\documentclass[12pt]{article} 

\usepackage{amsfonts}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{fullpage}                    % If you want 1 inch margins
\usepackage{papermacs}

\pagestyle{headings}


%-------------------------------------------
%        Begin Topmatter infomation
%-------------------------------------------

\title{Sample \LaTeX\ document} 
\author{Thomas R. Shemanske\\Dartmouth College}
\date{\today}


%-------------------------------------------
%        End Topmatter infomation
%-------------------------------------------


%-------------------------------------------
%       Begin Local Macros
%-------------------------------------------

\DeclareMathOperator{\Sh}{\mathbf{S}}
               

%-------------------------------------------
%       End Local Macros
%-------------------------------------------

\emergencystretch=20pt

\begin{document}
\maketitle
%-------------------------------------------
%        Other Topmatter Information
%-------------------------------------------

\dedication{This is dedicated to the one I love}

\subjclass {Primary 11Fxx; Secondary 11Fxx}

\keywords{Maximal Order, Central Simple Algebra, Bruhat--Tits
 Building}

\begin{abstract}
  This is a great paper.  Read no further, because I don't want you to
  hurt yourself, but if you can't help yourself, better strap in.  It
  gets bumpy from here on in.
\end{abstract}


%-------------------------------------------
%       End Other Topmatter Information
%-------------------------------------------



\section[intro]{Introduction}

The results which follow will dwarf all others that have come before.
It amazes me that I have been able to write them down.  I know that
you too will be duely impressed.

\section{Preliminaries}

What! You don't know what I'm talking about!!

Let's try a little fraktur $\mathfrak {A B C}$. Let's try a little
black board bold $\mathbb {Z, P, Q, R, C}$.  Let's try some other
symbols like $ x \gg 0$ or $\otimes$.  How about $M \otimes_{\mathbb
  Z} N$ or ${\mathbb Z}^{{\mathbb Z}^{\mathbb Z}}$?

How about $p \nmid N$ or $\boxplus$?


\section{Some sample theorems}



\begin{lemma}
\label{lemma:squareclasses} 
Let's put here exactly what we need to prove the next theorem.
\end{lemma}

\begin{thm}\label{thm:nonzerolifts}  Let $f$ be a nonzero element of $S_{k/2}(4N,\psi)$.  Then
  there exist an infinite number of square-free positive integers $t$
  such that $\Sh_t(f) \ne 0$.
\end{thm}

\dr{The proof which is below is correct, but unmotivated.
  Perhaps we can find an alternate proof which provides more insight}



\begin{proof}  If $\Sh_t(f) = 0$ for all but a finite number of square-free
  positive integers $t$, then by \lemref{lemma:squareclasses} the
  Fourier coefficients of $f$ are supported on only a finite number of
  square classes.  By Theorem 3 of \cite{SerreStark} the weight of $f$
  must be $1/2$ of $3/2$ and at weight $3/2$ must be in the span of
  the theta series $h_\psi$, contrary to assumption.
\end{proof}




By \thmref{thm:nonzerolifts}, we see that it we can always find
nonzero Shimura lifts.



\begin{defn}
\label{defn:horse}   
A horse is a horse of course, of course, but noone can talk to a horse
of course \dots.
\end{defn}


Here we have some displayed and aligned equations.

Here is an unnumbered displayed equation:
\begin{equation*}
  T(m) T(n) = \sum_{d \mid (m,n)} d^{k-1}\chi(d) T(mn/d^2).
\end{equation*}

Here is a numbered displayed equation:
\begin{equation}
  T(m) T(n) = \sum_{d \mid (m,n)} d^{k-1}\chi(d) T(mn/d^2).
\end{equation}

Here is the same expression, but inline and not displayed.  Notice it
is set smaller and the summation indeices are placed differently:
$T(m) T(n) = \sum_{d \mid (m,n)} d^{k-1}\chi(d) T(mn/d^2).$ Note I
need to use \$ to surround my formula when in an inline mode.

