\input amssym.def
\input amssym.tex
%\def\Bbb{\bf}
\nopagenumbers
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\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.}
\rm
\medskip
\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).} $Y^2 = X^3 + 2 X$.
\ \ \ \ \ {\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(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(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 $p+1$.
\medskip
\noindent {\bf 5(a).}
Show that the point $(2,0)$ is of
order 2 on $Y^2 = (X-2)(X^2 + X + 1)$.
\par\noindent {\bf (b).} Find all $\bbQ$-rational points of order~2
and all $\bbC$-rational points of order~2
on each of the following elliptic curves: $Y^2 = X(X^2-3)$,
$Y^2 = X^3 - 7$ and $Y^2 = X(X-1)(X-7)$. In each case, find the
group structure (expressed as a product of cyclic groups)
of the $\bbQ$-rational 2-torsion group (that is, the group
of all $\bbQ$-rational points~$P$ such that $2P = {\bf o}$).
\medskip
\par\noindent {\bf 6.} Show that the point $(0,2)$ is of
order 3 on $Y^2 = X^3 + 4$. 
\medskip
\noindent
{\bf 7(a).}
Let $Y^2 = (X-\alpha)(X^2 + aX + b)$
be an elliptic curve with $a,b,\alpha\in K$ (characteristic $\not= 2$),
and  ${\bf o} =$ point at infinity, as usual. Show that $(\alpha , 0)$
is a point of order~2. Let $x',y'$ be defined by: $(x',y')
= (x,y) + (\alpha , 0)$, and define $T: K\rightarrow K : x\mapsto x'$.
Find $t_{11},t_{12},t_{21},t_{22}$ in terms of $a,b,\alpha$ such that:
$x' = \mu(x) = (t_{11} x + t_{12})/(t_{21} x + t_{22})$.
Check that $\mu^2 : x\mapsto x$.
\smallskip
\par\noindent {\bf (b).} Consider~$Y^2 = (X-\alpha_1)(X-\alpha_2)
(X-\alpha_3)$, with $\alpha_1,\alpha_2,\alpha_3$ distinct,
and let $T_1,T_2,T_3$ be as in (a), but with $\alpha$ replaced
by $\alpha_1,\alpha_2,\alpha_3$, respectively. 
Express each $T_i$ in terms of $x,\alpha_1,\alpha_2,\alpha_3$.
Show, directly from expressions, that $T_1,T_2,T_3$
commute (i.e.\ $T_1 T_2 = T_2 T_1$, $T_1 T_2 = T_2 T_1$ and
$T_2 T_3 = T_3 T_2$), and that $T_1 T_2 T_3 : x\mapsto x$.
Find the fixed points of $T_1$
and show that they are permuted by $T_2$.
\medskip
\noindent {\bf 8.}
Let $K$ be any field with $\hbox{Char }K \not= 2,3$,
and let 
\par\centerline{${\cal E}: F(X_0,X_1,X_2)
= X_1^2 X_2 - (X_0^3 + A X_0 X_2^2 + B X_2^3), \hbox{ with }A,B \in K,$}
\noindent be an elliptic curve (N.B. This is just the standard projective 
form, but with $X,Y,Z$ replaced by $X_0,X_1,X_2$). Let~$P$ be a
point on~$\cal E$.
\smallskip
\par\noindent {\bf (a).} Show that $3P = {\bf o}$ iff.\ the tangent line
to $\cal E$ at~$P$ intersects $\cal E$ only at~$P$.
\smallskip
\par\noindent {\bf (b).} Show that if $3P={\bf o}$ then the $3\times 3$
matrix
$\bigl( \partial^2 F / \partial X_i \partial X_j (P) \bigr)$
has determinant~$0$. [This matrix is called the Hessian matrix].
\smallskip
\par\noindent {\bf (c).} Show that there are at most nine
$3$-torsion points over~$K$.
\medskip
\vfil \eject \end