\input amssym.def
\input amssym.tex
%\def\Bbb{\bf}
\nopagenumbers
\magnification=\magstep1
%\hoffset=1truecm
%\voffset=2truecm
\baselineskip = 5.2 true mm
\font\frkkk=eufm10
\font\twelverm=cmr12
\font\tenrm=cmr10
\font\ninerm=cmr9
\font\ninebf=cmbx9
\font\eightrm=cmr8
\font\sevrm=cmr7
\font\sixrm=cmr6
\font\scrpp=eusm10
\font\frkk=eufm10
\font\deffont=cmssi10
\font\chaptitle=cmbx10 at 14 pt
\tolerance=10000
\def\sqr{\ifmmode\square\else{$\square$}\fi}
\def\square{\vcenter{
\hrule height.1mm
\hbox{\vrule width.1mm height2.2mm\kern2.18mm\vrule width.1mm}
\hrule height.1mm}}                  % This is a slimmer sqr.
\null
\def\le{\leqslant}
\def\ge{\geqslant}
\def\etq{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q})}
\def\etqp{{\cal E}_{\lower 1pt\hbox{\eightrm tors}}({\Bbb Q}_p)}
\def\c{{\cal C}}
\def\d{{\cal D}}
\def\e{{\cal E}}
\def\pk{\phi _\kappa}
\def\im{{\hbox{\sl im}}}
\def\hs{H_{\varsigma}}
\def\hpk{\hat \phi _\kappa}
\font\sc=cmssqi8
\def\scc#1{\hbox{\sc #1}}
\def\sf{{\scc F}}
\def\pnbq{{\Bbb P}^n(\overline {\Bbb Q} )}
\def\hk{{\hat \kappa}}
\def\bq{{\overline {\Bbb Q}}}
\def\hq{{\hat q}}
\def\pv{\prod\limits_v }
\def\pnk{{\Bbb P}^n(K)}
\def\mnkvw{{\Bbb M}^n(K[{\bf v}^2,{\bf w}^2])}
\def\pnkv{{\Bbb P}^n(K[{\bf v}^2])}
\def\kj{\kappa (J)}
\def \qmods {{\Bbb Q}^*/({\Bbb Q}^*)^2}
\def \qmodss { {\Bbb Q}^*/({\Bbb Q}^*)^2 \times
{\Bbb Q}^*/({\Bbb Q}^*)^2 }
\def \qs{{\Bbb Q}^*}
\def \qss{({\Bbb Q}^*)^2}
\def\bbQ{\Bbb Q}
\def\bbF{\Bbb F}
\def\bbZ{\Bbb Z}
\def\bbR{\Bbb R}
\def\bbC{\Bbb C}
\def\notdiv{{\not\hskip-.5pt |\ }}
\def\Q{{\Bbb Q}}
\def\F{{\Bbb F}}
\def\Z{{\Bbb Z}}
\def\R{{\Bbb R}}
\def\C{{\Bbb C}}
%
\chaptitle
\noindent
\centerline{Elliptic Curves. MT 2017/18. Sheet 2.}
\rm
\bigskip
\noindent
{\bf 1.} Let $K$ be a field with
non-Archimedean valuation $|\ |$.
\smallskip
\par\noindent {\bf (a).} For any $x,y\in K$ show that, if $|x| \not= |y|$
then $|x \pm y | = \hbox{max}( |x|, |y| )$.
\par\noindent {\bf (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 {\bf (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~$0$.
\medskip
\noindent {\bf 2(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 {\bf (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 $\prod | x |_i$?
\medskip
\noindent {\bf 3.} Which of the following are convergent in $\Q_5$?
\par\ \ \ \ \ \ \
${\bf (a).}\ 1/5^n.\ \ {\bf (b).}\ n.\ \ {\bf (c).}\ n!
\ \ {\bf (d).}\ 3 + 10^n.
\ \ {\bf (e).}\ \sum_0^\infty 10^n.\ \ {\bf (f).}\ \sum_0^\infty 7^n.$
%$ 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).} $p=5, r=1/ 4, m=4$.
\medskip
\noindent {\bf 4.} 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
{\bf (a).} $p=5, r=-1, m=4$.\ \
{\bf (b).} $p=3, r=7/8, m=7$.\ \
{\bf (c).} $p=5, r=5/4, m=4$.\ \
\medskip 
\noindent {\bf 5.} 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\noindent {\bf 6.} Let $x\in \Q$. Show that
$x\in \Z \iff \bigl( x\in \Z_p \hbox{ for all }p\bigr)$. 
\medskip\noindent {\bf 7.} Decide whether there exists $x\in \Q_p$ such that
$x^2 = -28$ for each of: $p=2,3,5,7,11$.
\medskip\noindent {\bf 8.} 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\noindent {\bf 9.} Is~$4$ a cube in~$\Q_3$? Is~$28$ a cube
in~$\Q_3$? Is~$13$ a cube in~$\Q_7$?
\bigskip
\hrule
\medskip
{\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 $|x|_2 < {1\over 2}$.
\vfil \eject \end
