\section{Introduction}
\textbf{Notation:} i) We shall use the notations $K$, $E$, $\mathfrak{p}$ and $p$  in the sense of chapter 3 throughout this chapter except in section 2.\\
ii) The group law on $E$ shall be denoted by same $\pm$. Meaning will be clear from context. $O$ shall denote the identity element.\\~\\
In this chapter we shall give a proof of theorem $0.2$ .\\
Before going into the main computations we review the basic facts about the N\'eron - Tate height on elliptic curve and its decomposition in terms of local height.\\
\section{N\'eron - Tate height on elliptic curves}
\textbf{Notation:} In this section the notation $E$ will be used to denote a general elliptic curve. $F$ shall denote an arbitrary number field.\\~\\
Let $E$ be an elliptic curve over $F$ presented in a given Weierstrass form .\\~\\
\textbf{Proposition 4.2.1:} Let $P \in E(\overline{\mathbb{Q}})$. Define 
\[ h(P) =
\begin{cases}
0 & \text{if} \, P = O, \\
\frac {h(x)} {2} & \text{if} \, P = ( x, y) \neq O. 
\end{cases}
\]
Then the limit \[\lim_{ n \to \infty}\frac {h([2^{n}]P)} {4^{n}} \]  exists.\\~\\
\textbf{Proof :} See \cite{sil1} chapter VIII, proposition 9.1.\\~\\
\textbf{Definition 4.2.2:} The N\'eron - Tate height on $E$ is the function $\widehat{h} : E(\overline{\mathbb{Q}}) \to \mathbb{R}$ defined by \[ \widehat{h}(P) = \lim_{n \to \infty} \frac {h([2^n]P)} {4^{n}}. \]\\
The following proposition lists the properties of N\'eron - Tate height which we shall use again and again.\\~\\
\textbf{Proposition 4.2.3:} Let $E$ be an elliptic curve defined over a number field $F$ and let $\widehat{h}$ be the N\'eron - Tate height defined on $E$.\\
Then the following holds:\\
i) Let $P, Q \in E(\overline{\mathbb{Q}})$. Then \[ \widehat{h}(P + Q) + \widehat{h}(P - Q) = 2 (\widehat{h}(P) + \widehat{h}(Q) ).\]
ii) $\widehat{h}([m]P) = m^{2}\widehat{h}(P)$ for all $ m \in \mathbb{Z}$ and for all $P \in E(\overline{\mathbb{Q}})$.\\
iii) $\widehat{h}$ is a quadratic form. In other words, $\widehat{h}$ is even and the pairing \[<,> : E(\overline{\mathbb{Q}})\times E(\overline{\mathbb{Q}}) \to \mathbb{R}\] given by $< P, Q > = \widehat{h}(P+Q) - \widehat{h}(P) - \widehat{h}(Q)$ is bilinear.\\
iv) Let $P \in E(\overline{\mathbb{Q}})$. Then $\widehat{h}(P) \geq 0$. Further $\widehat{h}(P) = 0$ if and only if $P$ is a torsion point.\\
v) Let $Q$ be a torsion point. Then $\widehat{h}(P + Q) = \widehat{h}(P)$ for all $P \in E(\overline{\mathbb{Q}})$.\\~\\
\textbf{Proof:} See \cite{sil1} chapter VIII theorem 9.3.\\~\\
\subsection{Local Height Functions}
In this subsection we briefly recall the facts about local height functions. For details one is referred to \cite{sil 2}, chapter VI.\\~\\
\textbf{Proposition 4.2.4:} Let $K$ be a field which is complete with respect to an absolute value $|\cdot|_{v}$ and let \[v(\cdot) = - \log |\cdot|_{v} \] denote the corresponding additive absolute value. Let $E/K$ be an elliptic curve. Choose a Weierstrass equation for $E/K$,
\[ E : y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, \] and let $\Delta$ be the discriminant of this equation.\\
a) There exists a unique function \[ \lambda : E(K) - \{O\} \to \mathbb{R} \] with the following properties:\\
i) $\lambda$ is continuous on $E(K) - \{O\}$ with respect to the $v$-adic topology and is bounded on the complement of any $v$-adic neighborhood of $O$.\\
ii) The limit \[ \lim_{ P \to O} \{\lambda(P) + \frac {1} {2}v(x(P)) \} \] exists where the limit is being taken with respect to $v$-adic topology.\\
iii) For all $P \in E(K)$ with $[2]P \neq O$, \[\lambda([2]P) = 4 \lambda(P) + v((2y + a_{1}x + a_{3})P) - \frac {1} {4}v(\Delta). \]     
b)$\lambda$ is independent of the choice of Weierstrass equation for $E/K$.\\
c) Let $L/K$ be a finite extension and $\overline{v}$ is the extension of $v$ to $L$. Then \[ \lambda_{\overline{v}}(P) = \lambda_{v}(P) \hspace{1cm} \text{for all } P \in E(K) - \{O\}.\]\\
\textbf{Proof :} See \cite{sil 2} Chapter VI, Theorem 1.1. \\~\\
\textbf{Remark 4.2.5 :} Let $\overline{K}$ denote a fixed algebraic closure of $K$ and let $\overline{v}$ denote a fixed extension of $v$ to $\overline{K}$.\\ 
