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\school{Final Honour School of Mathematics Part C}
\title{C3.7 Elliptic Curves\\
Victor Flynn \\
Checked by: Alan Lauder}
\date{20/1/17}
\message{Examinations are 1.5 hours in duration. Students may 
answer as many questions as they wish; the best two answers 
count for the total mark.}

\begin{document}

%produce the coverpage
\makecoverpage

\begin{questions}

\question
\begin{parts}
\part[10]
Let ${\mathcal C}$ be a non-singular cubic curve, defined
over a field~$K$, with a $K$-rational point~{\bf o}. Describe
the standard law for adding two points on~${\mathcal C}$, and 
prove that it is associative.
%
%\part[5]
%Show that, for any~$k \in \mathbb{N}$, the Elliptic Curve Method
%using ${\mathcal E} : Y^2 = X^3 - 8X + 8$ and $kP$ with $P = (1,1)$,
%will not factorise~$57$.

\part[7]
State Hensel's lemma. For which primes~$p$ do there exist
$x,y \in \mathbb{Z}_p$ such that $x^2 - 2y^2 = 3$\ ?
[{\it Hint: For $p > 3$, you may find it helpful to consider 
the sizes of the sets $\{ x^2 : x \in \mathbb{F}_p \}$ and 
$\{ 3 + 2y^2 : y \in \mathbb{F}_p \}$.}]

\part[8] Let $a,b\in \mathbb{Z}$ satisfy 
$a \equiv b \equiv 4$ (mod~$27$) and $b \equiv a + 3$ (mod~$7$).
Show that, for any prime~$p$, there exist $x,y,z \in \mathbb{Z}_p$
which satisfy $(x^3 - y^3 + a)(x^3 - z^3 + b)(y^3 - z^3 + b - a) = 0$.
Show that there are no solutions in~$\mathbb{Z}$.
\end{parts}


\question
\begin{parts}
\part
Let~$R$ be any ring (commutative, with~1), and let~$F,G$ be
formal groups over~$R$.
\begin{subparts}
\subpart[7]
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 $a\in R$.

\subpart[6]
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~$F$].
\end{subparts}

\part
Let ${\mathcal C} : Y^2 = X(X + \alpha)(X + \beta)$,
where $\alpha, \beta \in \mathbb{Q}$, satisfying $\alpha \not= 0$,
$\beta \not= 0$ and $\alpha \not= \beta$.
\begin{subparts}
\subpart[6]
Show that $2(x,y)$ has $x$-coordinate: 
\begin{equation*}
\frac{(x^2 - \alpha \beta)^2}{4x(x+\alpha)(x+\beta)}.
\end{equation*}
Show that ${\mathcal C}(\mathbb{Q})$ has a point of order~$4$
if and only if:
\begin{equation*} 
\begin{split}
&\Bigl( \alpha \in (\mathbb{Q}^*)^2 \hbox{ and } 
               \beta \in (\mathbb{Q}^*)^2 \Bigr)\\
\hbox{ or }
&\Bigl( -\alpha \in (\mathbb{Q}^*)^2 \hbox{ and } 
               \beta - \alpha \in (\mathbb{Q}^*)^2 \Bigr)\\
\hbox{ or }
&\Bigl( -\beta \in (\mathbb{Q}^*)^2 \hbox{ and } 
               \alpha - \beta \in (\mathbb{Q}^*)^2 \Bigr).
\end{split}
\end{equation*}
\subpart[6]
Now assume that $\alpha\in (\mathbb{Q}^*)^4$ and 
$\beta \in (\mathbb{Q}^*)^4$, so that $\alpha = c^4, \beta = d^4$,
for some $c,d \in \mathbb{Q}^*$, and further assume
that $c^2 + d^2 \in (\mathbb{Q}^*)^2$.
Show that ${\mathcal C}(\mathbb{Q})$ has a point of order~$8$.
[{\it Hint: You may find it helpful to define a curve
$\mathcal D$ such that there are $2$-isogenies
$\phi : {\mathcal C} \longrightarrow {\mathcal D}$
and $\hat\phi : {\mathcal D} \longrightarrow {\mathcal C}$,
with $\hat\phi \circ \phi = [2]$, the multiplication by~$2$
map on~${\mathcal C}$.}]
\par\noindent
Find an example of such an elliptic curve~${\mathcal C}$ 
for which ${\mathcal C}(\mathbb{Q})$ has a point of order~$8$.
\end{subparts}
\end{parts}

\eject
\question
Let ${\mathcal{C}} : Y^2 = X(X^2 + aX + b)$
and ${\mathcal{D}} : Y^2 = X(X^2 + a_1 X + b_1)$,
where $a,b\in \Z$ with $b(a^2-4b)\not= 0$ and $a_1 = -2a$,
$b_1 = a^2 - 4b$. Let the map~$\phi$ [which you may assume to
be a homomorphism] be defined as usual by 
$$ \phi : {\mathcal{C}} (\Q ) 
      \rightarrow {\mathcal{D}} (\Q) : (x,y)
\mapsto \Bigl( \frac{y^2}{x^2} ,\ y - \frac{by}{x^2} \Bigr) 
= \Bigl( \frac{x^2 + ax + b}{x},\ y - \frac{by}{x^2} \Bigr).$$
\noindent Let~$q$ be defined as usual by
$$ q : {\mathcal{D}} (\Q ) \rightarrow \Q^* / (\Q^*)^2 : (u,v) \mapsto u
\hbox{ when }u\not= 0,$$
$$ q : (0,0) \mapsto b_1,\,\,\, q: {\bf o} \mapsto 1. $$
\begin{parts}
\part[5]
Show that $q$ is a homomorphism.
\par\noindent
[You are only required to show that $q(P+Q) = q(P)q(Q)$ in the typical case 
when none of $P,Q,P+Q$ are $(0,0)$ or {\bf o}.]
\part[10]
Let~$p$ be prime, satisfying $p\equiv 2$ (mod~$3$)
and $p \equiv 5$ (mod~$8$). Show that the elliptic
curve $Y^2 = X(X^2 - pX + p^2)$ has rank~$0$.
\part[10]
Let $p_1,p_2$ be primes, satisfying $p_1 \equiv 3$ (mod~$8$),
$p_2 \equiv 1$ (mod~$8$), such that $p_1$ is not a quadratic
residue modulo~$p_2$. Show that the elliptic curve
$Y^2 = X(X^2 - p_1 p_2)$ has rank~$0$.
\end{parts}

\end{questions}

\end{document}