\input amssym.def
\input amssym.tex
%\def\Bbb{\bf}
\nopagenumbers
\magnification=\magstep1
%\hoffset=1truecm
%\voffset=2truecm
\baselineskip = 5.2 true mm
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\font\tenrm=cmr10
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\font\deffont=cmssi10
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\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 1. To be handed in during 2nd Week.}
\rm
\bigskip
\noindent
\noindent {\bf 1.} For each of the following elliptic curves,
find all the points (including, as always,
the point at infinity) over ~$\bbF_5$.
Draw a complete
group table in each case and describe each group as a product of
cyclic groups.
\par\noindent
{\bf (a).} $Y^2 = X^3 + 2 X$.
\smallskip
\par\noindent
{\bf (b).} $Y^2 = X^3 + 1$.
\medskip
\noindent {\bf 2.} Show that the point~$(2,4)$ is of order~4
on $Y^2 = X^3 + 4X$, defined over~$\bbQ$.
\medskip
\par\noindent {\bf 3.}
\par\noindent {\bf (a).} Let $m\in {\Bbb N}$ be odd or $f_m\in \qss$ (or both).
Show that the curve
\par
\centerline{$Y^2 = f_mX^m + f_{m-1}X^{m-1} + \ldots + f_0$, where
all $f_i\in \bbQ$ and $f_m\not= 0$,}
\par\noindent
can be birationally transformed over~$\bbQ$ to 
a curve of the form
\par
\centerline{$Y^2 = X^m + g_{m-1}X^{m-1} + \ldots+ g_0$, with
all $g_i\in \bbZ$.}
\smallskip
\par\noindent {\bf (b).} Birationally
transform over $\bbQ$ the curve $Y^2 = {1\over 5}X^3 + 3 X^2
+ 1$ to a curve of the form~$Y^2 = X^3 + AX + B$, where $A,B\in \bbZ$.
\medskip
\par\noindent {\bf 4.}
\par\noindent {\bf (a).} Let $p \equiv 2$~(mod~$3$) be prime
and let $A \in \bbF_p^*$.
Show that the number of points (including the point
at infinity) on the curve $Y^2 = X^3 + A$ over $\bbF_p$
is exactly $p+1$.
\smallskip
\par\noindent {\bf (b).} Let $p \equiv 3$~(mod~$4$) be prime
and let $B \in \bbF_p^*$.
Show that the number of points (including the point
at infinity) on the curve $Y^2 = X(X^2 + B)$ over $\bbF_p$
is exactly $p+1$.
\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
\par\noindent {\bf 5.}
\par\noindent {\bf (a).} Let $\c : Y^2 = Q(X) = X^4 + f_3X^3
+ f_2X^2 + f_1X + f_0$, where all $f_i\in \bbQ$. Show that
the curve $\c$ can be birationally transformed over $\bbQ$
to a curve of the form $Y^2 = X^3 + AX + B$, where $A,B\in \bbZ$.
[Begin by finding $G(X)$ and $H(X)$ such that $Q(X) = G(X)^2 + H(X)$,
with $G$ quadratic and $H$ linear, and let $T=Y+G(X)$,
$S=X\bigl(Y+G(X)\bigr)$.]
\smallskip
\par\noindent {\bf (b).} Birationally
transform over $\bbQ$ the curve $\c_1: Y^2 = 2X^4 + 9$
to the standard form $Y^2 = X^3 + AX + B$,
with $A,B\in \bbZ$. Show that $\c_2: Y^2 = 2X^4 + 7$ can
also be birationally
transformed over $\bbQ$ to the same form.
[Hint: In each case, first transform to the
form $Y^2 = Q(X)$ of part (a).]
\smallskip
\par\noindent {\bf (c).} Birationally transform over $\bbQ (i)$ the
curve $Y^2 = -X^4 - 1$ to the standard form $Y^2 = X^3 + AX + B$,
with $A,B\in \bbZ$. Is this possible over $\bbR$? Is this
possible over $\bbQ (\sqrt{-2})$?
\medskip
\vfil \eject \end