% MARKS ALLOCATED AS FOLLOWS:
% 1: 10. 2: 10. 3: 10. 4: 10. 5: 10. 6: 20. 7: 30. Total: 100.
% In more detail:
% 1: 10. 2: 10. 3: 10. 4: 10. 5: 10. 6: each part 5. 7: each part 5.
\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. Sheet 4. To be handed in during 5th Week.}
\rm
\bigskip
\noindent {\bf 1.} 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 2.} Show that, for all~$p$, there
%exist~$x,y\in \Z_p$ such that
%$y^2 = x^3 + x - 3$.
\medskip\noindent {\bf 2.} 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 3.} Is~$4$ a cube in~$\Q_3$? Is~$28$ a cube
in~$\Q_3$? Is~$13$ a cube in~$\Q_7$?
\medskip\noindent {\bf 4.} Show that the curve $2 Y^2 = X^4 - 17$
has points in $\R$ and every $\Q_p$, but not in~$\Q$. 
\par\noindent
[Hint: For showing that there are points in every $\Q_p$,
it is helpful to use Theorem~1.15 (note also that
the curve is birationally equivalent to $V^2 = 2 X^4 - 34$,
where $V = 2Y$). For showing there are
no points in~$\Q$,
first show that, if there were points in~$\Q$, then there would exist
$r,s,t\in \Z$ with $\hbox{gcd}(r,t) = 1$ such
that $2 s^2 = t^4 - 17 r^4$, and then show that any prime dividing
$s$ is a quadratic residue modulo~$17$].
\medskip\noindent {\bf 5.} 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 $x^3 = a$.
\medskip\noindent {\bf 6.} Let $K$ be any field with a
non-Archimedean valuation $|\ |$, and
let $ R = \{ x\in K : |x| \leqslant 1\}$.
Let $f(X) \in R[x]$ have discriminant~$D$, and let
$a_0 \in R$ satisfy $|f(a_0)| < |D|^2$. Show that $f(X)$
has a root $a\in R$.
\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
\medskip\noindent {\bf 7.} Show that $f(X) = 5 X^3 - 7 X^2 + 3 X + 6$
has a root $\alpha \in \Z_7$ with $|\alpha - 1|_7 < 1$.
Find $a\in\Z$ such that $|\alpha - a|_7 \leqslant 7^{-4}$.
\vfil\eject\end