\documentclass{amsart}
\input ellc/emac
\title{Take Home Exam 1}
\author{Topics in Algebraic Geometry: Elliptic Curves}
\begin{document}
\maketitle
\begin{enumerate}
\item Every year, the Sunday Telegraph in London has a New Year's Quiz.
In 1995, two of the questions were the following:
\begin{enumerate}
\item Solve the equation $A^3/B^3 + C^3/D^3 = 6$,
where $A, B, C, D$ are all positive whole numbers below $100$.
\item A special question with a \pounds $450$ prize. Either
give a second solution to the above equation where the four
variables are all whole numbers above $100$ ($A$, $B$ and
$C$, $D$ relatively prime), or demonstrate that no such
second solution can exist.
\end{enumerate}
It's too late to earn the \pounds $450$ (sorry!), but using
pari you can solve the problem.
\bigskip
\item Consider the cubic $y^2 = x^3$ over some field $K$ of
characteristic $\ne 2, 3$. Show that $O = [0:0:1]$ is the
only singularity, and define an addition on
$E_\ns(K) = E(K) \setminus \{O\}$ by declaring that
$P+Q+R = \cO = [0:1:0]$ if and only if $P, Q, R$ are collinear.
\begin{enumerate}
\item Parametrize $E_\ns(K)$ using lines with slope $t$ through $O$;
show that the points corresponding to the parameters
$t_1, t_2, t_3$ are collinear if and only if
$\frac1{t_1} + \frac1{t_2} + \frac1{t_3} = 0$.
\item Show that $E_\ns(K) \simeq (K,+)$, the additive group of $K$.
\item Consider the parabola $C:y^2 = x$ with neutral element $O$;
every line through $O$ with slope $t$ intersects $E$ in some
point $P$ and $C$ in some point $Q$; describe the map sending
$P$ to $Q$ in coordinates and show that it induces a group
homomorphism $E_\ns(K) \lra C(K)$.
\end{enumerate}
\bigskip
\item Let $E: y^2 = x^3+ax+b$ be an elliptic curve defined over a
finite field $\F_p$. Let $d$ be an integer not divisible by $p$
and consider the quadratic twist $E_d: dy^2 = x^3+ax+b$.
Show that if $\#E(\F_p) = p+1-a_p$, then
$$\#E_d(\F_p) = \begin{cases}
p+1-a_p & \text{if}\ (d/p) = +1, \\
p+1+a_p & \text{if}\ (d/p) = -1.
\end{cases} $$
Also show that $E(\F_p) \simeq E_d(\F_p)$ if $(d/p) = +1$.
(Hint: all you need is elementary number theory.)
\vfill \eject
\item Consider the family of all elliptic curves $E: y^2 = x^3+ax+b$
over $\F_p$ ($p > 2$) with discriminant $\Delta(E) = -4a^3-27b^3=1$.
Its number of points can be written as $N_p = p+1 - a_p(E)$,
where $|a_p(E)| < 2\sqrt{p}$. Now form the sum over all $a_p(E)$
with $\Delta(E) = 1$.
For a fixed prime $p$, the following pari program computes this sum:
\begin{verbatim}
{p=5:s=0:for(a=0,p-1,for(b=0,p-1,d=Mod(-4*a^3-27*b^2,p):
d=lift(d):if(d-1,,e=ellinit([0,0,0,a,b]):s=s+ellap(e,p):
print(a," ",b," ",Mod(d,p)," ",s))))}
\end{verbatim}
The sum of all the $a_p$ is the last number in the output.
Look at the output for several small primes $p \equiv 3 \bmod 4$,
make a conjecture and prove it. For getting more data on the sums
when they are nonzero, modify the program
slightly:
\begin{verbatim}
{forstep(p=5,100,4,if(isprime(p),
s=0:for(a=0,p-1,for(b=0,p-1,d=Mod(-4*a^3-27*b^2,p):
d=lift(d):if(d-1,,e=ellinit([0,0,0,a,b]):s=s+ellap(e,p)))):
print(p," ",s),))}
\end{verbatim}
Recall that primes $p \equiv 1\bmod 4$ can be written
as a sum of two squares; the same holds for $p^2$,
by the way.
What happens if you replace the elliptic curves with
discriminant $1$ by curves with discriminant $2$ (or $3$)?
Note: for primes $p \equiv 1 \bmod 4$, these conjectures
(apparently due to N. Katz) have not yet been proved.
\bigskip
\item What does the Hasse bound tell you about the number of points
on elliptic curves $E_{a,b}: y^2 = x^3+ax+b$ over $\F_5$?
\begin{enumerate}
\item Use pari to do a complete search over all elliptic
curves and list the orders of $E(\F_p)$ that occur.
\item Use the fact that $-1$ is a square mod $5$ to explain
why the elliptic curves $E_{a,b}$ and $E_{a,-b}$ have
the same number of points (and, as a matter of fact,
the same group structure).
\item For squarefree orders, the group structure of
$E(\F_p)$ is uniquely determined. I mentioned
that we know $E(\F_p) = \Z/n_1\Z \oplus \Z/n_2\Z$
with $n_2 \mid n_1$ and $n_2 \mid (p-1)$. Use this
result to determine the group structure for
the curves with $\# E(\F_5) = 9$.
\item Use explicit calculations to determine the group
structure for the curves with $\# E(\F_5) = 8$.
\end{enumerate}
\end{enumerate}
\end{document}