\documentclass{beamer}

%\usepackage{beamerthemesidebar, fancybox}
\usepackage{beamerthemesplit,fancybox}
\usepackage{graphicx,pgfarrows,pgfnodes}
\newtheorem{thm}{Theorem}[section]
\theoremstyle{definition}
\newtheorem*{ex}{Example}
\newtheorem{cor}{Corollary}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{conjec}[thm]{Conjecture}
\theoremstyle{remark}
\newtheorem*{rem}{Remark}
\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\theoremstyle{definition}
\newtheorem*{ack}{Acknowledgements}
\def\Q{\mathbb{Q}}
\def\F{\mathbb{F}}
\def\Gal{{\rm Gal}}
\newcommand{\legen}[2]{\genfrac{(}{)}{}{}{#1}{#2}}
\def\ord{{\rm ord}}
\def\Tr{{\rm Tr}}
\def\d{d}
\def\const{\text{const}}
\newcommand{\leg}[2]{\genfrac{(}{)}{}{}{#1}{#2}}
\newcommand{\bfrac}[2]{\genfrac{}{}{}{0}{#1}{#2}}
\newcommand{\sm}[4]{\left(\begin{smallmatrix}#1&#2\\ #3&#4 \end{smallmatrix} \right)}
\newcommand{\mfG}{\mathfrak{G}}
\newcommand{\mfF}{\mathfrak{F}}
\newcommand{\pr}{\text {\rm pr}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calW}{\mathcal{W}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\GG}{\mathcal{G}}
\newcommand{\FF}{\mathcal{F}}
\newcommand{\QQ}{\mathcal{Q}}
\newcommand{\Mp}{\text {\rm Mp}}
\newcommand{\frakG}{\mathfrak{G}}
\newcommand{\Qmd}{\mathcal{Q}_{m,d}}
\newcommand{\la}{\lambda}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Qd}{\mathcal{Q}_d}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\SL}{{\text {\rm SL}}}
\newcommand{\GL}{{\text {\rm GL}}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\PSL}{{\text {\rm PSL}}}
\newcommand{\Op}{\mathcal{O}_K}
\newcommand{\h}{\mathfrak{h}}
\newcommand{\G}{\Gamma}
\newcommand{\g}{\gamma}
\newcommand{\zaz}{\Z / a\Z}
\newcommand{\znz}{\Z / n\Z}
\newcommand{\ve}{\varepsilon}
\newcommand{\tr}{{\text {\rm tr}}}
\newcommand{\odd}{{\text {\rm odd}}}
\newcommand{\bk}{B_k}
\newcommand{\rr}{R_r}
\newcommand{\sump}{\sideset{}{'}\sum}
\newcommand{\gkr}{\mathfrak{g}_{k,r}}
\newcommand{\re}{\textnormal{Re}}
\newcommand{\im}{\textnormal{Im}}
\def\H{\mathbb{H}}


\definecolor{Dblue}{rgb}{.255,.41,.884}

\title[The $\omega(q)$ mock theta function]{The $\omega(q)$ mock theta function and vector-valued Maass-Poincar\'e series}
\author{Sharon Anne Garthwaite}
\date{October 7, 2006}

\begin{document}


\frame{\titlepage}

%\frame{\tableofcontents}

\section[Introduction]{Introduction}
\subsection[History]{}


\frame{ \frametitle{History}

Let $p(n)$ denote the number of \alert{partitions} of $n$.
\vspace{.2 in}

\onslide<2->Hardy-Ramanujan-Rademacher formula (1917,1922):
\begin{equation*}
p(n)= 2 \pi (24n-1)^{-\frac{3}{4}} \sum_{k =1}^{\infty}
\frac{A_k(n)}{k}\cdot  I_{\frac{3}{2}}\left( \frac{\pi
\sqrt{24n-1}}{6k}\right).
\end{equation*}
\begin{itemize}
\item<2-> $I_{s}(z)$ is an $I$-Bessel
function.
\item<2-> $A_k(n)$ is a ``Kloosterman-type'' sum.
\end{itemize}

