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\begin{document}
\title[Explicit Heegner points]
{Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the
Shafarevich-Tate group}
%
\author{Dimitar Jetchev}
\email{jetchev@math.berkeley.edu}
\address{Department of Mathematics\\University of California at Berkeley\\Berkeley, CA 94720}
%
\author{Kristin Lauter}
\email{klauter@microsoft.com}
\address{Microsoft Research\\One Microsoft Way\\Redmond, WA 98052}
%
\author{William Stein}
\email{wstein@math.washington.edu}
\address{Department of Mathematics\\University of Washington\\Seattle, WA 98195}
%
%\dedication{A dedication can be included here.}
\classification{11G05}
\keywords{Elliptic curves, Heegner points, Selmer groups}
\thanks{The first author was supported by summer internships at Microsoft Research as
well as by graduate fellowships from Microsoft Research and University of California at
Berkeley. The third author was partially supported by NSF grant 0555776.}
\begin{abstract}
Kolyvagin used Heegner points to associate a system of cohomology
classes to an elliptic curve over $\Q$ and conjectured that the
system contains a nontrivial class. His conjecture has profound
implications on the structure of Selmer groups. We provide new
computational and theoretical evidence for Kolyvagin's
conjecture. More precisely, we apply results of Zhang and others to
deduce that Kolyvagin classes are computable, then explicitly study
Heegner points over ring class fields and Kolyvagin's conjecture for
specific elliptic curves of rank two. We explain how Kolyvagin's
conjecture implies that if the analytic rank of an elliptic curve is
at least two then the $\Z_p$-corank of the corresponding Selmer
group is at least two as well. We also use explicitly computed
Heegner points to produce non-trivial classes in the
Shafarevich-Tate group.
\end{abstract}
\maketitle
% \vspace*{6pt}\tableofcontents % for this guide only.
\section{Introduction}\label{sec:intro}
Let $E_{/F}$ be an elliptic curve over a number field $F$.
The analytic rank $r_\an(E/F)$ of $E$ is
the order of vanishing of the $L$-function
$L(E_{/ F}, s)$ at $s = 1$.
The Mordell-Weil rank $r_{\MW}(E/F)$ is the rank of the
Mordell-Weil group $E(F)$.
The conjecture of Birch and Swinnerton-Dyer asserts that $r_\an(E/F) = r_{\MW}(E/F)$.
Kolyvagin constructed explicit cohomology classes from Heegner points over certain
abelian extensions of quadratic imaginary fields and used these classes to bound
the size of the Selmer groups for elliptic curves over $\Q$ of analytic rank at
most one (see~\cite{kolyvagin:euler_systems},~\cite{kolyvagin:mordellweil} and~\cite{gross:kolyvagin}). His results, together with the Gross-Zagier formula (see~\cite{gross-zagier}), imply the following theorem:
\begin{thm}[(Gross-Zagier, Kolyvagin)]\label{thm:GZK}
Let $E_{/\Q}$ be an elliptic curve which satisfies $r_{\an}(E/\Q) \leq 1$. Then the
Shafarevich-Tate group $\Sha(E/\Q)$ is finite and $r_{\an}(E/\Q) = r_{\MW}(E/\Q)$.
\end{thm}
Unfortunately, very little is known about the Birch and Swinnerton-Dyer conjecture
for elliptic curves $E_{/\Q}$ with
$r_{\an}(E/\Q) \geq 2$. Still, it implies the following conjecture:
\begin{conj}\label{conj:an-MW}
If $r_{\an}(E/\Q) \geq 2$ then $r_{\MW}(E/\Q) \geq 2$.
\end{conj}
As far as we know, nothing has been proved towards the above assertion.
A weaker conjecture can be formulated in the language of Selmer coranks.
The Selmer corank $r_p(E/F)$ of $E_{/F}$ is the $\Z_p$-corank of the
Selmer group $\Sel_{p^\infty}(E/F)$. Using Kummer theory, one shows
that $r_p(E/\Q) \geq r_{\MW}(E/\Q)$ with an equality occuring if and only
if the $p$-primary part of the Shafarevich-Tate group $\Sha(E/\Q)$ is
finite. Thus, one obtains the following weaker conjecture:
\begin{conj}\label{conj:an-corank}
If $r_{\an}(E/\Q) \geq 2$ then $r_{p}(E/\Q) \geq 2$.
\end{conj}
For elliptic curves $E$ of arbitrary analytic rank, Kolyvagin was able
to explain the exact structure of the Selmer group $\Sel_{p^\infty}(E/\Q)$
in terms of
Heegner points and the associated cohomology classes under a conjecture
about
the non-triviality of these classes (see \cite[Conj.A]{kolyvagin:structure_of_selmer}). Unfortunately, Kolyvagin's conjecture appears to be extremely difficult to prove.
Until the present paper,
there has been no example of an elliptic curve over~$\Q$ of rank at least two for
which the conjecture has been verified.
In this paper, we present a complete algorithm to compute
Kolyvagin's cohomology classes by explicitly computing the corresponding Heegner points
over ring class fields. We use this algorithm to verify Kolyvagin's conjecture for the
first time for elliptic curves of analytic rank two. We also explain
(see Corollary~\ref{cor:large-selmer}) how Kolyvagin's conjecture implies
Conjecture~\ref{conj:an-corank}. In addition, we use methods of Cornut
(see~\cite{cornut:compte-rendus}) to provide theoretical evidence for Kolyvagin's conjecture. As a separate application of the explicit computation of Heegner
points, we construct nontrivial cohomology classes in the
Shafarevich-Tate group $\Sha(E/K)$ of elliptic curves $E$ over certain quadratic
imaginary fields. One of the main contributions of this paper is that by establishing
certain height bounds, we prove that there exists an algorithm
which {\em provably} computes the correct Heegner points over ring class fields.
The paper is organized as follows. Section~\ref{sec:heegpts}
introduces Heegner points over ring class fields and Kolyvagin
cohomology classes. We explain the methods of computation and
illustrate them with several examples. In Section~\ref{sec:kolconj} we
state Kolyvagin's conjecture, discuss Kolyvagin's work on Selmer
groups and establish Conjecture~\ref{conj:an-corank} as a corollary.
Moreover, we present a proof of the theoretical evidence following
closely Cornut's arguments.
Section~\ref{subsec:389a1} contains the essential examples for which
we manage to explicitly verify the conjecture. Finally, in
Section~\ref{sec:expl-sha} we apply our computational techniques to
produce explicit non-trivial elements in the Shafarevich-Tate
groups for specific elliptic curves. Finally, the appendix establishes
certain bounds on the logarithmic heights of the Heegner points over
ring class fields.
\section{Heegner points over ring class fields}\label{sec:heegpts}
We discuss Heegner points over ring class fields in Section~\ref{subsec:heegdefs} and describe a method for computing them in Section~\ref{subsec:heegptscomp}. Height
estimates for these points are given in the appendix. We illustrate the method with some examples in Section~\ref{subsec:ptsexamples}. The standard references are \cite{gross:kolyvagin}, \cite{kolyvagin:euler_systems} and~\cite{mccallum:kolyvagin}.
\subsection{Heegner points over ring class fields}\label{subsec:heegdefs}
Let $E$ be an elliptic curve over $\Q$ of conductor $N$ and let $K =
\Q(\sqrt{-D})$ for some fundamental discriminant $D > 0$, $D \neq 3, 4$, such that
all prime factors of $N$ are split in $K$. We refer to such a discriminant as a
\emph{Heegner discriminant} for $E/\Q$. Let $\cO_K$ be the ring of integers of $K$.
It follows that $N\cO_K = \cN\bar{\cN}$ for an ideal $\cN$ of $\cO_K$ with
$\cO_K / \cN \simeq \Z / N\Z$.
By the modularity theorem (see~\cite{breuil-conrad-diamond-taylor}), there exists
an optimal (having minimal degree) modular parameterization $\varphi : X_0(N) \ra E$.
Let $\cN^{-1}$ be the
fractional ideal of $\cO_K$ for which $\cN \cN^{-1} = \cO_K$.
We view $\cO_K$ and $\cN$ as
$\Z$-lattices of rank two in $\C$ and observe that $\C / \cO_K \ra \C /
\cN^{-1}$ is a cyclic isogeny of degree $N$ between the elliptic
curves $\C / \cO_K$ and $\C / \cN^{-1}$. This
isogeny corresponds to a complex point $x_1 \in X_0(N)(\C)$. According to
the theory of complex multiplication~\cite[Ch.II]{silverman:aec2}, the
point $x_1$ is defined over the Hilbert class field $H_K$ of $K$.
More generally, for an integer $c$,
let $\cO_c = \Z + c\cO_K$ be the order of conductor $c$ in
$\cO_K$ and let $\cN_c = \cN \cap \cO_c$, which is an invertible ideal of $\cO_c$.
Then $\cO_c / \cN_c \simeq \Z / N\Z$ and the map $\C / \cO_c \ra \C / \cN_c^{-1}$
is a cyclic isogeny of degree $N$. Thus, it defines a point $x_c \in X_0(N)(\C)$.
By the theory of complex multiplication, this point is defined
over the ring class field $K[c]$ of conductor $c$ over $K$ (that is, the
unique abelian extension of $K$ corresponding to the norm subgroup
$\widehat{\cO_c}^\times K^\times \subset \widehat{K}^\times$; e.g., if $c = 1$ then
$K[1] = H_K$).
We use the parameterization $\varphi : X_0(N) \ra E$ to obtain points
$$
y_c = \varphi (x_c) \in E(K[c]).
$$
Let $y_K = \Tr_{H_K / K}(y_1)$. We refer to $y_K$ as the
{\em Heegner point} for the discriminant~$D$, even though it is only
well defined up to sign and torsion (if $\cN'$ is another ideal with
$\cO / \cN' \simeq \Z / N\Z$ then the new Heegner point differs from $y_K$ by at most
a sign change and a rational torsion point).
