\mychapter{Introduction}

\setcounter{page}{1}

In this chapter, we will describe modular forms and related objects, concluding with a statement of this thesis' main results. 

\section{Classical Modular Forms}
We begin by defining \emph{modular forms}. These are complex-analytic functions with strong transformation properties. More precisely, from now on we let $\mathbb{H}$ denote the \emph{upper half-plane}
\[\mathbb{H}:=\{z\in\C\colon\Im z>0\}.\]
We will also write throughout $z=x+iy$. The upper half-plane comes endowed with a natural hyperbolic metric, $d\mu:=\frac{dx\, dy}{y^2}$.
\\
\indent 
The key group that describes the symmetries of modular forms is $\operatorname{SL}_2(\Z)$, the group of $2\times2$ matrices with integer entries and determinant $1$. We have that $\operatorname{SL}_2(\Z)$ acts on $\H$ by M\"obius transformations
\[\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\cdot z:=\frac{az+b}{cz+d}.\]
That this is indeed a group action follows from a routine calculation which shows that 
\[\Im(\gamma\cdot z)=\frac{\Im z}{|cz+d|^2}\]
for $\gamma=\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\Gamma$.
We can now define modular forms.
\begin{definition}
For any $k\in\Z$ we say that a  holomorphic function \\$f\colon\mathbb{H}\rightarrow\C$ is a weight $k$ modular form on $\Gamma=\operatorname{SL}_2(\Z)$ if
\begin{enumerate}
\item $f(\frac{az+b}{cz+d})=(cz+d)^kf(z)$ for all $\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\operatorname{SL}_2(\Z),$
\item $\displaystyle\lim_{z\rightarrow i\infty}f(z)<\infty$.
\end{enumerate}
\end{definition}
The second condition, which states that $f$ is holomorphic at the ``cusp'' infinity will later be weakened in more general spaces of ``modular'' objects, and we will also consider ``higher level'' examples later where the first condition is only required to hold for a subgroup of $\operatorname{SL}_2(\Z)$. By considering the action of $\bigl(\begin{smallmatrix}-1&0\\0&-1\end{smallmatrix} \bigr)$ in the first condition, we see that a modular form of odd weight is automatically zero. 
\\
\indent
A very useful classical fact is that $\Gamma$ is generated by just two matrices, namely 
\[T:=\bigl(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix} \bigr),\ \ \ \ \ \ \ S:=\bigl(\begin{smallmatrix}-1&0\\0&1\end{smallmatrix} \bigr).\]
Thus, the modular transformations are equivalent to the two transformations $f(z+1)=f(z)$ and $f(-1/z)=z^kf(z)$. Throughout, we will denote the $\C$-vector space of modular forms of weight $k$ by $M_k$. In particular, modular forms are periodic, holomorphic functions on $\mathbb{H}$, which implies that they have Fourier expansions. Throughout, we denote $q:=e^{2\pi i z}$. Thus, for any modular form $f$ there are complex numbers $a_n$ such that 
\[f(z)=\sum_{n\geq0} a_nq^n.\]
The fact that there are no negative powers of $q$ in this sum comes from the growth condition (2) in the above definition of modular forms. Such a Fourier expansion will often be referred to as a \emph{$q$-series}. Much of the classical interest in modular forms come from studying their Fourier expansions, providing an analytic tool for studying sequences which often encode number theoretic, combinatorial, or geometric data.
\\
\indent

It is not immediately clear from the definition that the spaces $M_k$ are ever non-zero, but we can show this with our first canonical family of modular forms known as \emph{Eisenstein series}. For $k>2$, we define the series
\[G_{k}:=\displaystyle\sum_{(0,0)\neq(m,n)\in\Z^2}\frac{1}{(m+nz)^k}\]
For $k>2$, this series is absolutely convergent, and it is a simple exercise to show that $G_k$ satisfies the modularity transformations by checking the transformations under $S$ and $T$. Holomorphicity at $\infty$ is also clear from the definition. As noted above, the modularity transformations for $k$ odd imply that $G_{2k+1}\equiv0$. 
\\
\indent
For $k$ even, $G_k$ is non-zero and we can write down its Fourier expansion explicitly. When given a $q$-series, it is customary to ``normalize'' so that the leading coefficient is $1$, and in this case it will even turn out that the Fourier coefficients become rational. Namely, we let $E_k:=\frac{G_k}{2\zeta(k)}$ where $\zeta(s)$ is the Riemann-zeta function. Then a standard trick shows that 
\[E_k=1+\frac{-2k}{B_k}\displaystyle\sum_{n\geq1}\sigma_{k-1}(n)q^n,\]
where $B_k$ is the $k^{\text{th}}$ Bernoulli number, and $\sigma_k(n):=\sum_{d|n}d^k$ is the $k^{\text{th}}$ divisor sum. Thus, for example, we have the following:
\[E_4=1+240\sum_{n\geq1}\sigma_3(n)q^n,\]
\[E_6=1-504\sum_{n\geq1}\sigma_5(n)q^n,\]
\[E_8=1+480\sum_{n\geq1}\sigma_7(n)q^n.\]

One of the salient features of modular forms is that $M_k$ forms a finite-dimensional vector space. Thus, proving equalities (and it turns out, congruences) between modular forms only requires checking finitely many coefficients. In order to describe this result, we require the important notion of a \emph{fundamental domain}. This is a subset $\mathcal{F}$ of the upper half-plane such that every point in $\H$ is $\operatorname{SL}_2(\mathbb{Z})$-equivalent to exactly one point of $\mathcal{F}$. Thus, by the modularity transformations, any modular form is uniquely determined by its values on a fundamental domain. The standard choice for $\mathcal{F}$ in this case is given by
\[\mathcal{F}=\{z\in\H\ \colon |z|\geq1,\ -\frac12\leq \Re z<\frac12, \  \Re z\leq0\text{ for }|z|=1\}.\]
The fundamental result we now state is known as the \emph{valence formula}.
\begin{proposition}
Let $0\neq f(z)\in M_k(\Gamma)$. Let $\nu_{\infty}$ be the exponent of the first non-zero term in the Fourier expansion of $f(z)$, and let $\nu_z(f)$ for a point $z\in\H$ be the order of vanishing of $f$ at $z$. Then we have that
\[\nu_{\infty}(f)+\frac12\nu_i(f)+\frac13\nu_{\omega}(f)+\sum_{z\in\mathcal{F},\ z\neq i,\omega}\nu_{z}(f)=\frac{k}{12},\]
where $\omega=e^{\frac{2\pi i}{3}}$.
\end{proposition}
This formula easily yields that $M_k$ is finite dimensional as a $\C$-vector space, and moreover gives us explicit dimension formulas.

