}_0$ for $H_1(X^\mathrm{an},\mathbf Z)$ discussed above). Then it is straightforward linear algebra to show that $A_n = QB_nQ^{-1}$. 

\subsection{Computing the denominator $d_{g,n}$}\label{denom}
The final ingredient to be computed is the denominator $d_{g,n}$, or the smallest positive
 integer such that $d_{g,n}T_{g,n} \in \mathbf T_{\mathbf Z}$. This can be accomplished
  by computing a $\mathbf Z$-basis for the ($\mathbf Z$-finite free) Hecke algebra 
  $\mathbf T_{\mathbf Z}$ as a subring of $M_{2t}(\mathbf Q)$, where $t$ is the 
  genus of $X_0(N)$, by identifying $\mathbf T$ with an algebra of endomorphisms
   of the $(2t)$-dimensional $\mathbf Q$-vector space of cuspidal modular symbols
    of weight 2 and level $N$. As $\mathbf T_{\mathbf Z}$ is generated as an abelian
     group by $T_i$ for $1 \leq i \leq \frac{m}{6} - \frac{m-1}{N}$, where 
     $m = [\Gamma(1) : \Gamma_0(N)]$, this is a finite computation. 
Once $\mathbf T_{\mathbf Z}$ has been computed it is a simple matter to find
 the matrix representation of $T_{g,n}$ and  compute the smallest $d_{g,n}$ such that $d_{g,n}T_{g,n} \in \mathbf T_{\!
 mathbf Z}$.
 
 For  more details on modular symbols and generating the Hecke algebra, see \cite{Stn}. 

\subsection{Remarks on complexity}\label{complexity}
The complexity of the computations we have described
is primarily determined by
 the {\em number} $n_D$ of Fourier coefficients required
to compute $J_{\omega,\alpha}(\gamma_f)$ to a given number $D$ of digits of accuracy.
In this subsection we sketch a method for obtaining a bound on $n_D$ in terms of $N$.

\subsubsection{}
Write the Fourier expansion of $u$ as 
$$u(\tau) = \sum_{n\geq -n_0} c_n q^n, \qquad q = e^{2\pi i \tau}, \tau \in \mathfrak H.$$ 
Let the principal part of $u$ at $\infty$ be
\[\mathrm{pp}_\infty (u) = \sum_{-n_0 \leq -m \leq 0} \frac{d_m}{m} q^{-m},\qquad d_m = mc_{-m}.\]
In \cite{ono}, Bringmann and Ono prove an exact formula for the Fourier coefficients of harmonic Maass forms, of which weakly holomorphic modular functions such as $u$ are examples. To avoid introducing unnecessary notation, we state only the very special case of their result applicable to our situation. We remark that long ago Rademacher used the circle method to prove a similar exact formula for the coefficients of the $j$-function \cite{rad}, and a modification his argument would probably yield a simpler and more direct proof of the special case we require.
Using \cite[Thm. 1.1]{ono}, one can express the coefficients $c_n, n>0$ in terms of the coefficents $d_m$,
 the order-1 $I$-Bessel function $I_1(z)$,  and the Kloosterman sum
\[K(-m,n,c):=\sum_{\substack{0<v< c\\ (v,c)=1}} \exp\left(\frac{2\pi i}{c}(nv + m\bar v)\right),\]
where $\bar v$ denotes the multiplicative inverse of $v$ modulo $c$.
Namely, \textit{loc. cit.} yields the formula
\begin{equation}\label{coeff}
  c_n = 2\pi  \sum_{-n_0\leq -m \leq 0} d_m \sum_{\substack{c>0\\ c\equiv 0\pmod N}} (\tfrac{m}{n})^{1/2} \frac{K_0(-m,n,c)}{c}I_1\left(\frac{4\pi\sqrt{|mn|}}{c}\right),\qquad n>0.
\end{equation}
By Remark \ref{ueff}, we have 
\begin{equation}\label{n0est}
  n_0 = -\ord_\infty(u) \leq A_1 N^{2 \sigma_0(N) + 2},
\end{equation}
 where the constant $A_1$ is absolute and $\sigma_0$ denotes the divisor function.
We can trivially bound the numbers $d_m$ as follows.
Let $\xi_r(x) = re^{2\pi i x}$ for $0<r<1$
and set $y=-\frac{1}{2\pi}\log r>0$.
The Cauchy integral formula applied to the meromorphic function $U(q) = \sum c_n q^n$
of $q$ in the unit disk gives
\[\frac{d_m}{m} = \frac{1}{2\pi i} \int_{\xi_r} \frac{U(q)}{q^{m+1}}\operatorname{d}q = e^{2\pi m y} \int_0^1 
u(x+iy)e^{-2\pi inx}\operatorname{d}x.\]
Taking $y=1$, say, we thus have
\[|d_m| \leq m e^{2\pi m}\int_0^1 |u(x+i)|\operatorname{d}x
= m e^{2\pi i m} \int_0^1 \prod_{d|N} |\eta(dx+id)|^{r_d}\operatorname{d}x,\]
where $\eta$ is the Dedekind eta function and $(r_d)_{d|N}$ is the vector giving rise to $u$.
Recall that $\eta(z)$ is a nonvanishing holomorphic function on $\mathfrak H$,
and satisfies $|\eta(z+1)|=|\eta(z)|$. Thus there is an absolute upper bound $B$
for $\max(|\eta(z)|,|\eta(z)|^{-1})$
on the strip  $\{1-\epsilon \leq \mathrm{Im}(z) \leq N+\epsilon\}\subset\mathfrak H$.
It follows that 
\[|d_m| \leq me^{2\pi m} B^{\sum |r_d|}.\]
To control this quantity we must bound both $B$ and $\sum |r_d|$ in terms of $N$.

As regards $B$, note that since $\eta(z)$ is holomorphic on $\mathfrak H^*$,
the upper bound for $|\eta(z)|$ on the strip of interest poses no problem;
it is a constant independent of $N$.
To find a lower bound for $|\eta(z)|$ on our strip, we can make use of the Euler formula
\[ \eta(z) = e^{\pi i z/12}\left(1+\sum_{m\geq 1} q^{\frac{3m^2-m}{2}}+q^{\frac{3m^2+m}{2}}\right), \qquad q = e^{2\pi i z}.\]
From this it is straightforward to deduce a bound
$|\eta(z)|^{-1} = O(e^{\pi N/12})$, for $z$ in the strip in question.

To handle $\sum |r_d|$, we recall some results from numerical analysis.
Let $A$ be an invertible $\sigma\times\sigma$ matrix of real numbers.
We denote by $||A||_2$ the quantity $\displaystyle\sup_{x\in \mathbf R^\sigma, |x|=1} |A x|$
and by $||A||_{\max}$ the maximum of the absolute values of the matrix entries $a_{ij}$.
A result in numerical linear algebra \cite[Thm. 2.1]{M} asserts
that when $A$ has \textit{integer} entries, one can control its so-called ``condition number''
and obtain the estimate
\[||A^{-1}||_2\leq \sigma^{2+\sigma/2}||A||_{\max}^{\sigma-1}.\]
We apply this result to the matrix $A_N$ from Definition \ref{etadef}
with $\sigma = \sigma_0(N)$.
By Newman and Ligozat's result, the vector $\mathbf r = (r_d)_{d|N}$
and the vector $\mathbf n = (24\ord_{a/d}(u))_{d|N}$
(where $a/d$ is any cusp of $X_0(N)$ of denominator $d$)
satisfy $ \mathbf r = A_N^{-1}\mathbf n$.
Moreover since $u$ has a pole of order $n_0$ at infinity and no other poles,
the fact that $\div(u)$ has degree 0 implies $|\mathbf n| = O(n_0 \sqrt{\sigma_0(N)})$.
Finally, the formula for the entry of $A_N$ indexed by $(d,d')$
entails $||A_N||_{\max} \leq N^3$.
We thus have, using  \eqref{n0est},
\[\sqrt{\sum_{d|N} |r_d|} \leq \sqrt{\sum_{d|N} |r_d|^2} = |\mathbf r| \leq ||A_N^{-1}||_2 |\mathbf n|
\leq \sigma_0(N)^{\frac{\sigma_0(N)+5}{2}}n_0 N^{3(\sigma_0(N)-1)}= O(N^{\frac{11\sigma_0(N)+1}{2}}).\]
Consequently, for an absolute constant $C$
we have
\begin{eqnarray*}
  |d_m|&\ll& me^{2\pi m} e^{\pi N \sum |r_d|/12} \leq m e^{2\pi m} e^{C N^{11\sigma_0(N) + 2}}
\\ &\leq& n_0 e^{2\pi n_0} e^{C N^{11\sigma_0(N) + 2}} \\
& =& O(n_0\exp(2\pi N^{2\sigma_0(N)+2}+C N^{11\sigma_0(N)+2}))
\end{eqnarray*}
From this we deduce
\begin{equation}\label{dest}
|d_m| =O(n_0 \exp(A_0 N^{11\sigma_0(N)+2})),   
\end{equation}
for an absolute constant $A_0$.

From \eqref{coeff} and \eqref{dest}, standard estimates for Kloosterman sums, and asymptotics for the $I$-Bessel function, one obtains by the method of \cite[\S \S 5.1-2]{BrPh} the estimate
\[c_n = O(N^{5/4}n_0^{7/4} n^{-3/4} \exp(A_0 N^{11\sigma_0(N)+2} + N^{-1}4\pi \sqrt{nn_0})).\]
In light of \eqref{n0est} this yields
\begin{equation}\label{cnest}
  c_n = O(n^{-3/4} \exp(A_2 N^{11\sigma_0(N)+2}+\tfrac{4\pi\sqrt{A_1}}{N} N^{\sigma_0(N)+1} \sqrt n)),
\end{equation}
for absolute constants $A_1,A_2$.

\subsubsection{} 
The coefficients $c_n$ determine the Fourier coefficients of the 1-forms occuring in the formula \eqref{pgf}
for $P_{g,f,n}$. Unfortunately the relationship is indirect, as the construction of the 1-forms $\omega_{g,i}$ and $\alpha_{\omega_{g,i},\omega_{g,j}}$ involves multiplying $u$ against a basis of cusp forms for $\Gamma_0(N)$
and then performing a lot of linear algebra. By Deligne's proof of the Ramanujan-Petersson conjecture,
the cusp forms have $n$th coefficient of size $O_\delta(n^{\frac{1}{2}+\delta})$.
It follows that $n$th Fourier coefficient of an element of the basis $\mathcal B$
for $H^1_{\mathrm{dR}}(X/\mathbf Q)$ computed in \S \ref{symplecticbasis} has size at most
\[O(P(n) \exp(A_2 N^{11\sigma_0(N)+2}+\tfrac{4\pi\sqrt{A_1}}{N} N^{\sigma_0(N)+1} \sqrt n)),\]
for absolute constants $A_1,A_2$ and a universal polynomial $P(n)$.
To compute the 1-forms $\omega_{g,i}$ and $\alpha_{\omega_{g,i},\omega_{g,j}}$, linear algebra operations
are performed on this basis, which spans a vector space of dimension $\text{genus}(X) = O(N\log\log N)$.
It thus seems likely that a careful analysis of the linear algebra operations performed
would yield a bound 
\begin{equation}\label{final}
  O(Q(n) \exp(A_3 N^{11\sigma_0(N)+2}+\tfrac{4\pi\sqrt{A_1}}{N} N^{\sigma_0(N)+1} \sqrt n)),
\end{equation}
for the $n$th Fourier coefficient of \textit{any} 1-form integrated in the course of computing \eqref{pgf}.
Here $A_1,A_3$ are absolute constants and $Q(X)$ is a universal polynomial independent of $N$.

Suppose $\lambda$ is such a 1-form (on $X$ or $\tilde X$),
and consider the problem of integrating the pullback of $\lambda$ to $\mathfrak H^*$
along a path from $\tau_1$ to $\tau_2$. 
By the method explained in \S \ref{integration},
we can assume that $\text{im} \tau_1,\text{im}\tau_2 \geq (c_N^*\cdot N)^{-1}$,
where $c^*_N\cdot N$ is the largest of the lower-left entries
of the elements $\gamma^{(j)}\in \Gamma_0(N)$ introduced at the beginning of \S \ref{computepoincaredual}.
Recall that these consisted of a collection of elements that span $H_1(X^\mathrm{an},\mathbf Z)$
and have lower-left entries as small as possible. We do not know how to bound $c^*_N$ in terms of $N$,
although in practice it seems to be very small.

If the Laurent expansion for $\lambda$ about $\infty$ (or a lift of $\infty$ to $\tilde X$)
is $\lambda = \sum a_\lambda(n) \frac{\operatorname{d}q}{q}$,
then setting $\tau_j = x_j + iy_j$ for $j=1,2$ (where $y_j \geq (c^*_N N)^{-1}$), we have
\[\int_{\tau_1}^{\tau_2} \lambda = \sum_{n\gg -\infty} \frac{a_\lambda(n)}{n} (e^{2\pi i n x_2}e^{-2\pi n y_2}
- e^{2\pi i n x_1}e^{-2\pi n y_1}).\]
Our problem is to determine $n_D$
such that the tails of these sums are bounded by the requisite precision, say $10^{-D}$.
It clearly suffices to take for $n_D$ any $m$ such that
\[S(m):=\sum_{n\geq m} n^{-1} |a_\lambda(n)| e^{-2\pi n y} \leq 10^{-D}.\]
Granting \eqref{final}, we have
\[S(m) \ll \sum_{n\geq m} n^{-1}Q(n) \exp(A_3 N^{11\sigma_0(N)+2}+\tfrac{4\pi\sqrt{A_1}}{N} N^{\sigma_0(N)+1} \sqrt n - \tfrac{2\pi n}{Nc_N^*}).\]
For a suitable constant $A_4$, we can assume 
$$n^{-1}Q(n)\exp(\frac{4\pi\sqrt{A_1}}{N} N^{\sigma_0(N)+1} \sqrt n)\leq \exp( \frac{A_4}{N} N^{\sigma_0(N)+1} \sqrt n).$$
Thus
\begin{eqnarray*} S(m) &\ll& \exp(A_3 N^{11\sigma_0(N)+2}) \sum_{n\geq m} \exp(\frac{1}{Nc_N^*}(A_4 c_N^* N^{\sigma_0(N)+1} \sqrt n - 2\pi n))\\
&\leq&\exp(A_3 N^{11\sigma_0(N)+2})\sum_{n\geq m} e^{- n/ Nc^*_N} = \exp(A_3 N^{11\sigma_0(N)+2}) \frac{e^{- m/ N c^*_N}}{1-e^{-1/Nc^*_N}},\end{eqnarray*}
provided $m$ is large enough that $A_4 c_N^* N^{\sigma_0(N)+1} \sqrt n - 2\pi n \leq - n$
for $n\geq m$.
The latter can be ensured by requiring $m\geq A_5 c_N^{*2} N^{2\sigma_0(N)+2}$ for an absolute constant $A_5$.
It follows that provided \eqref{final} holds, we have the following estimate for $n_D$ in terms of $D$ and $N$:
\begin{equation}\label{nDest}
  n_D = O(\max\{Nc^*_N(D + N^{11\sigma_0(N)+2}),  c^{*2}_N N^{2\sigma_0(N)+2}\}),
\end{equation}
where the implied constant is absolute, as always.