Then using part c of proposition 4.2.4 one can $\lambda$ extend uniquely on $E(\overline{K}) - \{O\}$.\\~\\
\textbf{Proposition 4.2.6 :} Let $F$ be a number field. Let $M_{F}$ denote the set of normalized absolute values on $F$. Put $d_{v} = [F_{v} : \mathbb{Q}_{v}]$ for each $v \in M_{F}$. Let $E/F$ be an elliptic curve. For each $v \in M_{F}$, let $\lambda_{v}: E(F_{v}) - \{O\} \to \mathbb{R}$ be the local height function as described in proposition 4.2.4. Then \[ \widehat{h}(P) = \frac {1} {[F: \mathbb{Q}]} \sum_{v \in M_{F}} d_{v}\lambda_{v}(P) \] for all $P \in E(F) - \{O\}$.\\~\\
\textbf{Proof :} See \cite{sil 2} chapter VI, theorem 2.1. \\~\\
\textbf{ Lemma 4.2.7 :} Let $E$ and $F$ be as in statement of proposition 4.2.6. Let $v$ be a finite place of $E$. Assume that $E$ has a good reduction at $v$. Fix a minimal Weierstrass equation for $E$ at $v$. Then $\lambda_{v}(P) = \frac {1} {2} \max \{ - v(x(P)), 0\}$ where $x(P)$ denotes the x- coordinate of $P$ with respect to the Weierstrass presentation we have fixed. \\~\\
\textbf{Proof :} See \cite{sil 2} chapter VI, theorem 4.1.\\
\section{First height estimates}
\textbf{Notation :} i) Let $l$ be a prime. We use the notation $E[l^{\infty}]$ to denote all points on $E$ (defined over $\overline{\mathbb{Q}}$) which are annihilated by a power of $l$.\\
ii) $N$ shall denote a positive integer. We shall write $N = p^n M$ where $n$ is an integer $\geq 0$ and $M$ is a positive integer such that $p \nmid M$.\\
iii) We fix an embedding of $K$ in $\overline{\mathbb{Q}}_p$ by completion at $\mathfrak{p}$. Thus $E$ can be thought as an elliptic curve over $\mathbb{Q}_{w}$ as before.\\
iv) $|\cdot|_p$ on $\mathbb{Q}_{w}(N)$ determines a place of $K(N)$ for each $N$. We shall call this place $v$.\\
v)Let $\widetilde{E}$, $\widetilde{\phi}$ be as in chapter 2. $\widetilde{\phi}$ acts on $\widetilde{E}$ as multiplication by an integer. We shall denote this integer by $a_w$.\\
vi) For an extension of local fields $L/K$, $e(L/K)$ shall denote the ramification index of the extension.\\
vii) Let $L$ be a finite extension of $K$. Let $u$ be a fixed place of $K$. For any place $\overline{u}$ of $L$ which extends $u$ we put $d_{\overline{u},u} = \frac{d_{\overline{u}}}{d_{u}}$ and define \[ \widehat{h}_{u}(P) = \frac {1} {[L : K]} \sum_{\overline{u} | u} d_{\overline{u}, u} \lambda_{\overline{u}}(P) \hspace{1 cm} \text{for all}\, P \in E(L). \] With this notation one has \[ \widehat{h}(P) = \frac {1} {[K : \mathbb{Q}]}\sum_{u \in M_{K}} d_{u} \widehat{h}_{u}(P) \hspace{1 cm} P \in E(L).\] \\~\\
The goal of this section is to prove a series of lemmas which will eventually lead to the proof of theorem 0.2.\\
These lemmas are similar to the results proved in chapter 3 and in most cases are elliptic analogues of them.\\
We assume that $E$ is presented in a Weierstrass form over $K$ such that it is minimal thought as a curve in $\mathbb{Q}_w$. By hypothesis, it has a good supersingular reduction. \\
\subsection{Lemmas on torsion points}
The next two lemmas are analogues of lemma 2.4.3 and 2.4.4 respectively :\\~\\
\textbf{Lemma 4.3.1 :} $E(\mathbb{Q}_{w}(N)) \cap E[p^{\infty}] = E[p^n].$\\~\\
\textbf{Proof :} Clearly $E[p^{n}] \subseteq E(\mathbb{Q}_{w}(N)) \cap E[p^{\infty}]$. We would like to show the reverse inclusion.\\
Let $T \in E(\mathbb{Q}_{w}(N)) \cap E[p^{\infty}]$ . Assume that $T$ has order $p^{m}$ for some integer $m \geq 0$. We would like to show $m \leq n$.\\
Consider the case $n = 0$. Then $\mathbb{Q}_{w}(N)/\mathbb{Q}_{w}$ is an unramified extension. Hence by lemma 2.3.3 we conclude that $m\geq1$ cannot hold.\\
So $m =0 = n$ and we are done in this case.\\
Now let $n \geq 1$. If already $m \leq n$, then we are done. Assume $ m > n$.\\
Using proposition 2.4.2 and theorem 2.1.1 we conclude that $e(\mathbb{Q}_{w}(N)|\mathbb{Q}_{w}) = q^{n - 1}(q - 1)$.\\
But by lemma 2.3.3 $e(\mathbb{Q}_w(T)|\mathbb{Q}_w) = q^{m - 1}(q - 1)$.\\
Since $\mathbb{Q}_w(T) \subseteq \mathbb{Q}_{w}(N)$ we conclude that $m \leq n$, a contradiction !