 }



\frame {
  \frametitle{History}

   In 1920 Ramanujan wrote about his discovery
 of ``very interesting functions,'' such as
\begin{align*}\label{f(q)}
f(q) &:=1+\sum_{n=1}^{\infty}\frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots
(1+q^n)^2} \\
\notag &=1+q-2q^2+3q^3-3q^4+3q^5-5q^6+\cdots; \\
\omega(q) &:=\sum_{n=0}^{\infty}
\frac{q^{2n^2+2n}}{(1-q)^2(1-q^3)^2\cdots(1-q^{2n+1})^2}\\
\notag &= 1+2q+3q^2+4q^3+6q^4+8q^5+10q^6+\cdots.
\end{align*}

Here $q:=e^{2\pi i z}$. }


\frame{ \frametitle{History}

\onslide<1-> Define $\alpha_f(n)$ and $\alpha_\omega(n)$ by
\[
f(q)=\sum_{n\geq 0}\alpha_f(n)q^n; \quad \omega(q)=\sum_{n\geq
0}\alpha_\omega(n)q^n.
\]

\onslide<2-> Andrews-Dragonette Conjecture (1952,1966,2003):
\vspace{.1in}
 {\footnotesize
\begin{equation*}
 \alpha_f(n) = \pi
(24n-1)^{-\frac{1}{4}}\sum_{k=1}^{\infty} \frac{ (-1)^{\lfloor
\frac{k+1}{2}\rfloor}A_{2k}\left(n-\frac{k(1+(-1)^k)}{4}\right)}{k}
\cdot I_{1/2}\left(\frac{\pi \sqrt{24n-1}}{12k}\right).
\end{equation*}
}

%{\footnotesize
%\begin{align*}
% \alpha_f(n)&= \sum_{k=1}^{\lfloor
%\sqrt{n}\rfloor}\frac{(-1)^{\lfloor \frac{k+1}{2}\rfloor}
%A_{2k}\left(n -\frac{k\left(1+(-1)^{k}\right)}{4}\right)
%\exp\left\{\frac{\pi\sqrt{(n-\frac{1}{24})}}{\sqrt{6}k}\right\}}{2\sqrt{k}\sqrt{(n-1/24)}}+O(n^\epsilon);\\
%\alpha_\omega(n)&=\sum_{\substack{k=1
%\\(k,2)=1}}^{\lfloor
%\sqrt{n}\rfloor}\frac{(-1)^{\frac{k-1}{2}}
%A_k\left(\frac{n(k+1)}{2}-\frac{3(k^2-1)}{8}\right)
%\exp\left\{\frac{\pi\sqrt{(n+\frac{2}{3})}}{\sqrt{3}k}\right\}}{2^2\sqrt{k}\sqrt{(n+2/3)}}+O(n^\epsilon).
%\end{align*}}

\begin{itemize}
\item<2-> $A_k(n)$ is the $p(n)$ ``Kloosterman-type'' sum.
\item<2-> $I_{1/2}(z)$ satisfies {\scriptsize\[
I_{1/2}(z)=\left(\frac{2}{\pi z}\right)^\frac{1}{2}\sinh(z).\]}
\end{itemize}
}

\subsection{Recent work}
 \frame{ \frametitle{Recent work}
 \onslide<1->  Zwegers (Contemp. Math., 2003) :
\begin{itemize}
\item Vector-valued modular forms
\end{itemize}

\onslide<2-> Bringmann and Ono (Invent. Math., 2006) :
\begin{itemize}
\item <2->
Weak Maass forms\\
\item<3->Andrews-Dragonette conjecture
\end{itemize}

}




\subsection{Main Theorem}{}
%\begin{beamercolorbox}{yellow}
\frame{ \frametitle{Main Theorem} \onslide<1->
\begin{theorem}[G.]
{ The coefficients $\alpha_\omega(n$) of $\omega(q)$ are
{\footnotesize \begin{equation*}
 \frac{\pi(3n+2)^{-1/4}}{2\sqrt{2}}\sum_{\substack{k=1
\\(k,2)=1}}^\infty\frac{(-1)^{\frac{k-1}{2}}A_k\left(\frac{n(k+1)}{2}-\frac{3(k^2-1)}{8}\right)}{k}I_{1/2}\left(\frac{\pi\sqrt{3n+2}}{3k}\right).
\end{equation*}}}
\end{theorem}
%\end{beamercolorbox}


\onslide<2->
 Define $c(n,m)$ by formula for $\alpha_\omega(n)$ truncated at $k = 2m-1$.