\section{Numerical examples}\label{eg}
\subsection{Example: {\bf 37a1}}\label{37}

 Take $N=37$ in the setup of our algorithm. In this setting, the space of regular 
 differentials on $X=X_0(37)$ is spanned by $\omega_f$ and $\omega_g$,
 which are associated to elliptic curves over $\mathbf Q$ (labeled {\bf 37a1} and {\bf 37b1} in Cremona's database) of ranks $1$ and $0$, respectively. The elliptic curve $37a1$ has minimal Weierstrass equation given by
 $$ 37a1:  y^2+y = x^3-x,$$
 and its Mordell-Weil group is generated by the point $(0,0)$. 

 By computing the periods attached to $\omega_f$ and $\omega_g$, it can be checked that the
  classes of the matrices
 $$
 \gamma_1=\begin{pmatrix}2&-1\\37 &-18\end{pmatrix}, \,\gamma_2= \begin{pmatrix}3&-1\\37 &-12\end{pmatrix}, \,\gamma_3= \begin{pmatrix}5&2\\37&15\end{pmatrix} ,\, \gamma_4=
 \begin{pmatrix}14 & 3\\ 37 &  8\end{pmatrix} 
 $$
  generate the rational homology of $X$.
These are a ``nice'' basis for the homology in the sense of the first paragraph of \S \ref{computepoincaredual}; that is, the lower left entries are exactly 37 (rather than $37c$ for $|c|>1$), so the integral $\int_\tau^{\gamma_i\tau}\lambda$ can be evaluated efficiently for any meromorphic differential 1-form $\lambda$ on $X_0(37)$ or its universal cover regular away form $\infty$, by the method of \S \ref{integration}.

 To obtain differentials of the second kind representing classes in the deRham cohomology, we consider the elements of the form 
 $$ \eta_1 = u\cdot \omega_f, \qquad \eta_2 = u\cdot \omega_g, \qquad u = \eta(q)^2\eta(q^{37})^{-2} = q^{-3} \prod_{n=1}^\infty (1-q^n)^2 (1-q^{37n})^{-2},$$
where $\eta(q)$ is the Dedekind eta function. The modular function $u$ is an example of an eta product with its only pole at $\infty$,  as considered in \S \ref{eta}.
 It is not hard to check directly (by calculating the Poincar\'e pairing on all pairs of elements) 
 that the classes of $\omega_f, \omega_g, \eta_1$ and $\eta_2$ generate the de Rham cohomology
 of $X$; alternatively one could apply Lemma \ref{h1basis}.

After computing the matrix $M$ of the Hecke operator $T_2$
acting on $H^1_{\dR}(X_0(37))$ with respect to the basis $\omega_f,\omega_g,\eta_1,\eta_2$, and then determining the eigenspaces of $M$, one finds that
 \begin{eqnarray*}  
 \eta_f &=&  \frac{1}{4}(-37\omega_g+4\eta_1 -8  \eta_2), \\
 \eta_g &=&  \frac{1}{4}(37\omega_f-6\eta_1 + 10 \eta_2)
 \end{eqnarray*}
 are in  the $f$ and $g$ isotypic components of the deRham cohomology respectively. In addition these linear combinations of 1-forms have been chosen so that
$\{\omega_f,\eta_f\}$ and $\{\omega_g,\eta_g\}$ form symplectic bases for the components with respect to the Poincar\'e pairing.

When one computes the Poincar\'e dual $\gamma_f$ of $\omega_f$ as in \S \ref{computepoincaredual},
one finds (with our normalization) that it is 
\[\gamma_f = \frac{1}{2\pi i}\left(A\left([\gamma_2]-[\gamma_3]+[\gamma_4]\right)-B\left(-[\gamma_1]+2[\gamma_2]\right)\right).\]
Here 
\[A \approx (2.4513893\ldots)i,\qquad B \approx 2.9934586\ldots\]
are certain linear combinations of the periods of $f$ against a basis of $H_1(X)[f]$; see \S \ref{computepoincaredual} for a more exact description.

The method of \S \ref{computealpha} can be used to compute $\alpha_{\omega_g,\eta_g}$. However, in this case, it is easy to find $\alpha_{\omega_g,\eta_g}$ by inspection. Working with principal parts, one finds that $\mathrm{pp}_\infty(\omega_gF_{\eta_g})
\equiv \mathrm{pp}_\infty(\frac{1}{4}(\eta_1-\eta_2))\mod \frac{\mathrm{d}q}{q}$.
Thus we may take $\alpha_{\omega_g,\eta_g}=\frac{1}{4}(\eta_1-\eta_2)$.
Integrating this over $\gamma_f$ yields the rational number $\int_{\gamma_f} \alpha_{\omega_g,\eta_g}=-\frac{1}{2}$ (to many digits of precision).

Since $g$ is a rational newform, then by Remark \ref{pointrelations}, we can find all the points $P_{g,f,n}$ by only computing $P_{g,f}$. According to Remark \ref{Tg}, this amounts to computing the complex number $z_{g,f} : = \int_{\gamma_f}(\omega_g\cdot\eta_g-\eta_g\cdot\omega_g - 2\alpha_{\omega_g,\eta_g})$. The method in \S \ref{integration}, coupled with the previous paragraph, yields
\[z_{g,f} = -0.4093610\ldots + (1.2256946\ldots)i.\]
Let $W$ be the Weierstrass uniformization of $E$. Then the point $W(z_{g,f}) \in E(\mathbf C)$ does not necessarily lie in $E(\mathbf Q)$. This is because $T_{g}$ is a \textit{rational} combination of cycles, and so $W(z_{g,f})$ is a $\mathbf Q$-linear combination of points in $E(\mathbf Q)$. Thus, the image of $W(z_{g,f})$ in $E(\mathbf C) \otimes \mathbf Q$ lies in the subspace $E(\mathbf Q)\otimes \mathbf Q$. So in order to write $P_{g,f}$ as an element of this space, we must compute the ``denominator'' of $T_{g}$. As in \S \ref{denom}, one can compute using the first few Fourier coefficients of $f$ and $g$ that  the idempotent $e=(0,1)\in \mathbf Q\times \mathbf Q \simeq \mathbf T$ $(\star)$ does not belong to $\mathbf T_{\mathbf Z} \subset \mathbf T$ but $2e$ does. Here, the identification $(\star)$ associates $T_n\otimes 1\in \mathbf T$ to $(a_n(f),a_n(g))\in \mathbf Q\times \mathbf Q$. By definition, $T_{g}$ corresponds to $e$ as an element of the Hecke algebra, so it's denominator is 2. 
Thus, we can write $P_{g,f} = W(2z_{g,f}) \otimes \frac{1}{2} \in E(\mathbf Q) \otimes \mathbf Q$.
  One finds that $W(2z_{g,f})$ agrees with the global point  $ (\frac{1357}{841}:\frac{28888}{24389}:1)$ to within 13 digits of accuracy
   using 350 Fourier coefficients,  so  we are led to conclude that
$$ P_{g,f} = \left(\frac{1357}{841}:\frac{28888}{24389}:1\right) \otimes \frac{1}{2}  = 12(0:0:1) \otimes \frac{1}{2} = 6(0:0:1) 
\in E(\mathbf Q)\otimes \mathbf Q  $$



\subsection{Example: {\bf 43a1}}\label{43}
Let $N=43$ and let $E$ be the elliptic curve labeled {\bf 43a1} in Cremona's database. The modular curve $X=X_0(43)$ has genus 3. There are two isotypic components of $H^1_{\dR}(X)$, one of dimension 2 corresponding to the modular form $f$ that parametrized $E$,
and another of dimension 4 corresponding to a newform $g$
with Fourier coefficients in $\mathbf Q(\sqrt2)$, associated to an abelian surface quotient of $J_0(43)$.

In this case, a linear programming algorithm identifies the eta-quotient $u$
that is modular for $\Gamma_0(43)$ of weight 0, holomorphic away from the cusp $\infty$, and with minimal pole order at $\infty$, as
\[u = \frac{\eta(q)^{4}}{\eta(q^{43})^{4}} =  q^{-7}-4q^{-6}+2q^{-5}+8q^{-4}-5q^{-3}-4q^{-2}-10q^{-1}+8+9q+14q^3+O(q^4).\]
Computing the residue pairing shows that for a basis of cuspforms with rational Fourier coefficients, corresponding to holomorphic 1-forms $\omega_f, \omega_{g,1},\omega_{g,2}$ on $X$, the collection
\[\omega_f,\omega_{g,1},\omega_{g,2},u\omega_f,u\omega_{g,1},u\omega_{g,2}\]
forms a basis for $H^1_{\dR}(X/\mathbf Q)$. By finding the matrices of a few Hecke operators with respect to this basis, one can as in the case $N=37$ produce symplectic bases
\[\omega_f,\eta_f,\qquad \text{and}\qquad \omega_{g,1},\omega_{g,2},\eta_{g,1},\eta_{g,2}\]
for $H^1_{\dR}(X)[f]$ and $H^1_{\dR}(X)[g]$ respectively.

We can compute the Poincar\'e dual $\gamma_f$ and the iterated integrals
\[\int_{\gamma_f}(\omega_{g,i} \cdot \omega_{g,j} - \alpha_{\omega_{g,i},\omega_{g,j}}), \int_{\gamma_f}(\omega_{g,i} \cdot \eta_{g,j} - \alpha_{\omega_{g,i},\eta_{g,j}}), \int_{\gamma_f}(\eta_{g,i} \cdot \omega_{g,j} - \alpha_{\eta_{g,i},\omega_{g,j}})\]
in the same manner as in the case $N=37$ with one exception. One simply cannot find $\alpha_{\omega_{g,i},\eta_{g,j}}$ by inspection. No linear combination of our chosen basis has the same principal part as $\omega_{g,i}F_{\eta_{g,j}}$, however some linear combination is cohomologous to such a form. The techniques from \S \ref{computealpha} can be used to find such a form.

Each $T_{g,n}$ gives rise to an element of $\End(H^1_{\dR}(X)[g]) \otimes \mathbf Q$. The collection of elements arising from $T_{g,n}$, $n \geq 1$ generate a subspace of dimension 2, generated by $T_{g,1}$ and $T_{g,2}$. Thus, we can effectively compute $P_{g,f,n}$ for all $n$ simply by computing $P_{g,f}$ and $P_{g,f,2}$. The formula for $P_{g,f}$ is the one given in Remark \ref{Tg}, so we have
\begin{align*}
z_{g,f} & = \int_{\gamma_f} (\omega_{g,1} \cdot \eta_{g,1} - \eta_{g,1} \cdot \omega_{g,1} - 2\alpha_{\omega_{g,1},\eta_{g,1}} + \omega_{g,2} \cdot \eta_{g,2} - \eta_{g,2}\cdot \omega_{g,2} - 2\alpha_{\omega_{g,2},\eta_{g,2}}) \\
& = -2.0768300\ldots + (2.7263648\ldots)i\end{align*}
The Hecke algebra $\mathbf T$ can be identified with $\mathbf Q \times \mathbf Q(\sqrt{2})$ via $T_n \otimes 1 \mapsto (a_n(f),a_n(g))$. Under this identification, $T_{g,1}$ corresponds to $e_1=(0,1)$, and an examination of the Fourier coefficients of $f$ and $g$ shows that $e_1$ does not lie in the image of $\mathbf T_{\mathbf Z}$, but $2e_1$ does. So, we have
$$P_{g,f} = W(2z_{g,f}) \otimes \frac{1}{2} = \left(\frac{11}{49} : -\frac{363}{343} : 1\right) \otimes \frac{1}{2} \in E(\mathbf Q) \otimes \mathbf Q.$$

Finding $P_{g,f,2}$ is a little more involved, as we must compute the matrix of $T_2$ acting on $\omega_{g,1},\omega_{g,2},\eta_{g,1},\eta_{g,2}$. Two methods for doing this were discussed in \S \ref{symplecticbasis} and \S \ref{Tnmatrix}, and either shows that $T_2 \omega_{g,1} = 2\omega_{g,2}$, $T_2 \omega_{g,2} = \omega_{g,1}$, $T_2 \eta_{g,1} = -\frac{97997}{132319}\omega_{g,2} + \eta_{g,2}$ and $T_2 \eta_{g,2} = \frac{97997}{132319} \omega_{g,1} + 2\eta_{g,1}$. So the matrix $A_2$ is given by
\[\left(\begin{array}{cccc}
0 & 2 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & -\frac{97997}{132319} & 0 & 1 \\
\frac{97997}{132319} & 0 & 2 & 0 \end{array}\right).\]
Combining this with Lemma \ref{hodgeclass} and remembering that $\alpha_{\omega_{g,i},\omega_{g,j}} = 0$ by Remark \ref{alpha=0}, we find that
\begin{align*}
z_{g,f,2} & = -\frac{97997}{132319}\int_{\gamma_f} (\omega_{g,1}\cdot\omega_{g,2} - \omega_{g,2}\cdot\omega_{g,1}) + \int_{\gamma_f}(\omega_{g,1}\cdot\eta_{g,2} - \eta_{g,2}\cdot\omega_{g,1} - 2\alpha_{\omega_{g,1},\eta_{g,2}}) & \\
& \hspace{.45 cm}+ 2\int_{\gamma_f} (\omega_{g,2}\cdot\eta_{g,1} - \eta_{g,1}\cdot\omega_{g,2} - 2\alpha_{\omega_{g,2},\eta_{g,1}}) \\
& = 2.4055874\ldots - (1.0710898\ldots)i. \end{align*}
The cycle $T_{g,2}$ corresponds to the element $e_2 = (0,\sqrt{2})$ in $\mathbf Q \times \mathbf Q(\sqrt{2})$, which belongs to $\mathbf T_{\mathbf Z}$ by inspection of the Fourier coefficients of $f$ and $g$. By evaluating the Weierstrass uniformisation
on $z_{g,f,s}$ we find:
$$P_{g,f,2} = W(z_{g,f,2}) \otimes 1 = (-1 : 0 : 1) \otimes 1 \in E(\mathbf Q) \otimes \mathbf Q.$$


\subsection{Table}\label{table}
In Table \ref{tab} we report some Chow-Heegner points that lie on elliptic  
 curves over $\mathbf Q$ of rank 1 and conductor $<100$. 
The format of the table is as follows. We list the strong Weil curve $E$ in each isogeny class of rank $1$.
Let $N$ be the conductor of the curve $E$ in question, and $f\in S_2(\Gamma_0(N))$ the newform corresponding to $E$. 
The first two columns of the table list the label for $E$ in Cremona's database
and a choice of generator for $E(\mathbf Q)$, using the canonical minimal Weierstrass equation for $E$
with invariants $a_1,a_3\in \{0,1\}$ and $a_2\in \{-1,0,1\}$.

