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  \[
    |\kappa a_i^{2d - 1}|
    \le \max_{j = 1, 2} \ub{|f_j(a_1, a_2)|}_{\le
    \kappa \height(F(P))} \ub{\sum_{j = 1}^2
    |g_{ij}(a_1, a_2)|}_{\le \gamma_i \height(D)^{d - 1}}
  \]
  where
  \[
    \gamma_i
    = \sum_{j = 1}^2 \text{(sum of absolute values
    of coefficients of $g_{ij}$)}
  .\]
  Therefore
  \[
    \kappa|a_i|^{2d - 1}
    \le \kappa \height(F(P)) \gamma_i \height(P)^{d - 1}
  \]
  so
  \[
    \height(P)^{2d - 1}
    \le \max(\gamma_1, \gamma_2) \height(F(P))
    \height(P)^{d -
    1}
  \]
  so
  \[
    \ub{\frac{1}{\max(\gamma_1, \gamma_2)}}_{=
    c_1} \height(P)^d
    \le \height(F(P))
    .
    \qedhere
  \]
\end{proof}

\begin{notation*}
  For $x \in \Qbb$, $\height(X) = \height((x : 1)) = \max(|u|,
  |v|)$, where $x = \frac{u}{v}$, $u, v \in \Zbb$
  coprime.
\end{notation*}

\begin{fcdefnstar}[Height of a point]
  \glssymboldefn{ellh}%
  Let $E / \Qbb$ be an \gls{ellc}, $y^2 = x^3 + ax
  + b$.

  Define the \emph{height}
  \begin{align*}
    H : E(\Qbb)
    &\to \Rbb_{\ge 1} \\
    P
    &\mapsto \begin{cases}
      H(x)
      & \text{if $P = (x, y)$} \\
      1
      & \text{if $P = 0_E$}
    \end{cases}
  \end{align*}
  Alsdefine \emph{logarithmic height}
  \begin{align*}
    h : E(\Qbb)
    &\to \Rbb_{\ge 0} \\
    P
    &\mapsto \log H(P)
  \end{align*}
\end{fcdefnstar}

\begin{fclemma}
  \label{lemma:13.2}
  Assuming:
  - $E$, $E'$ are \glspl{ellc} over $\Qbb$
  - $\phi : E \to E'$ an \gls{isog} defined over
  $\Qbb$
  Then:
  there exists $c > 0$ such that
  \[
    |\ellh(\phi(P)) - (\deg\phi) \ellh(P)|
    \le c
    \qquad \forall P \in E(\Qbb)
  .\]
\end{fclemma}

\begin{note*}
  $c$ depends on $E$ and $E'$, but \emph{not} $P$.
\end{note*}

\begin{proof}
  Recall (\cref{lemma:5.4})
  \begin{picmath}
    \begin{tikzcd}
      E
      \ar[r, "\phi"]
      \ar[d, "x"]
      & E'
      \ar[d, "x"] \\
      \Pbb^1
      \ar[r, "\xi"]
      & \Pbb^1
    \end{tikzcd}
  \end{picmath}
  $\deg \phi = \deg \xi$ ($= d$ say).
  \cref{lemma:13.1} tells us that there exists
  $c_1, c_2 > 0$ such that
  \[
    c_1 \ellH(P)^d
    \le \ellH(\phi(P))
    \le c_2 \ellH(P)^d
    \qquad \forall P \in E(\Qbb)
  .\]
  Taking logs gives
  \[
    |\ellh(\phi(P)) - d\ellh(P)|
    \le \ub{\max(\log c_2 - \log c_1)}_{=c}
    \qedhere
  \]
\end{proof}

\begin{example*}
  $\phi = [2] : E \to E$. Then there exists $c >
  0$ such that
  \[
    |\ellh(2P) - 4\ellh(P)|
    \le c
    \qquad \forall P \in E(\Qbb)
  .\]
\end{example*}

\begin{fcdefnstar}[Canonical height of a point]
  \glssymboldefn{canh}%
  The \emph{canonical height} is
  \[
    \hat{h}(P)
    = \lim_{n \to \infty} \frac{1}{4^n} \ellh(2^n
    P)
  .\]
\end{fcdefnstar}

We check convergence:

Let $m \ge n$. Then
\begin{align*}
  \left| \frac{1}{4^m} \ellh(2^m P) -
  \frac{1}{4^n} \ellh(2^n P) \right|
  &\le \sum_{r = n}^{m - 1} \left| \frac{1}{4^{r +
  1}} \ellh(2^{r + 1} P) - \frac{1}{4^r} \ellh(2^r
  P) \right| \\
  &= \sum_{r = n}^{m - 1} \frac{1}{4^{r + 1}}
  |\ellh(2(2^r P)) - 4\ellh(2^r P)| \\
  &< c \sum_{r = n}^\infty \frac{1}{4^{r + 1}} \\
  &= \frac{c}{4^{n + 1}} \frac{1}{1 - \quarter} \\
  &= \frac{c}{3 \times 4^n} \\
  &\to 0
\end{align*}
as $n \to \infty$. So the sequence is Cauchy,
$\canh(P)$ exists.