\vspace{.1 in} {\footnotesize
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline $n$ & $\alpha_\omega(n)$ & $c(n,1)$ & $c(n,2)$   & $c(n,1000)$\\
\hline  1& 2 & 1.9949  & 2.2428 & 1.9963
\\
5 & 8 & 7.8769 & 8.0420  & 7.9958\\
10 & 29 & 28.6164  & 29.0178  & 29.0000\\
100 & 1995002 & 1994993.7262  & 1995001.6972  &
 1995001.9987\\
\hline
\end{tabular}

\end{centering}}


}

\section[Proof of Main Theorem]{Proof of Main Theorem}
\subsection{Notation \& Background}{}

 \frame {
  \frametitle{Real analytic vector-valued modular forms}

  Define the following:

\[ F(z):=\left(q^{-\frac{1}{24}}f(q), \
2q^{\frac{1}{3}}\omega(q^{\frac{1}{2}}),\
2q^{\frac{1}{3}}\omega(-q^{\frac{1}{2}})\right)^{T}. \]
\[G(z):=2i\sqrt{3}\int_{-\overline{z}}^{i\infty} \frac{(g_1(\tau),\
g_0(\tau),\ -g_2(\tau))^{T}}{ \sqrt{-i(\tau+z)}} \ d\tau.
\]
The $g_i(\tau)$ are the cuspidal weight $3/2$ theta functions
\[
\alert<2-> {H(z) := (H_0(z), H_1(z), H_2(z)) = F(z)-G(z)}
\]
}

\frame {
  \frametitle{Real analytic vector-valued modular forms}




\begin{theorem}[Zwegers]
The function $H(z)$ is a vector-valued real analytic modular form of
weight $1/2$ satisfying
\begin{displaymath}
\begin{split}
H(z+1)&=\left(\begin{matrix}e(-1/24) & 0 & 0\\
0 & 0 & e(1/3)\\
0 &e(1/3)& 0\end{matrix}
\right) H(z),\\
\ \ \\
H(-1/z)&= \sqrt{-iz} \cdot \left( \begin{matrix}
0&1&0\\
1&0&0\\
0&0&-1\end{matrix} \right) H(z),
\end{split}
\end{displaymath}
where $e(x):=e^{2\pi i x}$.
\end{theorem}

  }



 \frame {
  \frametitle{Weak Maass forms}

\begin{theorem}[Bringmann-Ono]
\begin{itemize}
\item<1->  $H_0(24z)$ is a weak Maass form of weight $1/2$ on
$\Gamma_0(144)$ with Nebentypus  $\leg{12}{\bullet}$.
\item<2->$H_0(24z) =
P_\frac{1}{2}\left(\frac{3}{4};24z\right)$, where
\begin{equation*}
\alert{P_k(s;z):=\frac{2}{\sqrt{\pi}}
\sum_{M=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in
\Gamma_{\infty}\backslash \Gamma_0(2)}
\chi(M)^{-1}(cz+d)^{-k}\varphi_{s,k}(Mz)}.
\end{equation*}
Here
\[
\varphi_{s,k}(Mz) = |y|^{-\frac{k}{2}}
M_{\frac{k}{2}\sgn(y),\,s-\frac{1}{2}}(|y|)\left(-\frac{\pi
y}{6}\right)e\left(-\frac{x}{24}\right).
\]
%{\footnotesize \[\chi_0(M) :=
%\begin{cases}
%i^{-1/2}(-1)^{\frac{1}{2}(c+ad+1)}e\left(\frac{3dc}{8}-\frac{(a+d)}{24c}-\frac{a}{4}
%\right)
%\omega_{-d,c}^{-1} &\text{if $c>0$, $2|c$,}\\
%e\left(\frac{-b}{24}\right) &\text{if $c=0$.}
%\end{cases}\]}
\end{itemize}
\end{theorem}
}