This contradiction proves the lemma.$\square$\\~\\
\textbf{Lemma 4.3.2 :} Assume that $n \geq 1$ and $\psi \in \text{Gal}(\mathbb{Q}_{w}(N)|\mathbb{Q}_{w}(N/p))$. Let $A \in E(\mathbb{Q}_{w}(N))$ such that $\psi(A) - A \in E_{\text{tor}}$. Then,
\[\psi(A) - A \in E[Q(n)]\] where $Q(n)$ is as in lemma 2.4.4.\\~\\
\textbf{Proof :} Put $B = \psi(A) - A$ and assume that order of $B$ is $N' = p^{n'}M'$ where $n' \geq 0$ and $p\nmid M'$.\\
Let $T= [p^{n'}](B)$ and note that its order is $M'$ which is coprime to $p$. Arguing exactly like lemma 2.4.3 (here we need to use proposition 2.4.2 instead of theory of cyclotomic extensions) we conclude that $T \in E[M]$, in particular $\psi(T) = T$.\\
The order of $[M'](B)$ is $p^{n'}$. So by lemma 4.3.2 $[M'](B) \in E[p^n]$. Then $[pM'](B) \in E[p^{n - 1}]$.\\
Therefore $\psi([pM'](B)) = [pM'](B)$.\\
Now one can imitate the proof of lemma 2.4.4 to get $[pt](B) = O$ where $t$ is order of $\psi \in \text{Gal}\,(\mathbb{Q}_{w}(N)|\mathbb{Q}_{w}(N/p))$.\\
From here the lemma follows easily. $\square$\\~\\

\subsection{Local estimates}
In this subsection we shall consider $E$ and $\widetilde{E}$ as projective curves. So without loss of generality, the points can be assumed to have primitive integral coordinates and we have a well defined reduction map.\\~\\ 
\textbf{Lemma 4.3.3:} Say $p \nmid N$ and $A \in E(\mathbb{Q}_{w}(N))$.\\
Then $A \in E(\mathbb{Q}_{p}^{ur})$. If additionally we have $\phi_{w}(A) \neq [a_{w}]A$ then, \[\lambda_{v}(\phi_{w}(A) - [a_w](A)) \geq \frac {1} {2}\log p .\]\\~\\
\textbf{Proof :} The first part of the assertion follows from proposition 2.4.1.\\
Note that $v$ extends the place $\mathfrak{p}$. Thus it is a place of good reduction.\\
Since $\widetilde{\phi} = [a_w]$ on $\widetilde{E}$, $\phi_{w}(A) - [a_w](A) \equiv 0 \mod p$.\\
The extension $\mathbb{Q}_{w}(N)/\mathbb{Q}_{w}$ is unramified.
Now the lemma follows from lemma 4.2.7.$\square$ \\~\\
\textbf{Lemma 4.3.4:} Let $n \geq 1$ and $\psi \in \text{Gal}\,(\mathbb{Q}_{w}(N)/\mathbb{Q}_{w}(N/p)$.\\
Assume that $A \in E(\mathbb{Q}_{w}(N))$ with $\psi(A) \neq A$. Then \[ \lambda_{v}(\psi(A) - A) \geq \frac {\log p} {2(q - 1)}.\]\\
\textbf{Proof :} As in proof of lemma 3.2.2 $\psi \in G_{i}(\mathbb{Q}_{w}(N)|\mathbb{Q}_{w})$ for $ i = q^{n - 1} - 1$.\\
Let $\mathfrak{P}$ the unique maximal ideal of the ring of integers of $\mathbb{Q}_{w}(N)$.\\
So $\psi(A)$ and $A$ are same element of $E$ reduced modulo $\mathfrak{P}^{q^{n - 1}}$. Let $x$ denote the first coordinate of $\psi(A) - A$. Then $\log |x|_p  \leq - \frac {q^{n - 1}} {e} \log p$ where $e = e(\mathbb{Q}_{w}(N)|\mathbb{Q}_{w}) = q^{n - 1}(q - 1)$ using theorem 2.1.1 and proposition 2.4.2.\\
Now the lemma follows from lemma 4.2.7. $\square$\\~\\

\subsection{Lower bound on $\widehat{h}_{\mathfrak{p}}$ }
\textbf{Lemma 4.3.5:} Assume that $n = 0$. If $A \in E(K(N)) - E_{\text{tor}}$ there is a nontorsion point $B \in E(\overline{\mathbb{Q}})$ with $\widehat{h}(B) \leq 2( w+ 1)\widehat{h}(A)$ such that \[ \widehat{h}_{\mathfrak{p}}(B) \geq \frac {1} {2}\log p.\]\\
\textbf{Proof :} Since $p \nmid N$, $K(N)/K$ is unramified at $\mathfrak{p}$ and $\phi_w$ acts on $K(N)$. Further since completion of $K$ at $\mathfrak{p}$ is $\mathbb{Q}_{p^f} \subseteq \mathbb{Q}_{w}$ we conclude that $\phi_w$ is trivial on $K$. Thus $\phi_w \in \, \text{Gal}\,(K(N)|K)$. As in proof of lemma 3.3.1 it lies in the center of the Galois group.\\
Put $B = \phi_{w}(A) - [a_w]A$. First note that $B$ is not a torsion point. Indeed, if it were a torsion point then $\widehat{h}(A) = \widehat{h}(\phi_{w}(A)) = w \widehat{h}(A)$ (recall $a_w = \pm p^f$) which implies $\widehat{h}(A) = 0$ contradicting our hypothesis $A \notin E_{\text{tor}}$.\\
Using parallelogram identity 
\[
\begin{split}
\widehat{h}(B) & \leq \widehat{h}(\phi_{w}(A) - [a_w](A)) + \widehat{h}(\phi_{w}(A) + [a_w](A)) \\
                         & = 2 (\widehat{h}(\phi_w(A)) + \widehat{h}([a_w](A))) \\
                         & = 2 ( w + 1) \widehat{h}(A). \\
\end{split}
\] 
