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\school{Final Honour School of Mathematics Part C}
\title{Course Title: C3.7. Elliptic Curves\\
Lecturer: Prof Minhyong Kim}
\date{16/2/15}

\def\cE{{\cal E}}
\def\F{{\mathbb F}}
\begin{document}

%produce the coverpage
\makecoverpage

\begin{questions}

\question

\begin{parts}


\part

[10 marks] State and prove Hensel's lemma.

\bigskip

\part

[5 marks] Which of the following numbers has a cube root in $\Q_3$? 
Be sure to justify your answer.
$$(i) \ \ \ \ 35 \ \ \ \ \ \  (ii) \ \ \ \ 132 \ \ \ \ \ \  
(iii) \ \ \ \ 10/459.$$
\bigskip

\part

[10 marks] Prove that $2y^2=x^4-17$ has a solution in $\R$ and 
in $\Q_p$ for all primes $p$.
\medskip

[{\em You may use any of the theorems stated during lectures.}]
\bigskip

\end{parts}

\question

\begin{parts}

\part 

[10 marks] For the elliptic curve  $\cE$ with equation
$y^2=x^3+1$
compute the groups
\medskip

(i) $\cE(\F_5)$

(ii) $\cE(\F_7)$
\medskip

For case (i) describe directly  the group law on the points of the curve.
\bigskip

\part 

[10 marks] Compute the torsion subgroups of the rational points of the elliptic curves over $\Q$ given by the equations
$$D_1: \ \ \  y^2=x^3+x^2+2x+1;  \ \ \ \ D_2: y^2=x^3+64.$$
\medskip


[{\em You may use any of the theorems stated during lectures.}]
\bigskip

\part 

[5 marks] Let $\cE$ be the elliptic curve over $\Q$ given by the equation $$y^2=x^3+8.$$
 Prove or disprove: $\cE$ has infinitely many rational points.
 
\medskip
 [{\em You may use any of the theorems stated during lectures.}]

\end{parts}
\pagebreak

\question

\begin{parts}

\part

[15 marks] Find the structure of the Mordell-Weil group 
of $${\cal C}: \ \ y^2=x^3-10x.$$
\bigskip
	
\part

[7 marks ] Let $p$ be a prime
such that $p\equiv 5 \mod 8$. Show that
$$\cE_p: \ \ y^2=x^3+px$$ has rank at most 1.

\bigskip

\bigskip

\part

[3 marks] What is the rank of $\cE_p$ for $p=5$, $p=13$, and $p=29$?

\bigskip

\bigskip

\end{parts}
\end{questions}

\end{document}