In particular, we find that if $k\in\{4,6,8,10,14\}$ then $M_k$ is one-dimensional, and hence spanned by the Eisenstein series $E_k$. As an amusing consequence of finite dimensionality, we prove an otherwise highly non-obvious combinatorial divisor sum formula. Namely, it is clear from the definition of a modular form that if $f\in M_{k_1}$ and $g\in M_{k_2}$, then $f\cdot g\in M_{k_1+k_2}$. Thus, we have that $E_4^2\in M_8$. But the dimension of $M_8$ is $1$ and $E_8\in M_8$ as well. Thus, $E_4^2$ and $E_8$ are scalar multiples of one another. But we normalized them so their Fourier expansions started with $1$, so in fact $E_4^2=E_8$. Comparing Fourier expansions and rearranging yields the following curious combinatorial identity:
\[\sigma_7(n)=\sigma_3(n)+120\sum_{i=1}^{n-1}\sigma_3(i)\sigma_3(n-i).\]
We would also like to remark that although there are no modular forms of weight $2$ and the series defining $G_2$ is not absolutely convergent, it is still convergent and gives rise to a function $E_2$ which has a Fourier expansion in terms of the divisor function $\sigma_{k-1}(n)$ as in the weight $k>2$ case, namely 
\[E_2=1-24\sum_{n\geq1}\sigma_1(n)q^n.\]
Thus, $E_2$ is still periodic, but it has a slightly more complicated transformation under $S$, namely
\[z^{-2}E_2(-1/z)=E_2(z)+\frac{12}{2\pi i z}.\]
We call $E_2$ ``quasimodular'', and we can ``fix'' the modularity by setting $E_2^*(z):=E_2(z)-\frac{3}{\pi y}$. Although it is nonholomorphic, $E_2^*(z)$ satisfies the modularity properties in condition (1) of the definition of a modular form. We will also see that one can build ``higher level" weight $2$ modular forms out of $E_2$.
\\
\indent 
Besides the space of Eisenstein series, there is another distinguished set of modular forms which we will often need.
\begin{definition}
We say that a modular form $f(z)=\sum_{n\geq0}a_nq^n$ is a \emph{cusp form} if $a_0=0$. 
\end{definition}
The term ``cusp form'' will become apparent when we discuss cusps and higher level forms below. We denote the space of cusp forms of weight $k$ by $S_k(\Gamma)=S_k$. 
We can also compute the dimensions of the space of cusp forms.
Namely, we have that $S_k=0$ if $k<12$ or $k=14$, as the dimension of $M_k$ is $1$ and we have the Eisenstein series, which are not cusp forms. For $k=12$, there is a distinguished cusp form called the discriminant modular form or delta function.
\begin{proposition}
Let $\Delta(z):=q\prod_{n\geq1}(1-q^n)^{24}$. Then $\Delta(z)$ is a weight $12$ cusp form.
\end{proposition}

It is customary to denote the Fourier coefficients of $\Delta(z)$ by $\tau(n)$; i.e. $\Delta(z):=\sum_{n\geq1}\tau(n)q^n$. These coefficients, which were studied by Ramanujan, have very special properties, as we will see in our discussion of Hecke operators below. 
\\
\indent 
One of the key properties of $\Delta(z)$ is that it is non-vanishing on the upper-half plane, which is apparent from its product definition. Thus, the map $f\colon M_k\rightarrow S_{k+12}$ defined by $f\mapsto f\cdot\Delta$ is an isomorphism. In many general situations, we may decompose spacese of modular forms into Eisenstein series and cusp forms. The philosophy is that any modular form has an ``Eisenstein part'' which is easily describable, with a much smaller contribution to the coefficients coming from the ``cusp part''. The coefficients of cusp forms are much more mysterious. For example, a famous unsolved conjecture of Lehmer states that that $\tau(n)\neq0$ for all $n$.
\\
\indent 
Ramanujan's study of the $\tau$ function leads us to an important family of operators on modular forms, the \emph{Hecke operators}. They can be used to easily prove a conjecture of Ramanujan that the $\tau$ function is multiplicative, i.e. $\tau(mn)=\tau(m)\tau(n)$ for $(m,n)=1$. For every $m>1$, we define the $m^{\text{th}}$ Hecke operator $T_m$ applied to a weight $k$ modular form $f$ by
\[T_mf:=m^{k-1}\displaystyle\sum_{\gamma=\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix} \bigr)\in\operatorname{SL}_2(\Z)\backslash \mathcal{M}_m}(cz+d)^{-k}f(\gamma\cdot z),\]
where $\mathcal{M}_m$ is the group of $2\times2$ integer matrices of determinant $m$. The constant $m^{k-1}$ is chosen so that Hecke operators preserve integrality of coefficients. The quotient is under the action of $\Gamma$ on $M_m$ by left multiplication. For convenience, we define the \emph{Petersson slash operator} $|_k\gamma$ by $f|_k\gamma:=(\det\gamma)^{k/2}(cz+d)^{-k}f(\gamma\cdot z)$, so that $T_k f=m^{k-1}\sum_{\operatorname{SL}_2(\Z)\backslash \mathcal{M}_m}f|_k\gamma$. It is elementary to show that $T_m\colon M_k\rightarrow M_k$.  A very useful fact about the Hecke operators is that their action on the Fourier coefficients of $f$ is easily described. 
\begin{lemma}
Let $f(z)=\sum_{n\geq0}a_nq^n\in M_k$. Then the Fourier expansion of $T_k f$ is given by 
\begin{equation}\label{HeckeExp}T_m f(z)=\displaystyle\sum_{n\geq0}\left(\sum_{d|\operatorname{gcd}(m,n)}d^{k-1}a_{mn/d^2}\right)q^n.\end{equation}
\end{lemma}
It also turns out that for $(m,n)=1$, the Hecke operators commute: \begin{equation}\label{HeckeCommute}T_mT_n=T_nT_m=T_{mn}.\end{equation} We define a Hecke eigenform to be a modular form which is an eigenfunction $T_mf=\lambda_mf$ for all $m>1$. Although this may seem at first like a strong condition, there are many natural examples of Hecke eigenforms. It can be shown that every Eisenstein series $E_k$ is a Hecke eigenform, and we will show momentarily that $\Delta(z)$ is one. The key arithmetic property of a Hecke eigenform is that its coefficients are multiplicative (up to a constant). This is because for a Hecke eigenform $f$ whose Fourier coefficient of $q^1$ is $1$, (1.1) implies that the the $m^{\text{th}}$ Fourier coefficient of $f$ is equal to the eigenvalue of $f$ by $T_m$ and by (\ref{HeckeCommute}).
\\
\indent
We are now in a position to prove Ramanujan's conjecture.
\begin{corollary} The $\tau$ function is multiplicative. That is, if $(m,n)=1$, we have $\tau(mn)=\tau(m)\tau(n)$. 
\end{corollary}
\begin{proof}
By the discussion above, it suffices to show that $\Delta$ is a Hecke eigenform. By (\ref{HeckeExp}), it follows that $T_m$ takes cusp forms to cusp forms. As the dimension of $S_{12}$ is $1$, $T_m\Delta$ is a multiple of $\Delta$, as desired.
\end{proof}
We conclude this section with a discussion of several generalizations of the definition of modular forms above. Firstly, we often do not require the full strength of modular transformations on all of $\operatorname{SL}_2(\Z)$. Namely, we may only require that $f$ has modular transformations for matrices $\gamma$ in a \emph{congruence subgroup} of $\operatorname{SL}_2(\Z)$. The groups of primary interest for us are the following congruence subgroups of level $N$, for $0<N\in\Z$:
\[\Gamma(N):=\left\{\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\operatorname{SL}_2(\Z)\mid a\equiv d\equiv1,\, b\equiv c\equiv0\pmod N\right\},\]
\[\Gamma_1(N):=\left\{\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\operatorname{SL}_2(\Z)\mid a\equiv d\equiv1,\, c\equiv0\pmod N\right\},\]
\[\Gamma_0(N):=\left\{\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\operatorname{SL}_2(\Z)\mid c\equiv0\pmod N\right\}.\]
In order to describe modularity on congruence subgroups, we first need to define \emph{cusps}. We have $\mathbb{P}_1(\Q)=\Q\cup\{\infty\}$. Then $\operatorname{SL}_2(\Z)$ clearly acts on $\mathbb{P}_1(\Q)$ by M\"obius transformations, and for any congruence subgroup $\Gamma$ we call the set of $\Gamma$-orbits of $\mathbb{P}_1(\Q)$ the set of cusps of $\Gamma$. For $\operatorname{SL}_2(\Z)$, one can show there is only one cusp using the Euclidean algorithm, and we usually choose $\infty$ as a representative. The set of cusps of any congruence subgroup is always finite. We then modify the definition of modular forms given above for $\operatorname{SL}_2(\Z)$ to modular forms $f$ on an arbitrary congruence subgroup $\Gamma$ by changing condition (1) to require modular transformations to hold only for $\gamma\in\Gamma$ and by requiring that $f$ be holomorphic at all of the cusps. We say that $f$ is a cusp form on $\Gamma$ if $f$ also vanishes at all the cusps.
We denote the space of modular forms of weight $k$ on $\Gamma$ by $M_k(\Gamma)$, and the subspace of cusp forms by $S_k(\Gamma)$. It is also customary to denote $M_k(\Gamma_0(N))=M_k(N)$ and $S_k(\Gamma_0(N))=S_k(N)$. We note that as $T\in\Gamma_0(N)$ and $T\in\Gamma_1(N)$, modular forms on these congruence subgroups are periodic and hence still have Fourier expansions as above. 
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\indent
We will also frequently require the theory of half-integral weight modular forms as developed by Shimura. For the remainder of this section, we assume that $k\in\frac12+\Z$. Due to the non-uniqueness of square roots, we need to be more careful when defining modular forms of half-integral weight. We first define 
\[\epsilon_d=\begin{cases}1&\text{ if }d\equiv1\pmod4\\ i&\text{ if }d\equiv3\pmod4\end{cases},\]
and we recall the notation for the \emph{Kronecker symbol} $\left(\frac{c}{d}\right)$. We also choose a branch of the square root having argument in $(-\pi/2,\pi/2]$. 
\begin{definition}
Let $N$ be a positive integer and $k\in\frac12+\Z$. Then we say a holomorphic function $f$ on $\mathbb{H}$ is a \emph{holomorphic half-integral weight modular form} of weight $k$ if
\begin{enumerate}
\item $f\left(\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\cdot z\right)=\left(\frac cd\right)^{2k}\epsilon_d^{-2k}f(z)$ for all $\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\Gamma_0(4N)$,
\item $f$ is holomorphic at the cusps of $\Gamma_0(4N)$.
\end{enumerate}
\end{definition}
Important examples of half-integral weight modular forms are provided by the theory of theta functions, which inspired much early interest in modular forms. These are modular forms defined by summing over a lattice. The prototypical example is given by Jacobi's theta function
\[\theta(z):=\sum_{n\in\Z}q^{n^2},\]
which was of classical interest as the Fourier coefficients of $\theta(z)^r$ count the number of representations of integers as the sum of $r$ squares.
In general, we have a theory of weight $1/2$ \emph{unary theta functions}. For a Dirichlet character $\chi$ mod $N$ (resp. $4N$) on a congruence subgroup $\Gamma$ of level $N$, we define the space of modular forms of integer (resp. half-integral) weight $k$ with \emph{Nebentypus} $\chi$ by modifying the definition of the modular transformations as follows:
\begin{equation}
f(\gamma\cdot z)=\begin{cases}\chi(d)(cz+d)^kf(z)&\text{ if }k\in\Z\\ \chi(d)\left(\frac cd\right)^{2k}\epsilon_d^{-2k}(cz+d)^kf(z)&\text{ if }k\in\frac12+\Z.\end{cases}
\end{equation}
We denote the space of modular forms with Nebentypus $\chi$ and weight $k$ on a subgroup $\Gamma$ by $M_k(\Gamma,\chi)$, and similarly we denote the subspace of cusp forms by $S_k(\Gamma,\chi)$. For a Dirichlet character $\chi$, we may define a theta function by 
\begin{equation}\theta_{\chi}(z)=\begin{cases}\sum_{n\in\Z}\chi(n)q^{n^2}&\text{ if }\chi\text{ is even}\\ \sum_{n\in\Z}\chi(n)nq^{n^2}&\text{ if }\chi\text{ is odd}.\end{cases}\end{equation}
As with the Jacobi theta function, one can show that these theta functions are modular forms. More specifically, we have the following.
\begin{theorem}[See \cite{cbms}]
Suppose that $\chi$ is a primitive Dirichlet character of conductor $r(\chi)$. 
\begin{enumerate}
\item If $\chi$ is even, then $\theta_{\chi}(z)\in M_{\frac12}(\Gamma_0(4\cdot r(\chi)^2),\chi).$
\item If $\chi$ is odd, then $\theta_{\chi}(z)\in S_{\frac 32}(\Gamma_0(4\cdot r(\chi)^2,\chi\chi_{-4})$, where $\chi_{-4}$ is the nontrivial Dirichlet character modulo $4$. 
\end{enumerate}
\end{theorem}

