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\begin{document}

\noindent
\centerline{\bf Elliptic Curves. MT 2025. Sheet 2.}

\noindent {\it Section A}
\medskip

\begin{enumerate}
\item Let $K$ be a field with
non-Archimedean valuation $|\ |$.
%% \smallskip
\par\noindent {(a)} For any $x,y\in K$ show that, if $|x| \not= |y|$
then $|x \pm y | = \hbox{max}( |x|, |y| )$.
\smallskip
\par\noindent {(b)}
If $x_1, \ldots , x_n \in K$ and
if there exists $\ell$ such that
$|x_\ell| > |x_i|$ for all $i\not= \ell$,
then show that $|x_1 + \ldots + x_n| = |x_\ell|$.
\smallskip
\par\noindent {(c)} Suppose that $s_n \rightarrow s$
in $K,|\ |$. Show that $|s_n| \rightarrow |s|$
in $\bbR, |\ |_\infty$. 
%Suppose
%that $s_n \rightarrow s \not= 0$ in~$\bbQ_p$; show
%that there exists $N$ such that, for all $n > N$, $|s_n|_p = |s|_p$.
When $s \not= 0$, show
that there exists $N$ such that, for all $n > N$, $|s_n| = |s|$.
%\smallskip
%\par\noindent {\bf (d).}
%Show that if $\sum_{n=1}^\infty x_n$
%converges to $x \in K, |\ |$, then
%the set $\{ | x_i | : i \geqslant 1\} \subset \bbR$ 
%has a maximum element, and $|x| \leqslant \hbox{max}_i |x_i|$.
%Show that, if there exists $\ell$ such that
%$|x_\ell| > |x_i|$ for all $i\not= \ell$,
%then $\sum_{n=1}^\infty x_n$ does not converge to~$0$.


\item {(a)} Find: $| 3/50 |_5$, $| 3/50 |_3$, $|3/50 |_7$,
$d_5(2/3 , 1/5)$, $d_7(2/3 , 1/5)$, $d_{11}(2/3, 1/5)$.
\smallskip
\par\noindent {(b)} Describe $| 3/7 |_p$ for all~$p$. What
is the product $\prod | 3/7 |_i$, taken over $i=p$, for all primes $p$,
and $i=\infty$? Given any $x\in\Q$ ($x\not= 0$), what is $\prod | x |_i$?

\item Which of the following are convergent in $\Q_5$?
\par\ \ \ \ \ \ \
${(a)}\ 1/5^n.\ \ {(b)}\ n.\ \ {(c)}\ n!
\ \ {(d)}\ 3 + 10^n.
\ \ {(e)}\ \sum_0^\infty 10^n.\ \ {(f)}\ \sum_0^\infty 7^n.$
%$ a_n = 1/5^n,\ \ a_n = n,\ \ a_n = n!,\ \ a_n = 3 + 10^n.$
%\medskip
%\noindent {\bf 4.} Which of the following are convergent in $\Q_5$?
%$\sum_0^\infty 10^n , \ \ \sum_0^\infty 7^n.$
%\medskip
%\noindent {\bf 5.} For each $p,m,r$, either find an $x\in \Z$ such that
%$|x-r|_p \leqslant p^{-m}$ or show that no such~$x$ exists.
%\smallskip
%\par\noindent
%{\bf (a).} $p=257, r=1/ 2, m=1$.\ \
%{\bf (b).} $p=3, r=7/ 9, m=7$.\ \
%{\bf (c).} $p=5, r=1/ 4, m=4$.
\end{enumerate}
\medskip
\noindent{\it Section B}

\begin{enumerate}[resume]
\item For each $p,m,r$, either find an $x\in \Z$ such that
$|x^2-r|_p \leqslant p^{-m}$ or show that no such~$x$ exists. 
\smallskip
\par\noindent
{(a)} $p=5, r=-1, m=4$.\ \
{(b)} $p=3, r=7/8, m=7$.\ \
{(c)} $p=5, r=5/4, m=4$.

\item Find the $7$-adic expansion of each of: $200$ and $3/14$.
Determine the member of~$\Q$ expressed by
the $5$-adic expansion $2,\overline{34}$.
\medskip


\item For which primes $p$ does there exist $x\in \Q_p$ such that
$x^2 = -28$? Your answer should be given in terms of a congruence condition on $p$.
\medskip

\item Show that $(X^2 - 2)(X^2-17)(X^2-34)$
has a root in $\R$ and in every $\Q_p$, but not in $\Q$.
\medskip

\item Is~$4$ a cube in~$\Q_3$? Is~$28$ a cube
in~$\Q_3$? Is~$13$ a cube in~$\Q_7$?
\medskip

\item Let $p\equiv 2$ mod~$3$. For any
$a\in \Z$ such that $p\notdiv a$, show that there exists
$x\in \Z_p$ with $x^3 = a$.
\end{enumerate}
\medskip

\pagebreak
\noindent {\it Section C}

\begin{enumerate}[resume]
\item For any odd prime $p$ and any positive integer $m$ not divisible by $p$, show that there is a primitive $m$th root of unity in $\bbQ_p$ if and only if $m | (p-1)$. 

(\emph{In fact, the roots of unity in $\bbQ_p^\times$ are a cyclic subgroup of order $(p-1)$}.)

\medskip
\item {(a)}
Show that $|n!|_p = p^{-M}$ where $M = \sum_{i=1}^\infty 
\bigl[ {n\over p^i} \bigr]$ (where $[ x ]$ denotes the
greatest integer $\leqslant x$).

\smallskip\noindent{(b)}
Let $K$ be a field containing $\Q_p$,
let $|\ |$ be a non-Archimedean valuation on~$K$ which extends
$|\ |_p$, and assume that~$K$ is complete with respect
to this valuation. For any $x\in K$, show that 
$\hbox{exp}_p(x) = \sum_{n=0}^\infty {x^n\over n!}$
converges if and only if
$|x| < p^{-{1\over p-1}}$. 

\smallskip\noindent{(c)} For $p \ne 2$, show that $\exp_p$ defines a continuous group isomorphism
\[\exp_p: \{x \in \bbQ_p: |x|_p < 1\} \to \{x \in \bbQ_p: |x-1|_p < 1\} \] where the group law on the left hand side is addition, whilst on the right hand side it is multiplication.
\end{enumerate}
\end{document}


{\it The following question is compulsory for students taking
the MSc in MFoCS (Mathematics and the Foundations of Computer
Science). For everyone else, it is optional.}
\medskip
\noindent {\bf 10.}
Show that $|n!|_p = p^{-M}$ where $M = \sum_{i=1}^\infty 
\bigl[ {n\over p^i} \bigr]$ (where $[ x ]$ denotes the
greatest integer $\leqslant x$).
Let $K$ be a field containing $\Q_p$,
let $|\ |$ be a non-Archimedean valuation on~$K$ which extends
$|\ |_p$, and assume that~$K$ is complete with respect
to this valuation. For any $x\in K$, show that 
$\hbox{exp}_p(x) = \sum_{n=0}^\infty {x^n\over n!}$
converges if and only if
$|x| < p^{-{1\over p-1}}$. 
When $K = \Q_p$ ($p\not= 2$), show that $\hbox{exp}_p(x)$ converges
if any only if $|x|_p < 1$. When $K = \Q_2$, show that
$\hbox{exp}_2(x)$ converges
if any only if $|x|_2 < {1\over 2}$.