\subsection{Explicit computation of the points $y_c$}\label{subsec:heegptscomp}
Significant work has been done on explicit calculations of Heegner points on
elliptic curves (see \cite{cohen:nt},~\cite{delauney:phd},
~\cite{elkies:heegner},~\cite{watkins:heegner}). Yet, all of these only compute
the points $y_1$ and $y_K$. In~\cite{ejl:cassels} explicit computations of the
points $y_c$ were considered in several examples and some difficulties were outlined.
However, there has been no algorithm which provably computes the points $y_c$. One
of the main contributions of this paper is the description of such an algorithm.
To compute the point $y_c = [\C / \cO_c \ra \C / \cN_c^{-1}] \in E(K[c])$ we let $f \in S_2(\Gamma_0(N))$ be the newform corresponding to the elliptic curve $E$ and $\Lambda$ be
the complex lattice (defined up to homothety), such that $E \isom \C / \Lambda$. Let
$\mathfrak h^\times = \mathfrak h \cup \mathbb{P}^1(\Q) \cup \{i \infty\}$, where $\mathfrak h = \{z \in \C : \Im(z) > 0\}$ is the upper-half plane equipped with the
action of $\Gamma_0(N)$ by linear fractional transformations. The modular
parametrization $\vphi : X_0(N) \ra E$ is then given by the function
$\varphi : \h^\times \ra \C / \Lambda$
\begin{equation}\label{eqn:modparam}
\varphi(\tau) = \int_{\tau}^{i \infty} f(z) dz
= \sum_{n \geq 1} \frac{a_n}{n}e^{2\pi i n \tau},
\end{equation}
where $\ds f = \sum_{n = 1}^\infty a_n q^n$ is the Fourier expansion of the modular
form $f$.
We first compute ideal class representatives $\mathfrak a_1, \mathfrak a_2, \dots, \mathfrak a_{h_c}$ for the Picard group
$\Pic(\cO_c) \isom \Gal(K[c]/K)$, where $h_c = \# \Pic(\cO_c)$. Let $\sigma_i \in \Gal(K[c]/K)$ be the image of the ideal class of $\mathfrak a_i$ under the Artin map.
We use the ideal $\mathfrak a_i$ to compute a complex number
(a {\em quadratic surd})$\tau_i \in \h$ representing the CM point $\sigma_i(x_c)$
for each $i = 1, \dots, h_c$ (since
$X_0(N) = \Gamma_0(N) \backslash \h^\times$). Explicitly, the Galois conjugates of
$x_c$ are
$$
\sigma_i(x_c) = [\C / \mathfrak a_i^{-1} \ra \C / \mathfrak a_i^{-1} \cN_c^{-1}],\ i = 1, \dots, h_c.
$$
Next, we use (\ref{eqn:modparam}) to approximate $\vphi(\sigma_i(x_c))$ as an element
of $\C / \Lambda$ by truncating the infinite series up to sufficiently many terms whose
number is determined precisely by the results of the appendix. Finally, the image of $\vphi(\tau_i) + \Lambda$ under the Weierstrass $\wp$-function
gives us an approximation of the $x$-coordinate of the point $y_c$ on the Weierstrass model of the elliptic curve $E$. On the other hand,
this coordinate is $K[c]$-rational. Thus, if we compute the map (\ref{eqn:modparam}) with
sufficiently many terms and up to high enough floating point accuracy, we must
be able to recognize the correct $x$-coordinate of $y_c$ on the Weierstrass model as an element of
$K[c]$.
To implement the last step, we use the upper bound established in the appendix
on the logarithmic height of the Heegner point $y_c$. The bound on the logarithmic
height comes from a bound on the canonical height combined with bounds on the height difference (see the appendix for complete details).
Once we have a height bound, we estimate the floating point accuracy required for the computation. Finally, we estimate the number of terms
of (\ref{eqn:modparam}) necessary to compute the point $y_c$ up to the corresponding
accuracy (see~\cite[p.591]{cohen:nt} for more details).
\begin{rem}
In practice, there are two ways to implement the above algorithm. The first approach is to compute an approximation $x_i$ of the $x$-coordinates of
$y_c^{\sigma_i}$ for every $i = 1, \dots, c$ and form the polynomial $F(z) = \prod_{i = 1}^{h_c} (z - x_i)$.
The coefficients of this polynomial are very close to the rational coefficients of the minimal polynomial of the actual $x$-coordinate of $y_c$.
Thus, one can try to recognize the coefficients of $F(z)$ by using the continued fractions method. The second approach is to search for the $\tau_i$ with
the largest imaginary part (which will make the convergence of the corresponding series (\ref{eqn:modparam}) defining the modular parametrization fast) and then
try to search for an algebraic dependence of degree $[K[c] : K]$ using standard algorithms implemented in PARI/GP. Indeed, computing a conjugate with a smaller imaginary part
might be significantly harder since the infinite series in (\ref{eqn:modparam}) will
converge slower and one will need more terms to compute the image up to the
required accuracy.
\end{rem}
\begin{rem}
We did {\em not} actually implement an algorithm for computing bounds
on heights of Heegner points as described in the appendix of this
paper. Thus, the computations in the specific examples below are not
provably correct, though we did many consistency checks and we are
convinced that our computational observations are correct. The
primary goal of the examples and practical implementation of our
algorithm is to provide tools and data for improving our theoretical
understanding of Kolyvagin's conjecture, and not making the computations
below provably correct does not detract from either of these goals.
\end{rem}
\subsection{Examples}\label{subsec:ptsexamples}
We compute the Heegner points $y_c$ for specific elliptic curves and choices of quadratic imaginary fields.
\vspace{0.1in}
\noindent {\bf \rm{\bf 53a1}:} Let ${E}_{/\Q}$ be the elliptic curve with label \textrm{\bf 53a1} in Cremona's database (see~\cite{cremona:onlinetables}). Explicitly, $E$ is
the curve
{\small
$y^2 + xy + y = x^3 - x^2.$
}
\noindent Let $D = 43$ and $c = 5$. The conductor of $E$ is $53$ which is split
in $K = \Q(\sqrt{-D})$, so $D$ is a Heegner discriminant for $E$. The modular form associated
to $E$ is
{\small
$
f_{E}(q) = q - q^2 - 3q^3 - q^4 + 3q^6 - 4q^7 + 3q^8 + 6q^9 + \cdots .
$
}
\noindent One applies the methods from Section~\ref{subsec:heegptscomp} to compute the minimal polynomial of the $x$-coordinate
of $y_5$ for the above model
{\small
$$
F(x) = x^6 - 12x^5 + 1980x^4 - 5855x^3 + 6930x^2 - 3852x + 864.
$$
}
\noindent Since $F(x)$ is an irreducible polynomial over $K$, it generates the ring class field $K[5]/K$, i.e.,
$K[5] = K[\alpha] \isom K[x] / \langle F(x) \rangle$, where $\alpha$ is one of the roots. To find the $y$-coordinate of $y_5$ we substitute $\alpha$ into the equation of
$E$ and factor the resulting quadratic polynomial over $K[5]$
to obtain that the point $y_5$ is equal to
{\small
$$
\left( \alpha , -4/315\alpha^5 + 43/315 \alpha^4 - 7897/315\alpha^3 + 2167/35 \alpha^2
- 372/7\alpha + 544/35 \right ) \in E(K[5]).
$$
}
\vspace{0.1in}
\noindent {\bf \rm{\bf 389a1:}}
The elliptic curve with label \textrm{\bf 389a1} is
{\small
$
y^2 + y = x^3 + x^2 - 2x
$
}
and the associated modular form is
{\small
$
f_{E}(q) = q - 2q^2 -2q^3 +2q^4 -3q^5 + 4q^6 -5q^7 +q^9 + 6q^{10} + \cdots.
$
}
Let $D = 7$ (which is a Heegner discriminant for $E$) and $c = 5$. As above, we
compute the minimal polynomial of the $x$-coordinate of $y_{5}$:
{\small
$$
F(x) = x^6 + \frac{10}{7}x^5 - \frac{867}{49}x^4 - \frac{76}{245}x^3 + \frac{3148}{35}x^2 -
\frac{25944}{245}x + \frac{48771}{1225}.
$$
}
If $\alpha$ is a root of $F(x)$ then $y_5 = (\alpha,\beta)$ where
{\small
\begin{eqnarray*}
\beta &=& \frac{280}{7761}\sqrt{-7}\alpha^5 + \frac{1030}{7761}\sqrt{-7}\alpha^4 -
\frac{12305}{36218}\sqrt{-7}\alpha^3 - \frac{10099}{15522}\sqrt{-7}\alpha^2 \\
&+& \frac{70565}{54327}\sqrt{-7}\alpha + \frac{- 18109 -33814\sqrt{-7} }{36218}.
\end{eqnarray*}
}
\vspace{0.1in}
\noindent {\bf \rm{\bf 709a1:}} The elliptic curve \textrm{\bf 709a1} with equation
{\small
$
y^2 + y = x^3 - x^2 - 2x
$
}
has an associated modular form
{\small
$
f_{E}(q) = q - 2q^2 - q^3 + 2q^4 - 3q^5 + 2q^6 - 4q^7 - 2q^9 + \cdots.