Theta functions are particularly useful in the case of weight $1/2$ modular forms, thanks to the \emph{Serre-Stark basis theorem} which states that any weight $1/2$ modular form is a linear combination of weight $1/2$ theta functions.
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\indent
Another important modular form is the \emph{Dedekind eta-function}, defined by the infinite product
\begin{equation}\eta(z):=q^{\frac{1}{24}}\prod_{n\geq1}(1-q^n)\end{equation}
The eta function serves as a basic building block in the theory of classical modular forms, and it has the following modular transformation properties
\begin{equation}\eta(z+1)=e^{\frac{\pi i}{12}}\eta(z),\end{equation}
\begin{equation}\eta(-1/z)=(-iz)^{\frac12}\eta(z).\end{equation}
In particular, this implies that $\eta(24z)\in S_{\frac12}(\Gamma_0(576),\chi_{12})$. 
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\indent
Our final generalization of the spaces of classical modular forms comes from weakening the condition at the cusps. Namely, we say that $f$ is a \emph{weakly holomorphic modular form} of weight $k$ for a congruence subgroup $\Gamma$ if the same conditions as above hold, except that we only require $f$ to be \emph{meromorphic} at the cusps instead of holomorphic. We denote the space of such forms by $M_k^!(\Gamma)$. The first canonical example of a weakly holomorphic modular form is the $j$-invariant, given by 
\begin{equation}j(z):=\frac{E_4^3}{\Delta}=q^{-1}+744+196884q+\ldots \end{equation}
Thus, $j(z)\in M_0^!(\operatorname{SL}_2(\Z))$. The $j$-function plays a beautiful role in class field theory, as its values at quadratic irrationalities describe the abelian extensions of imaginary quadratic fields, and it serves as a parameterization of elliptic curves over $\C$. 
\\
\indent
 One of the combinatorial applications of the Dedekind eta-function stems from the fact that its reciprocal, which is essentially a modular form of weight $-\frac12$, is closely related to the \emph{partition function}. Recall that a partition of a positive integer $n$ is a non-decreasing sequence of positive integers which sum to $n$, and we define $p(n)$ to be the number of partitions of $n$. If $n=0$, by convention we write $p(0)=1$. This function has many deep arithmetic properties which were in particular studied by Ramanujan. The role of modular forms in studying this function comes from the following elementary identity due to Euler:
 \begin{equation}\sum_{n\geq0}p(n)q^n=\prod_{n\geq1}\frac{1}{1-q^n}.\end{equation}




