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\school{Final Honour School of Mathematics Part C}
\title{Course Title: C9.1b. Elliptic Curves\\
Lecturer: Prof Minhyong Kim}
\date{16/2/14}

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

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\makecoverpage

\begin{questions}

\question

\begin{parts}


\part
Let $\cE$ be an elliptic curve over $\mathbb{Q}_p$ with affine 
equation $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}_p$. Prove 
that $(x,y)$ reduces to the origin modulo $p$ if and only if
$|x|_p>1$ or $|y|_p>1$.

\part

Prove or disprove:
If $a\in \Q^*$ has a square root in $\Q_p$ for all primes $p$, then 
it has a square root in $\Q$.

[Warning: We are {\em not} assuming it has a square root in $\mathbb{R}$.]
\part
State Hensel's lemma.
\part

Let $a\in\Q_p^*$. If $p\neq 2$, show that $a$ has a square root if 
and only if $a=p^nu$ where $n$ is  even and $u\in \Z_p^*$ is such 
that $u \mod p$ has a square root.
State and prove the analogous criterion for $p=2$.

\part

Prove that $(x^2+7)(x^2-11)(x^2+77)$ has a root in 
each $\Q_p$ and $\mathbb{R}$.


\end{parts}

\question

Let $\cE$ be an elliptic curve with affine equation
$y^2=x^3+Ax+B$
where $A,B\in \Z_p$.


\begin{parts}

\part

 Give the definition of  a {\em formal group} over a ring $R$. 
 State precisely the theorem regarding the isomorphism
$$\Phi: (p\Z_p, F_{\cE})\simeq (\cE_1(\Q_p), +_{\cE}),$$
where $+_{\cE}$ refers to the elliptic curve group law and $F_{\cE}$ 
is a formal group law.
(Along the way, you should define $\cE_1(\Q_p)$.)



\part
Given a formal group law $F$ over $\Z_p$, show that there is a unique 
normalized invariant differential form $\omega\in \Z_p[[T]]dT$.

\part
Given a formal group law $F$ over $\Z_p$, for each $m
\in  \Z$, define the homomorphism $[m]$ from $F$ to itself.
Show that $$[p](T)=pf(T)+g(T^p)$$  for $f,g\in T\Z_p[[T]]$, 
where $f(T)=T+\mbox{terms of degree $\geq 2$}.$ 

[You may assume that the power series $[m]$ defined previously are 
all homomorphisms.]
\part
Prove that
$$\cE_1(\Q_p)$$ is torsion-free.

\part
Let $\cE$ be the elliptic curve $$y^2=x^3+3.$$
 Prove or disprove: $\cE$ has infinitely many rational points.

\end{parts}
\pagebreak

\question

Let $$\mathcal{E}: y^2=x(x^2+ax+b)$$ be an elliptic curve, with
$a,b\in\mathbb{Z}$, and let $\mathcal{C}=\mathcal{E}(\mathbb{Q}).$
 Define the map 
$q:\mathcal{C}\rightarrow \mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^2$ 
by setting
$$q(x,y)=x(\mathbb{Q}^{\times})^2$$ if $x\neq 0$ and 
$$q(x,y)=b(\mathbb{Q}^{\times})^2$$ otherwise.  
\bigskip

\begin{parts}

\part

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



Now let $\mathcal{E}'$ be the elliptic curve $y^2=x(x^2+a_1x+b_1)$,
with $a_1=-2a$ and $b_1=a^2-4b$, and let $\mathcal{D}=\mathcal{E'}(\mathbb{Q})$.
Stating clearly any properties you may require concerning the
2-isogeny $\phi:\mathcal{E}\rightarrow
\mathcal{E'}$ and its dual $\hat{\phi}$, show that
$\mathcal{C}/2\mathcal{C}$ is finite.

\bigskip



[You may assume that the map $q$ above is a homomorphism, with kernel
 $\hat{\phi}(\mathcal{D})$.] 
\bigskip

\part



In the special case $\mathcal{E}: y^2=x(x^2-p^2)$, where $p$ is an odd
prime, show that the rank is at most 2.

\bigskip



[You may use without proof the fact that the order of
  $\mathcal{C}/2\mathcal{C}$ is $2^{R+s}$ where $R$ is the rank of
  $\mathcal{C}$ and $2^s$ is the order of its rational 2-torsion
  subgroup.]
\bigskip

\part



Show that if $x^4+p^2y^4=2z^2$ has a non-trivial integer
solution, then $(\tfrac{2}{p})=1$. 
\bigskip

\part



Deduce that, for $\mathcal{E}: y^2=x(x^2-p^2)$ as before, the rank is at most
1 if $p\equiv 3$ or $5\pmod {8}$.  [You may use the fact that $(\tfrac{2}{p})=1 $ if and only if
$p\equiv \pm 1 \pmod{8}.$]
 For $p=5$ display a non-integral
point $(x,y)$ on the curve. [It may help to consider points for which
  $x$ is a square.]  Deduce that for $p=5$ the rank is
exactly 1. 

\end{parts}
\end{questions}

\end{document}