The next column indicates a Galois orbit of normalized Hecke eigenforms $g \in S_2(\Gamma_0(N))$.
The column gives the index of the orbit in question when the orbits are ordered lexicographically by the sequence of traces of $T_1$ and $T_p$ for all primes $p$, starting with index 0. In particular, the trace of $T_1$ is equal to twice the dimension of $S_2(N)[g]$, so the components are sorted first by dimension, and rational newforms are always listed first. As a result, the $f$-isotypic component, which is omitted from the table, is typically the index 0 component.
In parentheses we give the integer $N_g | N$ such that $g$ is associated to a newform of level $N_g$.

The fourth column gives an integer $n\geq 1$ corresponding to a Hecke operator $T_n$.
The table includes $[K_g:\mathbf Q]\cdot\sigma(N/N_g)$ values of $n$ such that $T_g\cdot T_n$ form a $\mathbf Q$-basis for
for the direct factor $K_g$ of $\mathbf T$.
The fifth column lists $P_{g,f,n}$ as an element of $E(\mathbf Q)\otimes_{\mathbf Z}\mathbf Q$, in terms of the generator $P$ in the second column.
The sixth column gives the denominator $d_{g,n}$ defined above.




\begin{table}[ht]
 \caption{Chow-Heegner points on curves of rank 1 and conductor $<100$}
 \label{tab}
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|m{1.3cm}|m{1.75cm}|r|r|r|r|r|r|}
\hline
Curve $E = E_f$ &generator $P\in E(\mathbf Q)$ &
$g$ ($N_g$) & $n$ & $P_{g,f,n}$ & $d_{g,n}$ % & $d_{g,n}P_{g,f,n}$ % & $I_{g,f,n}$
\\
\hline\hline
{\bf 37a1}& $(0,-1)$& 1 (37) & 1& $-6P$&2 %& $(\frac{1357}{841},\frac{28888}{24389})$ %& 12
\\
\hline
{\bf 43a1}& $(0,-1)$& 1 (43)& 1 & $4P$& 2 %& $(\frac{11}{49}, -\frac{363}{343})$%& 8 
\\         &       &   &  2 & $2P$& 1% & $(-1,0)$                           %& 2 
\\
\hline
{\bf 53a1}& $(0,-1)$& 1 (53)& 1& $-2P$ & 2
\\
&&&2 & $-8P$& 2
\\
&&&3&$4P$&2 
\\
\hline
{\bf 57a1}& $(2,1)$ & 1 (57) & 1 & $\frac43P$& 12 % & $(-\frac{1976}{7569}, \frac{750007}{658503})$ %& 16 
\\
          &       & 2 (57) & 1 & $-\frac{16}{3}P$ &3 %& $(-\frac{1976}{7569},-\frac{1408510}{658503})$ %&16 
\\
          &       & 3 (19) & 1 & $-4P$ & 2 %& $(\frac{13}{9}, -\frac{28}{27})$          %& 8 
  \\
          &       &   & 3& $-4P$ & 2 %& $(\frac{13}{9}, -\frac{28}{27})$          %& 
   \\
\hline
{\bf 58a1}& $(0,-1)$& 1 (58)& 1& $4P$ & 4  %& $(\frac{50959}{12769}, -\frac{12940767}{1442897})$ %&16 
\\
          &       & 2 (29)&  1 & 0   & 2 %& O %& 8
\\
           &      &   &  2 & $4P$ & 2 %& $(15,-64)$%&
 \\
           &      &   &  3& $4P$ & 2 %& $(15,-64)$%& 
\\
           &      &   &  4& $4P$ & 2 %& $(15,-64)$%& 
\\
 \hline
{\bf 61a1}& $(1,-1)$& 1 (61) & 1 & $-2P$ & 2 %& $(\frac34, -\frac58)$ %&4
\\
 & & & 2 & $4P$ & 2 %& $(-\frac{111}{64}, \frac{475}{512})$ %&
\\
 &&& 3 & $-4P$ & 1 %& $(\frac34,-\frac58)$ %&
\\
\hline
{\bf 65a1}& $(-1,1)$& 1 (65)& 1& $P$ & 2 %& $(4,-10)$ %& 2
\\
          &       & & 2 & $P$ & 2 %& $(4,-10)$ %&
  \\
 && 2 (65)& 1 & $P$ & 2 %& $(4,-10)$ %& 2
\\
 &&&2& $3P$ & 2 %& $(\frac{498436}{540225},-\frac{74002214}{397065375})$ %&
\\
\hline
{\bf 77a1} & $(2,3)$ & 1 (77) & 1 & $\frac{12}{5}P$ & 20 
\\
 && 2 (77)& 1 & $-\frac43P$ & 6
\\
&& 3 (11)& 1 & $\frac43P$ & 6
\\
&&& 7 & $\frac{44}{3}P$ & 6
\\
&& 4 (77)& 1 & $-\frac{12}{5}P$ & 10
\\
&&& 2 & $-4P$ & 2
\\
\hline
{\bf 79a1} & $(0,0)$& 1 (79)& 1 & $-4P$ &2 
\\
&&& 2 & $-4P$& 2 
\\
&&&3 &$-4P$ & 2 
\\
&&&4 & $0$ &2
\\
&&&5 &$0$& 2
\\
\hline
{\bf 82a1}& $(0,0)$ & 1 (82)& 1 & $0$ & 4 %& O %&4 
\\
          &       &  & 3 & $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %&
\\
          &       & 2 (41)& 1 & $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %& 4
\\
         &       &  & 2& $0$ & 2 %& O %& 4
\\
 &       &  & 3& $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %& 4
\\
 &       &   & 4& $0$ & 2 %& O
\\
 &       &  & 5 & $4P$ & 2 %& $(\frac{1693226}{133225},\frac{1881510029}{48627125})$ %& 4
\\
 &       &  & 6& $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %& 4
\\ 
\hline

{\bf 83a1} & $(0,0)$& 1 (83)&1 & $0$ & 2
\\
&&&2&$2P$& 2
\\
&&&3&$4P$& 2
\\
&&&4&$-4P$& 2
\\
&&&5&$-4P$& 1
\\
&&&7& $0$ &2
\\
\hline
\end{tabular}
}
\hfill
\parbox{.45\linewidth}{
\centering
\begin{tabular}{|m{1.3cm}|m{1.75cm}|r|r|r|r|r|}
\hline
Curve $E = E_f$ &generator $P\in E(\mathbf Q)$ &
$g$ ($N_g$) & $n$ & $P_{g,f,n}$ & $d_{g,n}$ % & $d_{g,n}P_{g,f,n}$ % & $I_{g,f,n}$
\\
\hline\hline

{\bf 88a1} &$(2,-2)$ & 1 (88)& 1 & 0 & 16
\\
&&& 3 & 0& 16
\\
&&2 (44)&1 & 0& 8
\\
&&& 2 & $8P$ & 2
\\
&&3 (11)& 1 & 0 & 2
\\
&&& 2 & $8P$ & 2
\\
&&& 4& $16P$ & 1
\\
&&& 8 & $16P$ & 1 
\\
\hline
{\bf 89a1}&$(0,-1)$& 1 (89) & 1 & $\frac85P$ & 5
\\[.1cm]
&& 2 (89) & 1 &$\frac25P$ &10 \\[.1cm]
&&& 2 &$\frac{22}{5}P$&10 \\[.1cm]
&&& 3&$-\frac{16}{5}P$& 10\\[.1cm]
&&& 4 &$-\frac25P$& 10\\[.1cm]
&&& 6 &$\frac{24}5P$&10\\[.1cm]
\hline
{\bf 91a1} & $(0,0)$ & 1 (91) & 1 & $2P$ & 4
\\
 & & 2 (91)& 1 & $2P$ & 4
\\
 &&& 2 & $-2P$ & 2
\\
&& 3 (91)& 1 & $4P$ & 4
\\
&&& 2 & $2P$ & 2
\\ 
&&& 3 & $6P$ & 2
\\
\hline
{\bf 91b1} & $(-1,3)$ & 0 (91)& 1 & $0$ & 4
\\
&& 2 (91) & 1 & $0$ & 4
\\
&&  & 2& $0$ & 2
\\
&& 3 (91) & 1 & $0$ & 4
\\
&&& 2& $0$ & 2
\\
&&& 3& $0$& 2
\\
\hline
{\bf 92b1} & $(1,1)$ & 1 (92) & 1 & $0$& 2
\\
&& 2 (46) &1 & $0$ & 15
\\
&&& 2 & $0$ & 5
\\
&&3 (23)&1&$0$ & 20
\\
&&&2& $0$& 5
\\
&&&3& $0$& 4
\\
&&&4&$0$& 6
\\
&&&6&$0$&1
\\
&&&8&$0$&5
\\[.1cm]
\hline 
{\bf 99a1}& $(2,0)$ & 1 (99)& 1 & $-\frac23 P$ & 12 %& $(\frac{252307233}{12503296}, \frac{3456859444039}{44211654656})$ %&8
\\[.1cm]
          &       & 2 (99)& 1 & $0$ & 12 %&O %& 
 \\[.1cm]
          &       & 3 (99)& 1 & $\frac23P$ & 6 %& $(\frac{105}{64}, \frac{455}{512})$ %& 4
 \\[.1cm]
          &       & 4 (33)& 1 & $\frac23P$ & 12 %& $(\frac{252307233}{12503296},-\frac{4329322974583}{442116511654656})$%&8 
\\[.1cm]
       &       &  &3 & $\frac83P$ & 3 %&$(\frac{252307233}{12503296},-\frac{4329322974583}{442116511654656})$%&8 
\\[.1cm]
&& 5 (11) & 1 &  $-\frac23P$ & 6 %& $(\frac{105}{64},-\frac{897}{512})$ %&2
 \\[.1cm]
&&&  3 & $-\frac23P$ &3 %& $(3,2)$ %&
\\[.1cm]
&&& 9 & $-\frac{22}{3}P$ & 3 %& * %& 
\\[.1cm]\hline
\end{tabular}
}
\end{table}

% \begin{table}[ht]
% \caption{Chow Heegner points on curves of rank 1 and conductor $<100$}
% \label{tab}

% \begin{tabular}{|m{1.8cm}|m{1.8cm}|r|r|r|r|r|}
% \hline
% Curve $E = E_f$ &generator $P\in E(\mathbf Q)$ &
% $g$ & $n$ & $P_{g,f,n}$ & $d_{g,n}$ % & $d_{g,n}P_{g,f,n}$ % & $I_{g,f,n}$
% \\[.3cm]
% \hline\hline
% % \end{tabular}

% % \begin{tabular}[h]{|c||m{1.8cm}||rl||r|r|}
% % \hline

% {\bf 37a1}& (0,-1)& 1 & 1& $-6P$&2 %& $(\frac{1357}{841},\frac{28888}{24389})$ %& 12
% \\[.1cm]
% \hline
% {\bf 43a1}& (0,-1)& 1 & 1 & $4P$& 2 %& $(\frac{11}{49}, -\frac{363}{343})$%& 8 
% \\[.1cm]
%            &       &   &  2 & $2P$& 1% & $(-1,0)$                           %& 2 
% \\[.1cm]
% \hline
% {\bf 53a1}& (0,-1)& 1& 1& $-2P$ & 2
% \\[.1cm]
% &&&2 & $-8P$& 2
% \\[.1cm]
% &&&3&$4P$&2 
% \\[.1cm]
% \hline
% {\bf 57a1}& (2,1) & 1 & 1 & $\frac43P$& 12 % & $(-\frac{1976}{7569}, \frac{750007}{658503})$ %& 16 
% \\[.1cm]
%           &       & 2  & 1 & $-\frac{16}{3}P$ &3 %& $(-\frac{1976}{7569},-\frac{1408510}{658503})$ %&16 
% \\[.1cm]
%           &       & 3  & 1 & $-4P$ & 2 %& $(\frac{13}{9}, -\frac{28}{27})$          %& 8 
%   \\[ .1cm]
%           &       &   & 3& $-4P$ & 2 %& $(\frac{13}{9}, -\frac{28}{27})$          %& 
%    \\[ .1cm]
% \hline
% {\bf 58a1}& (0,-1)& 1& 1& $4P$ & 4  %& $(\frac{50959}{12769}, -\frac{12940767}{1442897})$ %&16 
% \\[.1cm]
%           &       & 2 &  1 & 0   & 2 %& O %& 8
% \\[.1cm]
%            &      &   &  2 & $4P$ & 2 %& $(15,-64)$%&
%  \\[.1cm]
%            &      &   &  3& $4P$ & 2 %& $(15,-64)$%& 
% \\[.1cm]
%            &      &   &  4& $4P$ & 2 %& $(15,-64)$%& 
% \\[.1cm]
%  \hline
% {\bf 61a1}& (1,-1)& 1 & 1 & $-2P$ & 2 %& $(\frac34, -\frac58)$ %&4
% \\[.1cm]
%  & & & 2 & $4P$ & 2 %& $(-\frac{111}{64}, \frac{475}{512})$ %&
% \\[.1cm]
%  &&& 3 & $-4P$ & 1 %& $(\frac34,-\frac58)$ %&
% \\[.1cm]
% \hline
% {\bf 65a1}& (-1,1)& 1& 1& $P$ & 2 %& $(4,-10)$ %& 2
% \\[.1cm]
%           &       & & 2 & $P$ & 2 %& $(4,-10)$ %&
%   \\[.1cm]
%  && 2 & 1 & $P$ & 2 %& $(4,-10)$ %& 2
% \\[.1cm]
%  &&&2& $3P$ & 2 %& $(\frac{498436}{540225},-\frac{74002214}{397065375})$ %&
% \\[.1cm]
% \hline
% {\bf 77a1} & (2,3) & 1 & 1 & $\frac{12}{5}P$ & 20 
% \\[.1cm]
%  && 2 & 1 & $-\frac43P$ & 6
% \\[.1cm]
% && 3 & 1 & $\frac43P$ & 6
% \\[.1cm]
% &&& 7 & $\frac{44}{3}P$ & 6
% \\[.1cm]
% && 4 & 1 & $-\frac{12}{5}P$ & 10
% \\[.1cm]
% &&& 2 & $-4P$ & 2
% \\[.1cm]
% \hline
% {\bf 79a1} & & 1 & 1 & $-4P$ &2 
% \\[.1cm]
% &&& 2 & $-4P$& 2 
% \\[.1cm]
% &&&3 &$-4P$ & 2 
% \\[.1cm]
% &&&4 & $0$ &2
% \\[.1cm]
% &&&5 &$0$& 2
% \\[.1cm]
% \hline

% {\bf 106b1}& (2,1) & $\frac{12}{5}P$ & 10& 24&38&1800\\
%            &       & $-\frac43 P$ & 6 & 8 && \\
%            &       & $-\frac{11}{3}P$&48& 176& & \\
%            &       & $P$ & 16 & 16& & \\
%            &       & $\frac{28}{5}P$ & 10 & 56 && \\[.1cm]
% \hline
% {\bf 122a1}& (1,-3)& $-\frac{16}{13}P$ &  26 & 32&  28& 1600\\
%            &       & $-P$ & 16 & 16 & & \\
%            &       & $P$  & 16 & 16 & & \\
%            &       & $ -\frac{36}{13}P$ & 26 & 72 & & \\[.1cm]