For an aligned display we have

\begin{align*}
  \Lambda_N(s;f) &= \left(\frac{2\pi}{\sqrt N}\right)^{-s} \Gamma(s)
  L(s;f)\\ \Lambda_M(s;g) &= \left(\frac{2\pi}{\sqrt M}\right)^{-s}
  \Gamma(s) L(s;g)
\end{align*}

A numbered version is given by
\begin{align}
  \Lambda_N(s;f) &= \left(\frac{2\pi}{\sqrt N}\right)^{-s} \Gamma(s)
  L(s;f)\\ \Lambda_M(s;g) &= \left(\frac{2\pi}{\sqrt M}\right)^{-s}
  \Gamma(s) L(s;g)
\end{align}


A version with only one number associated to the group of equations is
given by
\begin{align}
\begin{split}
  \Lambda_N(s;f) &= \left(\frac{2\pi}{\sqrt N}\right)^{-s} \Gamma(s)
  L(s;f)\\ \Lambda_M(s;g) &= \left(\frac{2\pi}{\sqrt M}\right)^{-s}
  \Gamma(s) L(s;g)
\end{split}
\end{align}


Something with cases
\begin{equation*}
  \phi_p(s) =
\begin{cases}
  \left(\frac{1 - b(p) p^{-s} + \psi(p) p^{k-1-2s}} {1 - a(p) p^{-s} +
      \chi(p) p^{k-1-2s}}\right) &\qquad \text{if } p\mid L\\ 
  1&\qquad\text{if } p \nmid L.
\end{cases}
\end{equation*}

\dr{This should be more than enough displayed equations for the
  average person.  Gosh, I sure hope this paper gets accepted.  More
  remarks of little permanent consequence.}

\bigbreak
\begin{thm}  Suppose that $N$ is an odd positive integer and $\psi$ is an
  even Dirichlet character defined modulo $4N$.  Let $F \in
  S^+_{k-1}(N,\psi^2) \cup S^+_{k-1}(2N,\psi^2)$ be a normalized
  newform, and suppose that $S_{k/2}(4M,\psi,F) \ne 0$ for some $M
  \mid N$.  Then
\begin{enumerate}
\item $M = N$
\item $S_{k/2}^-(4N,\psi) \cap S_{k/2}(4N,\psi,F) = \{0\}$, and so
  $S_{k/2}(4N,\psi,F) \subset S_{k/2}^+(4N,\psi)$.
\item If $N$ is square-free and $\psi^2 = 1$, then
  $S^+_{k/2}(4N,\psi)_K \subset S^+_{k/2}(4N,\psi)$.
\end{enumerate}
\end{thm}

Let's get the other references in now.  See \cite{Cipra} and
\cite{Koblitz}.



%-------------------------------------------
%        Begin References
%-------------------------------------------

\begin{thebibliography}{99}

\bibitem{Cipra} B. Cipra, On the Niwa-Shintani Theta-Kernel Lifting of
  Modular Forms, {\em Nagoya Math. J.}, {\bf 91}, (1983), 49--117.

\bibitem{Koblitz} N. Koblitz, ``Introduction to Elliptic Curves and
  Modular Forms", Springer-Verlag, New York, 1984.

\bibitem{SerreStark} J.-P. Serre and H. Stark, Modular Forms of Weight
  1/2, In Lecture Notes in Math. {\bf 627}, Springer-Verlag, Berlin
  and New York, (1977), 27--67.

\end{thebibliography}

%-------------------------------------------
%        End References
%-------------------------------------------


%-------------------------------------------
%        Begin Closing Infomation
%-------------------------------------------

\address{Department of Mathematics, Dartmouth College, Hanover, New
Hampshire 03755} 

\email {Jumpin'.Jack.Flash@dartmouth.edu}

\URL{http://www.math.dartmouth.edu/\symbol{126}trs/}

%-------------------------------------------
%        End Closing Infomation
%-------------------------------------------



\end{document}