\documentclass[12pt]{article} 

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

\pagestyle{headings}

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

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\emergencystretch=20pt

\begin{document}
\maketitle


\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}


\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 mathematical expressions}


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*}



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



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\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}

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