\section{Harmonic Maass Forms}
We now turn our study to \emph{harmonic Maass forms} and more generally \emph{weak Maass forms}. These are also functions which map from $\H$ to $\C$, but they are no longer required to be holomorphic, or even meromorphic. Instead, we demand that they satisfy a certain differential equation. More specifically, we define the weight $k$ hyperbolic Laplacian by 
\begin{equation}
\Delta_k:=-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+iky\left(\frac{\partial}{\partial y}+i\frac{\partial}{\partial y}\right).
\end{equation}
We note that $\Delta_k$ is an elliptic differential operator, and hence its eigenfunctions are real-analytic. Then we define a weak Maass form as an eigenfunction of $\Delta_k$ which transforms like a modular form and has a suitable growth condition.
\begin{definition}\label{MaassDefn}
Let $k\in\frac12\Z$. We say that a $\mathcal{C}^2$ function $f\colon \H\rightarrow\C$ is a \emph{weak Maass form} of weight $k$ on a congruence subgroup $\Gamma$ if the following are satisfied.
\begin{enumerate}
\item For all $\gamma\in\Gamma$, we have \[M(\gamma\cdot z)=\begin{cases}(cz+d)^kM(z)&\text{ if }k\in\Z\\ \left(\frac cd\right)^{2k}\epsilon_d^{-2k}(cz+d)^{k}M(z)&\text{ if }k\in\frac12+\Z.\end{cases}\]
\item There is a complex number $\lambda$ for which $\Delta_k f(z)=\lambda f(z).$
\item There is a finite Fourier polynomial $P_{\infty}=\sum_{n\leq0}c^+(n)q^n$ and a constant $C>0$ for which $f(z)-P_{\infty}=O(e^{-Cy})$ as $y\rightarrow\infty$. The analogous condition is required at all cusps of $\Gamma$. 
\end{enumerate}
\end{definition}
Note that the factor which appears in the transformations in half-integral weight is consistent with the transformation of classical modular forms of half-integral weight. We also remark that sometimes condition (3) is weakened to say only that $f(z)=O(e^{cy})$ as $y$ approaches any cusp. The Fourier polynomial in (3) is called the \emph{principal part} of $f(z)$ at the corresponding cusp. 
\\
\indent
In the case that the eigenvalue in (2) is zero, we call the form a (weak) harmonic Maass form. We denote the space of such forms by $H_k$. Any weakly holomorphic modular form is automatically a harmonic Maass form. Maass forms are known to have deep analytic structure, and constructing explicit examples is a very important problem. In the sense of Maass cusp forms, which we do not consider here, this is a difficult problem; see \cite{Cohen} for an explicit example of a Maass cusp form constructed from $q$-hypergeometric series. In the next subsection, we will see that beautiful examples of Ramanujan provide nice examples of harmonic Maass forms, as shown by Zwegers \cite{Zwegers1,Zwegers2}. This has led to an explosion of research with numerous applications scattered throughout combinatorics, Lie theory, moonshine, and even black holes and theoretical physics. 

\section{Mock Theta Functions}
Some of the most interesting examples of Maass forms comes to us from the \emph{mock theta functions}. The story of the mock theta functions begins with the enigmatic ``deathbed'' letter of Ramanujan to Hardy, written just months before his untimely death. After a very successful several years in England with Hardy and Littlewood, Ramanujan became very ill and returned to his native India. Incredibly, in this state, and in mathematical isolation, Ramanujan began investigating a new class of strange $q$-series. Ramanujan had a strange notion of definition for his forms and could not even show that they satisfied his own definition, even classifying his forms into categories which he never defined. If Ramanujan's notion of what he was defining was vague, and his death came before he had time to develop any substantial theory of them, Ramanujan's legacy has shown him to be a great anticipator with remarkable intuition. To describe what Ramanujan was talking about, we first describe some of his examples of ``Eulerian series''. These are $q$-series which are built out of $q$-hypergeometric terms. We define the \emph{$q$-Pochhammer symbol}
\begin{equation}
(a;q)_n:=(1-a)(1-aq)\ldots(1-aq^{n-1}), \ n\geq1
\end{equation}
and we set $(a;q)_0:=1$ by convention. Here we consider a typical mock theta function offered by Ramanujan called $f(q)$:
\begin{equation}
f(q):=\sum_{n\geq0}\frac{q^{n^2}}{(-q;q)_n^2}.
\end{equation}
Although the mock theta functions are not modular, Ramanujan noted that they have well-defined asymptotic expansions as $q$ tends to roots of unity. He also noted that near roots of unity, they ``behave'' like modular forms (or theta functions, in Ramanujan's language). Ramanujan then asked a question: must an Eulerian series which looks like a modular form at roots of unity be simply a modular form plus a function which is bounded at roots of unity? We will return to this question in $\S\ref{MainResults}$.


As it stood, little progress was made in proving Ramanujan's definition or in understanding where the mock theta functions come from or in putting them into a coherent theory, although many suspected that these functions should be important. A remarkable chain of events then led to the finding of the ``lost notebook'' by George Andrews, which had narrowly avoided destruction and been forgotten by history for decades (see \cite{RobertHistory}, for example, for an account with personal interviews of Andrews and others on this delightful tale). This notebook showed that Ramanujan had been working diligently on the mock theta functions in his last year of life, and they provided much insight and inspiration for a theory yet to come. As Freeman Dyson remarked in a talk at the centennial of Ramanujan's birth:
\\
\noindent
{\it ``The mock theta-functions give us tantalizing hints of a grand
synthesis still to be discovered....This remains a challenge for the
future.''}
In particular, Dyson called for a theory which would put the mock theta functions into a ``group-theoretic'' framework as for classical modular forms. 
\\
\indent
The future arrived in 2002, 82 years after Ramanujan's last letter to Hardy, when Zwegers proved that Ramanujan's mock theta functions can be explained and studied using the theory of harmonic Maass forms in his seminal Ph.D. thesis (see \cite{Zwegers1,Zwegers2}). Namely, Zwegers tied together clues from work of Watson and others, together with work by Appell and Lerch on certain generalized Lambert-type series, to prove the following.
\begin{theorem}[Zwegers \cite{Zwegers1,Zwegers2}]
For any of Ramanujan's mock theta functions $f$, there exist integers $\gamma$ and $\delta$ for which \[q^{\gamma}f(q^{\delta})\] is the holomorphic part of a weight $1/2$ harmonic Maass form on some $\Gamma_1(N)$.
\end{theorem}
This theorem led to a flood of applications to fields throughout number theory, combinatorics, topology, and even string theory and black holes. Thus, Watson was well-justified in proclaiming in \cite{Watson} that
\emph{ ``Ramanujan�s discovery of the mock theta functions makes it obvious that his skill and
ingenuity did not desert him at the oncoming of his untimely end. As much as any of his
earlier work, the mock theta functions are an achievement sufficient to cause his name to
be held in lasting remembrance. To his students such discoveries will be a source of delight
and wonder until the time shall come when we too shall make our journey to that Garden
of Proserpine (a.k.a. Persephone)...''}


\section{Quantum Modular Forms}
In a recent Clay lecture, Zagier defined a new type of automorphic object known as a ``quantum modular form'' \cite{ZagierClay}. These are functions which live not on $\H$ but on the cusps $\mathbb{P}_1(\Q)$. As a congruence subgroup actions on $\mathbb{P}_1(\Q)$ with only finitely many orbits, functions $\mathbb{P}_1(\Q)\rightarrow\C$ which transform like modular forms are not very interesting. However, we can obtain numerous interesting examples by relaxing the definition of our transformations slightly. Zagier then makes the following:
\begin{definition}
We say a function $f(z)$ from $\mathbb{P}_1(\Q)$ (or possibly an infinite subset of $\mathbb{P}_1(\Q)$) to $\C$ is a weight $k$ \emph{quantum modular form} if for all $\gamma=\bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \bigr)\in\operatorname{SL}_2(\Z)$ the function
\[h_{\gamma}:=f(z)-\epsilon(\gamma)(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right)\]
is ``sufficiently nice''.
\end{definition}
Here, sufficiently nice means some appropriate analyticity condition such as differentiable, $\mathcal{C}^k$, etc, and the $\epsilon(\gamma)$ are suitable complex numbers, for example those occurring in the multiplier system in the definition of half-integral weight modular forms. The definition is intentionally vague to allow flexibility for different types of quantum behavior. Zagier constructs several examples of quantum modular forms related to period integrals, Dedekind sums, and $q$-series connected to Maass cusp forms. 
\\
\indent
One particularly interesting example of a quantum modular form comes from the Kontsevich ``strange function''
\begin{equation}
F(q):=\sum_{n\geq0}(q;q)_n=1+(1-q)+(1-q)(1-q^2)+\ldots
\end{equation}
The reason this function is ``strange'' is that $F(q)$ is not convergent on any open subset of $\C$, and only makes sense when $q$ is a root of unity. Zagier then shows a beautiful $q$-series identity connecting $F(q)$ to a ``half-derivative'' of the Dedekind eta-function using a ``sum of tails'' identity \cite{Zagier-Vassiliev}. In particular, this implies that
\begin{equation}\label{StrangeHalfD}F(q)=-\frac12\sum_{n\geq1}n\chi_{12}(n)q^{\frac{n^2-1}{24}},\end{equation}
where $\chi_{12}(n)=\left(\frac{12}{n}\right)$. 
Even stranger is the fact that neither side of this identity makes sense simultaneously, as the right-hand side only makes sense for $|q|<1$. By the identity, we mean that the left-hand side at roots of unity agrees with the radial limit of the right-hand side. In fact, the equality is true in the sense of full asymptotic expansions as well (this is what Zagier refers to as a \emph{strong} quantum modular form). 
\\
\indent
Using the connection of the half-derivative to Eichler integrals, Zagier shows that if we define $\phi(z):=e^{\frac{2\pi i z}{24}}$, then $\phi$ is a weight $3/2$ quantum modular form, with the obstruction to modularity $h_{\gamma}$ being a smooth function on $\R$. 
\\
\indent
One application of quantum modular forms is in finding asymptotic expansions for $q$-series at roots of unity. For example, using (\ref{StrangeHalfD}), Zagier uses a standard Mellin transform argument to show the following identity

