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\school{Final Honour School of Mathematics Part C}
\title{C3.7 Elliptic Curves\\
Victor Flynn \\
Checked by: Alan Lauder}
\date{23/11/17}
\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~$K$ be a field, complete with respect to a non-Archimedean
valuation~$|\ \, |$, with valuation ring~$R = \{ x\in K : |x| \leqslant 1\}$.
Prove Hensel's Lemma, that if $f(x) \in R[x]$ and $a_0\in R$ satisfies
$| f(a_0) | < | f'(a_0) |^2$, then there exists a unique $a\in R$ such that
$f(a) = 0$ and $| a - a_0 | \leqslant | f(a_0) |/ | f'(a_0) |$.

\part[8]
Show that there does not exist $x \in \mathbb{Z}$
such that $|x^{10} - 2|_{11} \leqslant 11^{-3}$.
For any prime~$p$, show there does not exist $x\in \mathbb{Z}$
such that $|x^p - p|_p \leqslant p^{-2}$.
Find $x\in \mathbb{Z}$ such that $|x^2 + 2|_3 \leqslant 3^{-5}$.
Find $x\in \mathbb{Q}$ such that $|x^2 + 7|_2 \leqslant 2^{-8}$.

\part[7]
Let $p_1,p_2,p_3$ be distinct primes satisfying 
$p_1 \equiv p_2 \equiv p_3 \equiv 5$ (mod~$8$), such
that~$p_i$ is a quadratic residue modulo~$p_j$ for any $i \not= j$.
Show that, for any prime~$p$, there exist $x,y \in \mathbb{Q}_p$
such that $y^2 = p_1 x^4 - p_1 p_2^2 p_3^2$.
\end{parts}

\question
\begin{parts}
\part[8]
State and prove the Nagell-Lutz Theorem
for $\mathbb Q$-rational torsion points on 
the elliptic curve $y^2 = x^3 + Ax + B$, 
with $A,B\in {\mathbb Z}$.
[You may assume the result that any $\mathbb Q$-rational 
torsion point $(x,y)$
on such a curve satisfies $x,y\in {\mathbb Z}$. You may also use the
polynomial identity:
$\phi_1(X) \psi_1(X) + \phi_2(X) \psi_2(X) = 4A^3 + 27B^2$,
where $\phi_1(X)= 3X^2+4A$, $\psi_1(X) = (3X^2+A)^2$,
$\phi_2(X)= -27(X^3 + AX - B)$ and $\psi_2(X) = X^3 + AX + B$.]

\part[7]
Show that the torsion group over~$\mathbb{Q}$ of
the elliptic curve ${\mathcal E} : Y^2 = X^3 - 3$
is the trivial group.
Show that $3$ divides $\# \widetilde {\mathcal E} (\mathbb{F}_p)$
for every prime~$p > 3$. [{\it Hint: consider separately
$p \equiv 1$ and $p\equiv 2$ (mod~$3$)}.]

\part[6]
Find the torsion group over~$\mathbb{Q}$ of the elliptic
curve $Y^2 = X^3 + X^2 - X$.

\part[4]
Let $(x,y)$ be a $\mathbb{Q}$-rational torsion point on the
elliptic curve $Y^2 = X^3 + AX + B$, with $A,B\in \mathbb{Z}$,
and let $\Delta = 4 A^3 + 27 B^2$.
Show that $|x| \leqslant \hbox{max}\bigl( 2|A|, |\Delta| + |B| \bigr)$.
[In this part, $|\ \ |$ denotes the standard absolute
value $|\alpha| = \hbox{max}(\alpha,-\alpha)$.]
\end{parts}

\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[9]
Show that the image of~$q$ is a subset
of the finite set
\begin{equation*}
\{ r : r \hbox{ is a square free integer and } r | b_1 \}.
\end{equation*}
\part[7]
Find the rank of the elliptic curve $Y^2 = X(X^2 - X - 1)$.
\part[9]
Let $p$ be a prime satisfying $p \equiv 3$ (mod~$4$).
Show that the elliptic curve $y^2 = x^3 + px$ has
rank at most~$1$.
When $p \equiv 7$ (mod~$16$) show that the rank is~$0$.
\end{parts}

\end{questions}

\end{document}