\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 3.}
\rm
\medskip\noindent {\bf 1.} 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 2.} 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 3.} 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 $a\in R$.
\medskip\noindent {\bf 4.} Prove that, if $d\in\Z_p$
is non-square, then 
\par\centerline{$ | a + b\sqrt{d} |_p = | a^2 - b^2 d |_p^{1/2}, 
\hbox{ for any } a,b \in \Q_p, $}
\par\smallskip\par\noindent
defines a non-Archimedean valuation on $\Q_p(\sqrt{d})$ which extends
the usual $|\ |_p$ on $\Q_p$. 
\smallskip\par\noindent [Hint: First show that, for
any $\alpha \in \Q_p(\sqrt{d})$, $|\alpha|_p \leqslant 1
\Rightarrow |\alpha + 1 |_p \leqslant 1$].
\medskip\noindent {\bf 5.} Let $\e : Y^2 = X^3 + 17$, defined
over~$\Q$, and $\widetilde \e : Y^2 = X^3 + 2$, defined over
$\F_5$. What does $(-64/25 , 59/125) \in \e (\Q)$ map to
under the reduction map modulo~$5$?
\medskip\noindent {\bf 6.} Let $\e : Y^2  = X^3 + p$, defined
over~$\Q_p$, and $\widetilde \e : Y^2 = X^3$, defined over~$\F_p$,
where~$p\not= 2$. Show that~$(0,0)$ on~$\widetilde \e$ does not
lift to a point in~$\e (\Q_p)$.
\medskip\noindent {\bf 7.} Give examples of elliptic curves
defined over~$\Z_p$ ($p\not= 2$) such that $\widetilde \e$,
defined over~$\F_p$, has:
\par \noindent {\bf (a).} A cusp which lifts to a point
in~$\e (\Q_p)$.
\par \noindent {\bf (b).} A cusp which does not lift to a point
in~$\e (\Q_p)$.
\par \noindent {\bf (c).} A node which lifts to a point
in~$\e (\Q_p)$.
\par \noindent {\bf (d).} A node which does not lift to a point
in~$\e (\Q_p)$.
\medskip\noindent {\bf 8.} A {\it non-commutative formal group}
over a ring~$R$ is a power series $F(X,Y) \in R[[X,Y]]$
which satisfies:
\par $F(X,Y) = X + Y +$ terms of degree~$\geqslant 2$,
%\par and 
\par $F(X, F(Y,Z)) = F(F(X,Y),Z)$ [associativity],
\par\noindent but not $F(X,Y) = F(Y,X)$ [commutativity].
Let $R = \F_p[t]/I$, where $I = t^2 \F_p[t]$.
Find a non-commutative formal group over~$R$.
%of the form $F(X,Y) = X + Y + c X^i Y^j$, where $c$ is a constant.
\medskip
\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 9.} Let $\e : Y^2 = X^3 + AX$, where
$A \in \Z$ and $A \not= 0$. Let $F(X,Y)$ be the formal group
associated to~$\e$ [as in lectures] and let $F(X,Y) = \sum F_n(X,Y)$,
where each $F_n(X,Y)$ is homogeneous of degree~$n$.
Show that $F_n(X,Y) = 0$ unless $n\equiv 1$~(mod~$4$).
What is the similar result when the elliptic curve
is of the form $\e : Y^2 = X^3 + B$?
\vfil\eject\end