$
}
Let $D = 7$ (a Heegner discriminant for $E$) and $c = 5$. The minimal polynomial
of the $x$-coordinate of $y_5$ is
{\small
$$
F(x) = \frac{1}{5^2\cdot 7^2 \cdot 19^2} \left ( 442225x^6 - 161350x^5 - 2082625x^4 - 387380x^3 + 2627410x^2 + 18136030x + 339921 \right ),
$$
}
and if $\alpha$ is a root of $x$ then $y_5 = (\alpha, \beta)$ for
{\small
\begin{eqnarray*}
\beta &=& \frac{341145}{62822}\sqrt{-7}\alpha^5 - \frac{138045}{31411}\sqrt{-7}\alpha^4 - \frac{31161685}{1319262}\sqrt{-7}\alpha^3 +
\frac{7109897}{1319262}\sqrt{-7}\alpha^2 + \\
&+& \frac{39756589}{1319262}\sqrt{-7}\alpha + \frac{- 219877+ 4423733\sqrt{-7}}{439754}.
\end{eqnarray*}
}
\vspace{0.1in}
\noindent {\bf \rm{\bf 718b1:}} The curve \textrm{\bf 718b1} has equation
{\small
$
\ y^2 + xy + y = x^3 - 5x
$
}
with associated modular form
{\small
$
f_E(q) = q - q^2 - 2q^3 + q^4 - 3q^5 + 2q^6 - 5q^7 - q^8 + q^9 + 3q^{10} + \dots.
$
}
Again, for $D = 7$ and $c = 5$ we find
{\small
$
F(x) = \frac{1}{3^4\cdot 5^2}\left (2025x^6 + 12400x^5 + 32200x^4 + 78960x^3 + 289120x^2 + 622560x + 472896 \right )
$
}
and $y_5 = (\alpha, \beta)$ with
{\small
\begin{eqnarray*}
\beta &=& \frac{16335}{12271}\sqrt{-7}\alpha^5 + \frac{206525}{36813}\sqrt{-7}\alpha^4 + \frac{54995}{5259}\sqrt{-7}\alpha^3 +
\frac{390532}{12271}\sqrt{-7}\alpha^2 + \\
&+& \frac{- 36813 + 9538687\sqrt{-7}}{73626}\alpha + \frac{- 12271 + 4018835\sqrt{-7}}{24542}.
\end{eqnarray*}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Kolyvagin's conjecture: consequences and evidence}\label{sec:kolconj}
We recall Kolyvagin's construction of the cohomology classes in Section~\ref{subsec:kolclasses} and state
Kolyvagin's conjecture in Section~\ref{subsec:kolconj}. Section~\ref{subsec:selmer} is devoted to the proof of the promised
consequence regarding the $\Z_p$-corank of the Selmer group of an elliptic curve with
large analytic rank. In Section~\ref{subsec:cornut} we provide
Cornut's arguments for the theoretical evidence for Kolyvagin's conjecture and finally, in Section~\ref{subsec:389a1} we verify Kolyvagin's conjecture
for particular elliptic curves. Throughout the entire section we assume that
$E_{/\Q}$ is an elliptic curve of conductor $N$, $D$ is a Heegner discriminant for $E$ and $p \nmid ND$ is a prime such that the mod $p$ Galois representation
$\rhobar_{E, p} : \Gal(\Qbar / \Q) \ra \Aut(E[p])$ is surjective.
\subsection{Preliminaries}\label{subsec:prelim}
Most of this section follows the exposition in~\cite{gross:kolyvagin},~\cite{mccallum:kolyvagin} and~\cite{kolyvagin:structureofsha}.
\vspace{0.1in}
\noindent \textit{1. Kolyvagin primes.} We refer to a prime number $\ell$ as a \emph{Kolyvagin prime} if $\ell$ is inert in
$K$ and $p$ divides both $a_\ell$ and $\ell+1$). For a Kolyvagin prime $\ell$ let
$$
M(\ell) = \ord_p(\gcd(a_\ell, \ell+1)).
$$
We denote by $\Lambda^r$ the set of all square-free products of
exactly $r$ Kolyvagin primes and let
$\ds \Lambda = \bigcup_r \Lambda^r$.
For any $c \in \Lambda$, let
$\ds M(c) = \min_{\ell \mid c} M(\ell)$. Finally, let
$$
\Lambda^r_m = \{c \in \Lambda^r : M(c) \geq m\}
$$
and let $\ds \Lambda_m = \bigcup_{r} \Lambda_m^r$.
\vspace{0.1in}
\noindent \textit{2. Kolyvagin derivative operators.}
Let $\cG_c= \Gal(K[c]/K)$ and $G_c = \Gal(K[c]/K[1])$.
For each $\ell \in \Lambda^1$, the group $G_\ell$ is cyclic of order
$\ell + 1$. Indeed,
$$
G_\ell \simeq (\cO_K / \ell \cO_K)^\times / (\Z / \ell \Z)^\times \simeq \F_\lambda^\times / \F_\ell^\times.
$$
Moreover, $\ds G_c \isom \prod_{\ell \mid c} G_\ell$ (since
$\Gal(K[c]/K[c/\ell])\isom G_\ell$).
Next, fix a generator $\sigma_\ell$ of $G_\ell$ for each $\ell \in \Lambda^1$. Define
$D_\ell = \sum_{i = 1}^\ell i \sigma_\ell^i \in \Z[G_\ell]$ and let
$$
D_c = \prod_{\ell \mid c} D_\ell \in \Z[G_c].
$$
Note that $(\sigma_\ell - 1) D_\ell = 1 + \ell - \Tr_{K[\ell]/K[1]}$.
We refer to $D_c$ as the \emph{Kolyvagin derivative operators}. Finally, let $S$ be a set of coset representatives
for the subgroup $G_c \subseteq \cG_c$. Define
$$
P_c = \sum_{s \in S} s D_c y_c \in E(K[c]).
$$
The points $P_c$ are derived from the points $y_c$, so we will refer to them as \emph{derived Heegner points}.
\vspace{0.1in}
\noindent \textit{3. The function $m : \Lambda \ra \Z$ and the sequence $\ds \{m_r\}_{r \geq 0}$.}
For any $c \in \Lambda$ let $m'(c)$ be the largest positive integer such that
$P_c \in p^{m'(c)}E(K[c])$ (if $P_c$ is torsion then $m'(c) = \infty$).
Define a function $m : \Lambda \ra \Z$ by
$$
m(c) = \left \{
\begin{array}{ll}
m'(c) & \textrm{ if }m'(c) \leq M(c), \\
\infty & \textrm{ otherwise.}
\end{array} \right .
$$
Finally, let $\ds m_r = \min_{c \in \Lambda^r} m(c)$.
\begin{prop}\label{prop:inequal}
The sequence $\{m_r\}_{r \geq 0}$ is non-increasing, i.e.,
$m_r \geq m_{r+1}$ for every $r \geq 0$.
\end{prop}
\begin{proof}
This is proved in~\cite[Thm.C]{kolyvagin:structureofsha}.
\end{proof}
\subsection{Kolyvagin cohomology classes}\label{subsec:kolclasses}
Kolyvagin uses the points $P_c$ to construct classes $\kappa_{c, m} \in \H^1(K, E[p^m])$ for any $c \in \Lambda_m$. For the details of the construction,
we refer to~\cite[pp.241-242]{gross:kolyvagin}) and~\cite[\S 4]{mccallum:kolyvagin}.
The class $\kappa_{c, m}$ is explicit, in the sense that it is represented by the 1-cocycle
\begin{eqnarray}\label{eqn:cocycle}
\sigma \mapsto \sigma \left (\frac{P_c}{p^m} \right ) - \frac{P_c}{p^m} - \frac{(\sigma - 1)P_c}{p^m},
\end{eqnarray}
where $\ds \frac{(\sigma - 1)P_c}{p^m}$ is the unique $p^m$-division point of $(\sigma-1)P_c$ in $E(K[c])$ (see~\cite[Lem.~4.1]{mccallum:kolyvagin}).
The class $\kappa_{c, m}$ is non-trivial if and only if $P_c \notin p^m E(K[c])$
(which is equivalent to $m > m(c)$).
Finally, let $-\eps$ be the sign of the functional equation corresponding to $E$.
For each $c \in \Lambda_m$, let
$\eps(c) = \eps \cdot (-1)^{f_c}$ where $f_c = \# \{\ell : \ell \mid c\}$
(e.g., $f_1 = 0$). It follows from~\cite[Prop.5.4(ii)]{gross:kolyvagin} that
$\kappa_{c, m}$ lies in the $\eps(c)$-eigenspace for the action of complex conjugation on $\H^1(K, E[p^m])$.
\subsection{Statement of the conjecture}\label{subsec:kolconj}
We are interested in
$\ds m_\infty = \min_{c \in \Lambda} m(c) = \lim_{r \ra \infty} m_r$. In the case
when the Heegner point $P_1 = y_K$ has infinite order in $E(K)$, the Gross-Zagier formula
(see~\cite{gross-zagier}) implies that $r_\an(E/K) = 1$, so (by the results of
Kolyvagin) $r_{\MW}(E/K) = 1$. This means that $m_0 = \ord_p([E(K) : \Z y_K]) < \infty$.
In particular, $m_\infty < \infty$ which is equivalent to the system of cohomology classes
$$
T = \{\kappa_{c, m} : m \leq M(c)\}
$$
containing at least one non-zero class. A much more interesting and subtle is the case
of an elliptic curves $E$ over $K$ of analytic rank at least two. In this case,
Kolyvagin conjectured (see~\cite[Conj.C]{kolyvagin:structure_of_selmer}) that $T$
contains a non-trivial class as well.
\begin{conj}[(Kolyvagin's conjecture)]\label{conj:kolconj}
We have $m_\infty < \infty$, i.e., $T$ contains at least one class $\kappa_{c, m} \ne 0$.