\subsection{Sketch of Proof}{}
 \frame {
  \frametitle{Outline of Proof}

To prove the Main Theorem:
\begin{itemize}
\item<2-> Construct a real analytic weight 1/2 vector-valued modular
form reflecting transformations of $P_{\frac{1}{2}}(\frac{3}{4},z)$
on $\SL_2(\Z)$
\item<3-> Express the Fourier expansions of the component
functions
\item<4-> Use Bringmann-Ono and the constructed vector-valued modular
form to establish the coefficients of $\omega(q)$.
\end{itemize}
}


 \frame {
   \frametitle{Constructing the modular form}

\begin{definition}
If $M = \left(\begin{matrix}a&b\\c&d\end{matrix}\right) \in
\SL_2(\Z)$, define, {\footnotesize \begin{align*} &\chi_0(M) :=
\begin{cases}
i^{-1/2}(-1)^{\frac{1}{2}(c+ad+1)}e\left(\frac{3dc}{8}-\frac{(a+d)}{24c}-\frac{a}{4}
\right)
\omega_{-d,c}^{-1} &\text{ if $c>0$, $c$ even,}\\
e\left(\frac{-b}{24}\right) &\text{ if $c=0$;}
\end{cases}\\
&\chi_1(M) :=
i^{-1/2}(-1)^\frac{c-1}{2}e\left(\frac{3dc}{8}-\frac{(a+d)}{24c}\right)\omega_{-d,c}^{-1}
\qquad \quad \quad \text{if $c>0$, $d$ even,}\\
 &\chi_2(M) :=
%\begin{cases}
i^{-1/2}(-1)^\frac{c-1}{2}e\left(\frac{3dc}{8}-\frac{(a+d)}{24c}\right)\omega_{-d,c}^{-1}
\qquad \quad \quad \text{if $c>0$, $c,d$ odd,}
\end{align*}}
\end{definition}
}

\frame{ \frametitle{Constructing the modular form}

\begin{definition}
%\label{P} Define
\begin{align*}
\mathcal{P}(z)&: = \left(P_0(z), P_1(z),P_2(z)\right)^T,
\end{align*}
where, {\footnotesize \begin{align*}
 P_0(z)&:=\frac{2}{\sqrt{\pi}}&&\sum_{M=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right) \in
\Gamma_\infty\backslash\Gamma_0(2)}\chi_0(M)^{-1}(cz+d)^{-1/2}\varphi_{3/4,1/2}(Mz);\\
P_1(z)&:=
\frac{2}{\sqrt{\pi}}&&\sum_{\substack{M=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)=M'S\\M'
\in
\Gamma_\infty\backslash\Gamma_0(2)}}\chi_1(M)^{-1}(cz+d)^{-1/2}\varphi_{3/4,1/2}(Mz)
;\\
P_2(z)&:=
\frac{2}{\sqrt{\pi}}&&\sum_{\substack{M=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)=M'ST\\M'
\in
\Gamma_\infty\backslash\Gamma_0(2)}}\chi_2(M)^{-1}(cz+d)^{-1/2}\varphi_{3/4,1/2}(Mz).
\end{align*}}
\end{definition}


}

 \frame {
  \frametitle{Connection to $H(z)$}

 \begin{theorem}[G.]
\label{Poincare} The function $\mathcal{P}(z)$ is a vector-valued
real analytic modular form of weight $1/2$ satisfying
\begin{displaymath}
\begin{split}
\mathcal{P}(z+1)&=\left(\begin{matrix}e(-1/24) & 0 & 0\\
0 & 0 & e(1/3)\\
0 &e(1/3)& 0\end{matrix}
\right) \mathcal{P}(z),\\
\ \ \\
\mathcal{P}(-1/z)&= \sqrt{-iz} \cdot \left( \begin{matrix}
0&1&0\\
1&0&0\\
0&0&-1\end{matrix} \right) \mathcal{P}(z).
\end{split}
\end{displaymath}
\end{theorem}
}