% \hline
% {\bf 129a1}& (1,-5)& $-\frac{16}{15}P$& 15& 16&  28& 1600 \\
%  &&$-\frac43P$&12 &16&&\\
% &&$-\frac{20}{7}P$ & 14&40&&\\
% &&$ -\frac85P$& 40&64&&\\
% &&$-\frac87P$& 14&16&&\\[.1cm]
% \hline
% {\bf 153a1}& (0,1)& $P$  & 48 & 48& 24 & 1800  \\
%            &      & $2P$ & 24 &   48&  &        \\
%            &      & 0    & 24 &    $\infty$&&        \\
%            &      & $-P$ & 48 &    48&&       \\
%            &      & $-P$ & 16 &    16&&       \\
%            &      & $-2P$& 24 &    48&&       \\
%            &      & $P$  & 16 &    16&&       \\[.1cm]
% \hline
% \hline
% {\bf 106b1}& (2,1) & $\frac{12}{5}P$ & 10& 24&38&1800\\
%            &       & $-\frac43 P$ & 6 & 8 && \\
%            &       & $-\frac{11}{3}P$&48& 176& & \\
%            &       & $P$ & 16 & 16& & \\
%            &       & $\frac{28}{5}P$ & 10 & 56 && \\[.1cm]
% \hline
% {\bf 122a1}& (1,-3)& $-\frac{16}{13}P$ &  26 & 32&  28& 1600\\
%            &       & $-P$ & 16 & 16 & & \\
%            &       & $P$  & 16 & 16 & & \\
%            &       & $ -\frac{36}{13}P$ & 26 & 72 & & \\[.1cm]

% \hline
% {\bf 129a1}& (1,-5)& $-\frac{16}{15}P$& 15& 16&  28& 1600 \\
%  &&$-\frac43P$&12 &16&&\\
% &&$-\frac{20}{7}P$ & 14&40&&\\
% &&$ -\frac85P$& 40&64&&\\
% &&$-\frac87P$& 14&16&&\\[.1cm]
% \hline
% {\bf 153a1}& (0,1)& $P$  & 48 & 48& 24 & 1800  \\
%            &      & $2P$ & 24 &   48&  &        \\
%            &      & 0    & 24 &    $\infty$&&        \\
%            &      & $-P$ & 48 &    48&&       \\
%            &      & $-P$ & 16 &    16&&       \\
%            &      & $-2P$& 24 &    48&&       \\
%            &      & $P$  & 16 &    16&&       \\[.1cm]
% \hline
% \end{tabular}



% \begin{tabular}{|m{1.8cm}|m{1.8cm}|r|r|r|r|r|}
% \hline
%  Curve $E = E_f$ &generator $P\in E(\mathbf Q)$ &
%  $g$ & $n$ & $P_{g,f,n}$ & $d_{g,n}$ % & $d_{g,n}P_{g,f,n}$ % & $I_{g,f,n}$
%  \\[.3cm]
%  \hline\hline
% % \end{tabular}
% {\bf 82a1}& (0,0) & 1 & 1 & $0$ & 4 %& O %&4 
% \\[.1cm]
%           &       &  & 3 & $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %&
% \\[.1cm]
%           &       & 2 & 1 & $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %& 4
% \\[.1cm]
%          &       &  & 2& $0$ & 2 %& O %& 4
% \\[.1cm]
%  &       &  & 3& $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %& 4
% \\[.1cm]
%  &       &   & 4& $0$ & 2 %& O
% \\[.1cm]
%  &       &  & 5 & $4P$ & 2 %& $(\frac{1693226}{133225},\frac{1881510029}{48627125})$ %& 4
% \\[.1cm]
%  &       &  & 6& $2P$ & 2 %& $(\frac{26}{25},-\frac{164}{125})$ %& 4
% \\[.1cm] 
% \hline
% {\bf 83a1}
% \\[.1cm]
% \hline
% {\bf 88a1}
% \\[.1cm]
% \hline
% {\bf 91b1}
% \\[.1cm]
% \hline
% {\bf 92b1}
% \\[.1cm]
% \hline
% {\bf 99a1}& (2,0) & 1& 1 & $-\frac23 P$ & 12 %& $(\frac{252307233}{12503296}, \frac{3456859444039}{44211654656})$ %&8
% \\[.1cm]
%           &       & 2 & 1 & $0$ & 12 %&O %& 
%  \\[.1cm]
%           &       & 3 & 1 & $\frac23P$ & 6 %& $(\frac{105}{64}, \frac{455}{512})$ %& 4
%  \\[.1cm]
%           &       & 4 & 1 & $\frac23P$ & 12 %& $(\frac{252307233}{12503296},-\frac{4329322974583}{442116511654656})$%&8 
% \\[.1cm]
%           &       &  &3 & $\frac83P$ & 3 %&$(\frac{252307233}{12503296},-\frac{4329322974583}{442116511654656})$%&8 
% \\[.1cm]
% && 5 & 1 &  $-\frac23P$ & 6 %& $(\frac{105}{64},-\frac{897}{512})$ %&2
%  \\[.1cm]
% &&&  3 & $-\frac23P$ &3 %& $(3,2)$ %&
% \\[.1cm]

% &&& 9 & $-\frac{22}{3}P$ & 3 %& * %& 
% \\[.1cm]\hline

% \end{tabular}

% \end{table}
% \begin{table}
% \begin{threeparttable}
% \caption{Some elliptic curves with only torsion Chow-Heegner points}
% \begin{tabular}[h]{|c||m{1.8cm}||r|l|c||r|r||c|}
% \hline
% Curve $E=E_f$ &generator $P\in E(\mathbf Q)$ & $P_{g,f}$& $d_g$ & $E(\mathbf Q)_{\mathrm{tor}}$ & \multicolumn{2}{m{2.1cm}|}{accuracy and \# coeff.}&reason\\[.3cm]
% \hline
% \hline
% {\tt 33a1}& rank 0& 0 & 3 & $\mathbf Z/2\oplus \mathbf Z/2$ &53& 800& OK\\[.1cm]
% \hline
% {\tt 34a1}& rank 0 &$(2,-3)$\tnote{a}& 2& $\mathbf Z/6$& 56 & 800 & OK\\[.1cm]
% \hline
% {\tt 35a1}& rank 0 &$(1,-4)$\tnote{a} & 2 & $\mathbf Z/3$ &54 & 800 & OK\\[.1cm]
% \hline
% {\tt 88a1}& (2,-2)& 3 points, all zero & -\tnote{b} & 0& 19&1600& ()\\[.1cm]
% %  $0$ & 16& 19& 1600\\
% %           &       & $0$ & 8 &   &     \\
% %           &       & $0$ & 2 &   &     \\[.1cm]
% \hline
% {\tt 92b1}& (1,1) & 3 points, all zero & -\tnote{b} & 0 & 30 &1200& ()\\[.1cm]
% % $0$ & 2& 30 & 1200\\
% %           &       & $0$ & 15 &   &     \\
% %           &       & $0$ & 20 &   &     \\[.1cm]
% \hline
% {\tt 112a1}& (0,-2)& 4 points, all zero & -\tnote{b} & $\mathbf Z/2$ &  23& 2400&()\\
%  && $\frac{1}{2}\cdot$(irrational 2-torsion point)\tnote{c}& 16& & &&\\[.1cm]
% % $0$ & 8&23&2400 \\
% %   & & $0$ & 16 & & \\
% %   & & $0$ & 8 & & \\
% %    & & $0$ & 16 & & \\  
% % & & $0$ & 16 & & \\[.1cm]
% \hline

% {\tt 117a1}& $(-\frac14,\frac{15}8)$ & 4 points, all zero & -\tnote{b} & $\mathbf Z/4$ & 23 & 1200&()\\[.1cm]
% % & $0$ & 4 & 23&1200 \\
% %            &                         & $0$ & 8 &   &     \\
% %            &                         & $0$ & 8 &   &     \\
% %            &                         & $0$ & 8 &   &     \\[.1cm]
% \hline
% {\tt 124a1}& (-2,1)& 4 points, all zero & -\tnote{b} & 0 & 30 & 1600 & ()\\[.1cm] 
% % $0$& 6 & 30 &1600 \\
% % &&0&6&&\\
% % &&0&66&&\\
% % &&0&44&&\\[.1cm]
% \hline
% {\tt 136a1}& (2,-2)& 6 points, all zero & -\tnote{b} & $\mathbf Z /2$ & 29 & 3600 & () \\[.1cm]
% % $0$& 8& 29& 3600\\
% % &&0&8&&\\
% % &&0&8&&\\
% % &&0&24&&\\
% % &&0&8&&\\
% % &&0&12&&\\[.1cm]
% \hline
% {\tt 148a1}& (-1,2) & 5 points, all zero & -\tnote{b} & 0 & 25 & 1600&()\\[.1cm]
% % &$0$& 12 & 25 & 1600 \\
% %  && 0 & 20 && \\
% %  && 0 & 12 && \\
% %  && 0 & 6 && \\
% %  && 0 & 30 && \\[.1cm]

% \hline
% {\tt 152a1}& (-1,-2)& 6 points, all zero & -\tnote{b} & 0 & 10 & 1200\\[.1cm]
% % 0 & 8 &10&1200  \\
% % && 0 & 24 &&\\
% % && 0 & 24 &&\\
% % && 0 & 16 &&\\
% % && 0 & 24 &&\\
% % && 0 & 6 &&\\
%  \hline

% {\tt 153b1}& (5,-14)& 7 points, all zero & -\tnote{b} & 0 & 25& 1800\\[.1cm]
% % $0$ & 24 & 25 & 1800 \\
% %            &        & $0$ & 24 &    &       \\
% %            &        & $0$ & 24 &    &       \\
% %            &        & $0$ & 48 &    &       \\
% %            &        & $0$ & 16 &    &       \\
% %            &        & $0$ & 24 &    &       \\
% %            &        & $0$ & 16 &    &       \\[.1cm]
% \hline
% {\tt 171b1}& (2, -5)& 8 points, all zero & -\tnote{b} & 0 & 15 & 1200\\[.1cm]
% % 0 & 24& 15 & 1200\\
% %  && 0 & 96 && \\
% % && 0 & 32 && \\
% % && 0 & 32 && \\
% % && 0 & 96 && \\
% % && 0 & 12 && \\
% % && 0 & 24 && \\
% % && 0 & 12 && \\[.1cm]

% \hline
% {\tt 172a1}&(-3,6)& 5 points, all zero &-\tnote{b}& $\mathbf Z/3$& 18 & 1400 & ()\\[.1cm]
% %$0^{12},\quad 0^{20},\quad 0^{42}$, $\quad 0^{30},\quad 0^{28}$& 16&1200\\
% \hline
% {\tt 176c1}&(1,-2)& 7 points, all zero & -\tnote{b} & 0 & 9 & 1600 \\[.1cm]
% %$0^{2^{13}},\quad 0^{2^{11}\cdot 3}, \quad 0^{2^{11}\cdot 3},\quad 0^{2^{14}}$, $\quad 0^{2^{10}\cdot 3},\quad 0^{2^{14}},\quad 0^{2^{13}}$&9&1600\\
% \hline
% {\tt 184a1}&(0,1)& 8 points, all zero & -\tnote{b} & 0 & 30 & 2400 &\\[.1cm]
% %&$0^{2^{11}},\quad0^{96},\quad0^{2^{11}\cdot 3},\quad 0^{2^8\cdot 3^2},\quad 0^{2^{11}}$, $\quad 0^{2^8},\quad 0^{2^5\cdot 3^2\cdot 5},\quad 0^{2^{12}\cdot 5}$&30&2400\\
% \hline


% \end{tabular}
% \begin{tablenotes}
% \item [a] 3-torsion point
% \item [b] various Hecke indices $d_g$ depending on $g$
% \item [c] Namely, ``$(1,1)$'' if we identify $\mathbf E[2](\mathbf C)\cong \mathbf Z/2\oplus\mathbf Z/2$.
% \end{tablenotes}
% \end{threeparttable}
% \end{table}


\subsection{Further observations}
We conclude with a few remarks.

\subsubsection{}
Table \ref{tab} includes several examples where the submodule $\underline{P}_{\mathbf T[g],f}\subset \underline{P}_{g,f}\subset E(\mathbf Q)\otimes_\mathbf Z\mathbf Q$ vanishes:
namely, this occurs when $E$ has label {\bf 88a1}
and $g$ has index 1, for all $g$ when $E$ is labeled {\bf 91b1} or {\bf 92b1}, and when $E$ has label {\bf 99a1} and $g$ has index 2,.
This raises the question of whether such vanishing can be explained via Theorem \ref{DRS1.4}. 

%Unfortunately the hypothesis of that theorem concerning the local signs of the triple product
%$L$-function is not entirely straightforward to verify.
%However, 

At primes $p$ which exactly divide the level $N$,
the local sign depends only on $E$ and can be computed easily from the Fourier expansion of $f$ as $\epsilon_p(g,g,f) = -a_p(f)=1$.

Thus, for example, one can check that for the curve labeled {\bf 91b1} we have $\epsilon_7(g,g,f) = -a_7(f) = -1$
and $\epsilon_{13}(g,g,f) = -a_{13}(f) = -1$. Hence Theorem \ref{DRS1.4} does not apply; nevertheless, in this case we expect the special value  $L(f\otimes \operatorname{Sym}^2(g),2)$ not to vanish --a claim which should be in principle numerically verifiable by means of Dokchitser's L-function calculator (cf.\,\cite{sage})-- and non-torsion points on $E(\Q)$ to arise as the Chow-Heegner points associated to the Shimura curve of full level structure on the quaternion algebra of discriminant $91$. 

Similarly, one can explain the vanishing of $\underline{P}_{\mathbf T[g],f}$ on the elliptic curve {\bf 92b1} because the prime $p=23$ exactly divides $92$ and $\epsilon_{23}(g,g,f)=-a_{23}(f) = -1$.
%One might hazard the guess, based on such examples, that the hypothesis on local signs
%in Theorem \ref{DRS1.4} is actually a necessary condition for the modules $\underline{P}_{g,f}$
%to be nonzero.

As for the examples in Table \ref{tab} of torsion Chow-Heegner points at levels 88 and 99, we have $\epsilon_{11}(g,g,f) = -a_{11}(f) = +1$ in both cases, but the local sign at $2$ (respectively, at $3$) is somewhat more dificult to compute: the reader interested in pursuing this computation will find it helpful to follow the recipe for that sign given by Prasad in \cite{Pr}, which can be approached numerically thanks to the work of Loeffler and Weinstein \cite{LoWe}.