Note that $\text{Gal}\,(K(N)|K)$ acts transitively on the places lying over $\mathfrak{p}$. Since $\phi_w$ lies in the center of the Galois group we have \[\lambda_{\sigma^{ - 1}v}(B) = \lambda_{v}(\phi_{w}(\sigma(A)) - [a_w](\sigma(A)))\]
for all $\sigma \in \text{Gal}\,(K(N)|K)$.\\
Clearly $|\text{Gal}\,(K(N)|K)v| = \sum_{u| \mathfrak{p},\mathfrak{p}\in M_{K(N)}} d_{u,\mathfrak{p}} $.\\
Using the estimate given in lemma 4.3.3 we conclude that the present lemma follows for this choice of $B$. $\square$\\~\\
\textbf{Lemma 4.3.6 :} Assume that $n \geq 1$. If $A \in E(K(N))$ satisfies $[Q(n)](A) \notin E(\mathbb{Q}_{w}(N/p))$, there is a nontorsion point $B \in E(\overline{\mathbb{Q}})$ with $\widehat{h}(B) \leq 4\widehat{h}(A)$ and
\[\widehat{h}_{\mathfrak{p}}(B) \geq \frac {\log p} {2p^6}. \]\\
\textbf{Proof :} It follows from hypothesis that there is a $\psi \in \text{Gal}\,(\mathbb{Q}_{w}(N)/\mathbb{Q}_{w}(N/p))$ such that $\psi([Q(n)](A)) \neq [Q(n)](A)$. Fix such a $\psi$. We shall use the same notation for its restriction to $K(N)$.\\
Let $B = \psi(A) - A \in E(\overline{\mathbb{Q}})$. By choice of $\psi$ it follows that $[Q(n)](B) \neq O$.\\
Using lemma 4.3.2 we conclude that $B \notin E_{\text{tor}}$.\\
An application of parallelogram law as in the previous lemma gives \[\widehat{h}(B) \leq 2(\widehat{h}(\psi(A)) + \widehat{h}(A)) = 4\widehat{h}(A).\]
Let $G_{\psi}$ denote the centralizer of $\psi$ in the group $ G = \text{Gal}(K(N)|K)$.\\
By lemma 3.3.2 we have \[|G_{\psi}v| \geq p^{- 4} \frac {|G|} {d_{v, \mathfrak{p}}}.\tag{4.3.1}\] 
For any $\sigma \in G_{\psi}$ we have $\lambda_{\sigma^{- 1}v} (B) = \lambda_v(\sigma(B)) = \lambda_v(\psi(\sigma(A)) - \sigma(A))$ and $\psi(\sigma(A)) \neq \sigma(A)$.\\
Applying lemma 4.3.4 we conclude that \[\lambda_{\sigma^{- 1}v}(B) \geq \frac {\log p} {2( q - 1)}\tag{4.3.2}\] for all $\sigma \in G_{\psi}$.\\
Note that if $u \in G_{\psi}v$ then $d_{u, \mathfrak{p}} = d_{v, \mathfrak{p}}$.\\
$\mathfrak{p}$ is a place of good reduction. Hence for each $u | \mathfrak{p}$, $u$ is a place of good reduction and $\lambda_{u}(B) \geq 0$.\\
 
Now 
\[
\begin{split}
\widehat{h}_{\mathfrak{p}}(B) & = \frac {1} {[K(N):K]} \sum_{u | \mathfrak{p}} d_{u, \mathfrak{p}}\lambda_{u}(B) \\
                                                   & \geq \frac {1} {[K(N):K]} \sum_{u \in G_{\psi}v} d_{u, \mathfrak{p}} \lambda_{u}(B)\\
                                                   & \geq \frac {\log p} {2(q - 1)[K(N):K]}\, |G_{\psi}v| d_{v, \mathfrak{p}}  \\
                                                   & \geq  \frac {\log p} {2p^4(q - 1)}\\
                                                   & \geq \frac {\log p} {2p^6}
\end{split}
\] 
where we are using the estimates in $(4.3.1)$ and $(4.3.2)$ and the fact that $|G| = [K(N):K]$.\\
Therefore this choice of $B$ does the job.$\square$\\~\\ 
\textbf{Lemma 4.3.7 :} Assume that $ n \leq 1$. If $A \in E(K(N)) - E_{\text{tor}}$, there exists a nontorsion point $B \in E(\overline{\mathbb{Q}})$ with $\widehat{h}(B) \leq 4p^4( w + 1)( q - 1)^2\widehat{h}(A)$ and \[ \widehat{h}_{\mathfrak{p}}(B) \geq \frac {\log p} {2p^6}.\]
\textbf{Proof :} Without loss of generality we can assume that $n = 1$. Now consider two cases :\\~\\
\textbf{Case I :} There is a $\sigma \in \text{Gal}\,(K(N)|K)$ such that $[Q(1)](\sigma(A)) (= \sigma([Q(1)](A)) ) \notin E(\mathbb{Q}_{w}(N/p))$.\\
Apply lemma 4.3.6 to $\sigma(A)$ to get a $B \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}(B) \leq 4\widehat{h}(\sigma(A)) = 4\widehat{h}(A)$ and \[\widehat{h}_{\mathfrak{p}}(B) \geq \frac {\log p} {2p^6}. \]
This $B$ has the desired properties.\\~\\
\textbf{Case II :} For all $\sigma \in \text{Gal}\,(K(N)|K)$, $\sigma([Q(1)](A)) \in E(\mathbb{Q}_{w}(N/p))$ . Then by lemma 3.4.2 we have $\sigma([Q(1)](A)) \in E(K(N/p))$. Now $p \nmid N/p$. So we can use lemma 4.3.5 to get a $B \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}(B) \leq 2(w + 1)\widehat{h}([Q(1)](A)) = 2p^4(w + 1)(q - 1)^2\widehat{h}(A)$ and \[ \widehat{h}_{\mathfrak{p}}(B) \geq \frac {1} {2} \log p .\]
Clearly this $B$ does the job. Hence we are done in this case also.\\~\\
