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\school{Final Honour School of Mathematics Part C}
\title{Course Title: C9.1b. Elliptic Curves\\
Lecturer: Prof Roger Heath Brown}
\date{16/2/12}

\begin{document}

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\begin{questions}

\question

\begin{parts}

\part

Find an integer $m$ such that $|m^2-2|_7\le 7^{-3}$, and show that
there is no integer $n$ with $|n^3-2|_7\le 7^{-3}$. 

If $p$ is a prime
and the integer $k$ is a multiple of $p$, show that the sum
$1+k+k^2+\ldots$ converges in $\mathbb{Q}_p$, and find its limit. 

Show that
\[3^{2^n}\rightarrow 1\]
in $\mathbb{Q}_2$. Deduce that for any integers $a,b\in\mathbb{Z}$ there 
is a sequence $c_n\in\mathbb{Z}$ such that $c_n\rightarrow a$ in 
$\mathbb{Q}_2$ and $c_n\rightarrow b$ in $\mathbb{Q}_3$.

\part

Let $N\in\mathbb{N}$ be a product of two primes $p$ and $q$, and let
$P$ be a point modulo $N$ on an elliptic curve $\mathcal{E}$
defined over
$\mathbb{Z}/N\mathbb{Z}$. If one calculates $kP$ modulo $N$, under what
condition will the Elliptic Curve Method reveal the factors $p$ and
$q$ of $N$?  

Let $P$ be the point $(3,4)$ on the elliptic curve $y^2=x^3-11$.
Using the Elliptic Curve Method, factorize 391 by calculating $3P$.



\end{parts}

\question

\begin{parts}

\part

Let $R$ be a commutative ring with a 1.  What does it mean to say that 
$F(X,Y)\in R[[X,Y]]$ is a formal group defined over $R$? Show that
there is a unique power series $i(T)\in R[[T]]$ such that
$F(T,i(T))=0$.  

Let $P(T)\d T$ be a differential form for the formal group.  Write down the
condition on $P$ which must be satisfied for $P(T)\d T$ to be an invariant
differential.  Given that $F(0,T)=T$, show that the choice 
$P(T)=F_X(0,T)^{-1}$ produces an invariant differential.


\part

Let $p$ be an odd prime.
Suppose that $\mathcal{E}:y^2=x^3+Ax+B$ is an elliptic curve over 
$\mathbb{Q}_p$, with good
reduction at $p$.  What can you say about torsion elements in the
kernel of the reduction map $\mathcal{E}(\mathbb{Q}_p)\rightarrow
\tilde{\mathcal{E}}(\mathbb{F}_p)$ ? (Proof not required.)
Deduce that $\mathcal{E}_{\rm tors}(\mathbb{Q}_p)$ is finite.

\part

Let $n$ be a positive integer.  By considering primes $\equiv
5\pmod{6}$ show that the curve $y^2=x^3+n^2$ has at most 6 rational 
torsion points.

[{\em You may assume that for any integer $m$, the congruence 
$x^3\equiv m\pmod{p}$ has exactly one solution $x$ modulo $p$, 
provided that $p$ is a prime $\equiv 5\pmod{6}$. You may also
assume that for any positive integer $k$ there are
    infinitely many primes $p\equiv 5\pmod{6}$ for which $(p+1)/6$ is
    coprime to $k$.}]


\end{parts}

\question

Let $a,b\in\mathbb{Z}$ with $b(a^2-4b)\not=0$, and write
$\mathcal{C}: y^2=x(x^2+ax+b)$ for the corresponding elliptic curve.
Let $\mathcal{D}:
v^2=u(u^2+a_1u+b_1)$ be the dual curve, where $a_1=-2a$ and
$b_1=a^2-4b$.  Suppose that $\phi:\mathcal{C}\rightarrow\mathcal{D}$ and
$\widehat{\phi}:\mathcal{D}\rightarrow\mathcal{C}$ are the associated
isogenies, and let $\mathcal{G}=\mathcal{C}(\mathbb{Q})$ and
$\mathcal{H}=\mathcal{D}(\mathbb{Q})$ be the associated Mordell--Weil
groups.  

Explain, without proofs, how one defines a non-trivial homomorphism
$q$ from $\mathcal{H}$ to $\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^2$,
with kernel $\phi(\mathcal{G})$. (Proof not required.)

Show that if $(u,v)\in\mathcal{H}$ with $u\not=0$, then one may 
write $u=rw^2$ with $w\in\mathbb{Q}$ and with $r$ being a square-free 
integer dividing $b_1$.

Specifying any properties of $\phi$ and $\widehat{\phi}$ you may
require, show that 
\[\#\left(\frac{\mathcal{G}}{2\mathcal{G}}\right)\le
\#\left(\frac{\mathcal{G}}{\widehat{\phi}(\mathcal{H})}\right)
\#\left(\frac{\mathcal{H}}{\phi(\mathcal{G})}\right).\]
\bigskip

Find a set of coset representatives for 
$\mathcal{G}/\widehat{\phi}(\mathcal{H})$ and for
$\mathcal{H}/\phi(\mathcal{G})$ when $\mathcal{C}$ is the curve
$y^2=x(x^2+5x+3)$, and hence find the order of
$\mathcal{G}/2\mathcal{G}$.

\end{questions}





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