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  TODO
\end{example*}

\begin{example*}[Lindemann]
  $E$: $y^2 = x^3 + 17x$.
  \[
    \Im(\alpha_E)
    = \langle 17 \rangle
    \subset \Qbb^* / (\Qbb^*)^2
  .\]
  $E'$: $y^2 = x^3 - 68x$.
  \[
    \Im(\alpha_{E'})
    \subset \langle -1, 2, 17 \rangle
    \subset \Qbb^* / (\Qbb^*)^2
  .\]
  $b_1 = 2$, $w^2 = 2u^4 - 34v^4$.

  Replacing $w$ by $2w$ and dividing by $2$ gives
  \[
    C:~
    2w^2 = u^4 - 17v^4
  .\]

  \begin{notation*}
    $C(K) = \{(u, v, w) \in K^3 \setminus \{0\}
    \text{satisfying equation above}\} / \sim$,
    where $(u, v, w) \sim (\lambda u, \lambda v,
    \lambda^2 w)$ for all $\lambda \in K^*$.
  \end{notation*}

  $C(\Qbb_2) \neq \emptyset$ since $17 \in
  (\Zbb_2^*)^4$.

  $C(\Qbb_{17}) \neq \emptyset$ since $2 \in
  (\Zbb_{17}^*)^2$.

  $C(\Rbb) \neq \emptyset$ since $\sqrt{2} \in
  \Rbb$.

  Therefore $C(\Qbb_v) \neq \emptyset$ for all
  places $v$ of $\Qbb$.

  Suppose $(u, v, w) \in C(\Qbb)$. Without loss
  of generality say $u, v, w \in \Zbb$,
  $\gcd(u,, v) = 1$ and $w > 0$.

  If $17 \mid w$ then $17 \mid u$ and then $17
  \mid v$, contradiction

  So if $p \mid w$ then $p \neq 17$ and $\left(
  \frac{17}{p} \right) = + 1$. Therefore $\left(
  \frac{p}{17} \right) = \left( \frac{17}{p}
  \right) = 1$ (using quadratic reciprocity).
  (If $p$ is odd, also $\left( \frac{2}{17}
  \right) = +1$).

  Therefore $\left( \frac{w}{17} \right) = +1$.
  But $2w^2 \equiv u^4 \pmod{17}$, so $2 \in
  (\Fbb_{17}^*)^4 = \{\pm 1, \pm 4\}$,
  contradiction.

  So $C(\Qbb) = \emptyset$, i.e. $C$ is a
  counterexample to the Hasse Principle. It
  represents a non-trivial element of $\TS(E /
  \Qbb)$.
\end{example*}

\subsection{Birch Swinnerton Dyer Conjecture}

Let $E / \Qbb$ be an \gls{ellc}.

\begin{fcdefnstar}[$L(E, s)$]
  Define
  \[
    L(E, s)
    = \prod_p L_p(E, s)
  \]
  where
  \[
    L_p(E, s)
    = \begin{cases}
      (1 - a_p p^{-s} + p^{1 - 2s})^{-1}
      & \text{if \gls{gred}} \\
      (1 - p^{-s})^{-1}
      & \text{if split multiplicative reduction}
      \\
      (1 + p^{-s})^{-1}
      & \text{if nonsplit multiplicative
      reduction} \\
      1
      & \text{if additive reduction}
    \end{cases}
  \]
  where $\#\tilde{E}(\Fbb_p) = 1 + p - a_p$.
\end{fcdefnstar}

Hasse's Theorem implies $|a_p| \le 2\sqrt{p}$,
which shows that $L(E, s)$ converges for $\Re(s) >
\frac{3}{2}$.

\begin{theorem*}[Wiles, Breuil, Conrad, Diamon, Taylor]
  $L(E, s)$ is the $L$-function of a weight $2$
  modular form, and hence has an analytic
  continuation to all of $\Cbb$ (and a function
  equation relating $L(E, s) = L(E, 2 - s)$).
\end{theorem*}

\begin{conjecture*}[Weak Birch Swinnerton-Dyer Conjecture]
  $\ord_{s = 1} L(E, s) = \rank E(\Qbb)$.
\end{conjecture*}

($= r$ say).

\begin{conjecture*}[Strong Birch Swinnerton-Dyer Conjecture]
  $\ord_{s = 1} L(E, s) = \rank E(\Qbb)$, which we
  shall call $r$, and
  \[
    \lim_{s \to 1} \frac{1}{(s - 1)^r} L(E, s)
    = \frac{\Omega_E \Reg E(\Qbb) |\TS(E / \Qbb)|
    \prod_p c_p(E)}{|E(\Qbb)_{\text{tors}}|^2}
  ,\]
  where
  \begin{itemize}
    \item $c_p$ is given by:
      \begin{align*}
        c_p(E)
        &= \text{Tamagawa number of $E / \Qbb_p$} \\
        &= [E(\Qbb_p) : E_0(\Qbb_p)]
      \end{align*}
    \item If $E(\Qbb) / E(\Qbb)_{\text{tors}} =
      \langle P_1, P_2, \ldots, P_r \rangle$, then
      $\Reg E(\Qbb) = \det([P_i, P_j])_{i, j = 1,
      \ldots, r}$, where
      \[
        [P, Q]
        = \canh(P + Q)
        - \canh(P) - \canh(Q)
      .\]
    \item $\Omega_E$ is given by:
      \[
        \Omega_E
        = \int_{E(\Rbb)} \left| \frac{\dd x}{2y +
        a_1 x + a_3} \right|
      ,\]
      where $a_1, \ldots, a_6 \in \Zbb$ are
      coefficients of a globally \gls{min}
      \gls{weq} for $E / \Qbb$.
  \end{itemize}
\end{conjecture*}

\begin{theorem*}[Kolyvagin]
  If $\ord_{s = 1} L(E, s)$ is $0$ or $1$, then
  Weak Birch Swinnerton-Dyer holds, and also
  $|\TS(E / \Qbb)| < \infty$.
\end{theorem*}