\begin{equation}
e^{-\frac{t}{24}}\sum_{n\geq0}(1-e^{-t})(1-e^{-2t})\ldots(1-e^{-nt})=\sum_{n\geq0}\frac{T_n}{n!}\cdot\left(\frac{t}{24}\right)^n.
\end{equation}
Here $T_n$ is the $n^{\text{th}}$ \emph{Glaisher number}, which is also the algebraic part of $L(\chi_{12},2n+2)$.



\section{Main Results}\label{MainResults}
Classically, the term ``singular modulus'' refers to a value of the modular $j$-invariant at quadratic irrational points in the upper half plane. These are well-known to be algebraic integers, and they play a beautiful role in complex multiplication and explicit class field theory for imaginary quadratic fields. In \cite{Zagier}, Zagier initiated the study of ``traces'' of singular moduli. He proved that the generating function associated to these numbers is a modular form of weight $3/2$.
\\
\indent
Recall that a \emph{weakly holomorphic modular form} of weight $k$, where $k\in2\Z$, is a holomorphic function on the upper half plane $\mathbb{H}$ such that 
\[f\left(\frac{az+b}{cz+d}\right)=(cz+d)^kf(z)\]
 for all $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2(\Z)=:\Gamma$ and all $z\in\mathbb{H},$ which is meromorphic ``at infinity''.  We let $M_k^!$ denote the space of all weakly holomorphic modular forms of weight $k$. One can also define the space of modular forms of weight $k$ for any $k\in\frac{1}{2}\Z$. Recall also that any modular form has a $q$-expansion \[f(z)=\displaystyle\sum_{n\gg-\infty}^{\infty}a(n)q^n,\] where $q:=e^{2\pi i z}$. Suppose $d\equiv 0,1\pmod 4$ and consider any fundamental discriminant $D$ with $dD<0$. Further suppose that $F\in M_0^!$. Zagier defined the \emph{twisted trace of singular moduli} by 
\begin{equation}
\label{twtrace}\operatorname{Tr}_{d,D}(F):=\displaystyle\sum_Qw_Q^{-1}\chi(Q)F(z_Q),
\end{equation} %12 little q to big Q
where the sum is indexed over a complete set of $\Gamma$-inequivalent positive-definite, integral quadratic forms $Q(x,y)=ax^2+bxy+cy^2$ with discriminant $dD=b^2-4ac$. Here 
\begin{equation}z_Q:=\frac{-b+\sqrt{dD}}{2a}\in\mathbb{H}\end{equation}
 is the associated CM point, and $w_Q=1$ unless $Q\sim a(x^2+y^2)$ or $Q\sim a(x^2+xy+y^2)$, in which case $w_Q=2$ or $3$ respectively. Here also
 \[\chi(Q):=\chi(a,b,c):=\begin{cases}\chi_D(r) \mbox{ if } (a,b,c,D)=1 \text{ and } Q \text{ represents } r, \\ \text{ where } (r,D)=1;\\ 0  \mbox{if } (a,b,c,D)>1,\end{cases}\]
where $\chi_D$ is the Kronecker symbol $\left(\frac{D}{\cdot}\right)$. It is a classical fact that $\chi$ is well-defined on $\Gamma$-classes of binary quadratic forms with fixed discriminant.
\\
\indent 
To illustrate Zagier's general theory, let $j(z)$ be the usual modular $j$-invariant and consider the Hauptmodul for $\operatorname{SL}_2(\Z)$,
 \[J(z):=j(z)-744=q^{-1}+196884q+21493760q^2+\ldots\] 
 Then for the weight $3/2$ modular form $g$ defined by 
\[g(z):=\theta_1(z)\cdot\frac{E_4(4z)}{\eta(4z)^6}:=\displaystyle\sum_{n\geq-1}B(n)q^n,\]%6 changed d to n
 Zagier proved the following theorem. 

\begin{thm*}
[Zagier \cite{Zagier}, Theorem 1] Let $d$ be any positive integer such that $d\equiv0,3\pmod4$. Then  \[\operatorname{Tr}_{-d,1}\left(J(z)\right)=-B(d).\]
\end{thm*}

Zagier also considered examples of trace generating functions associated to modular forms of negative even weight. In order to define traces of singular moduli, we need a modular \emph{function}, and the theory of Maass forms provides us with the raising operators to increase the weight. Thus, we define the ``traces of singular moduli'' of a non-positive weight modular form $f\in M_{-2k}$ to be the traces of the modular function  \begin{equation}
\partial f:=R^kf
\end{equation}
in the sense of (\ref{twtrace}). 
 Here $R^k$ is the \emph{iterated Maass raising operator} defined as the composition of $k$ raising operators of the appropriate weights.
 

\begin{remark}
In general it seems that these ``singular moduli'' for a fixed $d$ are all Galois conjugates, though this remains to be proven.
\end{remark}%7 added "for fixed d"

Zagier also showed that these traces are the coefficients of certain half-integral weight modular forms.
Here we consider the general question of integrality for symmetric functions of these singular moduli. For a modular form $F$, define the \emph{Hilbert class polynomial} as the product 
\begin{equation}
H_d(F;x):=\displaystyle\prod_Q\left(x-\frac{F(z_Q)}{w_Q}\right).
\end{equation}
 Zagier notes that although the traces are algebraic integers for a form $F$ with integral coefficients, the other terms of $H_d(F;x)$ need not be. For example, consider the weight -2 modular form 
\[F_2(z):=\frac{E_4(z)E_6(z)}{\Delta}=q^{-1}-240-141444q-85292800q^2-238758390q^3+\ldots,\]
 and define the weight $0$ non-holomorphic derivative 
\[K(z):=\partial F_2=R_{-2}F_2=\frac{E_2^*(z)E_4(z)E_6(z)+3E_4^3(z)+2E_6(z)^2}{6\Delta(z)}.\]
Then one can compute that $H_{-31}(K;x)$ does not have integral coefficients. To illustrate the general phenomenon, consider the above table for $H_d(K;x)$ (we have chosen the first few negative discriminants of class number at least 3). 
\vspace{0.3in}%8 changed several to few

\begin{table}\small\caption{``Class Polynomials'' of $K(z)$}
\begin{tabular}{c l}  % centered columns (4 columns)
\hline\hline                        %inserts double horizontal lines
$d$&$H_d(K;x)$\\ [0.7ex] % inserts table
%heading
\hline                    % inserts single horizontal line
\\
$-23$&$x^3-23\cdot141826x^2-\frac{3945271661}{23}x -7693330369871$\\
&\\
$-31$&$ x^3-31\cdot 1201149x^2-\frac{61346290410}{31}x+1143159756791823$\\
&\\
$-39$& $x^4-39\cdot 8067588x^3+\frac{8602826222178}{39}x^2-84029669803810035x$\\ &$\hspace{0.2in}+\frac{95749227855890319016073}{39^2}$\\
\hline     %inserts single line
\end{tabular}
\end{table}


