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\begin{document}
% Insert your course title in the braces
\topictitle{C9.1b Elliptic Curves\\ %Insert the name of your checker below
Checked by: Victor Flynn}
% Insert your name in the braces
\author{Roger Heath-Brown}
% Insert the date of submission of this draft in the braces
\date{24/02/2011}
\maketitle

\begin{question}{1}

Find an $m\in\N$ such that $|m^3-7|_5<5^{-1}$.  Show that there
is no $n\in\N$ such that $|n^3-7|_3<3^{-1}$.  
\bigskip

Show that the equation $x^3+2y^3+4z^3=0$, with $x,y$ and $z$ all in
$\Q_2$, has only the solution $x=y=z=0$. 
\bigskip

Determine necessary and sufficient conditions on $x\in\Q_2$ for the 
sum $\sum_{n=0}^{\infty}x^n/n!$ to be convergent in $\Q_2$

[{\it You may use without proof the facts that $2^{n-1}\mid n!$ whenever
  $n$ is a power of $2$, and that $2^n$ never divides $n!$.}]
\bigskip

State Hensel's Lemma. 
\bigskip

Let $a$ and $b$ be integers.  Show that the elliptic curve $y^2=x^3+ax+b$ has
infinitely many points $(x,y)$ in $\Z_p$, for any prime $p\ge 5$ of 
good reduction.

[{\it You may use any standard facts about elliptic curves that you
  require, providing that they are clearly stated.}]  
\bigskip

Let $a$ and $b$ be integers as before.   If the elliptic curve $y^2=x^3+ax+b$ has bad reduction at a prime $p\ge 5$, what can
you say about the polynomial $x^3+ax+b$, considered modulo $p$?  Show that
$y^2=x^3+ax+b$ has infinitely many points $(x,y)$ in $\Z_p$, for any 
prime $p\ge 5$ of bad reduction.

[{\it You may wish to consider separately the two cases which arise.}] 

\end{question}


\begin{question}{2}

Let $R$ be a commutative ring with a 1.  Define a {\it Formal Group}
$F(X,Y)$ over $R$.  
\bigskip

If $R[[T]]$ denotes the power series ring over
$R$, show that there is a unique element $i(T)\in TR[[T]]$ such that
$F(T,i(T))=0$.  Show further that we have $F(T,0)=T$. 
\bigskip

Let $E:y^2=x^3+Ax+B$ be an elliptic curve over $\Q_p$, with
$A,B\in\Z_p$.  Using any facts about formal groups that you require,
show that if $E_1(\Q_p)$ is the kernel of the reduction map
$E_0(\Q_p)\rightarrow E_{{\rm ns}}(\mathbb{F}_p)$ then $E_1(\Q_p)$ is
torsion-free.  
\bigskip

Deduce that if $y^2=x^3+ax+b$ is an elliptic curve over $\Q$, with
$a,b\in\Z$, then any non-trivial torsion point $P=(x,y)$ has
$x,y\in\Z$.  Using reduction into $\mathbb{F}_3$, show that there are at most 
6 non-trivial torsion points with $3\nmid y$. 


\end{question}




\begin{question}{3}

Let $D:V^2=U(U^2+a_1U+b_1)$ be an elliptic curve, with $a_1,b_1\in\Z$ and
$a_1(a_1^2-4b_1)\not=0$.  Define $q:D(\Q)\rightarrow \Q^*/(\Q^*)^2$ by
setting $q(u,v)=u$ for $u\not=0$ and $q(u,v)=b_1$ for
$u=0$. Define also $q(\underline{0})=1$ for the point at infinity.
Show that $q$ is a homomorphism.  

[{\it You need only verify the homomorphism condition for pairs of points $P,Q$ 
for which none of $P,Q$ or $P+Q$ is $(0,0)$ or the point at infinity.}]
\bigskip

Show further that the image of $q$ is finite.  
\bigskip

Suppose $A,B,C,D\in\N$ are such that $A^2,B^2,C^2,D^2$ form a 4-term
arithmetic progression. Show that the curve $E:y^2=x(x^2+5x+4)$ has a
rational point $(x,y)$ with $x=-2B^2C^{-2}$. 
\bigskip

Use descent via 2-isogeny to show that $\# \left(E(\Q)/2E(\Q)\right)\le 4$. 
\bigskip

Find the rank of $E$.

[{\it You may use without proof the fact that if ${\mathcal E}$ is an 
elliptic curve, then
\[\#\left({\mathcal E}(\Q)/2{\mathcal E}(\Q)\right)=
\#\left({\mathcal E}_{{\rm tors}}(\Q)/2{\mathcal E}_{{\rm tors}}(\Q)\right)\times 2^r,\]
where $r$ is the rank of ${\mathcal E}$.  You may also use without 
proof that $E_{{\rm tors}}(\Q)\cong C_4\times C_2$ and that the torsion 
points have $x=-4,-2,-1,0$ or $2$.}]
\bigskip

Deduce that in fact $A,B,C,D$ are all equal. 

\end{question}

\end{document}