In the tables of Stein (see Section~\ref{sec:yzz})
there is an example of a torsion Chow-Heegner point corresponding to two newforms of level 158.
Since this level is squarefree, the hypotheses of Theorem \ref{DRS1.4} can be checked,
and since the rank of the corresponding elliptic curve (labeled {\bf 158b1})
is exactly 1, it would appear that $L(f\otimes \operatorname{Sym}^2(g),2)=0$ in this example.


\subsubsection{}
Examining Table \ref{tab}, we see that in each example in which $f$ and $g$ are both newforms of level $N$
(such as when $E$ has label {\bf 37a1},
or some of the examples when $E$ has label {\bf 57a1}, {\bf 77a1}, {\bf 99a1}, etc.)
 the
index $[E(\mathbf Q):\mathbf Z d_{g,1} P_{f,g,1}]$ is divisible by 4.
See Section~\ref{sec:appdiscuss}
for further discussion of this phenomenon.


\subsubsection{}
Let $E=E_f$ be the elliptic curve with Cremona label {\bf 99a1}.
As $g$ runs through the Galois orbits of eigenforms of level 99,
let the corresponding newforms of levels $N_g|99$ be
\[g_{99}^{(1)},g_{99}^{(2)},g_{99}^{(3)},g_{33},g_{11}.\]
The newforms $g_{99}^{(1)}$ and 
$g_{99}^{(3)}$ are twists by the quadratic character of conductor 3
of, respectively, $g_{33}$ and $g_{11}$.
Table \ref{tab} shows that in $E(\mathbf Q)\otimes\mathbf Q$ we have
\[P_{f,g_{99}^{(1)},1} = - P_{f,g_{33},1} = -\tfrac23P\]
\[P_{f,g_{99}^{(3)},1} = - P_{f,g_{11},1} = \tfrac23P,\]
where $P=(2:0:1)\in  E(\mathbf Q)$ is a generator. 
These observations are consistent with the expectation that not only the height
(cf. Theorem \ref{DRS1.4})
but even the exact index $[E(\mathbf Q):\mathbf Z d_{g,n} P_{g,f,n}]$
can be extracted from $L$-values of the form $L(f\otimes\operatorname{Sym}^2g,2)$
which are invariant under quadratic twists of $g$.



\newpage
\section[Appendix by William Stein]{ Appendix: Chow-Heegner points associated to pairs of elliptic curves}
\label{sec:app}

\newcommand{\ncisom}{\approx}   % noncanonical isomorphism
\newcommand{\isom}{\cong}
\newcommand{\CR}[1]{#1}
\newcommand{\tor}{\rm tor}

\newcommand{\tensor}{\otimes}
\newcommand{\h}{\mathfrak{h}}

\begin{center}
by William Stein
\end{center}

\markboth{WILLIAM STEIN} {APPENDIX: CHOW-HEEGNER POINTS ASSOCIATED TO PAIRS OF ELLIPTIC CURVES}

\medskip
In this appendix, we consider a special case of Chow-Heegner points
that has a simple concrete description due to S.~Zhang.  Given a
pair $E$, $F$ of nonisogenous elliptic curves, and  surjective
morphisms $\varphi_E: X_0(N) \to E$ and $\varphi_F: X_0(N) \to
F$ of curves over $\Q$, we associate a rational point $P\in E(\Q)$.
We describe a numerical approach to computing $P$, state some
motivating results of Zhang et al. about the height of $P$, and
present a table of data.

\subsection{Introduction}
Consider a pair $E,F$ of nonisogenous elliptic curves over $\Q$ and
fix surjective morphisms from $X_0(N)$ to each curve.  We do {\em not}
assume that $N$ is the conductor of either $E$ or $F$, though $N$ is
necessarily a multiple of the conductor.
$$
\xymatrix{
  & {X_0(N)}\ar[ld]_{\varphi_E}\ar[rd]^{\varphi_F} & \\
 E & & F}
$$
Let $(\varphi_E)_*:\Div(X_0(N))\to \Div(E)$ and 
$\varphi_F^*:\Div(F) \to \Div(X_0(N))$ be the pushforward
and pullback maps on divisors on algebraic curves. 
Let $Q\in F(\C)$ be any  point, and let
$$
  P_{\varphi_E, \varphi_F,Q} = \sum (\varphi_E)_* \varphi_F^*(Q) \in E(\C),
$$
where $\sum$ means the sum of the points in the divisor using the
group law on $E$, i.e., given a divisor $D = \sum n_i P_i \in \Div(E)$, 
we have $(\sum D) - \infty \sim D - \deg(D)\infty$, which uniquely
determines $\sum D$.

\begin{proposition}
The point $P_{\varphi_E, \varphi_F, Q}$ does not depend on the choice of $Q$.
\end{proposition}
\begin{proof}
The composition $(\varphi_E)_* \circ \varphi_F^*$ induces
a homomorphism of elliptic curves
$$
 \psi: \Pic^0(F)=\Jac(F) \to \Jac(E)=\Pic^0(E).
$$
Our hypothesis that $E$ and $F$ are nonisogenous implies that $\psi=0$.
We denote by $[D]$ the linear equivalence class of a divisor in the
Picard group.
If $Q'\in F(\C)$ is another point, then under the above composition of maps,
$$
  [Q - Q']\mapsto \left[(\varphi_E)_* \varphi_F^*(Q) - (\varphi_E)_*  \varphi_F^*(Q')\right]
 = [P_Q - P_{Q'}].
$$
Thus the divisor $P_Q - P_{Q'}$ is linearly equivalent to $0$.
But $F$ has genus $1$, so there is no rational function on $F$ of degree 1, 
hence $P_Q = P_{Q'}$, as claimed.
\end{proof}

We let $P_{\varphi_E, \varphi_F} = P_{\varphi_E, \varphi_F,Q} \in E(\C)$, for
any choice of $Q$.  

\begin{corollary}
We have $P_{\varphi_E, \varphi_F} \in E(\Q)$.
\end{corollary}
\begin{proof}
Taking $Q=\mathcal{O} \in F(\Q)$, we see that the divisor
$(\varphi_E)_* \circ \varphi_F^*(Q)$ is rational, so its sum
is also rational.
\end{proof}
In the rest of this appendix, we write 
$P_{E,F} = P_{\varphi_E, \varphi_F}$
when $E$ and $F$ are both optimal curves of the same conductor $N$,
and $\varphi_E$ and $\varphi_F$ are as in Section~\ref{sec:numerical}.

In Section~\ref{sec:37} we discuss an example in which $E$ and $F$
both have conductor $37$.  Section~\ref{sec:yzz} is about a formula of
Yuan-Zhang-Zhang for the height of $P_{E,F}$ in terms of the
derivative of an $L$-function, in some cases.  In
Section~\ref{sec:integrals}, we discuss the connection between this
appendix and the rest of the paper.  The heart of the appendix
is Section~\ref{sec:numerical}, which describes our numerical approach
to approximating $P_{E,F}$.  Finally, Section~\ref{sec:data} presents
a table of points $P_{E,F}$.

\vspace{1em}\noindent{}{\bf Acknowledgements: } We would like to thank
Ralph Greenberg, Ken Ribet, Barry Mazur, Karl Rubin, Shouwu Zhang, and
Jon Bober for helpful conversations related to this paper, and
especially thank Xinyi Yuan for introducing us to this topic, Henri
Darmon for encouraging us to flesh out the details and write it up for
publication, and Victor Rotger for clarifying some issues in
Section~\ref{sec:yzz}. We would also like to thank John Cremona for
refereeing the corresponding Sage \cite{sage} code and Cremona,
Bas Edixhoven and Samit Dasgupta for contributions to
Section~\ref{sec:g0nequiv}.  

\subsection{Example: $N=37$}\label{sec:37}

The smallest conductor curve of rank $1$ is the curve $E$ with Cremona
label \CR{37a} (see \cite{cremona:onlinetables}).  The paper
\cite{mazur-swinnerton-dyer:arithmetic} discusses the modular curve
$X_0(37)$ in detail.  It gives the affine equation
$y^2=-x^6-9x^4-11x^2 +37$ for $X_0(37)$, and describes
how $X_0(37)$ is equipped with three independent involutions $w$, $T$ and
$S$.  The quotient of $X_0(37)$ by $w$ is $E$, the quotient by $T$ is
an elliptic curve $F$ with $F(\Q)\ncisom \Z/3\Z$ and Cremona label
\CR{37b}, and the quotient by $S$ is the projective line $\PP^1$.
$$
\xymatrix{
  & {X_0(37)}\ar[ld]_{\varphi_E}\ar[d]^{\varphi_F}\ar[rd] & \\
  E=X/w & F=X/T & {\PP^1}=X/S }$$ The maps $\varphi_E$ and $\varphi_F$
have degree $2$, by virtue of being induced by an
involution.  As explained in \cite{mazur-swinnerton-dyer:arithmetic},
the fiber over $Q=0\in F(\Q)$ contains 2 points:
\begin{enumerate}\setlength{\itemsep}{.5ex}
\item the cusp $[\infty] \in X_0(37)(\Q)$, and
\item the noncuspidal affine rational point $(-1,-4) = T(\infty) \in X_0(37)(\Q)$.  
\end{enumerate}
We have $\varphi_E([\infty])=0\in E(\Q)$, and 
\cite[Prop.~3, pg.~30]{mazur-swinnerton-dyer:arithmetic}
implies that
$$\varphi_F((-1,-4)) = (6,14) = -6(0,-1),$$ 
where $(0,-1)$ generates $E(\Q)$.
We conclude that
$$
  P_{E,F} = (6,14)\qquad\text{and}\qquad[E(\Q):\Z P_{E,F}] = 6.
$$
On \cite[pg.~31]{mazur-swinnerton-dyer:arithmetic}, they remark: ``It
would be of the utmost interest to link this index  to something else in the theory.''

This remark motivates our desire to compute more examples.
Unfortunately, it is very difficult to generalize the above approach
directly, since it involves computations with $X_0(37)$ and its
quotients that rely on explicit defining equations.  Just as there are
multiple approaches to computing Heegner points, there
are several approaches to computing $P_{E,F}$:
\begin{itemize}\setlength{\itemsep}{.5ex}
\item a Gross-Zagier style formula for the height of $P_{E,F}$ (see Section~\ref{sec:yzz}),
\item explicit evaluation of iterated integrals (see Section~\ref{sec:integrals}), and
\item numerical approximation of the fiber
  in the upper half plane over a point on $F$ using a
  polynomial approximation to $\varphi_F$ (see Section~\ref{sec:numerical}).
\end{itemize}
This appendix is mainly about the last approach listed above.

\subsection{The Formula of Yuan-Zhang-Zhang}\label{sec:yzz}

Consider a special case of the triple product $L$-function of \cite{gross-kudla}
\begin{equation}\label{eqn:tprod}
L(E,F,F,s) = L(E,s) \cdot L(E,\Sym^{2}(F),s),
\end{equation}
where $E$ and $F$ are elliptic curves of the same conductor $N$,
and all $L$-functions are normalized so that $1/2$ is the
center of the critical strip.
The following theorem is proved in
\cite{YZZ}:
\begin{theorem}[Yuan-Zhang-Zhang]\label{thm:yzz}
  Assume that the local root number of $L(E,F,F,s)$ at every prime of bad
  reduction is $+1$ and that the root number at infinity is $-1$.  Then
$
  \hat{h}(P_{E,F}) = (*) \cdot L'(E,F,F,\frac{1}{2}),
$
where $(*)$ is nonzero.
\end{theorem}

\begin{remark}
The above formula resembles the Gross-Zagier formula 
$$ 
\hat{h}(P_K) = (*) \cdot (L(E/\Q,s) \cdot L(E^K/\Q,s))'|_{s=\frac{1}{2}},
$$
where $K$ is a quadratic imaginary field satisfying certain hypotheses. 
\end{remark}

If one could evaluate $L'(E,F,F,\frac{1}{2})$, e.g., by applying the
algorithm of \cite{dokchitser:lfun}, along with the factor $(*)$ in
the theorem, this would yield an algorithm to compute $\pm
P_{E,F}\pmod{E(\Q)_{\tor}}$ when the root number hypothesis is
satisfied.
Unfortunately, it appears that nobody has numerically evaluated
the formula of Theorem~\ref{thm:yzz} in any interesting cases.  

When $E$ and $F$ have the same squarefree conductor $N$,
\cite[\S1]{gross-kudla} implies that the local root number of
$L(E,F,F,s)$ at $p$ is the same as the local root number of $E$ at $p$;
computing the local root number when
the level is not square free is more complicated.
\begin{proposition}\label{prop:esymf}
  Assume that $E$ and $F$ have the same squarefree conductor $N$,
 that the local root numbers of $E$ at primes $p\mid N$
  are all $+1$ (equivalently, that we have $a_p(E)=-1$) and that
  $r_{\an}(E/\Q)=1$.  Then $L(E, \Sym^2 F, \frac{1}{2})\neq 0$ if and
  only if $\hat{h}(P_{E,F})\neq 0$.
\end{proposition}
\begin{proof}
By hypothesis, we have $L(E,\frac{1}{2})=0$ and  $L'(E,\frac{1}{2})\neq 0$.  
%First suppose $\hat{h}(P_{E,F})\neq 0$.
Theorem~\ref{thm:yzz} and the factorization \eqref{eqn:tprod} imply that
$$
\hat{h}(P_{E,F}) = (*) \cdot L'(E,\frac{1}{2}) \cdot L(E, \Sym^2 F, \frac{1}{2}),
$$
from which the result follows.
\end{proof}
%We hope to carry out this approach (building on \cite{dokchitser:lfun})
%in future work.



Section~\ref{sec:data} contains numerous examples in which $E$ has
rank $1$, $F$ has rank $0$, and yet $P_{E,F}$ is a torsion point.  The
first example is when $E$ is \CR{91b} and $F$ is \CR{91a}.  Then
$P_{E,F}=(1,0)$ is a torsion point (of order 3).  In this case, we
cannot apply Proposition~\ref{prop:esymf} since $\eps_7=\eps_{13}=-1$
for $E$.  Another example is when $E$ is \CR{99a} and $F$ is \CR{99c},
where we have $P_{E,F}=0$, and $\eps_3=\eps_{11}=+1$, but
Proposition~\ref{prop:esymf} does not apply since the level is not
square free. Fortunately, we found an example with squarefree level
$158=2\cdot 79$: here $E$ is \CR{158b}, $F$ is \CR{158d}, we have
$P_{E,F}=0$ and $\eps_{2}=\eps_{79}=+1$, so Proposition~\ref{prop:esymf}
implies that $L(E,\Sym^2 F, \frac{1}{2})=0$.

%{\em Does this mean anything in terms of the arithmetic of $E$ and $F$?}

\subsection{Iterated Complex Path Integrals}\label{sec:integrals}

The body of this paper contains
a  general approach using iterated path integrals to compute
certain Chow-Heegner points, of which $P_{E,F}$ is  a
specific instance.  Comparing our data (Section~\ref{sec:data}) with
theirs, we find that if $E$ and $F$ are optimal elliptic curves over
$\Q$ of the same conductor $N\leq 100$, if $e, f\in
S_2(\Gamma_0(N))$ are the corresponding newforms, and if $P_{f,e,1} \in
E(\Q)\tensor_{\Q}\Q$ is the associated Chow-Heegner point (as in the rest of
this paper),
then $2P_{E,F} = P_{f,e,1}$.
This is (presumably) a consequence of \cite{DRS}.


