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\school{Final Honour School of Mathematics Part C}
\title{Course Title: C9.1b. Elliptic Curves\\
Lecturer: Prof Minhyong Kim}
\date{16/2/13}

\begin{document}

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\begin{questions}

\question
\begin{parts}

\part
State and prove Hensel's Lemma.


\part
\begin{subparts}
\subpart Which numbers $a\in \Q_2$ have cube roots  in $\Q_2$?

\subpart
Which numbers $a\in \Q_3$ have cube roots  in $\Q_3$?

\subpart
How many solutions are there to
$$x^{37}-x=37$$
in $\Q_{37}$?
\end{subparts}

\part
Consider the equation
$$y^2=x^3+6.$$
Find all $p$ for which the equation has a solution in $\Z_p$.

[\textit{Warning: You are not allowed to use the point at `infinity'.}]

[\textit{You may use standard facts from the lectures as long as they are stated clearly.}]



\end{parts}


\question
Let~$R$ be any ring (commutative, with~1), and let~$F,G$ be
formal groups over~$R$.
\begin{parts}



\part

Show that there exists a unique
normalised invariant differential for~$F$, which is given by
$\omega = F_X(0, T)^{-1}\hbox{d} T \in R[[T]] \hbox{d} T$,
and that every invariant differential for~$F$ is of the form $a\omega$
for some $a\in R$.
%\hfill {\bf [6~marks]}

\part

Let~$f$ be a homomorphism over~$R$ from~$F$
to~$G$. Let~$\omega_F, \omega_G$ be
the normalised invariant differentials on~$F,G$, respectively.
Show that $\omega_G \circ f = f'(0)\ \omega_F$.
Deduce that, for any prime~$p$, there
exist $f,g\in R[[T]]$
such that $[p](T) = p f(T) + g(T^p)$ [where~$[p]$ represents
the multiplication-by-$p$ map on~$F$].
%\hfill {\bf [6~marks]}

\part

Let ${\mathcal E}$ be an elliptic curve given by an equation
$${\mathcal E}: y^2=x^3+(3m+1)x+9n^2,$$
where $m,n \in \Z$ and the polynomial $x^3+(3m+1)x+9n^2$ has no rational root.
Prove that ${\mathcal E}(\Q)$ is infinite.


[\textit{You may use standard facts from the lectures as long as they are stated clearly.}]


\part

 
Let ${\mathcal E}$ be an  elliptic curve given by an equation
$${\mathcal E}: y^2=x^3+ax,$$
where $a$ is a non-zero square-free integer. Show that ${\mathcal E}(\Q)$ has no element of order 4.








\end{parts}


\question


Let $l$ be a prime and ${\mathcal E}_l$ be the elliptic curve
$$y^2=x^3-l^2x.$$

\begin{parts}

\part
Find the torsion subgroup ${\mathcal E}_{l, \mbox{tor}}(\Q)$ of the Mordell-Weil group.



\part

Show that the rank of
$${\mathcal E}_l: y^2=x^3-l^2x$$
is at most two.

\part

Compute the rank of
$${\mathcal E}_5: y^2=x^3-25x.$$

\part
For $l\equiv 3 \mod 8$, show that ${\mathcal E}_l$ has rank zero.


\end{parts}




\end{questions}

\end{document}