\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}