\subsection{A Numerical Approach to Computing $P_{E,F}$}\label{sec:numerical}

The numerical approach to computing $P$ that we describe in this
section uses relatively little abstract theory.  It is inspired by
work of Delaunay (see \cite{delaunay:thesis}) on computing the fiber
of the map $X_0(389)\to E$ over rational points on the rank $2$ curve
$E$ of conductor $389$.  We make no guarantee about how many digits of
our approximation to $P_{E,F}$ are correct, instead viewing this as an
algorithm to produce something that is useful for experimental
mathematics only.


Let $\h$ be the upper half plane, and let $Y_0(N) =
\Gamma_0(N)\backslash \h \subset X_0(N)$ be the affine modular curve.
Let $E$ and $F$ be nonisogenous optimal elliptic curve quotients of
$X_0(N)$, with modular parametrization maps $\varphi_E$ and $\varphi_F$,
and assume both Manin constants are $1$.  Let $\Lambda_E$ and
$\Lambda_F$ be the period lattices of $E$ and $F$, so $E\isom
\C/\Lambda_E$ and $F\isom \C/\Lambda_F$.  Viewed as a map $[\tau]
\mapsto \C/\Lambda_E$, we
have, using square brackets to denote equivalence classes, that
$$
\varphi_E([\tau]) = \left[\sum_{n=1}^{\infty} \frac{a_n}{n} e^{2\pi i n \tau}\right],
$$
and similarly for $\varphi_F$, where $a_n = a_n(E)$ are the $L$-series
coefficients of $E$ (see \cite[\S2.10]{cremona}, which uses the
oppositive sign convention).  For any positive integer $B$, define the
polynomial
$$\varphi_{E,B}=\sum_{n=1}^B \frac{a_n}{n} T^n \in \Q[T],$$
and similarly for $\varphi_{F,B}$.

To approximate $P_{E,F}$, we proceed as follows.  First we make some
choices, and after making these choices we run the algorithm, which
will either find a ``probable'' numerical approximation to $P_{E,F}$
or fail. 
\begin{itemize}\setlength{\itemsep}{.5ex}
\item $y\in \R_{>0}$ -- minimum imaginary part of points in fiber,
\item $d\in\Z_{>0}$ -- degree of the first approximation to $\varphi_F$ in Step~\ref{step:lowprecroots} below,
\item $r \in \R_{\neq 0}$ -- real number specified to $b$ bits of 
  precision that defines $Q \in \C/\Lambda$,
\item $b'$ -- bits of precision when dividing points into
$\Gamma_0(N)$ orbits, and
\item $n$ -- number of trials before we give up and output FAIL.
\end{itemize}

We compute $P_{E,F,Q}$ using an approach that will always
fail if $Q$ is a ramification point.  Our algorithm will also fail if
any points in the fiber over $Q$ are cusps.  This is why we do not
allow $r=0$.  One can modify the algorithm to work when $Q$ is an
unramified torsion point by using modular symbols and keeping track of
images of cusps.

To increase our confidence that we have computed the right point
$P_{E,F}$, we often carry out the complete computation with more than
one choice of $r$.  

\begin{enumerate}\setlength{\itemsep}{.5ex}
\item\label{step:lowprecroots} {\bf Low precision roots:} Compute all
  complex double precision roots of the polynomial $\varphi_{F,d} -
  r$. One way to do this is to use ``balanced QR reduction of the
  companion matrix'', as implemented in GSL.\footnote{GSL is the the
    GNU scientific library, which is part of Sage \cite{sage}. Rough
    timings of GSL for this computation: it takes less than a half
    second for degree 500, about 5 seconds for degree 1000, about 45
    seconds for degree 2000, and several minutes for degree 3000.}
  Record the roots that correspond to $\tau\in \h$ with $\Im(\tau)\geq
  y$.

\item\label{step:highprecroots} {\bf High precision roots:} Compute $B$ such that if
  $\Im(\tau)\geq y$, then
 $$\left|\sum_{n=B+1}^{\infty} \frac{a_n(F)}{n}\tau^n\right| < 2^{-b},$$
 where $b$ is the number of bits of precision of $r$.  
Summing the tail end of the series and using that $|a_n|\leq n$ (see
 \cite[Lem.~2.9]{bsdalg1}), we find that
$$
B = \left\lceil \frac{\log(2^{-(b+1)}\cdot (1-e^{-2\pi y_1}))}{-2\pi y}
       \right\rceil
$$
works.  Next, compute the polynomial $\varphi_{F,B} \in \Q[T]$, and
use Newton iteration to refine all roots saved
in Step~\ref{step:lowprecroots} to roots $\alpha$ of 
$f=\varphi_{F,B} - r \in \R[T]$ such that $|f(\alpha)|<2^{-b}$.  
%Do this by
%iteratively replacing $\alpha$ by $\alpha - \frac{f(\alpha)}{f'(\alpha)}$,
%and 
Save those roots that correspond to $\tau\in\h$ with $\Im(\tau)\geq y$.

\item\label{step:orbits} {\bf $\Gamma_0(N)$-orbits:} Divide the
  $\tau$'s from Step~\ref{step:highprecroots} into
  $\Gamma_0(N)$-equivalence classes, testing equivalence to the chosen
  bit precision $b'\leq b$, as explained in
  Section~\ref{sec:g0nequiv}.  It is easy to efficiently compute the
  modular degree $m_F = \deg(\varphi_F)$ (see \cite{watkins:moddeg}).
  If we find $m_F$ distinct $\Gamma_0(N)$ classes of points, we
  suspect that we have found the fiber over $[r]$, so we map each
  element of the fiber to $E$ using $\varphi_E$ and sum, then apply the
  elliptic exponential to obtain $P_{E,F}$ to some precision, then
  output this approximation and terminate.  If we find more than $m_F$
  distinct classes, there was an error in the choices of precision in
  our computation, so we output FAIL (and suggest either increasing
  $b$ or decreasing $b'$).

\item\label{step:increase} {\bf Try again:} We did not find enough
  points in the fiber.  Systematically replace $r$ by $r + m\Omega_F$,
  for $m=1,-1,2,-2,\ldots$, where $\Omega_F$ is the least real period of
  $F$, then try again going to Step~\ref{step:lowprecroots} and
  including the new points found.  If upon trying $n$ choices
  $r+m\Omega_F$ in a row we find no new points, we output FAIL
  and terminate the algorithm.

\end{enumerate}



\subsubsection{Determining $\Gamma_0(N)$ equivalency}\label{sec:g0nequiv}

The field of meromorphic functions invariant under $\Gamma_0(N)$ is
generated by $j(z)$ and $j(Nz)$, so if two points $z_1$ and $z_2$ in
the upper half plane are equivalent under $\Gamma_0(N)$, then $z_1$
and $z_2$ are equivalent under $\SL_2(\Z)$ and $N z_1$ and $N z_2$ are
also equivalent under $\SL_2(\Z)$.  Because of singularities in the
affine curve defined by $j(z)$ and $j(Nz)$, the converse is {\em not}
true: for example, $z_1 = (-2+i)/5$ and $z_2 = (2+i)/5$ are equivalent
under $\SL_2(\Z)$ as are $5z_1$ and $5z_2$, but $z_1$ and $z_2$ are
not equivalent under $\Gamma_0(5)$.  This is why the algorithm we give
below must take into account singularities.

Suppose we are given arbitrary $z_1$ and $z_2$ in the upper half
plane. We first find $g_1, g_2\in \SL_2(\Z)$ such that $w_i= g_i(z_i)$
is the canonical representative for $z_i$ in the standard fundamental
domain for $\SL_2(\Z)$, as explained in \cite[\S2.14]{cremona}
but using interval arithmetic to avoid rounding errors.  If $w_1\neq
w_2$, then $z_1$ and $z_2$ are not equivalent under $\SL_2(\Z)$, so
they cannot be equivalent under $\Gamma_0(N)$.  Thus let
$w=g_1(z_1)=g_2(z_2)$.  The elements of $\PSL_2(\Z)$ that send $z_1$
to $z_2$ are the finitely many elements $g_2^{-1} A g_1$, for
$A\in\Stab(w)$, so we check whether any $g_2^{-1} A g_1$ is in
$\Gamma_0(N)$.  The only elements of the standard fundamental domain
for $\SL_2(\Z)$ with nontrivial stabilizers are $w=i$, with stabilizer
generated by $S\in\PSL_2(\Z)$ of order $2$, and $w=e^{2\pi i/3}$ with
stabilizer generated by $ST$, where $T$ corresponds to $z\mapsto z+1$.


\subsubsection{Data}\label{sec:data}

We implemented the above algorithm in Sage \cite{sage}\footnote{See
  \url{http://trac.sagemath.org/sage_trac/ticket/11975}.}.  The
columns of Tables~\ref{tab:ch1}--\ref{tab:ch4} are as follows.  The
columns labeled $E$ and $F$ contain Cremona labels for elliptic
curves, and those labeled $r_E$ and $r_F$ contain the corresponding
ranks.  The column labeled $E(\Q)$ gives a choice of generators $P_1,
P_2, \ldots$ for the Mordell-Weil group, with $r_E$ points of infinite
order listed first, then 0, 1 or 2 torsion points listed with a
subscript of their order.  The column labeled $P_{E,F}$ contains a
rational point close to the numerically computed Chow-Heegner point,
represented in terms of the generators $P_i$ from the column labeled
$E(\Q)$, where $P_1$ is the first generator, $P_2$ the second, and so
on.  The columns labeled $m_E$ and $m_F$ give the modular degrees of
$E$ and $F$.  The column labeled $\eps$'s contains the local root
numbers of $L(E,s)$ at each bad prime.  The notes column refers to the
notes after the table, which give information about the input
parameters needed to compute $P_{E,F}$:
\begin{itemize}\setlength{\itemsep}{.5ex}
\item[(1)] We used $y=10^{-5}$ and $d=1500$, which takes a few minutes.
\item[(2)] We used $y=\frac{1}{2}\cdot 10^{-5}$ and $d=3000$, which takes over an hour.
\end{itemize}


We believe that the values of $P_{E,F}$ are ``likely'' to be correct,
but we emphasize again that {\em they are not proven correct}.  In the
table we give an exact point, but the algorithm computes a numerical
approximation $\tilde{P}_{E,F}$ to $P_{E,F} \in E(\Q)$.  We find what
we call $P_{E,F}$ in the table by running through several hundred low
height points in $E(\Q)$ and find the one closest to
$\tilde{P}_{E,F}$; in all cases, the coordinates of the point
we list are within $10^{-5}$ of the coordinates of
$\tilde{P}_{E,F}$.

The table contains {\em every} pair $E,F$ of nonisogenous optimal
elliptic curves of the same conductor $N\leq 184$ with $r_E=1$, and
{\em most} (but not all) with $N\leq 250$.  It also contains a few additional
miscellaneous examples, e.g., with $r_E=0$ and some of larger
conductor with $r_F=2$.  Most rows took only a few seconds to compute,
though ones with $m_F$ large in some cases took much longer; the total
CPU time to compute the entire table was about 8 hours.  Unless
otherwise noted, we used $y=10^{-4}$, $d=500$, $b'=20$, and $r=0.1$
with 53 bits of precision, as in Section~\ref{sec:numerical}.  We also
repeated all computations with $r=0.2$, and in every case got the
same result.

\subsubsection{Discussion}\label{sec:appdiscuss}

In Tables~\ref{tab:ch1}--\ref{tab:ch4} we always have $2\mid
[E(\Q)_{/\tor}: \Z P_{E,F}].$  In may be possible to prove this in some cases
by using that when $r_{\an}(E)=1$ then the sign in the functional
equation for $L(E,s)$ is $-1$, so at least one nontrivial Atkin-Lehner
involution $w_q$ acts as $+1$ on $E$, which means that the map $X_0(N)
\to E$ factors through $X_0(N)\to X_0(N)/w_q$. 
Also, there are four cases in which the index $[E(\Q)/_{\tor}: \Z
P_{E,F}]$ is divisible by a prime $\ell\geq 5$. They are
(\CR{106b}, \CR{106c}, $\ell=11$), (\CR{118a}, \CR{118d}, $\ell=7$),
(\CR{121b}, \CR{121d}, $\ell=7$), and (\CR{158b}, \CR{158c}, $\ell=7$).
These prime divisors do not appear to
have anything to do with the invariants of $E$ and $F$,
individually.