\indent Note that the third symmetric function is always integral. Based on numerics, it seems that the first (trace) and third symmetric functions are the only coefficients which are generically integral.
\\
\indent
We explain this phenomenon and give a bound on all other denominators of the coefficients of the class polynomials. This bound  appears to be sharp in general.
%changed out to our below... and added word eigenform
 Recall that for a weakly holomorphic modular form $f(z)=\sum a(n)q^n$, we define the \emph{principal part} of $f$ to be the sum $\displaystyle\sum_{n\leq0}a(n)q^n$. To state our theorem, we recall that a prime $p$ is said to be \emph{ordinary} for an cusp form $f=\displaystyle\sum a_nq^n$ which is an eigenform for the Hecke operators if $p\nmid a_p$ and is otherwise said to be \emph{non-ordinary}. 
 For convenience, will say that a prime $p$ is \emph{good} for a pair $(k,n)$, with $k$ a negative even integer and $n$ a positive integer if all eigenforms ina a basis of cusp forms of weight $2-\ell$ are $p$-ordinary for $\ell=-10,-14,\ldots,k\cdot n$. The following is our main result: 
\begin{thm}[\cite{Griffin-Rolen}]\label{mainthm1}
Let $f(z)\in M_{k}^!$ be a modular form of negative, even weight with integral principal part. Denote the $n^{th}$ symmetric function in the singular moduli of negative fundamental discriminant $d$ for the modular form $\partial f$ by $\mathbf{e}_{n;f}(d)$. If $(p,d)=1$, then $\mathbf{e}_{n;f}(d)$ is $p$-integral. Further let \[B(n,k):=\begin{cases}
\frac{-nk}{4}&\mbox{ if }nk\in4\Z\\
\frac{1}{4}(-nk+2k-2) &\mbox{ otherwise}.\end{cases}\]
Then if $d\neq -3$ or $4$, $p|d$, and $p$ is good for the pair $(k,n)$, we have that 
\[d^{B(n,k)}\cdot\mathbf{e}_{n;f}(d)\text{ is $p$-integral}\]
\end{thm}
In particular, this explains the integrality pattern in the computed examples for $K(z)$. As there are no cusp forms of weight less than $12$, we also have the following:
\begin{corollary}
For any $f(z)\in M_{-2}^!$ with integral principal part, we have that \[\mathbf{e}_{3;f}(d)\in\Z.\]
\end{corollary}
\begin{remarks}\hspace{2 in}
\\ \noindent (1) when $d=-3$ or  $-4$, we have that $3\mathbf{e}_{n;f}(d)$ or $2\mathbf{e}_{n;f}(d)$ must be integral. See the discussion at the end of Section \ref{IntegralityDukeJenkins}.
\\ \noindent (2). Although the theorem is only stated for $D=1$ and $d$ negative, it is clear from the proof that an analogous result for arbitrary ``twisted class polynomials'' holds in general.
\\ \noindent (3). By using the integrality results for $E_2^*(z)/\Delta(z)^{\frac{1}{6}}$, $E_4(z)/\Delta(z)^{\frac{1}{3}}$, and $E_6(z)/\sqrt{\Delta(z)}$ in \cite{Zagier}, one can prove Theorem \ref{mainthm1} directly in the case when $nk\in4\Z$.
\\ \noindent (4). Although the definition of good for the pair $(k,n)$ above includes all even weights in the given range, Theorem \ref{vanishing} we will show that certain weights automatically do not arise in the spectral decomposition and the proof of the theorem shows that these weights do not need to be checked.
\end{remarks}


These results will be proven in Chapter 3 following a review of the basic theory of harmonic Maass forms.
\\
\indent

Our second main result is the construction of a new ``strange'' vector-valued quantum modular form.
We revisit Zagier's construction using work of Andrews, Jim{\'e}nez-Urroz, and Ono on more general sums of tails formulae \cite{A-J-O} (see also \cite{Andrews-Advances}). We construct a natural $3$-dimensional vector-valued quantum modular form associated to tails of infinite products. Moreover, the components are analogous ``strange'' functions; they do not converge on any open subset of $\C$ but make sense for an infinite subset of $\Q$. We define:
\begin{equation}H(q)=\left(\begin{smallmatrix}\theta_1\\ \theta_2 \\ \theta_3\end{smallmatrix} \right):=\left(\begin{smallmatrix}\eta(z)^2/\eta(2z)\\ \eta(z)^2/\eta(z/2)\\ \eta(z)^2/\eta(\frac{z}{2}+\frac12) \end{smallmatrix} \right). \end{equation}
We also note that $\theta_3=\zeta_{48}^{-1}\cdot\frac{\eta(z/2)\eta(2z)}{\eta(z)}$ by the following identity which is easily seen by expanding the product definition of $\eta(z)$:
\begin{equation}\label{eta_identity1}
\eta(z+1/2)=\zeta_{48}\cdot\frac{\eta(2z)^3}{\eta(z)\cdot\eta(4z)},
\end{equation}
where $\zeta_k:=e^{2\pi i/k}$. From this it follows that if we let 
\begin{equation*}
\quad\quad F_9(z):=\eta(z)^2/\eta(2z),\quad\quad\quad\quad\quad\quad\quad\quad
F_{10}(z):=\eta(16z)^2/\eta(8z)
\end{equation*}
then 
\begin{equation}\label{F9F10}
H(q)=\bigl(\begin{smallmatrix}F_9(q)&F_{10}(q^{1/16})
&\zeta_{12}^{-1}F_{10}(\zeta_{16}\cdot q^{1/16})\end{smallmatrix} \bigr)^T\end{equation}
(the notations $F_9$ and $F_{10}$ come from \cite{A-J-O}).
For convenience, we recall the classical theta-series identities for $F_9$ and $F_{10}$:
\begin{equation}
F_9(q)= 1+2\displaystyle\sum_{n=1}^{\infty}(-1)^nq^{n^2},     \quad\quad\quad\quad\quad\quad\quad\quad F_{10}(q)=\displaystyle\sum_{n=0}^{\infty}q^{(2n+1)^2}
.\end{equation}
It is simple to check that $H(z)$ is a $3$-dimensional vector-valued modular form using basic properties of $\eta(z)$, as we describe in $\S\ref{ProofMainThm2}$. To each component $\theta_i$ we associate for all $n\geq0$ a finite product $\theta_{i,n}$:
\begin{equation}
\theta_{1,n}:=\frac{(q;q)_n}{(-q;q)_n},\,\,\, \theta_{2,n}:=q^{\frac{1}{16}}\cdot\frac{(q;q)_n}{(q^{\frac12};q)_{n+1}},\,\,\,\theta_{3,n}:=\frac{\zeta_{16}}{\zeta_{12}}\cdot q^{\frac{1}{16}}\cdot\frac{(q;q)_n}{(-q^{\frac12};q)_{n+1}},
\end{equation}
such that $\theta_{i,n}\rightarrow\theta_i$ as $n\rightarrow\infty$.
Next, we construct corresponding ``strange'' functions
$
\theta_i^S:=\sum_{n=0}^{\infty}\theta_{i,n}.
$
Note that these functions do not make sense on any open subset of $\C$, but that each $\theta_i^S$ is defined for an infinite set of roots of unity and, in particular, $\theta_2^S$ is defined for all roots of unity.
Our primary object of study will then be the vector of ``strange'' series $H_Q(z):=\bigl(\begin{smallmatrix}\theta_1^S(z)& \theta_2^S(z)& \theta_3^S(z)\end{smallmatrix} \bigr)^T.$ In order to obtain a quantum modular form, we first define $\phi_i(x):=\theta_i^S(e^{2 \pi i x})$ from a subset of $\Q$ to $\C$, and let $\phi(x):=\bigl(\begin{smallmatrix}\phi_1(x)& \phi_2(x)& \phi_3(x)\end{smallmatrix} \bigr)^T.$ We will then show the following, which is joint work with Robert Schneider.
\begin{theorem}[\cite{Rolen-Schneider}]\label{mainthm2} Assume the notation above. Then the following are true:
\begin{enumerate} %%%This statement needs work... also there is a funny parenthesis in G(q)%%%
\item There exist $q$-series $G_i$ (see $\S\ref{ProofMainThm2}$) which are well-defined for $|q|<1$, such that $\theta_i^S(q^{-1})=G_i(q)$ for any root of unity for which $\theta_i^S$ converges.
\item We have that $\phi(x)$ is a weight $3/2$ vector-valued quantum modular form. In particular, we have that 
\[
\phi(z+1)-\left(\begin{smallmatrix}1&0&0\\0&0&\zeta_{12}\\0&\zeta_{24}&0 \end{smallmatrix} \right)\phi(z)=0
,
\]
and we also have that
\[
\left(\frac{z}{-i}\right)^{-3/2}\phi(-1/z)+\left(\begin{smallmatrix}0&\sqrt{2}&0\\1/\sqrt{2}&0&0\\0&0&1\end{smallmatrix} \right)\phi(z)=\left(\begin{smallmatrix}0&\sqrt{2}&0\\1/\sqrt{2}&0&0\\0&0&1\end{smallmatrix} \right)g(z)
,
\]
where $g(z)$ is a $3$-dimensional vector of smooth functions defined as period integrals.
\end{enumerate}
\end{theorem}
In addition, we deduce the following corollary regarding generating functions of special values of zeta functions from the sums of tails identities. Let
\begin{equation}
H_9(t,\zeta):=-\frac14\displaystyle\sum_{n=0}^{\infty}\frac{(1-\zeta e^{-t})(1-\zeta^2e^{-2t})\cdots(1-\zeta^n e^{-nt})}{(1+\zeta e^{-t})(1+\zeta^2e^{-2t})\cdots(1+\zeta^ne^{-nt})},
\end{equation}
\begin{equation}
H_{10}(t,\zeta):= -2(\zeta e^{-t})^{1/8}\displaystyle\sum_{n=0}^{\infty}\frac{(1-\zeta e^{-2t})(1-\zeta^2e^{-4t})\cdots(1-\zeta^n e^{-2nt})}{(1-\zeta e^{-t})(1-\zeta^2e^{-3t})\cdots(1-\zeta^ne^{-(2n+1)t})}.
\end{equation}
\begin{rmk}
Note that there are no rational numbers for which all three components of $\phi$ make sense simultaneously. To be specific, $\phi_1(z)$ makes sense for rational numbers which correspond to primitive odd order roots of unity, $\phi_2(z)$ makes sense for all rational numbers, and $\phi_3(z)$ converges at even order roots of unity. Hence, by the equation in (2) of \ref{mainthm2}, we understand that each of the six equations of the vector-valued transformation laws is true where the corresponding component in the equation is well-defined; as there are no equations in which $\phi_1$ and $\phi_3$ both appear, then for all the equations there is an infinite subset of rationals on which this is possible.
\end{rmk}

