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\title{Homework 8: Elliptic Curves\\
{\sc\Large Due Wednesday, November 21}}
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\begin{document}
\maketitle
\noindent{There are {\bf six problems}. 
 Some of the problems involve tedious
algebra, and in all such cases you are allowed to do that algebra
using, e.g., PARI as long as you explain
{\em how} you used the system to do the algebra. 
Problems 4, 5, and 6 are from Silverman and Tate's
book {\em Rational Points on Elliptic Curves.}

\begin{problems}

\item (3 points) Consider the elliptic curve $y^2 + xy + y = x^3$ over $\Q$.
Find a linear change of variables that transforms this curve into
a curve of the form $Y^2 = X^3+a X + b$ for rational numbers~$a$ and~$b$.

\item (6 points) Let~$E$ be the elliptic curve over the finite 
field $K=\Z/5\Z$ defined by the equation 
$$
y^2 = x^3 + x +1.
$$  
\begin{subprob}
\item List all~$9$ elements of~$E(K)$.
\item What is the structure of the group 
$E(K)$, as a product of cyclic groups?
\end{subprob}

\item (8 points) Let~$E$ be an elliptic curve over~$\Q$.  
Define a binary operation $\boxplus$ on~$E$ as follows:
$$P \boxplus Q = -(P+Q).$$
Thus the $\boxplus$ of~$P$ and~$Q$ is the third
point of intersection of the line through~$P$ and~$Q$ 
with~$E$.  
\begin{subprob}
\item Lists the axiom(s) of a group that fail for
$E(\R)$ equipped with this binary operation.
(The group axioms are ``identity'', ``inverses'', and ``associativity''.)
\item Under
what conditions on $E(\Q)$ does this binary operation 
define a group structure on $E(\Q)$? (E.g., when $E(\Q)=\{\mathcal{O}\}$
this binary operation does define a group.)
\end{subprob}


\item (6 points) Let $g(t)$ be a quartic polynomial with distinct (complex) roots,
and let~$\alpha$ be a root of $g(t)$.  Let $\beta\neq 0$ be any number.
\begin{subprob}
\item Prove that the equations
$$
  x = \frac{\beta}{t-\alpha},
\qquad
 y = x^2 u = \frac{\beta^2 u}{(t-\alpha)^2}
$$
give an ``algebraic transformation'' between the curve $u^2=g(t)$
and the curve $y^2=f(x)$, where $f(x)$ is the cubic polynomial
$$
  f(x) = g'(\alpha) \beta x^3 + \frac{1}{2} g''(\alpha) \beta^2 x^2 + \frac{1}{6}g'''(\alpha) \beta^3 x + \frac{1}{24} g''''(\alpha)\beta^4.
$$
\item Prove that if~$g$ has distinct (complex) roots, then~$f$ also
has distinct roots, and so $u^2 = g(t)$ is an elliptic curve.

\end{subprob}



\item (8 points)
In this problem you will finally find out exactly 
why elliptic curves are called ``elliptic curves''!
Let $0<\beta\leq \alpha$, and let $C$ be the ellipse
$$\frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} = 1.$$
\begin{subprob}
\item Prove that the arc length of~$C$ is given 
by the integral
$$
  4\alpha\int_{0}^{\pi/2} \sqrt{1-k^2\sin^2\theta} d\theta
$$
for an appropriate choice of constant~$k$ depending on~$\alpha$
and~$\beta$.

\item 
Check your value for~$k$ in (i) by verifying that when $\alpha=\beta$,
the integral  yields the correct value for the arc length of a circle.

\item Prove that the integral in (i) is also equal to
$$
4\alpha\int_0^1 \sqrt{\frac{1-k^2t^2}{1-t^2}} dt
    = 4\alpha\int_0^1\frac{1-k^2t^2}{\sqrt{(1-t^2)(1-k^2t^2)}}dt.
$$
\item Prove that if the ellipse $E$ is not a circle, then the equation
$$
  u^2 = (1-t^2)(1-k^2t^2)
$$
defines an elliptic curve (cf. the previous exercise).  Hence
the problem of determining the arc length of an ellipse comes down
to evaluating the integral
$$
\int_0^1 \frac{1-k^2t^2}{u} dt
$$
on the ``elliptic'' curve $u^2=(1-t^2)(1-k^2t^2)$.

\end{subprob}

\item (8 points) 
Suppose that $P=(x,y)$ is a point on the cubic curve
$$
  y^2 = x^3 + ax + b.
$$
\begin{subprob}
\item Verify that the~$x$ coordinate of the point $2P$ 
is given by the duplication formula
$$
  x(2P) = \frac{x^4 - 2ax^2 -8bx +a^2}{4y^2}.
$$
\item Derive a similar formula for the $y$ coordinate of $2P$
in terms of~$x$ and~$y$.
\item Find a polynomial in~$x$ whose roots are the $x$-coordinates
of the points $P=(x,y)$ satisfying $3P=\mathcal{O}$. [Hint: The
relation $3P=\mathcal{O}$ can also be written $2P=-P$.]
\item For the particular curve $y^2 = x^3 + 1$, solve the equation
in (iii) to find all of the points satisfying $3P=\mathcal{O}$.
Note that you will have to use complex numbers.
\end{subprob}
\end{problems}

\end{document}