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\begin{document}
\title{An example of Latex in action}
\author{S. D. Galbraith}
\date{1 February 2005}
\maketitle
\begin{abstract}
This article gives an example of how to write mathematical documents
using the {\LaTeX} package.
\end{abstract}
\section{Introduction}
Everyone learns Latex by borrowing someone else's document.
That's what this is for. There are also books, articles and lots of
web pages which explain valuable things.
A good place to look for help on the web is here:
\noindent
\begin{verbatim}
http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/
\end{verbatim}
You can write text in \textbf{bold face} or \emph{italics (emphasised)}
or \textsf{sans serif font} or in \texttt{typewriter style}.
You can write text in {\large large letters} or {\Large larger letters} or
{\LARGE even larger letters } or {\huge the hugest} letters.
\section{Formulae}
The best thing about Latex is that it makes nice mathematical
formulae for you.
Possibly the three most important tools are superscripts,
subscripts and
fractions, for example:
\[
x_{1}^{77} \qquad a_{1, 2}^{2^{8}} \qquad \frac{ 2 + x}{ x^2 + 1 }
\qquad \tfrac{1}{2}.
\]
Formulae can be written as part of the line,
such as $\int_0^2 e^x dx$, or in display mode like
\[
\frac{\sin(x)}{x^2 + e^x + 23}.
\]
The above equation does not have an equation number.
Giving equations numbers is easy, and they can be
referred to in the following way:
see equation (\ref{eq-one}) below
\begin{equation}
\label{eq-one}
\sum_{i=0}^{N_3} \binom{N_4}{i} \frac{x^i}{i !}
\end{equation}
You can do equations on several lines, such as
\begin{eqnarray}
f(x) &=& (x + 1) (x + 2) (x + 3 ) \\
&=& x^3 + 6x^2 + 11x + 6
\end{eqnarray}
or without numbers as
\begin{eqnarray*}
f(x) &=& (x + 1) (x + 2) (x + 3 ) \\
&=& x^3 + 6x^2 + 11x + 6 .
\end{eqnarray*}
References are done like this \cite{cohen}.
Greek letters are obtained in mathematics mode, for example
$\alpha, \beta, \gamma, \Gamma, \delta, \Delta, \dots$.
Other fonts are available for mathematics, such as calligraphic
$\mathcal{A}, \mathcal{B}$ and blackboard bold $\mathbb{A}, \mathbb{R}$.
One can do underlining and overlining
\[
\underline{x} \in \overline{\mathbb{Q}}.
\]
There are lots of built-in symbols such as
$\Rightarrow, \rightarrow, \in, <, \le, \subset, \subseteq, \mid,
\dagger, \star, \oplus, \times, \pounds, \S, \perp$.
There are several ways to write modular arithmetic.
For example $a \equiv 23 \bmod{78}$ or $a \equiv 23 \pmod{78}$.
Operations can be negated, for example:
\[
a \not= b , \; \; a \not\equiv b \bmod{c}.
\]
The operations \verb|\left| and \verb|\right| are useful for making
braces the right size:
\[
\left\{ 0, \frac{1}{2}, 1 \right\}, \;
\left( \sum_{i=1}^3 (i^2 + 2) \right), \;
\left[ 1 + \frac{1}{2 + \frac{2}{4 + \frac{1}{5}}} \right].
\]
Here is a table:
\vskip 0.2cm
\begin{center}
\begin{tabular}{|l|l|}
\hline
$N$ & Information about $N$ \\
\hline
2 & A prime \\
3 & A prime \\
4 & A square \\
5 & A prime \\
6 & Half a dozen \\
\hline
\end{tabular}
\end{center}
\vskip 0.2cm
In the next section you will find Theorem \ref{main-thm}.
If you want to start on a new page then do this:
\newpage
\section{A theorem}
\begin{theorem}
\label{main-thm}
Let $E/F$ be an elliptic curve defined over a number field $F$.
Let $\End(E) = \OO$ be an order of discriminant $D$.
Let $p$ be a prime for which $E$ has good and supersingular reduction.
Let $\wp $ be a prime ideal of $F$ above $p$. Let $\tilde E$
over $k=\F_{p^m}$ be the reduction mod $\wp $ of $E$.
Let $\pi$ be the $p^m$-Frobenius map on $\tilde{E}$.
Suppose $r \mid \# \tilde{E}( \F_{p^m} )$ is a prime
such that $r > 3$ and $r \nmid pD$.
Let $d\in \N$ be such that $\sqrt{-d} \in \OO$.
Let $\Psi \in \End (E)$ satisfy $\Psi^2=-d$.
Let $\psi \in \End_{\F_p}(\tilde E)$ be the
reduction mod $\wp$ of $\Psi$.
Then $\psi$ is a suitable distortion map for points $P \in \tilde{E}[r]$
which lie in a $\pi$-eigenspace.
\end{theorem}
\begin{proof}
You don't want to see the proof. \hfill $\Box$
\end{proof}
\section{More things}
\subsection{Subsections}
\label{sub-sec}
This is subsection \ref{sub-sec}.
\subsection{Spot the difference}
Experts in Latex find that they like things a certain way,
for example:
\begin{itemize}
\item ``quotes'' rather than "quotes".
\item $a \mid b$ and $a \nmid b$ rather than
$a | b$ and $a \not| b$.
\end{itemize}
Doing references the right way is also important.
Some examples are given below.
\begin{thebibliography}{}
\bibitem{boneh} D. Boneh,
The decision Diffie-Hellman problem,
in J. Buhler (ed.), ANTS III, Springer LNCS 1423
(1998) 48--63.
\bibitem{cohen} H. Cohen,
{\it A course in computational algebraic number theory},
Springer GTM 138 (1993).
\bibitem{gross} B. H. Gross,
Heights and special values of $L$-series,
CMS proceedings, \textbf{7}, AMS (1986), 115--187.
\bibitem{velu} J. V\'elu,
Isog{\'e}nies entre courbes elliptiques,
C. R. Acad.\ Sci.\ Paris,
S{\'e}rie A, 273 (1971) 238--241.
\end{thebibliography}
\end{document}