Thus we have found a $B$ with desired properties in all cases .$\square$\\~\\
\textbf{ Lemma 4.3.8 :} If $A \in E(K(E_{\text{tor}})) - E_{\text{tor}}$, there is a nontorsion point $ B \in E(\overline{\mathbb{Q}})$ with $\widehat{h}(B) \leq 40p^4(w + 1)( q - 1)^2\widehat{h}(A)$ and \[ \widehat{h}_{\mathfrak{p}}(B) \geq \frac {\log p} {2p^6}.\]\\
\textbf{Proof :} Fix a positive integer $N$ such that $A \in E(K(N))$. Let $n , M$ be as before.\\
Use lemma 3.4.7 to get a $\sigma_0 \in \text{Gal}\,(K(N)|K)$ with properties as stated there.\\
Put $C = \sigma_{0}(A) - [2](A) \in E(K(N))$. Arguing as in proof of lemma 4.3.5 and using the hypothesis $A \notin E_{\text{tor}}$, we conclude that $C \notin E_{\text{tor}}$. \\
Using parallelogram law as before \[\widehat{h}(C) \leq 2(\widehat{h}(\sigma_{0}(A)) + \widehat{h}([2](A))) \leq 10 \widehat{h}(A) \tag{4.3.3}.\] 
Fix the least integer $n' \geq 0$ such that $C \in E(K(p^{n'}M))$. Clearly $n' \leq n$. Write $N' = p^{n'}M$. Consider two cases :\\~\\
\textbf{Case I:} $n' \leq 1$.\\
Apply lemma 4.3.7 to $C$ to get a $B \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}_{\mathfrak{p}}(B) \geq \frac {\log p} {2p^6}$ and it satisfies \[\widehat{h}(B) \leq 4p^{4}(w + 1)(q - 1)^2\widehat{h}(C) \leq 40p^{4}(w + 1)(q - 1)^{2}\widehat{h}(A)\] where we are using $(4.3.3)$.\\~\\
\textbf{Case II:} $n' \geq 2$.\\
Use lemma 3.4.2 to get a $\sigma \in \text{Gal}\,(K(N')|K)$ with $\sigma(C)( = C') \notin E(\mathbb{Q}_{w}(N'/p))$. Fix a $\psi \in \text{Gal}\,(\mathbb{Q}_{w}(N')/\mathbb{Q}_{w}(N'/p))$ such that $\psi(C') = C'$.\\
Set $A' = \sigma(A)$ to get \[ C' = \sigma_{0}(A') - [2](A') \in E(K(N')) \tag{4.3.4} \] since $\sigma_0$ lies in the center.\\
We would like to apply lemma 4.3.6 to $C'$. For that we need to verify $[Q(n')](C') = [q](C') \notin E(\mathbb{Q}_{w}(N'/p))$.\\
Assume the contrary. Write $T = \psi(C') - C'$.\\
Since $[q](C') \in E(\mathbb{Q}_{w}(N'/p))$ , $\psi([q](C')) = [q](C')$ and $T \in E[q]$.\\
By choice of $\psi$, $T \neq O$. Applying $\psi$ to both sides of $(4.3.4)$ and using the fact $\sigma_0$ lies in the center we have \[ C' + T =\psi(C') = \sigma_{0}(\psi(A')) - [2](\psi(A')).\tag{4.3.5} \] Put $P = \psi(A') - A' \in K(N)$. $(4.3.4)$ and $(4.3.5)$ gives $T = \sigma_{0}(P) - [2](P)$.\\
Since $T \in E_{\text{tor}}$, we have $\widehat{h}(P) = \widehat{h}(\sigma_{0}(P)) = \widehat{h}([2](P)) = 4\widehat{h}(P)$ and thus $P\in E_{\text{tor}}$.\\
Choose a positive integer $M_1$ such that it is coprime to $p$ and $[M_{1}](P) \in E[p^{\infty}]$. By lemma 4.3.1 $[M_1](P) \in E[p^n]$ .\\
By construction of $\sigma_0$, $\sigma_{0}([M_1](P)) = [2M_{1}(P)]$. Thus $[M_1](T) = O$ and $T \in E[M_1]$. But $T \in E[q]$ and $\gcd (M_1, q) = 1$. So $T = O$. A contradiction ! \\
So $[Q(n')](C') \notin \mathbb{Q}_{w}(N'/p)$. Now use lemma 4.3.6 to get a $B \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}_{\mathbb{p}}(B) \geq \frac {\log p} {2p^6}$ and \[\widehat{h}(B) \leq 4\widehat{h}(C') = 4\widehat{h}(C) \leq 40\widehat{h}(A)\]
where in the last step we are using $(4.3.3)$.\\~\\
Thus in all cases we have constructed a $B$ with desired properties. $\square$ \\~\\   
\section {Further estimates on $\widehat{h}_{v}$ }
For all $P \in E(\overline{\mathbb{Q}}) - \{O\}$ we have \[ \widehat{h}(P) = \frac {1} {[K :\mathbb{Q}]}\sum_{v \in M_K} d_{v}\widehat{h}_{v}(P). \tag{4.4.1} \]
We have computed some estimates for $\widehat{h}_{\mathfrak{p}}$. The goal of the current section is to find estimates for $\widehat{h}_{v}$ when $v \neq \mathfrak{p}$.\\
Habegger does it by using two equidistribution theorems. We shall follow his path.\\~\\
\subsection{Infinite places}
\textbf{Notation:} In this subsection $|\cdot|$ is the usual complex absolute value.\\
At first we recall some generalities.\\
Let $E$ be ab elliptic curve over a number field $F$ and $v$ be an infinite place of $F$.\\
Up to complex conjugation, $v$ determines an embedding of $F$ in $\mathbb{C}$. Fix one of the conjugate embeddings, call it $\sigma_{0}$.\\
We thus obtain an elliptic curve over $\mathbb{C}$ which we shall denote by $E_v$.\\