\end{conj}
\begin{rem}
Although Kolyvagin's conjecture is obvious in the case of elliptic curves of analytic
rank one over $K$, the number $m_\infty$ is still interesting. Indeed,
the $p$-part of the Birch and Swinnerton-Dyer conjectural formula
for $E_{/K}$ is equivalent to
$\ds m_\infty = \ord_p \left (\prod_{q \mid N} c_q \right )$, where
$c_q$ is the Tamagawa number of $E_{/\Q}$ at $q$. See~\cite{jetchev:tamagawa} for some
new results related to this question which imply (in many cases) the exact upper bound
on the order of the $p$-primary part of the Shafarevich-Tate group as predicted by the
conjectural formula.
\end{rem}
\subsection{A consequence on the structure of Selmer groups}\label{subsec:selmer}
Let $r_p^{\pm}(E/K) = \corank_{\Z_p} \Sel_{p^\infty}(E/K)^{\pm}$. Kolyvagin
(see~\cite{kolyvagin:structure_of_selmer}) proved the following:
\begin{thm}[(Kolyvagin)]\label{thm:kolselstruct}
Assume Conjecture~\ref{conj:kolconj} and let $f$ be the smallest nonnegative
integer for which $m_f < \infty$. Then
$$
\Sel_{p^\infty}(E/K)^{\eps(-1)^{f+1}} \isom (\Q_p / \Z_p)^{f+1}
\oplus \textrm{(a finite group)}
$$
and
$$
\Sel_{p^\infty}(E/K)^{\eps(-1)^f} \isom (\Q_p/\Z_p)^r \oplus
\textrm{(a finite group)}
$$
where $r \leq f$ and $f - r$ is even. In other words, $r_p^{\eps(-1)^f}(E/K) = r$ and
$r_p^{\eps(-1)^{f+1}}(E/K) = f+1$.
\end{thm}
The above structure theorem of Kolyvagin has the following consequence which strongly supports Conjecture~\ref{conj:an-corank}.
\begin{cor}\label{cor:large-selmer}
Assume Conjecture~\ref{conj:kolconj}. Then
\noindent (i) If $r_{\an}(E/\Q)$ is even and nonzero then
$$
r_{p}(E/\Q) \geq 2.
$$
\noindent (ii) If $r_\an(E/\Q)$ is odd and strictly larger than one then
$$
r_p(E/\Q) \geq 3.
$$
\end{cor}
\begin{proof}
(i) By~\cite{bump-friedberg-hoffstein} or~\cite{murty-murty}, one can choose a quadratic imaginary field $K = \Q(\sqrt{-D})$ with Heegner discriminant $D$, such
that $L'(E^D_{/\Q}, 1) \ne 0$, where $E^D$ is the twist of $E$ by the
quadratic character associated to $K$ (note that $D$ is a Heegner discriminant,
so the sign of the functional equation of $E^D$ is always odd since $E$ has even sign).
Hence, by Theorem~\ref{thm:GZK}, the Selmer group $\Sel_{p^\infty}(E^D/\Q)$ has
$\Z_p$-corank one, i.e., $r_p^{-}(E/K) = r_p(E^D/\Q) = 1$. We want to prove $r_p^{+}(E/K) = r_p(E/\Q) \geq 2$. Assume the contrary, i.e., $r_p^{+}(E/K) = r_p(E/\Q) \leq 1$.
This means (by Theorem~\ref{thm:kolselstruct}) that $r = f = 0$ (here, $r$ and $f$
are as in Theorem~\ref{thm:kolselstruct}). Therefore, $m_0 < \infty$ which means that
the Heegner point $y_K$ has infinite order in $E(K)$ and hence (by the Gross-Zagier formula), $L'(E_{/K}, 1) \ne 0$. But this is a contradiction since
$$
L'(E_{/K}, s) = L'(E_{/\Q}, s)L(E^D_{/\Q},s) + L(E_{/\Q}, s)L'(E^D_{/\Q},s),
$$
which vanishes at $s = 1$ since $L(E_{/\Q}, 1) = L'(E_{/\Q}, 1) = 0$.
Thus, $r_p(E/\Q) = r_p^{+}(E/K) \geq 2$. \\
\noindent (ii) It follows from the work of Waldspurger (see also~\cite[pp.543-44]{bump-friedberg-hoffstein}) that
one can choose a quadratic imaginary field $K = \Q(\sqrt{-D})$ with a Heegner
discriminant $D$, such that
$L(E^D_{/\Q}, 1) \ne 0$ (this uses the fact that $r_\an(E/\Q)$ is odd). Therefore, by Theorem~\ref{thm:GZK}, $r_p(E^D/\Q) = 0$, i.e., $r_p^-(E/K) = 0$. By
Theorem~\ref{thm:kolselstruct}, $r = 0$ and $f$ is even.
If $f \geq 2$ we are done since $r_p(E/\Q) = r_p^{+}(E/K) = f+1 \geq 3$.
If $f = 0$, we use the same argument as in (i) to arrive at a contradiction. Therefore, $r_p(E/\Q) = r_p^{+}(E/K) \geq 3$.
\end{proof}
\begin{rem}
The parity conjecture proved by Nekov\'a\v{r} (see~\cite{nekovar:parity2}) implies that
$$
r_p(E/\Q) \equiv r_{\an}(E/\Q) \mod 2.
$$
Yet, Nekov\'a\v{r}'s result does not imply in any obvious way the statements of the
above proposition.
\end{rem}
\subsection{Cornut's theoretical evidence for Kolyvagin's conjecture}\label{subsec:cornut}
The following evidence for Conjecture~\ref{conj:kolconj} was proven by Cornut.
\begin{prop}\label{prop:evidence}
For all but finitely many $c \in \Lambda$ there exists a set
$R$ of liftings for the elements of
$\Gal(K[1]/K)$ into $\Gal(K^{\ab}/K)$, such that if $P_c = D_0 D_c y_c$ is the derived Heegner point defined in terms of this choice of liftings (i.e, if
$\ds D_0 = \sum_{\sigma \in R} \sigma$), then $P_c$ is non-torsion.
\end{prop}
\begin{rem}
For a non-torsion point $P_c$, let $m'(c)$ be the function defined in
Section~\ref{subsec:prelim}.
Proposition~\ref{prop:evidence} provides little evidence towards Kolyvagin's conjecture.
The reason is that even if one gets non-torsion points $P_c$, it might
still happen that for each such $c$ we have $m'(c) > M(c)$ (i.e., $m(c) = \infty$) in
which case all classes $\kappa_{c, m}$ with $m \leq M(c)$ will be trivial.
\end{rem}
Let $\ds K[\infty] = \bigcup_{c \in \Lambda} K[c]$. The proof of the proposition
depends on the following two lemmas:
\begin{lem}\label{lem:fintors}
The group $E(K[\infty])_{\tors}$ is finite.
\end{lem}
\begin{proof}
Let $q$ be any prime which is a prime of good reduction for $E$, which is inert in $K$ and which is different from the primes in
$\Lambda^1$ (there are infinitely many such primes according to \v{C}ebotarev
density theorem). Let $\mathfrak q$ be the unique prime of $K$ over $q$. It follows from class field theory that the prime $\mathfrak q$ splits
completely in $K[\infty]$ since it splits completely in each of the finite
extensions $K[c]$.
Thus, the completion of $K[\infty]$ at any prime which
lies over $\mathfrak q$ is isomorphic to $K_{\mathfrak q}$ and therefore, $E(K[\infty])_{\tors} \hra E(K_{\mathfrak q})_{\tors}$. The last group is finite
since it is isomorphic to an extension of $\Z_q^2$ by a finite group (see~\cite[Lem.I.3.3]{milne:duality} or~\cite[p.168-169]{tate:p-div}).
Therefore, $E(K[\infty]_{\tors})$ is finite.
\end{proof}
Let $|E(K[\infty])_{\tors}| = M < \infty$ and let
$\ds d(c) = [K[c] : K[1]] = \prod_{\ell \mid c} (\ell+1)$ for any $c \in \Lambda$.
Let $m_E$ be the modular degree of $E$, i.e., the degree
of the fixed optimal modular parametrization $\vphi : X_0(N) \ra E$.
\begin{lem}\label{lem:D0}
Suppose that $c \in \Lambda$ satisfies $d(c) > m_E M$. There exists a set of lifting
$R$ of $\Gal(K[1] / K)$ into
$\Gal(K[c]/K)$, such that $D_0 y_c \notin E(K[c])_{\tors}$, where $\ds D_0 = \sum_{\sigma \in R} \sigma$.