 \frame {
  \frametitle{The coefficients of $\omega(q)$}

  \begin{itemize}
  \item<1->\footnotesize{ $
 \alert<3->{H_1(24z) =} (-i24z)^{-1/2}H_0\left(\frac{-1}{24z}\right) =
% (-i24z)^{-1/2}P_\frac{1}{2}\left(\frac{3}{4};
% \frac{-1}{24z}\right) =
(-i24z)^{-1/2}P_0\left(\frac{-1}{24z}\right)=\alert<3->{ P_1(24z)}.
 $}
 \item<2->
\footnotesize{$P_1(z) = \sum_{n \geq 0}
\alpha(n)q^{\frac{n}{2}+\frac{1}{3}}+\sum_{n <
0}\beta_y(n)q^{\frac{n}{2}+\frac{1}{3}}$},\\
 where, \footnotesize{\begin{align*}  \alert<3->{\alpha(n)=}&\alert<3->{
\frac{\pi}{\sqrt{2}}
(3n+2)^{-\frac{1}{4}}\sum_{\substack{k=1\\(k,2)=1}}^{\infty}
\frac{A_{k}\left(\frac{n(k+1)}{2}-\frac{3(k^2-1)}{8}\right) }{k}
\cdot I_{\frac{1}{2}}\left(\frac{\pi \sqrt{3n+2}}{3k}\right)},\\
 \beta_y(n)=& \frac{\pi^{\frac{1}{2}}}{\sqrt{2}} |3n+2|^{-\frac{1}{4}}\cdot \Gamma\left(\frac{1}{2}, \frac{\pi |3n+2|
\cdot y}{3}\right) \\
&\sum_{\substack{k=1\\(k,2)=1}}^{ \infty} \frac{A_{k}
\left(\frac{n(k+1)}{2}-\frac{3(k^2-1)}{8} \right)}{k}\cdot
 J_{\frac{1}{2}} \left(\frac{\pi
\sqrt{|3n+2|}}{3k} \right).
\end{align*}}
  \end{itemize}
}

\section{Maass-Poincar\'e series of all weights }

\frame{ \frametitle{Maass-Poincar\'e series of all weights }



Define {\footnotesize
\begin{equation*}
\alert{P(N,\chi,m,k,s;z) :=
\sum_{M=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)
\in
\Gamma_\infty\backslash\Gamma_0(N)}\chi(M)^{-1}(cz+d)^{-k}\varphi_{s,k,m}(Mz)}.
\end{equation*}}
\begin{itemize}
\item $k \in \frac{1}{2}\Z$, $N \in \mathbb{N}$, $ 0>m \in \Q$, $s
\in \C$, and $\chi$ is a multiplier system for $\Gamma_0(N)$.\\
\item $\varphi_{s,k,m}(z):=|y|^{-\frac{k}{2}}
M_{\frac{k}{2}\sgn(y),\,s-\frac{1}{2}}(|y|)\left(4m\pi
y\right)e\left(mx\right).$
\end{itemize}


}

\frame{ \frametitle{Properties of $P(N,\chi,m,k,s;z)$}
\begin{itemize}
\item<1->Absolutely convergent for $\Re(s)>1$.
\item<2->If $\Re(s)>1$ and $V = \left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right)\in\Gamma_0(N)$ then  \[P(N,\chi,m,k,s;Vz) =
\chi(V)(\gamma z+\delta)^kP(N,\chi,m,k,s;z).\]
\item<3->If $k < 0$ and $s = 1-k/2$ or $k > 2$
and $s =k/2$, then $P(N,\chi,m,k,s;z)$ is a weak Maass form of
weight $k$ on $\Gamma_0(N)$ with Nebentypus if $\chi$ is of the
correct form.
\item<4->We can express the Fourier expansion for
$P(N,\chi,m,k,s;Vz)$, $V\in \SL_2(\Z)$.
\end{itemize}
}

\section{Summary}
\frame{\frametitle{Summary}
\begin{itemize}
\item<1-> Mock theta functions are the holomorphic projection of
weight $1/2$ weak Maass forms
\item<2-> These Maass forms are weight $1/2$ vector-valued modular
forms
\item<3-> For $f(q)$ we can construct a Maass-Poincar\'e series
whose Fourier expansion yields $\alpha_f(n)$.
\item<4-> We can use the transformation properties of the Maass
form and Maass-Poincar\'e series to find $\alpha_\omega(n)$.
\item<5-> We can do this construction and express the Fourier
coefficients for the general $P(N,\chi,m,k,s;Vz)$, $V \in
\SL_2(\Z)$.
\end{itemize} }
\end{document}











\end{document}
