\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\\ #3 \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}