For a root of unity $\zeta$, we define the following two $L$-functions

\[L_1(s,\zeta):=\sum_{n=1}^{\infty}\frac{(-\zeta)^{n^2}}{n^s},\]

\[L_2(s,\zeta):=\sum_{n=1}^{\infty}\left(\frac2n\right)^2\cdot\frac{\zeta^{\frac{n^2}{8}}}{n^s}.\]
Then we have the following, which is also joint work with Robert Schneider.

\begin{corollary}[\cite{Rolen-Schneider}]\label{QuantumCor}
Let $\zeta=e^{2\pi i \alpha}$ be a primitive $k^{\operatorname{th}}$ order root of unity, $k$ odd for $H_9$ and $k$ even for $H_{10}$. Then as $t\searrow0$, we have as power series in $t$

\begin{equation}\label{Hurwitz1}  H_{9}(t,\zeta)=\sum_{n=0}^{\infty}\frac{L_1(-2n-1,\zeta)(-t)^n}{n!},
\end{equation}
\begin{equation}\label{Hurwitz2} H_{10}(t,\zeta)=\sum_{n=0}^{\infty}\frac{L_2(-2n-1,\zeta)(-t)^n}{8^nn!}.
\end{equation}
\end{corollary}

To illustrate our results by way of an application, we provide a  numerical example which gives finite evaluations of seemingly complicated period integrals. First define 
\begin{equation*}
\Omega(x):=\int_{x}^{i\infty}\frac{\theta_1(z)}{(z-x)^{3/2}}\, dz
\end{equation*} 
for $x\in\Q$, and consider $\theta_1^S(\zeta_k)$ for $k$ odd, which is a finite sum of $k^{\text{th}}$ roots of unity. Then the proof of Theorem \ref{mainthm2} will imply that $\Omega(1/k)=\pi i(1+i)\theta_1^S(\zeta_k)$ by showing that the period integral $\Omega(x)$ is a ``half-derivative'' which is related to $\theta_1^S$ at roots of unity by a sum of tails formula. Table 1.2 above gives finite evaluations of $\theta_1^S(\zeta_k)$ and numerical approximations to the integrals $\Omega(1/k)$.
\begin{table}\small\caption{Evaluation of $\theta_1^S(\zeta_k)$ and $\Omega(1/k)$}
\begin{center} 
\begin{tabular}{c c c c} % centered columns (4 columns)
\hline\hline                        %inserts double horizontal lines\

$k$&$\pi i(i+1)\theta_1^S(\zeta_k)$&$\int_{1/k+10^{-9}}^{10^9i}\frac{\theta_1(z)}{(z-1/k)^{\frac32}}\, dz$\\ [0.7ex] % inserts table
%heading
\hline                    % inserts single horizontal line
\\
$3$&$\pi i(i+1)(-2 \zeta_{3} + 3)\sim-7.1250+18.0078i$&$-7.1249+18.0078i$
\\
$5$&$\pi i(i+1)(-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 8 \zeta_{5} + 3)\sim12.078+35.7274i$&$12.078+35.7273i$
\\
$7$&$\pi i(i+1)(6 \zeta_{7}^{4} - 2 \zeta_{7}^{2} - 10 \zeta_{7} + 7)\sim52.0472+25.685i$&$52.0474+25.685i$
\\
$9$&$\pi i(i+1)(8 \zeta_{9}^{4} - 16 \zeta_{9} + 3)\sim76.4120-28.9837i$&$76.4116-28.9836i$
\\
\hline     %inserts single line
\end{tabular}
\end{center}

\end{table}

We will prove these results in Chapter 4.
\\
\indent


Our third main result addresses Ramanujan's deathbed letter \cite{RamaLetters}, which gave tantalizing hints of his theory of {\it mock theta functions}.
Thanks to Zwegers \cite{Zwegers1, Zwegers2}, it is now known that these functions are essentially the holomorphic parts of weight 1/2 {\it harmonic weak Maass forms} whose nonholomorphic parts are period integrals of weight 3/2
unary theta functions. This realization has many applications (e.g. \cite{OnoCDM, ZagierApp}).

