\documentclass[12pt]{amsart}
\usepackage{amssymb,amsmath}

% These settings adjust the size of the page
\setlength{\textwidth}{6in}
\setlength{\textheight}{8.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\evensidemargin}{0in}
\setlength{\topmargin}{0.25in}

% Use macros as follows to save yourself lots of typing
% and to help keep your notation consistent across your
% paper.  It makes it easy to change symbols.

% If you get a compiler error like ... already defined''
% then it's some default macro (like \P is the new paragraph
% symbol), so pick something else.

\newcommand{\R}{\mathbb{R}} %real number
\newcommand{\F}{\mathbb{F}} % fancy field name
\newcommand{\Z}{\mathbb{Z}}   %integers
\newcommand{\C}{\mathbb{C}}   %complex numbers

\newcommand{\diag}{\text{diag}}
\newcommand{\val}{\textrm{val}} %valuation in k
\newcommand{\A}{\mathcal{A}} % a fancy A

\theoremstyle{plain}
\newtheorem*{maintheorem}{Main Theorem}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\theoremstyle{remark}
\newtheorem{example}[theorem]{Example}
\newtheorem*{notation}{Notation}

\numberwithin{equation}{section}
\numberwithin{figure}{section}
\numberwithin{table}{figure}

\begin{document}
\author{You}
\thanks{Acknowledge external funding here}

\keywords{keywords}

\date{\today}

\begin{abstract}
Fill in when you know what you have.
\end{abstract}
\maketitle

\section{Introduction}

Give a brief statement of the problem, including its
historical context and an overview of the major developments
in its history, with references.  For instance,
the famous theorem in \cite{Nev} was stated as only a conjecture in
\cite{Car}, though in \cite{Cas}, Casselman claims it
was actually proven by Gauss in \cite{Gauss}.  All
references in the bibliography must be cited at some
point.

Provide a guide for the reader.  In Section~\ref{S:background}
we'll discuss some particular conventions for the writing of
mathematical articles.
In Section~\ref{S:problem} we'll define the problem
precisely    and give a proof of the main result.  We conclude
in Section~\ref{S:applications} with some examples and
applications.

\section{Mathematical Conventions} \label{S:background}

Most mathematical journals rely on authors to electronically
submit their articles in \LaTeX.  The journals usually provide
a \emph{style file} which allows an author to easily recreate
the look and feel'' of the particular journal.  For this report,
the default \texttt{amsart} documentclass will be fine.

We write math papers in first person plural, signifying the
author(s) and the reader together''.  Besides being a more natural
writing style than the stiff formality of one sees that''
or in this paper it is shown that'', it tends to be a bit
shorter.

Never start a sentence with a symbol.  $x$ is a vector.''  This
looks awful, is hard to read, and is never necessary.  We see
that $x$ is a vector.'' Let $x$ be a vector.'' Therefore $x$
is a vector.''

In the same vein, every sentence should contain a verb and be
readable out loud.  Theorems and definitions should be introduced
with complete sentences, not open-ended gaps like

\begin{lemma}
use English as their first language, and so might find
casual misuse of grammar or punctuation to be an obstacle in
\end{lemma}

Math papers don't have a conclusion, even though articles in most
other disciplines do.  Instead, the introduction takes up the slack.
So most articles will include in the introduction:
\begin{itemize}
\item a discussion
of the context of the work;
\item a statement of
the main theorem or theorems;
\item a discussion of related results in the literature; and
\item a roadmap'' of the paper, detailing what will be found in
each section.
\end{itemize}
For a paper whose goal is to present a new theorem, for example,
the main body of the paper will have minimal discussion, in general,
focussing instead on presenting the results and giving the proofs.
Whereas the introduction may be packed with several disjoint things,
the rest of the paper will be organized to flow as smoothly as possible.

When a new term is introduced, use \emph{italics} (command: emph),
not boldface or underlining.  \LaTeX has several features to help
typeset definitions or theorems consistently:  \texttt{theoremstyle}
commands.  (See the preamble of this document for one set-up.)

\begin{definition}
There are many \emph{environments} pre-defined in \LaTeX, and
many more you can create yourself.  Some favourites for lists
are
\begin{enumerate}
\item itemize,
\item enumerate, and
\item description.
\end{enumerate}
\end{definition}

\begin{theorem} \label{T:t1}
Footnotes are rarely used.  References to the literature are made
as they occur~\cite{Gauss}.  They should be made as specific as
possible~\cite[Thm3.3]{Gauss} if the work is a long one.
\end{theorem}

\begin{proof}
The bibliography should be typeset and arranged as uniformly as
possible.  Many different ways of citing a particular work may be
acceptable; the key points are to identify:  the author, the
title of the work, the publisher, and the year.  More detail
is usually needed for articles in journals, such as volume or issue
number.  Book and journal titles may be put in \emph{italics}, whereas
article titles are put in quotation marks''.
\end{proof}

Cross-referencing within your article using the command \texttt{ref}
can be very helpful to the
reader.  For example, refer to Theorem~\ref{T:t1}.
You might refer to an equation in the flow of the text.  For instance,
suppose we have established that
$$\label{E:first} 3x-4=2.$$
As we immediately deduce from Theorem~\ref{T:t1}, \eqref{E:first} implies
that
$$x=2.$$
Note that we didn't label this equation because we don't refer
back to it.

\section*{More on labelling}

Note that one can use *'' versions of most commands to
suppress labelling, such as for this section heading, or
for the first equation below.  Equations can be typeset
in line like $y=\int_a^b x\;dx$ or displayed like
\begin{equation*}
y = \int_a^b x\;dx;
\end{equation*}
\LaTeX\; automatically adjusts the symbols to be nice.

Other typical expressions in \LaTeX\; include $x \in \R$,
$\overline{x} \in \C$ and $\sin(x) = 1$.  Note that I
predefined $\R$ and $\C$ in the preamble to avoid extra
typing.

\section{Statement of the Problem} \label{S:problem}

The most important problem in mathematics involves
simplifying the expression
\begin{equation*}
\sum_{i=1}^n i,
\end{equation*}
a problem famously solved by Gauss~\cite{Gauss} at the age of six.
His solution
$$\label{E:gauss} \sum_{i=1}^n i = \frac{n(n+1)}{2}$$
is brilliant.

Next, we'd like to understand the $n$th term in a
sequence $x_0,x_1,x_2,\ldots$, where $x_i = 4i^2+2$.  We'll
use \eqref{E:gauss}.

\section{Examples} \label{S:applications}

There are many wonderful applications of number theory.

\begin{thebibliography}{999}

\bibitem[B]{Nev} R.~U.~Brilliant, The Quadratic Form'' in \emph{Journal of
Applied Number Theory}, Vol. 3, No. 2 (1998).

\bibitem[Car]{Car} Cartier, P. Representations of $p$-adic groups: a survey.  Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1,  pp. 111--155, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979. MR0546593 (81e:22029)

\bibitem[Cas]{Cas} Casselman, W., Introduction to the theory of admissible representations of $p$-adic reductive groups, preprint.

\bibitem[G]{Gauss} C.~F.~Gauss, \emph{Brilliant Math}, Princeton University Press, 2002.

\end{thebibliography}

\end{document}