Let $\lambda_v : E_v(\mathbb{C}) - \{O\} \to \mathbb{R}$ be the local height function. By Weierstrass uniformization there is a $\tau \in \mathbb{C}$ with $\text{Im}(\tau) > 0$ such that there is an analytic isomorphism $\mathbb{C}/\mathbb{Z} + \tau\mathbb{Z} \to E_{v}(\mathbb{C})$. Write $q = e^{2\pi i \tau}$ and note that $|q| < 1$.\\
If $A \in E_{v}(\mathbb{C}) - \{O\}$ is the image of $z \in \mathbb{C}$ and $q(z) = e^{2\pi i z}$ then,
\[
\lambda_{v}(A)\\
         = - \frac {1} {2}b_2(\frac {\text{Im}(z)} {\text{Im}(\tau)})\log |q| -\log |1 - q(z)| - \sum_{n\geq1}\log |(1 - q^nq(z))(1 - q^nq(z)^{- 1})|     
\tag{4.4.2} \]\\
where $b_2(X) = X^2 - X + \frac {1} {6}$ . (See \cite{sil 2} )\\
The group $E_v(\mathbb{C})$ endowed with complex topology is compact and let $\mu_{v, E}$ denote the unique Haar measure on it of total mass $1$.\\
Now we need an equidistribution theorem which is originally due to Szpiro, Ullmo, Zhang. We shall quote the simplified version given in $\cite{hab}$ :\\~\\
\textbf{Propositon 4.4.1:} Let $P_1, P_2, P_3 ... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ be a sequence of points such that $\widehat{h}(P_k) \to 0$. Let $f :E_{v}(\mathbb{C}) \to \mathbb{R}$ is a continuous function, then \[ \lim_{k \to \infty} \frac {1} {[F(P_k) : F]}\sum_{\sigma}f(\sigma(P_k)) = \int f\mu_{E, v},\] where $\sigma$ runs over all the all the field embeddings $F(P_k) \to F$ which extend $\sigma_{0}$.\\~\\
Now are back to our set up.\\
Say, $v$ is an infinite place of $K$. Call the corresponding embedding $\sigma_0$ as before.\\
Then \[\widehat{h}_{v}(P)  = \frac {1} {[K(P) : K]} \sum_{u | v, u \in M_{K(P)}} d_{u, v} \lambda_u(P)\] for $P \in E(\overline{\mathbb{Q}}) - \{O\}$.\\
Clearly, \[\widehat{h}_{v}(P) = \frac {1} {[K(P) : K]}\sum_{\sigma} \lambda_{v}(\sigma(P)) \] where the sum runs over all $\sigma : F(P) \to \mathbb{C}$ such that it extends $\sigma_{0}$.\\~\\
Let $P_1, P_2, P_3 ... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}(P_k) \to 0$ .\\
We would like to use proposition 4.4.1 to obtain an estimate for $\liminf \widehat{h}_{v}(P_k)$. One can not directly use proposition 4.4.1 since one can not continuously extend $\lambda_v$ to $O$.\\
So for each positive integer $m$ we define $\lambda_{v, m}(P) = \min \{ \lambda_{v}(P), m \} $ for all $P \in E(\mathbb{C}) - \{O\}$ and $\lambda_{v, m}(O) = m$.\\
It is easy to see that $\{\lambda_{v, m}\}_{m \geq 1}$ is a sequence of continuous function which increases pointwise to $\lambda_v$. By monotone convergence theorem, we have that $\lambda_v$ is measurable and $a_m \to \int \lambda_{v}\,\mu_{v, E}$ where $a_m = \int \lambda_{v, m}\mu_{v, E}$ for each $m \geq 1$.\\
Now note that \[\widehat{h}_{v}(P_{k}) \geq \frac {1} {[K(P_k) : K]} \sum_{\sigma} \lambda_{v, m}(\sigma(P))\] holds for each positive integer $k, m$.\\
At first let $k \to \infty$ and use proposition 4.4.1 to conclude that \[\liminf \widehat{h}_{v}(P_k) \geq a_m.\]
This holds for each positive integer $m$ . Letting $m \to \infty$ we have \[\liminf \widehat{h}_{v}(P_k) \geq \int \lambda_{v}\mu_{v, E}.\]
Now we have the following lemma:\\~\\
\textbf{Lemma 4.4.2:} $\int \lambda_v\mu_{v, E} = 0.$\\~\\
\textbf{Proof :} Let $\tau$ denote the corresponding parameter in Weierstrass uniformization of $E_v$. The points on $E_v$ are parameterized by the fundamental parallelogram \[ \{ x + y\tau \; | \; 0 \leq x, y < 1 \} \subseteq \mathbb{C}\] and for this parameterization the Haar measure is nothing but the usual measure on unit square.\\
We shall use the expression in $(4.4.1)$  and integrate term by term. This is allowed since the series converges absolutely ( follows from the fact that $|q| < 1$).\\
In the following computation $z = x + y\,\tau$. So $\text{Im}(z) = y\,\text{Im}(\tau)$.\\
Now \[\int_{0}^{1}\int_{0}^{1} - \frac {1} {2}b_{2}(\frac {\text{Im} (z)} {\text{Im} (\tau)}) \log |q| dy dx = - \frac {\log |q|} {2} \int_{0}^{1} (\int_{0}^{1} \, b_2(y) dy) dx.\]
Easy computation shows that the inner integral is zero.\\
Hence the first term integrates to $0$.\\
Now let $\epsilon \in \{\pm 1\}$ and $n \geq 1$.\\