\end{lem}
\begin{proof}
The $\Gal(K[c]/K[1])$-orbit of the point $x_c \in X_0(N)(K[c])$ consists of $d(c)$ distinct points (since $K[c] = K(j(\cO_c))$), so there are
at least $d(c) / m_E$ distinct points in the orbit $\Gal(K[c]/K[1])y_c$. Choose a set of representatives $R$
of $\Gal(K[c]/K) / \Gal(K[c]/K[1])$ which contains the identity element
$1 \in \Gal(K[c]/K)$. For $\tau \in \Gal(K[c]/K[1])$
define
$$
R_\tau = (R -\{1\}) \cup \{\tau\}.
$$
Let $\ds S = \sum_{\sigma \in R} \sigma y_c$ and $\ds S_\tau = \sum_{\sigma \in R_\tau} \sigma y_c$. Then
$$
S_\tau - S = \tau y_c - y_c,
$$
which takes at least $d(c) / m_E > M$ distinct values. Therefore, there exists an
automorphism
$\tau \in \Gal(K[c]/K[1])$, for which
$S_\tau \notin E(K[c])_{\tors}$, which proves the lemma.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:evidence}]
Suppose that $c \in \Lambda$ satisfies the statement of Lemma~\ref{lem:D0} for some
choice of liftings $R$ and the corresponding $\ds D_0 = \sum_{\sigma \in R}$, i.e.,
$D_0 y_{c} \notin E(K[c])_{\tors}$. For any ring class character $\chi : \Gal(K[c]/K) \ra \C^\times$,
let
$e_{\chi} \in \C[\Gal(K[c]/K)]$ be the eidempotent projector corresponding to $\chi$. Explicitly,
$$
e_{\chi} = \frac{1}{\# \Gal(K[c]/K)} \sum_{\sigma \in \Gal(K[c]/K)} \chi^{-1}(\sigma) \sigma \in \C[\Gal(K[c]/K)].
$$
Consider $V = E(K[c]) \otimes \C$ as a complex representation of $\Gal(K[c]/K)$. The
representation $V$ decomposes as
$$
V = \bigoplus_{\chi : \Gal(K[c]/K) \ra \C^\times} V_\chi,
$$
where $V_\chi$ is the one-dimensional subspace on which $\Gal(K[c]/K[1])$ acts via
the character $\chi$. Since the vector
$D_0 y_c \otimes 1 \in V$ is non-zero, there exists a ring class character $\chi$,
such that $e_\chi (D_0y_c \otimes 1) \ne 0$.
Next, we consider the point $D_0 D_c y_c \in E(K[c])$ and claim that
$D_0 D_c y_c \otimes 1 \in E(K[c]) \otimes \C$ is
non-zero, which is sufficient to conclude that
$P_c = D_0 D_c y_c \notin E(K[c])_{\tors}$. We will prove
that $e_\chi (D_0 D_c y_c \otimes 1) \ne 0$. Indeed,
\begin{eqnarray*}
e_\chi (D_0 D_c y_c \otimes 1) &=& e_{\chi} D_c (D_0 y_c \otimes 1) = \prod_{\ell \mid c}
\left ( \sum_{i = 1}^\ell i \sigma_\ell^i \right ) e_{\chi} (D_0y_c \otimes 1) = \\
&=& \prod_{\ell \mid c} \left ( \sum_{i = 1}^\ell i \chi ( \sigma_\ell )^i \right ) e_{\chi} (D_0y_c \otimes 1),
\end{eqnarray*}
the last equality holding since $\tau e_{\chi} = \chi(\tau) e_{\chi}$ in
$\C[\Gal(K[c]/K)]$ for all
$\tau \in \Gal(K[c]/K)$. Thus, it remains to compute $\ds \sum_{i=1}^\ell i \chi(\sigma_{\ell})^i$ for
every $\ell \mid c$. It is not hard to show that
$$
\sum_{i = 1}^\ell i \chi(\sigma_{\ell})^i = \left \{
\begin{array}{ll}
\frac{\ell + 1}{\chi(\sigma_\ell) - 1} & \textrm{if } \chi(\sigma_\ell) \ne 1 \\
\frac{\ell(\ell+1)}{2} & \textrm{if } \chi(\sigma_\ell) = 1.
\end{array}
\right.
$$
Thus, $e_\chi (D_0 D_c y_c \otimes 1) \ne 0$ which means that $P_c = D_0 D_c y_c \notin E(K[c])_{\tors}$
for any $c$ satisfying $D_0 y_c \notin E(K[c])_{\tors}$. To complete the proof, notice that for all, but finitely
many $c \in \Lambda$, the hypothesis of Lemma~\ref{lem:D0} will be satisfied.
\end{proof}
\subsection{Computational evidence for Kolyvagin's conjecture}\label{subsec:389a1}
Consider the example $E = \textrm{\bf 389a1}$ with equation
{\small $y^2 + y = x^3 + x^2 - 2x$.}
As in Section~\ref{subsec:ptsexamples}, let $D = 7$, $\ell = 5$, and $p = 3$. Using the algorithm of~\cite[\S 2.1]{bsdalg1} we verify
that the mod $p$ Galois representation $\rhobar_{E, p}$ is surjective. Next, we observe that $\ell = 5$ is a Kolyvagin prime for $E, p$ and $D$.
Let $c = 5$ and consider the class $\kappa_{5, 1} \in \H^1(K, E[3])$. We claim that $\kappa_{5, 1} \ne 0$, which will verify Kolyvagin's conjecture.
\begin{prop}\label{prop:389-verify}
The class $\kappa_{5, 1} \ne 0$. In other words, Kolyvagin's conjecture holds for
$E = \textrm{\bf 389a1}$,
$D = 7$ and $p = 3$.
\end{prop}
Before proving the proposition, we recall some standard facts about division polynomials (see, e.g.,~\cite[Ex.3.7]{silverman:aec}).
For an elliptic curve given in Weierstrass form over any field of characteristic different from 2 and 3,
{\small
$
y^2 = x^3 + Ax + B,
$
}
one defines a sequence of polynomials $\psi_m \in \Z[A, B, x, y]$ inductively as follows:
{\small
\begin{eqnarray*}
& & \psi_1 = 1, \ \psi_2 = 2y, \\
& & \psi_3 = 3x^4 + 6Ax^2 + 12Bx - A^2, \\
& & \psi_4 = 4y(x^6 + 5Ax^4 + 20Bx^3 - 5A^2 x^2 - 4ABx - 8B^2 - A^3), \\
& & \psi_{2m+1} = \psi_{m+2}\psi_m^3 - \psi_{m-1}\psi_{m+1}^3 \ \ \textrm{for } m \geq 2, \\
& & 2y\psi_{2m} = \psi_m(\psi_{m+2}\psi_{m-1}^2 - \psi_{m-2}\psi_{m+1}^2) \ \ \textrm{for } m \geq 3.
\end{eqnarray*}
}
Define also polynomials $\phi_m$ and $\omega_m$ by
{\small
$$
\phi_m = x\psi_m^2 - \psi_{m+1}\psi_{m-1}, \ 4y\omega_m = \psi_{m+2}\psi_{m-1}^2 - \psi_{m-2}\psi_{m+1}^2.
$$
}
After replacing $y^2$ by $x^3 + Ax+B$, the polynomials $\phi_m$ and $\psi_m^2$ can be viewed as polynomials in $x$ with leading
terms $x^{m^2}$ and $m^2 x^{m^2 - 1}$, respectively. Finally, multiplication-by-$m$ is given by
{\small
$$
mP = \left ( \frac{\phi_m(P)}{\psi_m(P)^2}, \frac{\omega_m(P)}{\psi_m(P)^3} \right ).
$$
}
\begin{proof}[Proof of Proposition~\ref{prop:389-verify}]
We already computed the Heegner point $y_5$ on the model
{\small
$
y^2 + y = x^3 + x^2 - 2x
$
}
in Section~\ref{subsec:ptsexamples}. The Weierstrass model for $E$ is
{\small
$
y^2 = x^3 - 7/3x + 107/108,
$
}
so $\ds A = -7/3$ and $\ds B = 107/108$.
We now compute the point $\ds P_5 = \sum_{i = 1}^5 i \sigma^i (y_5) \in E(K[5])$ on the
Weierstrass model, where $\sigma$ is a generator of $\Gal(K[5]/K)$.
To show that $\kappa_{5, 1} \ne 0$ we need to check that there is no point $Q = (x, y)$, such that $3Q = P_5$. For the verification of this fact, we use the division polynomial
$\psi_3$ and the polynomial $\phi_3$. Indeed, it follows from the recursive definitions
that
{\small
\begin{eqnarray*}
\phi_3(x) &=& x^9 - 12Ax^7 - 168Bx^6 + (30A^2 + 72B)x^5 - 168ABx^4 + \\
&+& (36A^3 + 144AB - 96B^2)x^3 + 72A^2Bx^2 + \\
&+& (9A^4 - 24A^2B + 96AB^2 + 144B^2)x + 8A^3B + 64B^3.
\end{eqnarray*}
}
Consider the polynomial $g(x) = \phi_3(x) - X(P_5)\psi_3(x)^2$,
where $X(P_5)$ is the $x$-coordinate of the point $P_5$ on the Weierstrass
model. We factor $g(x)$ (which has degree 9) over the number
field $K[5]$ and check that it is irreducible. In particular, there is no root
of $g(x)$ in $K[5]$, i.e., there is no $Q \in E(K[5])$, such that $3Q = P_5$.
Thus, $\kappa_{5,1} \ne 0$.
\end{proof}
\begin{rem}
Using exactly the same method as above, we verify Kolyvagin's conjecture for the
other two elliptic curves of rank two from Section~\ref{subsec:ptsexamples}.
For both $E = \textrm{\bf 709a1}$ and $E = \textrm{\bf 718b1}$ we use $D = 7$, $p = 3$
and $\ell = 5$ (which are valid parameters), and verify that $\kappa_{5,1} \ne 0$ in
the two cases. For completeness, we provide all the data of each computation
in the three examples in the files \verb,389a1.txt,,
\verb,709a1.txt, and \verb,718a1.txt,.
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-trivial elements in the Shafarevich-Tate group}\label{sec:expl-sha}
Throughout the entire section, let $E_{/\Q}$ be a non-CM elliptic curve,
$K = \Q(\sqrt{-D})$, where $D$ is a
Heegner discriminant for $E$ such that the Heegner point $y_K$ has infinite order
in $E(K)$ (which, by the Gross-Zagier formula and Kolyvagin's result, means that
$E(K)$ has Mordell-Weil rank one) and let $p$ be a prime, such that
$p \nmid DN$ and the mod $p$ Galois representation $\rhobar_{E, p}$ is surjective.