\begin{center}\small
\begin{table}[ht]
\caption{Chow-Heegner points associated to pairs of elliptic curves (part 1 of 4)\label{tab:ch1}}
\begin{tabular}{|c|c|c|c|c||c|c|c||c|c|}\hline
$E$& $\eps_p$'s & $r_E$ & $E(\Q)$ & $m_E$ & $F$& $r_F$ & $m_F$ & $P_{E,F}$ & Notes\\\hline\hline
\CR{37a}&$ + $& $1$ & $(0, -1)$ & $2$ & \CR{37b} & $0$ & $2$ & $-6P_{1}$ & \\\hline
\CR{37b}&$ - $& $0$ & $(8, 18)_{3}$ & $2$ & \CR{37a} & $1$ & $2$ & $P_{1}$ & \\\hline
\CR{57a}&$ ++ $& $1$ & $(2, 1)$ & $4$ & \CR{57c} & $0$ & $12$ & $8P_{1}$ & \\\hline
\CR{57a}&$ ++ $& $1$ & $(2, 1)$ & $4$ & \CR{57b} & $0$ & $3$ & $-8P_{1}$ & \\\hline
\CR{57b}&$ -+ $& $0$ & $(7/4, -11/8)_{2},(1, -1)_{2}$ & $3$ & \CR{57a} & $1$ & $4$ & $0$ & \\\hline
\CR{57b}&$ -+ $& $0$ & $(7/4, -11/8)_{2},(1, -1)_{2}$ & $3$ & \CR{57c} & $0$ & $12$ & $0$ & \\\hline
\CR{57c}&$ -+ $& $0$ & $(2, 4)_{5}$ & $12$ & \CR{57a} & $1$ & $4$ & $3P_{1}$ & \\\hline
\CR{57c}&$ -+ $& $0$ & $(2, 4)_{5}$ & $12$ & \CR{57b} & $0$ & $3$ & $P_{1}$ & \\\hline
\CR{58a}&$ ++ $& $1$ & $(0, -1)$ & $4$ & \CR{58b} & $0$ & $4$ & $8P_{1}$ & \\\hline
\CR{58b}&$ -+ $& $0$ & $(-1, 2)_{5}$ & $4$ & \CR{58a} & $1$ & $4$ & $3P_{1}$ & \\\hline
\CR{77a}&$ ++ $& $1$ & $(2, 3)$ & $4$ & \CR{77b} & $0$ & $20$ & $24P_{1}$ & (1) \\\hline % deg1=1500, min_imag=1e-5, equiv_prec=15
\CR{77a}&$ ++ $& $1$ & $(2, 3)$ & $4$ & \CR{77c} & $0$ & $6$ & $-4P_{1}$ & \\\hline
\CR{89a}&$ + $& $1$ & $(0, -1)$ & $2$ & \CR{89b} & $0$ & $5$ & $4P_{1}$ & \\\hline
\CR{91a}&$ ++ $& $1$ & $(0, 0)$ & $4$ & \CR{91b} & $1$ & $4$ & $4P_{1}$ & \\\hline
\CR{91b}&$ - - $& $1$ & $(-1, 3),(1, 0)_{3}$ & $4$ & \CR{91a} & $1$ & $4$ & $P_{2}$ & \\\hline
\CR{92b}&$ - - $& $1$ & $(1, 1)$ & $6$ & \CR{92a} & $0$ & $2$ & $0$ & \\\hline
\CR{99a}&$ ++ $& $1$ & $(2, 0),(-1, 0)_{2}$ & $4$ & \CR{99b}& $0$ & $12$ & $-4P_{1}$ & \\\hline
\CR{99a}&$ ++ $& $1$ & $(2, 0),(-1, 0)_{2}$ & $4$ & \CR{99c} & $0$ & $12$ & $0$ & \\\hline
\CR{99a}&$ ++ $& $1$ & $(2, 0),(-1, 0)_{2}$ & $4$ & \CR{99d}& $0$ & $6$ & $2P_{1}$ & \\\hline
\CR{102a}&$ +++ $& $1$ & $(2, -4),(0, 0)_{2}$ & $8$ & \CR{102b} & $0$ & $16$ & $-8P_{1}$ & (1) \\\hline  %deg1=1500, min_imag=1e-5
\CR{102a}&$ +++ $& $1$ & $(2, -4),(0, 0)_{2}$ & $8$ & \CR{102c} & $0$ & $24$ & $32P_{1}$ & \\\hline
\CR{106b}&$ ++ $& $1$ & $(2, 1)$ & $8$ & \CR{106a} & $0$ & $6$ & $-4P_{1}$ & \\\hline
\CR{106b}&$ ++ $& $1$ & $(2, 1)$ & $8$ & \CR{106c} & $0$ & $48$ & $-88P_{1}$ & \\\hline
\CR{106b}&$ ++ $& $1$ & $(2, 1)$ & $8$ & \CR{106d} & $0$ & $10$ & $12P_{1}$ & \\\hline
\CR{112a}&$ ++ $& $1$ & $(0, -2),(-2, 0)_{2}$ & $8$ & \CR{112b} & $0$ & $4$ & $0$ & \\\hline
\CR{112a}&$ ++ $& $1$ & $(0, -2),(-2, 0)_{2}$ & $8$ & \CR{112c} & $0$ & $8$ & $0$ & \\\hline
\CR{118a}&$ ++ $& $1$ & $(0, -1)$ & $4$ & \CR{118b} & $0$ & $12$ & $-8P_{1}$ & (1) \\\hline %deg1=1500, min_imag=1e-5
\CR{118a}&$ ++ $& $1$ & $(0, -1)$ & $4$ & \CR{118c} & $0$ & $6$ & $4P_{1}$ & \\\hline
\CR{118a}&$ ++ $& $1$ & $(0, -1)$ & $4$ & \CR{118d} & $0$ & $38$ & $-28P_{1}$ & \\\hline

\CR{121b}&$ + $& $1$ & $(4, 5)$ & $4$ & \CR{121a} & $0$ & $6$ & $4P_{1}$ & \\\hline
\CR{121b}&$ + $& $1$ & $(4, 5)$ & $4$ & \CR{121c} & $0$& $6$ & $4P_{1}$ & \\\hline

\CR{121b}&$ + $& $1$ & $(4, 5)$ & $4$ & \CR{121d} & $0$ & $24$ & $-28P_{1}$ & (2) \\\hline % actually: min_imag=1e-5, deg1=3000, and yes such a large deg1 needed - - finds all 24 points right off. - - takes 10 minutes.

\CR{123a}&$ - - $& $1$ & $(-4, 1),(-1, 4)_{5}$ & $20$ & \CR{123b} & $1$ & $4$ & $0$ & \\\hline
\CR{123b}&$ ++ $& $1$ & $(1, 0)$ & $4$ & \CR{123a} & $1$ & $20$ & $4P_{1}$ & \\\hline
\CR{124a}&$ - - $& $1$ & $(-2, 1),(0, 1)_{3}$ & $6$ & \CR{124b} & $0$ & $6$ & $0$ & \\\hline

\CR{128a}&$ + $& $1$ & $(0, 1),(-1, 0)_{2}$ & $4$ & \CR{128b} & $0$ & $8$ & $0$ & \\\hline
\CR{128a}&$ + $& $1$ & $(0, 1),(-1, 0)_{2}$ & $4$ & \CR{128c} & $0$ & $4$ & $0$ & \\\hline
\CR{128a}&$ + $& $1$ & $(0, 1),(-1, 0)_{2}$ & $4$ & \CR{128d} & $0$ & $8$ & $0$ & \\\hline

\CR{129a}&$ ++ $& $1$ & $(1, -5)$ & $8$ & \CR{129b} & $0$ & $15$ & $-8P_{1}$ & \\\hline


\CR{130a}&$ +- - $& $1$ & $(-6, 10),(-1, 10)_{6}$ & $24$ & \CR{130b} & $0$ & $8$ & $0$ & \\\hline  

\CR{130a}&$ +- - $& $1$ & $(-6, 10),(-1, 10)_{6}$ & $24$ & \CR{130c} & $0$ & $80$ & $0$ & \\\hline % a lot of rounds

\CR{135a}&$ ++ $& $1$ & $(4, -8)$ & $12$ & \CR{135b} & $0$ & $36$ & $0$ & (1) \\\hline%deg1=1500, min_imag=1e-5

\CR{136a}&$ - - $& $1$ & $(-2, 2),(0, 0)_{2}$ & $8$ & \CR{136b} & $0$ & $8$ & $0$ & \\\hline

\CR{138a}&$ +++ $& $1$ & $(1, -2),(-2, 1)_{2}$ & $8$ & \CR{138b} & $0$ & $16$ & $-16P_{1}$ & (1) \\\hline %actualy -- deg1=1000
\CR{138a}&$ +++ $& $1$ & $(1, -2),(-2, 1)_{2}$ & $8$ & \CR{138c} & $0$ & $8$ & $-8P_{1}$ & \\\hline
\CR{141a}&$ - - $& $1$ & $(-3, -5)$ & $28$ & \CR{141b} & $0$ & $12$ & $0$ & \\\hline
\CR{141a}&$ - - $& $1$ & $(-3, -5)$ & $28$ & \CR{141c} & $0$ & $6$ & $0$ & \\\hline
\CR{141a}&$ - - $& $1$ & $(-3, -5)$ & $28$ & \CR{141d} & $1$ & $4$ & $0$ & \\\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Chow-Heegner points associated to pairs of elliptic curves (part 2 of 4)\label{tab:ch2}}
\begin{tabular}{|c|c|c|c|c||c|c|c||c|c|}\hline
$E$& $\eps_p$'s& $r_E$ & $E(\Q)$ & $m_E$ & $F$& $r_F$ & $m_F$ & $P_{E,F}$ & Notes\\\hline\hline

\CR{141a}&$$ - - $$& $1$ & $(-3, -5)$ & $28$ & \CR{141e} & $0$ & $12$ & $0$ & \\\hline
\CR{141d}&$$ ++ $$& $1$ & $(0, -1)$ & $4$ & \CR{141a} & $1$ & $28$ & $-12P_{1}$ & \\\hline
\CR{141d}&$$ ++ $$& $1$ & $(0, -1)$ & $4$ & \CR{141b} & $0$ & $12$ & $4P_{1}$ & \\\hline
\CR{141d}&$$ ++ $$& $1$ & $(0, -1)$ & $4$ & \CR{141c} & $0$ & $6$ & $4P_{1}$ & \\\hline
\CR{141d}&$$ ++ $$& $1$ & $(0, -1)$ & $4$ & \CR{141e} & $0$ & $12$ & $4P_{1}$ & \\\hline

\CR{142a}&$$ - - $$& $1$ & $(1, 1)$ & $36$ & \CR{142b} & $1$ & $4$ & $0$ & \\\hline
\CR{142a}&$$ - - $$& $1$ & $(1, 1)$ & $36$ & \CR{142c} & $0$ & $9$ & $0$ & \\\hline

\CR{142a}&$ - - $& $1$ & $(1, 1)$ & $36$ & \CR{142d} & $0$ & $4$ & $0$ & \\\hline


\CR{142a}&$ - - $& $1$ & $(1, 1)$ & $36$ & \CR{142e} & $0$ & $324$ & $0$ & (2) \\\hline  % deg1=3000, min_imag=1e-5/2; takes 5196 seconds

\CR{142b}&$ ++ $& $1$ & $(-1, 0)$ & $4$ & \CR{142a} & $1$ & $36$ & $4P_{1}$ & (1) \\\hline  % actually, deg1=1000 
\CR{142b}&$ ++ $& $1$ & $(-1, 0)$ & $4$ & \CR{142c} & $0$ & $9$ & $-4P_{1}$ & \\\hline
\CR{142b}&$ ++ $& $1$ & $(-1, 0)$ & $4$ & \CR{142d} & $0$ & $4$ & $4P_{1}$ & \\\hline

\CR{142b}&$ ++ $& $1$ & $(-1, 0)$ & $4$ & \CR{142e} & $0$ & $324$ & $8P_{1}$ & (2) \\\hline % deg1=3000, min_imag=1e-5/2; takes 5380 seconds

\CR{152a}&$ ++ $& $1$ & $(-1, -2)$ & $8$ & \CR{152b} & $0$ & $8$ & $0$ & \\\hline
\CR{153a}&$ ++ $& $1$ & $(0, 1)$ & $8$ & \CR{153b} & $1$ & $16$ & $8P_{1}$ & \\\hline
\CR{153a}&$ ++ $& $1$ & $(0, 1)$ & $8$ & \CR{153c} & $0$ & $8$ & $8P_{1}$ & \\\hline
\CR{153a}&$ ++ $& $1$ & $(0, 1)$ & $8$ & \CR{153d} & $0$ & $24$ & $0$ & \\\hline
\CR{153b}&$ - - $& $1$ & $(5, -14)$ & $16$ & \CR{153a} & $1$ & $8$ & $0$ & \\\hline
\CR{153b}&$ - - $& $1$ & $(5, -14)$ & $16$ & \CR{153d} & $0$ & $24$ & $0$ & \\\hline
\CR{154a}&$ +++ $& $1$ & $(5, 3),(-6, 3)_{2}$ & $24$ & \CR{154b} & $0$ & $24$ & $-24P_{1}$ & \\\hline                                    
\CR{154a}&$ +++ $& $1$ & $(5, 3),(-6, 3)_{2}$ & $24$ & \CR{154c} & $0$ & $16$ & $16P_{1}$ & \\\hline                                     
\CR{155a}&$ - - $& $1$ & $(5/4, 31/8),(0, 2)_{5}$ & $20$ & \CR{155b} & $0$ & $8$ & $0$ & \\\hline                                        
\CR{155a}&$ - - $& $1$ & $(5/4, 31/8),(0, 2)_{5}$ & $20$ & \CR{155c} & $1$ & $4$ & $0$ & \\\hline                                        
\CR{155c}&$ ++ $& $1$ & $(1, -1)$ & $4$ & \CR{155a} & $1$ & $20$ & $-12P_{1}$ & \\\hline                                                
\CR{155c}&$ ++ $& $1$ & $(1, -1)$ & $4$ & \CR{155b} & $0$ & $8$ & $4P_{1}$ & \\\hline
\CR{156a}&$ -+- $& $1$ & $(1, 1),(2, 0)_{2}$ & $12$ & \CR{156b} & $0$ & $12$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{158a}&$ - - $& $1$ & $(-1, -4)$ & $32$ & \CR{158b} & $1$ & $8$ & $0$ & \\\hline
\CR{158a}&$ - - $& $1$ & $(-1, -4)$ & $32$ & \CR{158c} & $0$ & $48$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{158a}&$ - - $& $1$ & $(-1, -4)$ & $32$ & \CR{158d} & $0$ & $40$ & $0$ & \\\hline
\CR{158a}&$ - - $& $1$ & $(-1, -4)$ & $32$ & \CR{158e} & $0$ & $6$ & $0$ & \\\hline
\CR{158b}&$ ++ $& $1$ & $(0, -1)$ & $8$ & \CR{158a} & $1$ & $32$ & $-8P_{1}$ & \\\hline
\CR{158b}&$ ++ $& $1$ & $(0, -1)$ & $8$ & \CR{158c} & $0$ & $48$ & $-56P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{158b}&$ ++ $& $1$ & $(0, -1)$ & $8$ & \CR{158d} & $0$ & $40$ & $0$ & \\\hline
\CR{158b}&$ ++ $& $1$ & $(0, -1)$ & $8$ & \CR{158e} & $0$ & $6$ & $-8P_{1}$ & \\\hline
\CR{160a}&$ ++ $& $1$ & $(2, -2),(1, 0)_{2}$ & $8$ & \CR{160b} & $0$ & $8$ & $0$ & \\\hline
\CR{162a}&$ ++ $& $1$ & $(-2, 4),(1, 1)_{3}$ & $12$ & \CR{162b} & $0$ & $6$ & $0$ & \\\hline
\CR{162a}&$ ++ $& $1$ & $(-2, 4),(1, 1)_{3}$ & $12$ & \CR{162c} & $0$ & $6$ & $0$ & \\\hline
\CR{162a}&$ ++ $& $1$ & $(-2, 4),(1, 1)_{3}$ & $12$ & \CR{162d} & $0$ & $12$ & $0$ & \\\hline
\CR{170a}&$ +- - $& $1$ & $(0, 2),(1, -1)_{2}$ & $16$ & \CR{170d} & $0$ & $12$ & $0$ & \\\hline
\CR{170a}&$ +- - $& $1$ & $(0, 2),(1, -1)_{2}$ & $16$ & \CR{170e} & $0$ & $20$ & $0$ & \\\hline
\CR{171b}&$ - - $& $1$ & $(2, -5)$ & $8$ & \CR{171a} & $0$ & $12$ & $0$ & \\\hline
\CR{171b}&$ - - $& $1$ & $(2, -5)$ & $8$ & \CR{171c} & $0$ & $96$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{171b}&$ - - $& $1$ & $(2, -5)$ & $8$ & \CR{171d} & $0$ & $32$ & $0$ & \\\hline
\CR{175a}&$ - - $& $1$ & $(2, -3)$ & $8$ & \CR{175b} & $1$ & $16$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5