Then 
\[
\begin{split}
\int_{0}^{1}\int_{0}^{1} \log | 1 - q^{n}q(z)^{\epsilon} | dxdy & = \int_{0}^{1}( \int_{0}^{1} \log | e^{- 2\pi i \epsilon x} - e^{2\pi i \tau (n + \epsilon y)}| dx) dy \\
                                                                                                        & = \int_{0}^{1} \log^{+} (|\,\text{exp}\,(2\pi i \tau (n + \epsilon y))|) dy \\
\end{split}
\] 
where in the last step we are using the Jensen's formula. Since $\text{Im}(\tau) > 0$, $ n\geq 1$ and $ 0 \leq y < 1$ we conclude that $|\,\text{exp}\,(2\pi i \tau ( n+ \epsilon y))| < 1$. So the last integral is $0$.\\
Similar computation shows $| 1 - q(z)|$ integrates to zero.\\
Hence the lemma. $\square$ \\~\\
The discussion of this subsection can be summarized in the following lemma:\\~\\
\textbf{Lemma 4.4.3:} Let $v$ be an infinite place of $K$ and let $P_1, P_2, P_3 ... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}(P_k) \to 0$. Then \[ \liminf \widehat{h}_v(P_k) \geq 0.\]

\subsection{Finite places}
We shall recall some general facts about elliptic curves over non-Archimedean local fields. For details see \cite{sil 2}, chapter VI, section 4.\\
Let $E$ be an elliptic curve over a number field $F$. Assume that $v$ is a finite place of $F$. Consider $E$ as a curve in $F_v$.\\
If $v$ is a place of good reduction then for all $A \in E(\overline{F_v}) - \{O\}$, $\lambda_{v}(A) \geq 0$.\\
If $v$ is a place of split multiplicative reduction , then by Tate uniformization there is $q_v \in F_{v}^{\times}$ with $|q_v|_v < 1$ such that there is a surjective homomorphism $\phi : \overline{F_v}^{\times} \to E(\overline{F_v})$ with kernel $q_v^{\mathbb{Z}}$, the cyclic group generated by $q_v$. \\
It is easy to see that for any $A \in E(\overline{F_v}) - \{O\}$ there is an unique $q_{0}(A) \in \overline{F_v}^{\times}$ such that $ |q_v|_v < |q_{0}(A)|_v \leq 1$. Now the local height is given by
\[\lambda_v(A) = - \frac {1} {2}b_{2}(\frac {\log |q_0(A)|_v} {\log |q_v|_v})\log |q_v|_v - \log|1 - q_{0}(A)|_v \tag{4.4.3}\] 
Further using ultrametric triangle inequality $ |1 - q_0(A)|_v \leq 1$. Thus for all $A \in E(\overline{F_v}) - \{O\}$ we have \[ \lambda_v(A) \geq - \frac {1} {2}b_2(\frac {\log |q_0(A)|_v} {\log|q_v|_v}) .\tag{4.4.4} \]
It is known that $\phi$ commutes with the action of the absolute galois group of $F_v$. So $\lambda_v$ is invariant under action of $\text{Gal}(\overline{F}_v/F_v)$.\\
Note that we have a well defined map $l_v : E(\overline{F_v}) \to \mathbb{R}/\mathbb{Z}$ given by \[l_v(A) = \frac {\log |q(A)|_v} {\log |q_v|_v} + \mathbb{Z}\] where $q(A) \in \overline{F_v}^{\times}$ such that $\phi(q(A)) = A$.\\
We identify the topological group $\mathbb{R}/\mathbb{Z}$ with unit circle and equip it with the Haar measure $\mu_{\mathbb{R}/\mathbb{Z}}$ of lotal mass 1.\\
Now we have the following theorem which is due to Chambert-Loir. We quote the simplified version in \cite{hab} :\\~\\
\textbf{Proposition 4.4.4 :} Let $P_1, P_2, P_3, ... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ be a sequence of points such that $\widehat{h}(P_k) \to 0$.\\
If $f : \mathbb{R}/\mathbb{Z} \to \mathbb{R}$ is a continuous function, then 
\[ \lim_{k \to \infty} \frac {1} {[F(P_k) : F]} \sum_{\sigma} f(l_{v}(\sigma (P_k))) = \int f\mu_{\mathbb{R}/\mathbb{Z}} \]
where $\sigma$ runs over all field embeddings $F(P_k) \to \overline{F_v}$ which are identity on $F$.\\~\\
Now we are back to our set up.\\
By standard theory of elliptic curves we know that there is a finite, galois extension $L/K$ such that thought as an elliptic curve over $L$, $E$ has either good or split multiplicative reduction at finite places.\\
Let $\mathfrak{p}_1,...,\mathfrak{p}_s$ be the prime ideals of ring of integers of $K$ which appear in the denominator of $j$-invariant of $E$. (since $\mathfrak{p}$ is a place of good reduction we conclude that $\mathfrak{p} \neq \mathfrak{p}_i$ for any $i$ in the range)\\
The reduction type at some finite place $u \in M_L$ is given by the following rule :\\
if $u \nmid \mathfrak{p}_i$ for any $i$ then $u$ is a place of good reduction,\\