\subsection{Non-triviality in Kolyvagin classes.}
Under the above assumptions, the next proposition provides a criterion which
guarantees that an explicit class in the Shafarevich-Tate group
$\Sha(E/K)$ is non-zero.
\begin{prop}\label{prop:nontriv}
Let $c \in \Lambda_m$. Assume that the following hypotheses are satisfied:
\begin{enumerate}
\item{}[Selmer hypothesis]:\label{hyp:sel} The class $\kappa_{c, m}\in \H^1(K,E[p^m])$
is an element of the Selmer group $\Sel_{p^m}(E/K)$.
\vspace{0.1in}
\item{}[Non-divisibility]:\label{hyp:nondiv} The derived Heegner point $P_c$ is not divisible by $p^m$
in $E(K[c])$, i.e., $P_c \notin p^m E(K[c])$.
\vspace{0.1in}
\item{}[Parity]:\label{hyp:parity} The number $f_c = \# \{ \ell : \ell \mid c \}$ is odd.
\end{enumerate}
Then the image $\kappa'_{c, m}\in \H^1(K,E)[p^m]$
of $\kappa_{c,m}$ is a non-zero element of $\Sha(E/K)[p^m]$.
\end{prop}
\begin{proof}
The first hypothesis implies that the image $\kappa'_{c, m}$ of $\kappa_{c, m}$ in $\H^1(K, E)[p^m]$ is an element of
the Shafarevich-Tate group $\Sha(E/K)$. The second one implies that $\kappa_{c, m} \ne 0$. To show that $\kappa'_{c, m} \ne 0$
we use the exact sequence
$$
0 \ra E(K)/p^mE(K) \ra \Sel_{p^m}(E/K) \ra \Sha(E/K)[p^m] \ra 0
$$
which splits under the action of complex conjugation as
$$
0 \ra ( E(K)/p^mE(K) )^{\pm} \ra \Sel_{p^m}(E/K)^{\pm} \ra \Sha(E/K)^{\pm}[p^m] \ra 0.
$$
According to~\cite[Prop.5.4(2)]{gross:kolyvagin}, the class $\kappa_{c, m}$ lies in the $\eps_c$-eigenspace of the Selmer group $\Sel_{p^m}(E/K)$ for the
action of complex conjugation, where $\eps_c = \eps (-1)^{f_c} = -1$ ($f_c$ is odd by the third hypothesis and $\eps = 1$ since $-\eps$ is the sign of the functional equation
for $E_{/K}$ which is $-1$ by Gross-Zagier). On the other hand, the Heegner point $y_K = P_1$ lies
in the $\eps_1$-eigenspace of complex conjugation (again, by~\cite[Prop.5.4(2)]{gross:kolyvagin}) where $\eps_1 = \eps (-1)^{f_1} = 1$. Since $E(K)$ has rank one, the group
$E(K)^{-}$ is torsion and since $E(K)[p] = 0$, we obtain that $(E(K)/p^mE(K))^{-} = 0$. Therefore,
$$
\Sel_{p^m}(E/K)^{-} \isom \Sha(E/K)^{-}[p^m],
$$
which implies $\kappa'_{c, m} \ne 0$.
\end{proof}
\subsection{The example $E = \textrm{\bf 53a1}$.}
The Weierstrass equation for the curve $E=${\bf \textrm{53a1}} is
{\small
$
y^2 = x^3 + 405 x + 16038
$
}
and $E$ has rank one over $\Q$. The Fourier coefficient
$a_5(f) \equiv 5 + 1 \equiv 0 \mod 3$, so
$\ell = 5$ is a Kolyvagin prime for $E$, the discriminant $D = 43$ and the prime $p = 3$.
Kolyvagin's construction exhibits a class $\kappa_{5, 1} \in \H^1(K, E[3])$. We will
prove the following proposition:
\begin{prop}\label{prop:53a1}
The cohomology class $\kappa_{5, 1} \in \H^1(K, E[3])$ lies in the Selmer group $\Sel_3(E/K)$ and its image $\kappa'_{5, 1}$ in
the Shafarevich-Tate group $\Sha(E/K)$ is a nonzero $3$-torsion element.
\end{prop}
\begin{rem}
Since $E/K$ has analytic rank one, Kolyvagin's conjecture
is automatic (since $m_0 < \infty$ by Gross-Zagier's formula) and one
knows (see~\cite[Thm.~5.8]{mccallum:kolyvagin})
that there exist Kolyvagin classes $\kappa'_{c,m}$ which generate
$\Sha(E/K)[p^\infty]$. Yet, this result is not explicit in the sense that
one does not know any particular Kolyvagin class which is non-trivial.
The above proposition exhibits an explicit non-zero cohomology class
in the $p$-primary part of the Shafarevich-Tate group $\Sha(E/K)$.
\end{rem}
\begin{proof}
Using the data computed in Section~\ref{subsec:ptsexamples} for this curve, we
apply the Kolyvagin derivative to compute the point $P_5$. In order to do
this, one needs a generator of the Galois group $\Gal(K[5] / K)$.
Such a generator is determined by the image of $\alpha$, which will be
another root of $f(x)$ in $K[5]$. We check that the automorphism $\sigma$
defined by
{\small
\begin{eqnarray*}
\alpha & \mapsto & \frac{1}{1601320}(47343 + 54795\sqrt{-43})\alpha^5 +
\frac{1}{2401980}(- 614771 -936861\sqrt{-43})\alpha^4 + \\
& + & \frac{1}{600495}(34507457 + 40541607\sqrt{-43} )\alpha^3 +
\frac{1}{4803960}(102487877 -767102463\sqrt{-43} )\alpha^2 + \\
& + & \frac{1}{400330}(- 61171198 + 52833377\sqrt{-43})\alpha +
\frac{1}{200165}(18971815 -7453713\sqrt{-43})
\end{eqnarray*}
}
is a generator (we found this automorphism by factoring
the defining polynomial of the number field over the number
field $K[5]$). Thus, we can compute $\ds P_5 = \sum_{i = 1}^5 i \sigma^i(y_5)$.
Note that we are computing the point on the Weierstrass model of $E$
rather than on the original model. The cohomology class $\kappa_{5, 1}$
is trivial if and only if $P_5 \in 3 E(K[5])$. To show that
$P_5 \notin 3E(K[5])$, we repeat
the argument from Proposition~\ref{prop:389-verify} and verify (using
any factorization algorithm for polynomials over number fields) that
the polynomial $g(x) = \phi_3(x) - X(P_5)\psi_3(x)^2$ has no linear factors over
$K[5]$ (here, $X(P_5)$ is the $x$-coordinate of $P_5$). This means that there is no
point $Q = (x, y) \in E(K[5])$, such that $3Q = P_5$, i.e., $\kappa_{5, 1} \ne 0$.
Finally, using Proposition~\ref{prop:nontriv} we conclude that the class
$\kappa'_{5, 1} \in \Sha(E/K)[3]$ is non-trivial.
\end{proof}
\begin{rem}
For completeness, all the computational data is provided (with the appropriate explanations) in the file \verb,53a1.txt,.
We verified the irreducibility of $g(x)$ using MAGMA and PARI/GP independently.
\end{rem}
\appendix
\section{Upper bounds on the logarithmic heights of the Heegner points $y_c$}
We explain how to compute an upper bound on the logarithmic height $h(y_c)$. The method
first relates the canonical height $\widehat{h}(y_c)$ to special values of the first
derivatives of certain automorphic $L$-functions via Zhang's generalization of the Gross-Zagier formula. Then we either
compute the special values up to arbitrary precision using a well-known
algorithm (recently implemented by Dokchitser) or use effective asymptotic upper
bounds (convexity bounds) on the special values and Cauchy's integral formula.
Finally, using some known bounds on the difference between the canonical and the logarithmic heights, we obtain explicit upper bounds on the logarithmic height
$h(y_c)$. We provide a summary of the asymptotic bounds in
Section~\ref{subsec:asymptotic} and refer the reader to~\cite{jetchev:asymptotic}
for complete details.
\subsection{The automorphic $L$-functions $L(f, \chi, s)$ and
$L(\pi_{f \otimes \theta_\chi}, s)$}
\label{subsec:lfunc}
Let $d_c = c^2 D$ and let $\ds f = \sum_{n \geq 1} a_n q^n$ be the new eigenform of
level $N$ and weight two corresponding to $E$. Let $\chi : \Gal(K[c]/K) \ra \C^\times$
be a ring class character.
\vspace{0.1in}
\noindent \textit{1. The theta series $\theta_\chi$.} Recall that ideal classes
for $\Pic(\cO_c)$ correspond to primitive, reduced
binary quadratic forms of discriminants $d_c$. To each ideal class $\cA$ we
consider the corresponding binary quadratic form $Q_{\cA}$ and the theta series
$\theta_{Q_{\cA}}$ associated to it via
$$
\theta_{Q_{\cA}} = \sum_{M} e^{2\pi i z Q_{\cA}(M)}
$$
which is a modular form for $\Gamma_0(d_c)$ of weight one with character $\eps$ (the quadratic character of $K$)
according to Weil's converse theorem (see~\cite{shimura:jac_factors} for
details). This allows us to define a cusp form
$$
\theta_\chi = \sum_{\cA \in \Pic(\cO_c)} \chi^{-1}(\cA) \theta_{Q_{\cA}} \in S_1(\Gamma_0(d_c), \eps).
$$
Here, we view $\chi^{-1}$ as a character of $\Pic(\cO_c)$ via the
isomorphism $\Pic(\cO_c) \isom \Gal(K[c]/K)$. Let $\theta_\chi = \sum_{m \geq 0}
b_m q^m$ be the Fourier expansion. By $L(f, \chi, s)$ we will always mean the
Rankin $L$-function $L(f \otimes \theta_\chi, s)$ (equivalently, the $L$-function
associated to the automorphic representation $\pi = f \otimes \theta_\chi$ of
$\GL_4$).