\CR{175a}&$ - - $& $1$ & $(2, -3)$ & $8$ & \CR{175c} & $0$ & $40$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{175b}&$ ++ $& $1$ & $(-3, 12)$ & $16$ & \CR{175a} & $1$ & $8$ & $16P_{1}$ & \\\hline
\CR{175b}&$ ++ $& $1$ & $(-3, 12)$ & $16$ & \CR{175c} & $0$ & $40$ & $16P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{176c}&$ - - $& $1$ & $(1, -2)$ & $8$ & \CR{176b} & $0$ & $8$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Chow-Heegner points associated to pairs of elliptic curves (part 3 of 4)\label{tab:ch3}}
\begin{tabular}{|c|c|c|c|c||c|c|c||c|c|}\hline
$E$& $\eps_p$'s & $r_E$ & $E(\Q)$ & $m_E$ & $F$& $r_F$ & $m_F$ & $P_{E,F}$ & Notes\\\hline\hline
\CR{176c}&$ - - $& $1$ & $(1, -2)$ & $8$ & \CR{176a} & $0$ & $16$ & $0$ & \\\hline
\CR{176c}&$ - - $& $1$ & $(1, -2)$ & $8$ & \CR{176b} & $0$ & $8$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{184a}&$ - - $& $1$ & $(0, 1)$ & $8$ & \CR{184c} & $0$ & $12$ & $0$ & \\\hline
\CR{184a}&$ - - $& $1$ & $(0, 1)$ & $8$ & \CR{184d} & $0$ & $24$ & $0$ & \\\hline
\CR{184b}&$ ++ $& $1$ & $(2, -1)$ & $8$ & \CR{184a} & $1$ & $8$ & $0$ & \\\hline
\CR{184b}&$ ++ $& $1$ & $(2, -1)$ & $8$ & \CR{184c} & $0$ & $12$ & $0$ & \\\hline
\CR{184b}&$ ++ $& $1$ & $(2, -1)$ & $8$ & \CR{184d} & $0$ & $24$ & $0$ & \\\hline
\CR{185a}&$ ++ $& $1$ & $(4, -13)$ & $48$ & \CR{185b} & $1$ & $8$ & $8P_{1}$ & \\\hline
\CR{185a}&$ ++ $& $1$ & $(4, -13)$ & $48$ & \CR{185c} & $1$ & $6$ & $24P_{1}$ & \\\hline
%$\cdots$ &$  $& &  & &  & &  &&   \\\hline
\CR{185b}&$ - - $& $1$ & $(0, 2)$ & $8$ & \CR{185c} & $1$ & $6$ & $0$ & \\\hline
\CR{185c}&$ ++ $& $1$ & $(-5/4, 3/8),(-1, 0)_{2}$ & $6$ & \CR{185b} & $1$ & $8$ & $2P_{1}$ & \\\hline
\CR{189a}&$ ++ $& $1$ & $(-1, -2)$ & $12$ & \CR{189b} & $1$ & $12$ & $-12P_{1}$ & \\\hline
\CR{189a}&$ ++ $& $1$ & $(-1, -2)$ & $12$ & \CR{189c} & $0$ & $12$ & $12P_{1}$ & \\\hline
\CR{189b}&$ - - $& $1$ & $(-3, 9),(3, 0)_{3}$ & $12$ & \CR{189a} & $1$ & $12$ & $0$ & \\\hline
\CR{189b}&$ - - $& $1$ & $(-3, 9),(3, 0)_{3}$ & $12$ & \CR{189c} & $0$ & $12$ & $0$ & \\\hline
\CR{190a}&$ -+- $& $1$ & $(13, -47)$ & $88$ & \CR{190b} & $1$ & $8$ & $0$ & \\\hline
\CR{190a}&$ -+- $& $1$ & $(13, -47)$ & $88$ & \CR{190c} & $0$ & $24$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{190b}&$ +++ $& $1$ & $(1, 2)$ & $8$ & \CR{190c} & $0$ & $24$ & $16P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{192a}&$ ++ $& $1$ & $(3, 2),(-1, 0)_{2}$ & $8$ & \CR{192b} & $0$ & $8$ & $0$ & \\\hline
\CR{192a}&$ ++ $& $1$ & $(3, 2),(-1, 0)_{2}$ & $8$ & \CR{192c} & $0$ & $8$ & $0$ & \\\hline
\CR{192a}&$ ++ $& $1$ & $(3, 2),(-1, 0)_{2}$ & $8$ & \CR{192d} & $0$ & $8$ & $0$ & \\\hline
\CR{196a}&$ - - $& $1$ & $(0, -1)$ & $6$ & \CR{196b} & $0$ & $42$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{198a}&$ +- - $& $1$ & $(-1, -4),(-4, 2)_{2}$ & $32$ & \CR{198b} & $0$ & $32$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{198a}&$ +- - $& $1$ & $(-1, -4),(-4, 2)_{2}$ & $32$ & \CR{198c} & $0$ & $32$ & $0$ &  \\\hline
\CR{198a}&$ +- - $& $1$ & $(-1, -4),(-4, 2)_{2}$ & $32$ & \CR{198d} & $0$ & $32$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{198a}&$ +- - $& $1$ & $(-1, -4),(-4, 2)_{2}$ & $32$ & \CR{198e} & $0$ & $160$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{200b}&$ - - $& $1$ & $(-1, 1),(-2, 0)_{2}$ & $8$ & \CR{200c} & $0$ & $24$ & $0$ & \\\hline
\CR{200b}&$ - - $& $1$ & $(-1, 1),(-2, 0)_{2}$ & $8$ & \CR{200d} & $0$ & $40$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{200b}&$ - - $& $1$ & $(-1, 1),(-2, 0)_{2}$ & $8$ & \CR{200e} & $0$ & $24$ & $0$ & \\\hline
\CR{201a}&$ ++ $& $1$ & $(1, -2)$ & $12$ & \CR{201b} & $1$ & $12$ & $4P_{1}$ & \\\hline
\CR{201b}&$ - - $& $1$ & $(-1, 2)$ & $12$ & \CR{201a} & $1$ & $12$ & $0$ & \\\hline
\CR{201c}&$ ++ $& $1$ & $(16, -7)$ & $60$ & \CR{201a} & $1$ & $12$ & $-24P_{1}$ & \\\hline
\CR{201c}&$ ++ $& $1$ & $(16, -7)$ & $60$ & \CR{201b} & $1$ & $12$ & $8P_{1}$ & \\\hline
\CR{203b}&$ - - $& $1$ & $(2, -5)$ & $8$ & \CR{203a} & $0$ & $48$ & $0$ & \\\hline
\CR{203b}&$ - - $& $1$ & $(2, -5)$ & $8$ & \CR{203c} & $0$ & $12$ & $0$ & \\\hline
\CR{205a}&$ - - $& $1$ & $(-1, 8),(2, 1)_{4}$ & $12$ & \CR{205b} & $0$ & $16$ & $0$ & \\\hline
\CR{205a}&$ - - $& $1$ & $(-1, 8),(2, 1)_{4}$ & $12$ & \CR{205c} & $0$ & $8$ & $0$ & \\\hline
\CR{208a}&$ - - $& $1$ & $(4, -8)$ & $16$ & \CR{208c} & $0$ & $12$ & $0$ & \\\hline
\CR{208a}&$ - - $& $1$ & $(4, -8)$ & $16$ & \CR{208d} & $0$ & $48$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{208b}&$ ++ $& $1$ & $(4, 4)$ & $16$ & \CR{208a} & $1$ & $16$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{208b}&$ ++ $& $1$ & $(4, 4)$ & $16$ & \CR{208c} & $0$ & $12$ & $0$ & \\\hline
\CR{208b}&$ ++ $& $1$ & $(4, 4)$ & $16$ & \CR{208d} & $0$ & $48$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{212a}&$ - - $& $1$ & $(2, 2)$ & $12$ & \CR{212b} & $0$ & $21$ & $0$ & \\\hline
\CR{214a}&$ - - $& $1$ & $(0, -4)$ & $28$ & \CR{214b} & $1$ & $12$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{214a}&$ - - $& $1$ & $(0, -4)$ & $28$ & \CR{214d} & $0$ & $12$ & $0$ & \\\hline

\CR{214b}&$ ++ $& $1$ & $(0, 0)$ & $12$ & \CR{214a} & $1$ & $28$ & $-8P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{214b}&$ ++ $& $1$ & $(0, 0)$ & $12$ & \CR{214d} & $0$ & $12$ & $-4P_{1}$ & \\\hline
\end{tabular}
\end{table}

\begin{table}[ht]
\caption{Chow-Heegner points associated to pairs of elliptic curves (part 4 of 4)\label{tab:ch4}}
\begin{tabular}{|c|c|c|c|c||c|c|c||c|c|}\hline
$E$& $\eps_p$'s & $r_E$ & $E(\Q)$ & $m_E$ & $F$& $r_F$ & $m_F$ & $P_{E,F}$ & Notes\\\hline\hline
\CR{214c}&$ ++ $& $1$ & $(11, 10)$ & $60$ & \CR{214a} & $1$ & $28$ & $-4P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{214c}&$ ++ $& $1$ & $(11, 10)$ & $60$ & \CR{214d} & $0$ & $12$ & $16P_{1}$ & \\\hline
\CR{214c}&$ ++ $& $1$ & $(11, 10)$ & $60$ & \CR{214b} & $1$ & $12$ & $12P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{216a}&$ ++ $& $1$ & $(-2, -6)$ & $24$ & \CR{216b} & $0$ & $24$ & $0$ & \\\hline
\CR{219a}&$ ++ $& $1$ & $(2, -1)$ & $12$ & \CR{219c} & $1$ & $60$ & $-12P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{219a}&$ ++ $& $1$ & $(2, -1)$ & $12$ & \CR{219b} & $1$ & $12$ & $-4P_{1}$ & \\\hline
\CR{216a}&$ ++ $& $1$ & $(-2, -6)$ & $24$ & \CR{216d} & $0$ & $72$ & $0$ & \\\hline
\CR{219b}&$ - - $& $1$ & $(-3/4, -1/8),(0, 1)_{3}$ & $12$ & \CR{219a} & $1$ & $12$ & $0$ & \\\hline
\CR{219b}&$ - - $& $1$ & $(-3/4, -1/8),(0, 1)_{3}$ & $12$ & \CR{219c} & $1$ & $60$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{219c}&$ ++ $& $1$ & $(-6, 7),(10, -5)_{2}$ & $60$ & \CR{219a} & $1$ & $12$ & $-12P_{1}$ & \\\hline
\CR{219c}&$ ++ $& $1$ & $(-6, 7),(10, -5)_{2}$ & $60$ & \CR{219b} & $1$ & $12$ & $4P_{1}$ & \\\hline
\CR{220a}&$ - -+ $& $1$ & $(-7, 11),(15, 55)_{6}$ & $36$ & \CR{220b} & $0$ & $12$ & $0$ & \\\hline
\CR{224a}&$ ++ $& $1$ & $(1, 2),(0, 0)_{2}$ & $8$ & \CR{224b} & $0$ & $8$ & $0$ & \\\hline
\CR{225a}&$ ++ $& $1$ & $(1, 1)$ & $8$ & \CR{225b} & $0$ & $40$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{225e}&$ - - $& $1$ & $(-5, 22)$ & $48$ & \CR{225a} & $1$ & $8$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{225e}&$ - - $& $1$ & $(-5, 22)$ & $48$ & \CR{225b} & $0$ & $40$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{228b}&$ -+- $& $1$ & $(3, 6)$ & $24$ & \CR{228a} & $0$ & $18$ & $0$ & \\\hline
\CR{232a}&$ ++ $& $1$ & $(2, -4)$ & $16$ & \CR{232b} & $0$ & $16$ & $0$ & \\\hline
\CR{234c}&$ +++ $& $1$ & $(1, -2),(-2, 1)_{2}$ & $16$ & \CR{234b} & $0$ & $48$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{234c}&$ +++ $& $1$ & $(1, -2),(-2, 1)_{2}$ & $16$ & \CR{234e} & $0$ & $20$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{235a}&$ - - $& $1$ & $(-2, 3)$ & $12$ & \CR{235c} & $0$ & $18$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{236a}&$ - - $& $1$ & $(1, -1)$ & $6$ & \CR{236b} & $0$ & $14$ & $0$ & \\\hline
\CR{238a}&$ - -+ $& $1$ & $(24, 100),(-8, 4)_{2}$ & $112$ & \CR{238b} & $1$ & $8$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{238a}&$ - -+ $& $1$ & $(24, 100),(-8, 4)_{2}$ & $112$ & \CR{238c} & $0$ & $16$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{238a}&$ - -+ $& $1$ & $(24, 100),(-8, 4)_{2}$ & $112$ & \CR{238d} & $0$ & $16$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{238b}&$ +++ $& $1$ & $(1, 1),(0, 0)_{2}$ & $8$ & \CR{238a} & $1$ & $112$ & $12P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{238b}&$ +++ $& $1$ & $(1, 1),(0, 0)_{2}$ & $8$ & \CR{238c} & $0$ & $16$ & $-4P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{238b}&$ +++ $& $1$ & $(1, 1),(0, 0)_{2}$ & $8$ & \CR{238d} & $0$ & $16$ & $4P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{240c}&$ +++ $& $1$ & $(1, 2),(0, 0)_{2}$ & $16$ & \CR{240a} & $0$ & $16$ & $0$ & \\\hline
\CR{240c}&$ +++ $& $1$ & $(1, 2),(0, 0)_{2}$ & $16$ & \CR{240d} & $0$ & $16$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{243a}&$ + $& $1$ & $(1, 0)$ & $6$ & \CR{243b} & $0$ & $9$ & $0$ & (1) \\\hline% deg1=1500, min_imag=1e-5
\CR{245a}&$ - - $& $1$ & $(7, 17)$ & $48$ & \CR{245c} & $1$ & $32$ & $0$ & \\\hline
\CR{246d}&$ +++ $& $1$ & $(3, -6),(4, -2)_{2}$ & $48$ & \CR{246a} & $0$ & $84$ & $24P_{1}$ & (1) \\\hline% deg1=1500, min_imag=1e-5
%$\cdots$ &$  $& &  & &  & &  & $\cdots$  \\\hline
\CR{446a}&$ ++ $& 1 & $(4,-6)$ & $24$ & \CR{446d}& 2 & $88$ & 0  & (2)  \\\hline
\CR{446b}&$ - - $& 1& $(5,-10)$ & $56$ & \CR{446d} & 2 & $88$ & 0  & (2)  \\\hline
\CR{446d}&$+-$ & 2 & - & $88$ & \CR{446a} & 1 & 12 & 0 & (1) \\\hline
\CR{446d}&$+-$ & 2 & - & $88$ & \CR{446b} & 1 & 56 & 0 & (1) \\\hline
\CR{681a} & $++$ & $1$ & $(4, 4)$ & $32$ & \CR{681c} & $2$ & $96$ & $-24P_{1}$ & (2) \\\hline
\end{tabular}
\end{table}

\end{center}















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\end{document}