if $u \, | \mathfrak{p}_i$ for some $i$ then $u$ is a place of split multiplicative reduction.\\
Let $v$ be a finite place of $K$ such that $v \notin \{\mathfrak{p}, \mathfrak{p}_1,...,\mathfrak{p}_s \}$.\\
Then for any $u \in M_L$ with $u | v$, $u$ is a place of good reduction and hence $\lambda_u$ is nonnegative. Thus $\widehat{h}_v(P) \geq 0$ for all $P \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$.\\
Now let $v = \mathfrak{p}_i$ for some $i$.\\
Fix a place $u \in M_L$ such that $u |v$. Fix an algebraic closure $\overline{L_u}$ of $L_u$. Since $\lambda_u$ is invariant under the action of absolute galois group of $L_u$ we conclude from the definition of $\widehat{h}_{v}$ that \[\widehat{h}_{v}(A) = \frac {1} {[L:K]} \sum_{\tau}\frac {1} {[L(A) : L]} \sum_{\sigma} \lambda_{u}(\sigma(A)) \tag{4.4.5}\]
where the first sum runs over all the field embeddings $\tau : L \to \overline{L_u}$ such that $\tau|_K$ is the map $K \to K_v$ and the second sum runs over field embeddings $\sigma: L(A) \to \overline{L_u}$ and $A \in E(\overline{\mathbb{Q}}) - \{O\}$.\\
Fix a $\tau$ as above. So we have an elliptic curve $E$ over $\overline{L_u}$. \\
Let $q_u$ be the uniformizer for this curve.\\
Note that $b_2(0) = b_2(1)$. Thus it defines a continuous function on $\mathbb{R}/\mathbb{Z}$. Call it $B_2$.\\
Further it follows from definitions that \[b_2(\frac {\log |q_0(A)|_u} {\log |q_u|u}) = B_2(l_u(A))\] for all $A \in E(\overline{\mathbb{Q}}) - \{O\}$.\\  
Now let $P_1, P_2, P_3 ,... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ be a sequence of points such that $\widehat{h}(P_k) \to 0$. \\
Choose a $\tau_{1} \in \text{Gal}\,(\overline{\mathbb{Q}} | K)$ such that $\tau \circ \tau_{1} : L \to L_{u}$ is the identity mapping. Then the extensions of this map are given precisely by $\sigma \circ \tau_1$ where $\sigma$ is an extension of $\tau$.\\
Consider the sequence of points $\{\tau_1^{- 1}(P_k) \}_{k \geq 1}$. Clearly they are elements of $E(\overline{\mathbb{Q}}) - E_{\text{tor}}$. Further $\widehat{h}(\tau^{-1}(P_k)) \to 0$.\\
Using proposition 4.4.4 we have 
\[
\begin{split}
\lim_{k \to \infty} \frac {1} {[L(P_k) : L]}\sum_{\sigma} B_{2}(l_u(\sigma(P_k))) & = \lim_{k \to \infty} \frac {1} {[L(\tau_{1}^{- 1}P_k) :L]}\sum_{\sigma_{1}=\sigma \circ \tau_{1}} B_{2}(l_u(\sigma_{1}(\tau_1^{- 1}P_k)))\\
                                                                                                                                & = \int B_{2}\mu_{\mathbb{R}/\mathbb{Z}}
\end{split}
\] where we are using that $[L(P_k): L] = [L(\tau_1^{- 1}(P_k)) : L]$ for each $k$.\\
One can evaluate the integral easily as before, and see that it evaluates to $0$.\\
Use $(4.4.4)$ to conclude $\liminf \frac {1} {[L(P_k): L]}\sum_{\sigma}\lambda_{v}(\sigma(P_k)) \geq 0$.\\
True for each $\tau$ and there are only finitely many of them.\\
Using $(4.4.5)$ one has $\liminf \widehat{h}_v(P_k) \geq 0$.\\
The discussion of this subsection can be summarized as follows :\\~\\
\textbf{Lemma 4.4.5:} Let $v \neq \mathfrak{p}$ be a finite place of $K$. Let $P_1,...,P_k,... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ be such that $\widehat{h}(P_k) \to 0$. Then $\liminf \widehat{h}_v(P_k) \geq 0$.\\  

\section {Proof of theorem 0.2}
Assume that theorem 0.2 does not hold.\\
Then there is a sequence of distinct points $A_1, A_2, A_3 ... \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that $\widehat{h}(A_k) \to 0$.\\
Use lemma 4.3.8 to $A_k$ to get a $B_k \in E(\overline{\mathbb{Q}}) - E_{\text{tor}}$ such that \[\widehat{h}(B_k) \leq 40p^4(w + 1)(q - 1)^{2}\widehat{h}(A_k)\] and \[\widehat{h}_{\mathfrak{p}}(B_k) \geq \frac {\log p} {2p^6}\] for each $k \geq 1$. Clearly $\widehat{h}(B_k) \to 0$.\\
By lemma 4.4.3 and 4.4.5 we have $\liminf \widehat{h}_{v}(B_k) \geq 0$ for each $v \neq \mathfrak{p}$.\\
Use $(4.4.1)$ to conclude that $\liminf \widehat{h}(B_k) \geq \frac {d_{\mathfrak{p}}\log p} {2[K : \mathbb{Q}]p^6} > 0$.\\
A contradiction.\\
This contradiction proves the theorem. $\square$
{"config": "arxiv", "file": "1806.06587/4.tex"}