\vspace{0.1in}
\noindent \textit{2. The functional equation of $L(f, \chi, s)$.}
We recall some basic facts about the Rankin $L$-series $L(f \otimes \theta_\chi, s)$
following~\cite[\S III]{gross:heegner_points}. Since $(N, D)=1$, the {\em conductor} of
$L(f \otimes \theta_\chi, s)$ is
$Q = N^2 d_c^2$. The Euler factor at infinity (the gamma factor) is
$L_\infty(f \otimes \theta_{\chi}, s) = \Gamma_\C(s)^2$. If we set
$$
\Lambda(f\otimes \theta_{\chi}, s) = Q^{s/2} L_\infty(f \otimes \theta_\chi, s) L(f\otimes \theta_\chi, s)
$$
then the function $\Lambda$ has a holomorphic continuation to the entire complex plane
and satisfies the functional equation
$$
\Lambda(f\otimes \theta_\chi, s) = -\Lambda(f \otimes \theta_\chi, 2-s).
$$
In particular, the order of vanishing of $L(f\otimes \theta_\chi, s)$ at $s = 1$ is non-negative and odd, i.e., $L(f \otimes \theta_\chi, 1) = 0$.
\vspace{0.1in}
\noindent \textit{3. The automorphic $L$-function $L(\pi_{f \otimes \theta_\chi}, s)$.}
In order to center the critical line at $\ds \Re(s) = \frac{1}{2}$ instead of
$\ds \Re(s) = 1$ (which is consistent with Langlands convention), we consider
the $L$-function $L(\pi_{f \otimes \theta_\chi}, s)$ corresponding to the
automorphic representation attached to $f \otimes \theta_\chi$ for $\GL_4$.
This function satisfies
$$
L(\pi_{f \otimes \theta_\chi}, s) = L \left (f \otimes \theta_\chi, s+\frac{1}{2} \right )
$$
The function $L(\pi_{f \otimes \theta_\chi}, s)$ then satisfies a functional equation relating the values at $s$ and $1-s$.
For a general automorphic $L$-function $L(\pi, s)$, we consider the corresponding Dirichlet
series and Euler product
$$
L(\pi, s) = \sum_{n \geq 1} \frac{\lambda_\pi(n)}{n^{s}} =
\prod_p (1 - \alpha_{\pi, 1}(p)p^{-s})^{-1} \dots (1 - \alpha_{\pi, d}(p)p^{-s})^{-1},
$$
which are absolutely convergent for $\Re(s) > 1$.
\subsection{Zhang's formula}
For a character $\chi$ of $\Gal(K[c]/K)$, let
$$
e_\chi = \frac{1}{\# \Gal(K[c]/K)} \sum_{\sigma \in \Gal(K[c]/K)} \chi^{-1}(\sigma) \sigma \in \C[\Gal(K[c]/K)]
$$
be the associated eidempotent (see also Section~\ref{subsec:cornut}). The canonical
height $\widehat{h}( e_\chi y_c )$ is related via the generalized Gross-Zagier formula
of Zhang to a special value of the derivative of the Rankin-Selberg $L$-function
$L(f, \chi, s)$ at $s = 1$ (see~\cite[Thm.1.2.1]{zhang:gross-zagier}). More precisely,
\begin{thm}[(Zhang)]\label{thm:zhang}
If $(\,,)$ denotes the Petersson inner product on $S_2(\Gamma_0(N))$ then
$$
L'(f, \chi, 1) = \frac{4}{\sqrt{D}} (f, f) \widehat{h} (e_\chi y_c ).
$$
\end{thm}
\noindent Since $\langle e_{\chi'}y_c, e_{\chi''} y_c\rangle = 0$ whenever $\chi' \ne \chi''$ (here, $\langle \,,\rangle$ denotes the N\'eron-Tate
height pairing for $E$) and since $\widehat{h}(x) = \langle x, x\rangle $ then
\begin{equation}\label{eqn:height}
\widehat{h}(y_c) = \widehat{h} \left (\sum_{\chi} e_\chi y_c \right ) = \sum_{\chi} \widehat{h}(e_\chi y_c).
\end{equation}
Thus, we will have an upper bound on the canonical height $\widehat{h}(y_c)$ if
we obtain upper bounds on the special values $L'(f, \chi, 1)$ for every character
$\chi$ of $\Gal(K[c]/K)$.
\subsection{Computing special values of derivatives of automorphic $L$-functions}
For simplicity, let $\gamma(s) = L_\infty(f \otimes \theta_\chi, s+1/2)$ be the
gamma factor of the $L$-function $L(\pi, s)$. This means that if
$\Lambda(\pi, s) = Q^{s/2} \gamma(s) L(\pi, s)$ then $\Lambda(\pi, s)$ satisfies the
functional equation $\Lambda(\pi, s) = \Lambda(\pi, 1-s)$. We will describe a
classical algorithm to compute the value of $L^{(k)}(\pi, s)$ at $s = s_0$ up to arbitrary precision. The algorithm and its implementation is discussed in a greater generality in~\cite{dokchitser}. The main idea is to express $\Lambda(\pi, s)$ as an infinite series
with rapid convergence which is usually done in the following sequence of steps:
\begin{enumerate}
\item Consider the inverse Mellin transform of the gamma factor $\gamma(s)$, i.e.,
the function $\phi(t)$ which satisfies
$$
\gamma(s) = \int_0^\infty \phi(t) t^s \frac{dt}{t}.
$$
One can show (see~\cite[\S 3]{dokchitser}) that $\phi(t)$ decays exponentially for
large $t$. Hence, the sum
$$
\Theta(t) = \sum_{n = 1}^\infty \lambda_\pi(n) \phi \left (\frac{nt}{\sqrt{Q}} \right )
$$
converges exponentially fast. The function $\phi(t)$ can be computed numerically as
explained in~\cite[\S 3-5]{dokchitser}.
\item The Mellin transform of $\Theta(t)$ is exactly the function $\Lambda(\pi, s)$.
Indeed,
\begin{eqnarray*}
\int_0^\infty \Theta(t) t^s \frac{dt}{t} &=&
\int_0^\infty \sum_{n = 1}^\infty \lambda_{\pi}(n) \phi \left (\frac{nt}{\sqrt{Q}} \right )
t^s \frac{dt}{t} = \sum_{n=1}^\infty \lambda_{\pi}(n) \int_0^\infty
\phi \left (\frac{nt}{\sqrt{Q}} \right ) t^s \frac{dt}{t} = \\
& = & \sum_{n = 1}^\infty \lambda_{\pi}(n) \left ( \frac{\sqrt{Q}}{n} \right )^s \int_0^\infty \phi(t') t'^s \frac{dt'}{t'} = Q^{s/2} \gamma(s) L(\pi, s) = \Lambda(\pi, s).
\end{eqnarray*}
\item Next, we obtain a functional equation for $\Theta(t)$ which relates
$\Theta(t)$ to $\Theta(1/t)$. Indeed, since $\Lambda(\pi, s)$ is holomorphic, Mellin's
inversion formula implies that
$$
\Theta(t) = \int_{c - i \infty}^{c + i \infty} \Lambda(\pi, s) t^{-s} ds,\ \forall c.
$$
Therefore,
\begin{eqnarray*}
\Theta(1/t) &=& \int_{c - i\infty}^{c+i\infty} \Lambda(\pi, s) (1/t)^{-s} ds = -
t\int_{c - i\infty}^{c+i\infty} \Lambda(\pi, 1-s) t^{-(1-s)} ds = \\
&=& -t \int_{c - i\infty}^{c+i\infty} \Lambda(\pi, s') t^{-s'}ds' = -t \Theta(t).
\end{eqnarray*}
Thus, $\Theta(t)$ satisfies the functional equation $\Theta(1/t) = -t \Theta(t)$.
\item Next, we consider the incomplete Mellin transform
$$
G_s(t) = t^{-s} \int_t^\infty \phi(x) x^s \frac{dx}{x}, \ t > 0
$$
of $\phi(t)$. The function $G_s(t)$ satisfies $\ds \lim_{t \ra 0} t^s G_s(t) = \gamma(s)$
and it decays exponentially. Moreover, it can be computed numerically
(see~\cite[\S 4-5]{dokchitser}).
\item Finally, we use the functional equation for $\Theta(t)$ to obtain
\begin{eqnarray*}
\Lambda(\pi, s) &=& \int_0^\infty \Theta(t) t^s \frac{dt}{t} = \int_0^1
\Theta(t) t^s \frac{dt}{t} + \int_1^{\infty} \Theta(t) t^s \frac{dt}{t} = \\
&=& \int_1^\infty \Theta(1/t') t'^{-s} \frac{dt'}{t'} +
\int_1^\infty \Theta(t) t^s \frac{dt}{t} = \\
&=& -\int_1^\infty \Theta(t') t'^{1-s} \frac{dt'}{t'} + \int_1^\infty \Theta(t)t^s
\frac{dt}{t}.
\end{eqnarray*}
\item Finally, we compute
\begin{eqnarray*}
\int_1^\infty \Theta(t)t^{s} \frac{dt}{t} &=&
\int_1^\infty \sum_{n = 1}^\infty \lambda_{\pi}(n) \phi \left ( \frac{nt}{\sqrt{Q}}\right )
t^s \frac{dt}{t} =
\sum_{n=1}^\infty \lambda_{\pi}(n) \int_1^\infty \phi \left ( \frac{nt}{\sqrt{Q}}\right ) t^s
\frac{dt}{t} = \\
&=& \sum_{n=1}^\infty \lambda_{\pi}(n) \int_{\frac{n}{\sqrt{Q}}}^\infty \phi \left ( t'\right ) \left ( \frac{\sqrt{Q}t'}{n} \right )^s =
\sum_{n=1}^\infty \lambda_{\pi}(n) G_s \left (\frac{n}{\sqrt{Q}} \right).