\begin{fclemma}[]
  \label{lemma:13.3}
  $|\ellh(P) - \canh(P)|$ is \cloze{bounded for $P \in
  E(\Qbb)$.}
\end{fclemma}

\begin{proof}
  Put $n = 0$ in above calculation to get
  \[
    \left| \frac{1}{4^m} \ellh(2^m P) - \ellh(P)
    \right|
    < \frac{c}{3}
  .\]
  Take limit $m \to \infty$.
\end{proof}

\begin{fclemma}[]
  \label{lemma:13.4}
  Assuming:
  - $B > 0$
  Then:
  \[
    \#\{P \in E(\Qbb) \st \canh(P) \le B\}
    < \infty
  .\]
\end{fclemma}

\begin{proof}
  $\canh(P)$ bounded means we have a bound on
  $\ellh(P)$ (by \cref{lemma:13.3}). So only
  finitely many possibilities for $x$. Each $x$
  gives $\le 2$ choices for $y$.
\end{proof}

\begin{fclemma}[]
  \label{lemma:13.5}
  Assuming:
  - $\phi : E \to E'$ an \gls{isog} defined over
  $\Qbb$
  Then:
  \[
    \canh(\phi P)
    = (\deg \phi) \canh(P)
    \qquad \forall P \in E(\Qbb)
  .\]
\end{fclemma}

\begin{proof}
  By \cref{lemma:13.2}, there exists $c > 0$ such
  that
  \[
    |\ellh(\phi P) - (\deg \phi) \ellh(P)|
    < c
    \qquad \forall P \in E(\Qbb)
  .\]
  Replace $P$ by $2^n P$, divide by $4^n$ and take
  limit $n \to \infty$.
\end{proof}

\begin{remark*}
  \phantom{}
  \begin{enumerate}[(i)]
    \item Case $\deg \phi = 1$ shows that $\canh$
      (unlike $\ellh$) is independent of the
      choice of \gls{weq}.
    \item Taking $\phi = [n] : E \to E$ shows
    \[
      \canh(nP)
      = n^2 \canh(P)
      \qquad \forall n \in \Zbb, P \in E(\Qbb)
    .\]
  \end{enumerate}
\end{remark*}

\begin{fclemma}[]
  \label{lemma:13.6}
  Assuming:
  - $E / \Qbb$ an \gls{ellc}
  Then:
  there exists $c > 0$ such that for all $P, Q \in
  E(\Qbb)$ with $P, Q, P + Q, P - Q \neq 0$, we
  have
  \[
    \ellH(P + Q) \ellH(P - Q)
    \le c \ellH(P)^2 \ellH(Q)^2
  .\]
\end{fclemma}

\begin{proof}
  Let $E$ have \gls{weq} $y^2 = x^3 + ax + b$, $a,
  b \in \Zbb$. Let $P, Q, P + Q, P - Q$ have $x$
  coordinates $x_1, \ldots, x_4$. By
  \cref{lemma:5.8}, there exists $W_0, W_1, W_2
  \in \Zbb[x_1, x_2]$ of degree $\le 2$ in $x_2$
  such that
  \[
    (1 : x_3 + x_4 : x_3 x_4)
    = (\ub{W_0}_{= (x_1 - x_2)^2} : W_1 : W_2)
  .\]
  Write $x_i = \frac{r_i}{s_i}$ with $r_i, s_i \in
  \Zbb$ coprime.
  \begin{align*}
    \ub{(s_3 s_4 : r_3 s_4 + r_4 s_3 : r_3
    r_4)}_{\gcd = 1}
    &= ((r_1 s_2 - r_2 s_1)^2 : \cdots) \\
    \ellH(P + Q) \ellH(P - Q)
    &= \max(|r_3|, |s_3|) \max(|r_4|, |s_4|) \\
    &\le 2\max(|s_3 s_4|, |r_3 s_4 + r_4 s_3|, |r_3
    r_4|) \\
    &\le 2 \max(|r_1 s_2 - r_2 s_1|^2, \ldots) \\
    &\le c \max(|r_1|^2, |s_1|)^2 \max(|r_2|,
    |s_2|)^2 \\
    &= c\ellH(P)^2 \ellH(Q)^2
  \end{align*}
  where $c$ depends on $E$, but not on $P$ and
  $Q$.
\end{proof}

\begin{fcthm}[]
  \label{thm:13.7}
  $\canh : E(\Qbb) \to \Rbb_{\ge 0}$ is a
  \gls{qf}.
\end{fcthm}

\begin{proof}
  \cref{lemma:13.6} and $|\ellh(2P) - 4\ellh(P)|$
  bounded gives that there exists $c \in \Rbb$
  such that
  \[
    \ellh(P + Q) + \ellh(P - Q)
    \le 2\ellh(P) + 2\ellh(Q) + c
    \qquad \forall P, Q \in E(\Qbb)
  .\]
  Replacing $P$, $Q$ by $2^n P$, $2^n Q$, dividing
  by $4^n$ and taking the limit $n \to \infty$
  gives
  \[
    \canh(P + Q) + \canh(P - Q)
    \le 2 \canh(P) + 2\canh(Q)
  .\]
  Replacing $P, Q$ by $P + Q$, $P - Q$ and
  $\canh(2P) = 4\canh(P)$ gives the reverse
  inequality. Therefore $\canh$ satisfies the
  parallelogram law, and hence $\canh$ is a
  \gls{qf}.
\end{proof}