Here we revisit Ramanujan's original definition from his deathbed letter \cite{RamaLetters}. After a discussion of the asymptotics
of certain modular forms which are given as {\it Eulerian series}, he writes:

\medskip
\noindent
{\it ``...Suppose there is a function in the Eulerian form and suppose that all or an infinity of points $q=e^{2i \pi m/n}$ are exponential
singularities and also suppose that at these points the asymptotic form of the function closes as neatly as in the
cases of (A) and (B). The question is: - is the function taken the sum of two functions one of which is an ordinary theta function and the other
a (trivial) function which is $O(1)$ at \underbar{all} the points $e^{2 i \pi m/n}$? The answer is \underbar{it is not necessarily so.}
When it is not so I call the function a mock $\vartheta$-function. I have not proved rigorously that \underbar{it is not necessarily so}. But I have constructed
a number of examples in which it is inconceivable to construct a $\vartheta$-function to cut out the singularities of the original function.''}


 \begin{remark} By ``ordinary theta function'', Ramanujan meant a {\it weakly holomorphic modular form} with weight
 $k\in \frac{1}{2}\Z$ on some $\Gamma_1(N)$ (see \cite{cbms} for background). Recall that a weakly holomorphic modular form is a meromorphic modular form whose poles (if any) are supported at cusps.
 \end{remark}


Little attention has been given to Ramanujan's original
definition, prompting
Berndt to remark  \cite{Berndt}  that ``{\it it has not been {proved} that
{any} of Ramanujan's mock theta functions are really mock theta functions according to his definition."}
The following fact fills in this gap. This is joint work with Michael Griffin and Ken Ono.

 \begin{theorem}[\cite{Griffin-Ono-Rolen}]\label{mainthm3}
 Suppose that $f(z)=f^{-}(z)+f^{+}(z)$ is a  harmonic weak Maass form of weight $k\in \frac{1}{2}\Z$ on $\Gamma_1(N)$, where $f^{-}(z)$ {\text {\rm  (resp.
 $f^{+}(z)$)}} is the nonholomorphic {\text{\rm (}}resp. holomorphic{\text {\rm )}} part of $f(z)$. If $f^{-}(z)$ is nonzero and $g(z)$ is a weight $k$ weakly holomorphic modular form on any $\Gamma_1(N')$, then $f^{+}(z)-g(z)$ has exponential singularities
 as $q$ approaches infinitely many roots of unity $\zeta$.
 \end{theorem}



 As a corollary, we obtain the following fitting conclusion to Ramanujan's enigmatic question by proving that his alleged examples indeed satisfy his original definition. More precisely, we prove the following.

 \begin{corollary}[\cite{Griffin-Ono-Rolen}]\label{question}  Suppose that $M(z)$ is
 one of Ramanujan's mock theta functions, and let $\gamma$ and $\delta$ be integers for which $q^{\gamma}M(\delta z)$ is the holomorphic part of a weight 1/2 harmonic weak  Maass form. Then there does not exist a weakly holomorphic modular form $g(z)$ of any weight
 $k\in \frac{1}{2}\Z$ on any congruence subgroup $\Gamma_1(N')$
 such that for every root of unity $\zeta$ we have
 $$
 \lim_{q\rightarrow \zeta}\left(q^{\gamma}M(\delta z)-g(z)\right)=O(1).
 $$
 \end{corollary}
 \begin{remark} The limits in Corollary~\ref{question} are {\it radial} limits
 taken from within the unit disk.
 \end{remark}


As his letter indicates, Ramanujan was inspired by the intimate
relationship between the exponential singularities of modular forms at roots of unity and the asymptotics
of their corresponding Fourier coefficients.
As a toy model of his question, we begin by considering the following question whose
solution would have been clear to him:
 If $f(z)$ is a weight $k_1$ weakly holomorphic modular form which has some exponential singularities at cusps, then can there be another weakly holomorphic modular form of different weight $k_2$, say $g(z)$, that exactly cuts out its singularities at roots of unity?  The answer is {\it no}.
If such a $g(z)$ exists, then both $f(z)$ and $g(z)$ must have the same principal parts at all cusps, and at least one of these must be nonconstant. Without loss of generality, suppose that the principal part at the cusp infnity is nonconstant, and then consider the function $h(z):=f(z)-g(z)$. By hypothesis, $h(z)$  has
bounded radial limits as $q$ approaches every root of unity. Now, since $f(z)$ and $g(z)$ are modular on some common subgroup $\Gamma_1(N')$, then if we take $\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix} \bigr)\in \Gamma_1(N')$ with $cd\neq 0$, then we have \begin{equation}\label{simple} h\left(\tfrac{az+b}{cz+d}\right)=
f\left(\tfrac{az+b}{cz+d}\right)-g\left(\tfrac{az+b}{cz+d}\right)
=(cz+d)^{k_1}f(z)-(cz+d)^{k_2}g(z).
\end{equation}
 Letting $z\to i\infty$, we find that $f(z)$ and $g(z)$ cannot cut out the same exponential singularities at roots of unity because of the difference between the weights.

In the case of Ramanujan's examples, the situation is much more subtle, and this is the point of his last letter and this paper.

Although these functions cannot asymptotically match a modular form at all roots of unity, Ramanujan offers a tantalizing example of a ``near miss''. For his mock theta function 
\[f(q):=1+\frac{q}{(1+q)^2}+\frac{q^4}{(1+q)^2(1+q^2)^2}+\ldots,\]
he defines a function $b(q)$, which is modular up to multiplication by $q^{-\frac{1}{24}}$, and claims for a primitive even order $2k$ root of unity $\zeta$, that as $q$ approaches $\zeta$ radially inside the unit disk we have
\[f(q)-(-1)^kb(q)=O(1).\] 
In \cite{FOR} and \cite{FOR2}, Folsom, Ono, and Rhoades give two proofs that this is indeed the case by giving explicit formulas for the $O(1)$ numbers. Here we show another ``near miss'' example using a similar proof as that in \cite{FOR2}. Namely, we consider the mock theta function 

\begin{equation}\omega(q):=\sum_{n\geq0}\frac{q^{2n(n+1)}}{(q;q^2)_{n+1}^2}.\end{equation}
We let $\epsilon(k):=\begin{cases}-1&\text{ if }k\in2\Z+1\\1&\text{ if }k\equiv2\pmod4.\end{cases}$ We prove a similar result for Ramanujan's $\omega$ function.

\begin{theorem}\label{suppthm3} Let $0<k$ be an integer not divisible by $4$. Then for $\zeta$ a primitive $k^{\text{th}}$ root of unity, we have
\begin{equation}
\lim_{q\rightarrow\zeta} q\omega(q^2)+\epsilon(k)\frac12\cdot\left(\displaystyle\sum_{n\in\Z}q^{n(n+1)} \right)\cdot\prod_{n\geq0}(1+q^{2n})=
\epsilon(k)\sum_{n\geq0}(\epsilon(k)q)^n(-\epsilon(k)q;q^2)_n.
\end{equation}
\end{theorem}

Note that the right-hand side is a finite sum for any such $\zeta$. 
The proof uses a formula of Ramanujan and Watson, together with a result of Fine on hypergeometric series. 
Finally, we numerically illustrate this result. Define $W(q):=q\omega(q^2)$, $M(q):=\frac12\cdot\left(\displaystyle\sum_{n\in\Z}q^{n(n+1)} \right)\cdot\prod_{n\geq0}(1+q^{2n})$, and $N(q):=-\sum_{n\geq0}(-q)^n(q;q^2)_n$. Then for $\zeta_5:=e^{2\pi i/5}$, we consider Table 1.3 above.
\\
\begin{table}\caption{Radial Limits for $\omega(q)$}
\begin{center} 
\begin{tabular}{c c c c} % centered columns (4 columns)
\hline\hline                        %inserts double horizontal lines\
$q$&$W(q)$&$W(q)-M(q)$&$N(\zeta(5))$\\ [0.7ex] % inserts table
%heading
\hline                    % inserts single horizontal line
$\zeta_5\cdot(0.99)$&$-3.78+12.35  i$& $0.67-1.35 i$&$0.81-1.76 i$\\
$\zeta_5\cdot(0.995)$&$-31.89+98.87 i$&$0.72-1.51 i$&$0.81-1.76 i$\\
$\zeta_5\cdot(0.9975)$&$-1236.9+3807.7 i$&$0.76-1.62 i$&$0.81-1.76 i$\\
$\zeta_5\cdot(0.999)$&$-3.78\times10^7+1.16\times 10^8 i$&$0.79-1.70 i$&$0.81-1.76 i$
\\
\hline     %inserts single line
\end{tabular}
\end{center}
\end{table}

These results will be proven in Chapter 5. 



