\documentclass[a4paper]{article}

\usepackage{amsmath,amssymb,amsthm,enumerate,eucal,tikz,enumitem}

%\setlength\topmargin{-.1in}
%\setlength\headheight{0in}
%\setlength\headsep{0.2in}
%\setlength\textheight{8.5in}
%\setlength\textwidth{5.9in}
%\setlength\oddsidemargin{0.4in}
%\setlength\evensidemargin{0.4in}


%% \usepackage[T1]{fontenc}
%% \usepackage[scaled]{beramono,berasans}
%% \usepackage[charter]{mathdesign}
\usepackage{microtype}

\newfont{\Bb}{msbm10 scaled\magstep0}
\newfont{\Bbl}{msbm10 scaled\magstep1}
\newfont{\Bbs}{msbm10 scaled 800}




\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}}
%

\begin{document}


\centerline{\bf Elliptic Curves. MT 2025. Sheet 1.}

\noindent {\it Section A}
\medskip

\begin{enumerate}
\item { (a)} Find a birational transformation over~$\mathbb{Q}$
between the curves \[2X^2 - Y^2 = 1\] and \[X^2 + Y^2 - 6XY = 1.\]
\emph{Hint: both curves are birational to a line.}

\par\smallskip
\noindent {(b)} Find a birational transformation over~$\mathbb{Q}$
between the curves \[Y^2=(X+2)^6(X^3+1)\] and \[Y^2 = X^3 + 1.\]
\par\smallskip
\noindent { (c)} Find a birational transformation over~$\mathbb{C}$
between the curves~\[Y^2=2X^2\] and \[Y^2=X^2.\] Is there a birational
transformation over~$\mathbb{Q}$?

\item 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
{(a)} $Y^2 = X^3 + 2 X$.
\ \ \ \ \ {(b)} $Y^2 = X^3 + 1$.

\item Show that the point~$(2,4)$ is of order~4
on $Y^2 = X^3 + 4X$, defined over~$\bbQ$.
\end{enumerate}
\medskip

\noindent {\it Section B}

\begin{enumerate}[resume]
\item (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 {(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 $A,B\in \bbZ$.

\item
Consider the elliptic curve with projective equation 
\[X^3 + Y^3 + aZ^3 = 0\] for some $a \in \Q^\times$ and identity point $(1,-1,0)$. 

\par\noindent Show that the group law satisfies \[-(x,y,z) = (y,x,z)\] and \[[2](x,y,z) = (y(x^3-az^3),-x(y^3-az^3),z(y^3-x^3).\] (You might find Lemma 1.43 in lecture notes helpful.)

\newpage
%\par\noindent {(b)} \emph{(Section C)} Find a formula for the sum of two (distinct) points $(x_1,y_1,z_1)+(x_2,y_2,z_2)$.
\item (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 {(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 $p+1$.

\item (a)
Show that the point $(2,0)$ is of
order 2 on $Y^2 = (X-2)(X^2 + X + 1)$.
\par\noindent {(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 $2P = {\bf o}$).

\item Show that the point $(0,2)$ is of
order 3 on $Y^2 = X^3 + 4$. 
\end{enumerate}
\medskip


%{\bf 8.}(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 {(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 $T_2$.



\noindent {\it Section C}

\begin{enumerate}[resume]
\item
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 {(a)} Show that $3P = {\bf o}$ iff.\ the tangent line
to $\cal E$ at~$P$ intersects $\cal E$ only at~$P$.
\smallskip
\par\noindent {(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 {(c)} Show that there are at most nine
$3$-torsion points over~$K$.
\end{enumerate}


\end{document}
