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\school{Final Honour School of Mathematics Part C}
\title{Course Title: B3.7. Elliptic Curves.\\
Lecturer: Prof Victor Flynn\\
Checked by: Alan Lauder}
\date{20/2/16}

\begin{document}

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\makecoverpage

\begin{questions}

\begin{question}
\begin{parts}
\part[5]
Let~$K$ be a field, complete with respect
to a non-Archimedean valuation~${{|\ \, |}}$, and
let~$x_n$ be a sequence in~$K$.
Show that $x_n \rightarrow 0\hbox{ in }K$
if and only if $\sum x_n$ is convergent in~$K$.
For any $x,y \in K$, show that 
$|x| \not= |y| \implies |x + y| = \mbox{max}( |x|, |y| )$. 

\part[2]
For what prime~$p$ does 
$-{9\over 8} = 1 + 2p + 2p^2 + 2p^3 + \ldots$ in $\mathbb{Q}_p$?

\part[5]
Find an $x\in\mathbb{Z}$ such that
$| x^2 + 3 |_7 < 7^{-1}$. Show that there does not exist $x\in\mathbb{Z}$
such that $| x^2 + 14 |_7 < 7^{-1}$.

\part[6]
State Hensel's Lemma. Let~$p \equiv 1$~(mod~$3$) be prime.
Show that~$-3$ is a quadratic residue mod~$p$
[{\it Hint: consider separately the cases $p\equiv 1$~$($mod~$4)$
and $p\equiv 3$~$($mod~$4)$}].
Show that~$-3$ is a square in~$\mathbb{Q}_p^*$.

\part[7]
Let $q\equiv 1$~(mod~27) be prime.
Show that $(x^2 + 3)(x^3 - q) = 0$
has solutions in $\R$ and in $\mathbb{Q}_p$ for every prime~$p$, 
but has no solutions in~$\mathbb{Q}$.

\end{parts}
\bigskip

\end{question}

\begin{question}
\begin{parts}
\part[5]
Find the torsion group over~$\Q$ of the elliptic curve
$y^2 = x^3 + 4 x$.

\part[7]
Let $n\in\mathbb{Z}$ satisfy $n \equiv 2$~(mod~$5$).
Find the torsion group over~$\Q$ of the elliptic curve
$y^2 = x(x+1)(x+n^2)$.

\part[7]
State and prove the Nagell-Lutz Theorem
for $\mathbb{Q}$-rational torsion points on 
the elliptic curve $y^2 = x^3 + Ax + B$, 
with $A,B\in \mathbb{Z}$.

\par [{\it 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[6]
Let $k \in \Z$, $k \not= 0$, 
let $\cal{E}$ be the elliptic curve
$y^2 = x^3 - k^2 x + k^3$, and let $(x,y)$ be a torsion point
in~$\cal{E} (\mathbb{Q})$. Show that $ | y | \le 5 |k|^3$
and $| x | \le 3 |k|^2$. 

\par [{\it In this part, ${{|\ \, |}}$ denotes the
standard absolute value $|\alpha| = \mbox{max}\,(\alpha,-\alpha)$
for $\alpha\in\mathbb{Q}$}.]

\end{parts}
\bigskip

\end{question}

\begin{question}
\begin{parts}
\part[11]
Find the rank of the elliptic curve $Y^2 = X(X^2 + X + 2)$.
\par
[{\it Standard results may be used without proof, provided they
are accurately stated.}]

\part[9]
Let~$A$ be an Abelian group with
group operation~$+$, and let
$h : A \longrightarrow \R$ satisfy:
\par
(1) For any $Q\in A$, there exists $C_1 = C_1(Q)$
such that $h(P+Q) \le 2h(P) + C_1$ for all
\par\ \ \ \ \ $P\in A$.
\par
(2) There exists $C_2$, independent of~$P$, such that
$h(2P) \ge 4h(P) - C_2$ for all $P\in A$.
\par
(3) For any~$C_3$, the set $\{ P\in A : h(P) \le C_3\}$ is finite.
\par\noindent
Suppose also that $A/2A$ is finite. Prove that $A$ is
finitely generated.

\part[5]
Let $A$ and~$h$ be as in~(b).
Suppose that $P$ is a torsion element of~$A$
[that is: there exists an integer $N > 0$ such that
$NP$ is the identity element of~$A$]. Show that
$h(P) \le {1\over 3}{C_2}$.

\end{parts}

\end{question}

\end{questions}

\end{document}


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