\documentclass{amsart}
\input emac
\title{Homework 2}
\author{Topics in AG: Elliptic Curves}
\begin{document}
\maketitle
Due March 03, 2004
\bigskip
\begin{enumerate}
\item Compute the tangents to the hyperbola $X^2 - Y^2 = 1$
and to the parabola $Y = X^2$ (over the real numbers)
at their points at infinity.
Use the insight gained to give a definition of the
concept of an asymptote for algebraic curves defined
over arbitrary (e.g. finite) fields.
\bigskip
\item Let $K$ be a field of characteristic $\ne 2$, and $f \in K[X]$
a polynomial of degree $\ge 4$ without multiple roots. Show
that the projective closure of the hyperelliptic curve
$y^2 = f(x)$ has exactly one singular point.
\bigskip
\item Let $f, g, h \in K[x,y]$ be polynomials, and put $f = gh$.
Show that any point oof intersection of the curves
$g(x,y) = 0$ and $h(x,y) = 0$ is a singular point of the
curve $f(x,y) = 0$.
\bigskip
\item Show that the Klein quartic
$$X^3Y + Y^3Z + Z^3X = 0$$
defined over a field $K$ is smooth if and only if $K$
has characteristic $\ne 7$.
\bigskip
\item Determine the number of points at infinity of the projective
closure of the unit circle $x^2 + y^2 = 1$ over the finite
fields $\F_3$, $\F_5$ and $\F_9$.
\bigskip
\item Consider the parabola $\cC: y = x^2$ over some ring $R$. Show that
the geometric group law defined for conics specializes to
$$ (x_1,y_1) + (x_2,y_2) = (x_3,y_3) \quad \text{for} \
x_3 = x_1 + x_2.$$
Deduce that $\cC(R) \simeq (R,+)$, the additive group of $R$.
\bigskip
\item Consider the hyperbola $\cC: xy = 1$ over some ring $R$. Show that
the geometric group law defined for conics specializes to
$$ (x_1,y_1) + (x_2by_2) = (x_3,y_3) \quad \text{for} \ x_3 = x_1x_2.$$
Deduce that $\cC(R) \simeq R^\times$, the unit group of $R$.
\end{enumerate}
\end{document}