\end{eqnarray*}
Thus,
$$
\Lambda(\pi, s) = \sum_{n=1}^\infty \lambda_{\pi}(n) G_s \left (\frac{n}{\sqrt{Q}} \right)
- \sum_{n=1}^\infty \lambda_{\pi}(n) G_{1-s} \left (\frac{n}{\sqrt{Q}} \right)
$$
is the desired expansion. From here, we obtain a formula for the $k$-th derivative
$$
\frac{\partial^k}{\partial s^k} \Lambda(\pi, s) =
\sum_{n=1}^\infty \lambda_{\pi}(n) \frac{\partial^k}{\partial s^k}G_s \left (\frac{n}{\sqrt{Q}} \right)
- \sum_{n=1}^\infty \lambda_{\pi}(n) \frac{\partial^k}{\partial s^k}G_{1-s} \left (\frac{n}{\sqrt{Q}} \right).
$$
The computation of the derivatives of $G_s(x)$ is explained in~\cite[\S 3-5]{dokchitser}.
\end{enumerate}
\subsection{Asymptotic estimates on the canonical heights $\widehat{h}(y_c)$}\label{subsec:asymptotic}
In this section we provide an asymptotic bound on the canonical height
$\widehat{h}(y_c)$ by using convexity bounds on the special values of the
automorphic $L$-functions $L(\pi, s)$ defined in Section~\ref{subsec:lfunc}.
We only outline the basic techniques used to prove the asymptotic bounds and
refer the reader to~\cite{jetchev:asymptotic} for the complete details.
Asymptotic bounds on heights of Heegner points are obtained
in~\cite{ricotta-vidick}, but these bounds are of significantly different type than
ours. In our case, we fix the elliptic curve $E$ and let the fundamental discriminant
$D$ and the conductor $c$ of the ring class field both vary. We obtain the following:
\begin{prop}
Fix the elliptic curve $E$ and let the fundamental discriminant $D$ and the conductor
$c$ vary. For any $\eps > 0$ the following asymptotic bound holds
$$
\widehat{h}(y_c) \ll_{\eps, f} h_D D^{\eps} c^{2+\eps},
$$
where $h_D$ is the class number of the quadratic imaginary field $K = \Q(\sqrt{-D})$.
Moreover, the implied constant depends only on $\eps$ and the cusp form $f$.
\end{prop}
One proves the proposition by combining the formula of Zhang with convexity
bounds on special values of automorphic $L$-functions.
The latter are conveniently expressed in terms of a quantity
known as the {\em analytic conductor} associated to the automorphic representation
$\pi$ (see~\cite[p.12]{michel:parkcity}). It is a function $Q_\pi(t)$ over the real
line, which is defined as
$$
Q_\pi(t) = Q \cdot \prod_{i = 1}^d (1 + |it - \mu_{\pi, i}|), \ \forall t\in \R,
$$
where $\mu_{\pi, i}$ are obtained from the gamma factor
$$
L_\infty(\pi, s) = \prod_{i = 1}^d \Gamma_\R(s - \mu_{\pi, i}), \ \Gamma_\R(s) =
\pi^{-s/2}\Gamma(s/2).
$$
In our situation for $\pi = \pi_{f \otimes \theta_\chi}$, $d = 4$ and
$\mu_{\pi, 1} = \mu_{\pi, 2} = 0$, $\mu_{\pi, 3} = \mu_{\pi, 4} = 1$
(see~\cite[\S 1.1.1]{michel:parkcity} and~\cite[\S 3]{serre:zeta} for discussions
of local factors at archimedian places). Moreover, we let $Q_\pi = Q_\pi(0)$.
The main idea is to prove that for a fixed $f$,
$|L'(\pi_{f \otimes \theta_\chi}, 1/2)| \ll_{\eps, f} Q_{\pi_{f \otimes \theta_\chi}}^{1/4 + \eps}$, where the implied constant only depends on $f$ and $\eps$ (and is independent of $\chi$ and the
discriminant $D$). To establish the bound, we first prove an asymptotic bound for
the $L$-function $L(\pi_{f\otimes \theta_\chi}, s)$ on
the vertical line $\Re(s) = 1 + \eps$ by either using the Ramanujan-Petersson
conjecture or a method of Iwaniec (see~\cite[p.26]{michel:parkcity}). This gives us the
estimate
$|L(\pi_{f \otimes \theta_\chi}, 1+\eps + it)| \ll_{\eps, f} Q_{\pi_{f \otimes \theta_\chi}}(t)^\eps$. Then, by the functional equation
for $L(\pi_{f \otimes \theta_\chi}, s)$ and Stirling's approximation formula, we deduce an upper bound for the
$L$-function on the vertical line $\Re(s) = -\eps$, i.e.,
$|L(\pi_{f \otimes \theta_\chi}, -\eps + t)|
\ll_{\eps, f} Q_{\pi_{f \otimes \theta_\chi}}(t)^{1/2 + \eps}$. Next, we apply Phragmen-Lindel\"of's convexity principle
(see~\cite[Thm.5.53]{iwaniec-kowalski}) to obtain the bound
$|L(\pi_{f \otimes \theta_\chi}, 1/2 + it)| \ll_{\eps, f} Q_\pi(t)^{1/4+\eps}$ (also known as {\em convexity bound}). Finally, by applying Cauchy's integral formula for a
small circle centered at
$s = 1/2$, we obtain the asymptotic estimate
$|L'(\pi_{f \otimes \theta_\chi}, 1/2)| \ll_{\eps, f} Q_{\pi_{f \otimes \theta_\chi}}^{1/4+\eps}$.
Since $Q= N^2 d_c^2 = N^2 D^2 c^4$ in our situation and since
$[K[c]:K] = h_D \prod_{\ell \mid c} (\ell+1)$, Zhang's formula (Theorem~\ref{thm:zhang}) and equation (\ref{eqn:height}) imply that for any $\eps > 0$,
$$
\widehat{h}(y_c) \ll_{\eps, f} h_D D^{\eps} c^{2+\eps}.
$$
\begin{rem}
In the above situation (the Rankin-Selberg $L$-function of two cusp forms
of levels $N$ and $d_c = c^2D$), one can even prove a subconvexity bound
$|L'(\pi_{f \otimes \theta_\chi}, 1/2)| \ll_{f} D^{1/2 - 1/1057} c^{1 - 2/1057}$, where
the implied constant depends only on $f$ and is independent of $\chi$
(see~\cite[Thm.2]{michel:subconvexity}). Yet, the proof relies on much more involved
analytic number theory techniques than the convexity principle, so we do not discuss it here.
\end{rem}
\subsection{Height difference bounds and the main estimates}\label{subsec:heightbds}
To estimate $h(y_c)$ we need a bound
on the difference between the canonical and the logarithmic heights. Such a bound
has been established in~\cite{silverman:height} and~\cite{cremona-siksek} and is effective.
Let $F$ be a number field. For any non-archimedian place $v$ of $K$, let $E^0(F_v)$ denote
the points of $E(F_v)$ which specialize to the identity component of the N\'eron model of
$E$ over the ring of integers $\cO_v$ of $F_v$. Moreover, let $n_v = [F_v : \Q_v]$ and let
$M_F^\infty$ denote the set of all archimedian places of $F$. A slightly weakened
(but easier to compute) bounds on the height difference are provided by the
following result of~\cite[Thm.2]{cremona-siksek}
\begin{thm}[(Cremona-Prickett-Siksek)]
Let $P \in E(F)$ and suppose that $P \in E^0(F_v)$ for every non-archimedian place $v$
of $F$. Then
$$
\frac{1}{3[F : \Q]} \sum_{v \in M_F^\infty} n_v \log \delta_v \leq h(P) - \widehat{h}(P)
\leq \frac{1}{3[F : \Q]} \sum_{v \in M_F^\infty} n_v \log \eps_v,
$$
where $\eps_v$ and $\delta_v$ are defined in~\cite[\S 2]{cremona-siksek}.
\end{thm}
\begin{rem}
All of the points $y_c$ in our particular examples satisfies the condition
$y_c \in E^0(K[c]_v)$ for all non-archimedian places $v$ of $K[c]$. Indeed, according
to~\cite[\S III.3]{gross-zagier} (see also~\cite[Cor.3.2]{jetchev:tamagawa}) the
point $y_c$ lies in $E^0(K[c]_v)$ up to a rational torsion point.
Since $E(\Q)_{\tors}$ is trivial for all the curves that we are considering, the
above proposition is applicable. In general, one does not need this assumption in order
to compute height difference bounds (see~\cite[Thm.1]{cremona-siksek} for the
general case).
\end{rem}
\begin{rem}
A method for computing $\eps_v$ and $\delta_v$ up to arbitrary precision for real
and complex archimedian places is provided in~\cite[\S 7-9]{cremona-siksek}.
\end{rem}
\begin{acknowledgements}
We are indebted to Christophe Cornut for sharing his thoughts
on the Kolyvagin's conjecture. We would like to thank Jan Nekov\'{a}\v{r} for his
helpful discussions on height bounds, Stephen D. Miller and Mark Watkins for
pointing out analytic methods for estimating special values of derivatives of
automorphic $L$-functions, Henri Darmon and David Jao for useful conversations on
Heegner points computations, and Ken Ribet for reading the preliminary draft and for
useful conversations.
\end{acknowledgements}
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