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\title[Borcherds exponents]{Classical and $p$-adic modular forms arising from the Borcherds exponents of other modular forms}

\author{Jayce Getz \\ Senior Thesis} \address{
4404 South Ave. W \\ Missoula, MT
59804}\email{getz@fas.harvard.edu}

\date{\today}

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\begin{abstract}
Let $f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}$ be a modular form
on $\SL_2(\ZZ)$.  Formal logarithmic differentiation of $f$ yields
a $q$-series $g(z):=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n$ whose
coefficients are uniquely determined by the exponents of the
original form.  We provide a formula, due to Bruinier, Kohnen, and
Ono for $g(z)$ in terms of the values of the classical
$j$-function at the zeros and poles of $f(z)$. Further, we give a
variety of cases in which $g(z)$ is additionally a $p$-adic
modular form in the classical sense of Serre.  As an application,
we derive some $p$-adic formulae, due to Bruinier, Ono, and
Papanikolas, in which the class numbers of a family of imaginary
quadratic fields are written in terms of special values of the
$j$-function at imaginary quadratic arguments.
\end{abstract}
\thanks {The author would like to thank his family for their constant personal and financial support,
Particular thanks go to his little brother Joel, who is the
coolest person in the world.  This thesis is dedicated to them.}

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\section{Introduction}
\label{intro}

Suppose $f$ is a function on the upper half plane $\mathbb{H}$.
For each positive integer $k$, define an action $|_k$ of
$\textrm{GL}_2^+(\QQ)$ on the set of such $f$ by
\begin{equation}
f(z)|_k\gamma=\det(\gamma)^{k/2}(cz+d)^{-k}f\left(
\frac{az+b}{cz+d} \right).
\end{equation}
Here $\gamma = \left( \begin{smallmatrix} a & b \\ c & d
\end{smallmatrix} \right) \in \textrm{GL}_2^+(\QQ)$
(with the exception of the proof of Theorem \ref{bko}, in this
thesis we always use the symbol $\gamma$ in this sense).
 Suppose $\Gamma' \subset \Gamma:=\textrm{SL}_2(\ZZ)$
is a congruence subgroup. Let $\mathcal{M}_k^{\infty}(\Gamma')$
(resp., $\mathcal{M}_k^{\textrm{mero}}(\Gamma')$) denote the space
of holomorphic (resp., meromorphic) functions on the upper half
plane $\mathbb{H}$ that satisfy the functional equation
\begin{equation} \label{modfunc}
f(z)|_k\gamma:=f(z)
\end{equation}
for all $\gamma  \in \Gamma'$ and additionally are meromorphic at
the cusps of $\Gamma'$ (for a precise description of this
``meromorphic at the cusps" condition, see \cite[\S III.3,
p.~125]{k}). Such a function will be called a \emph{weakly modular
form of weight $k$} (resp., \emph{meromorphic modular form of
weight $k$}) following J-P. Serre's convention \cite[\S
VII.2]{S1}. We further define $M_k(\Gamma') \subset
\mathcal{M}_k^{\infty}(\Gamma')$ to be the space of weakly modular
forms that, additionally, are holomorphic at the cusps of
$\Gamma'$. Such a form will be called a \emph{holomorphic modular
form}, or, simply, a \emph{modular form}.  For any congruence
subgroup $\Gamma'$ containing the element
$\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1
\end{smallmatrix}\right)$, meromorphicity of $f$ at the cusps of $\Gamma'$ implies that
$f$ can be identified with a Fourier, or $q$-series, expansion
\begin{equation} \label{qseries}
f(z):=\sum_{n=n_0}^{\infty}a_nq^n
\end{equation}
where here, and throughout this thesis, $q:=e^{2 \pi i z}$.  In
the case $\Gamma'=\Gamma$, this is in fact equivalent to
meromorphicity at the cusps.  Holomorphicity at the cusps in the
case of $\Gamma'=\Gamma$ (which are all in the same orbit as
$\infty$ under the action of $\Gamma$) is equivalent to the
statement that $n_0 \geq 0$.  Finally, a holomorphic modular form
over $\Gamma'$ is said to be a \emph{cusp form} if it vanishes at
the cusps of $\Gamma'$; we denote the space of cusp forms of
weight $k$ over $\Gamma'$ by $S_k(\Gamma')$. In the case $f \in
M_k(\Gamma)$, this is simply the assertion that in the expansion
(\ref{qseries}) we have $n_0>0$. For convenience we define
$\mmer_k:=\mmer_k(\Gamma)$,
$\mathcal{M}_k^{\infty}:=\mathcal{M}_k^{\infty}(\Gamma)$ and
$M_k:=M_k(\Gamma)$.

We take the opportunity now to introduce the only congruence
subgroup we will explicitly use in this thesis, namely the
following level $N$ subgroup:
$$
\Gamma_0(N):=\left\{ \begin{pmatrix} a & b \\ c & d
\end{pmatrix} \in \Gamma : c \equiv 0 \pmod{N} \right\}.
$$
By convention, $\Gamma_0(1)=\Gamma$.
\begin{remark}  If $\left( \begin{smallmatrix} -1 & 0 \\ 0 & -1
\end{smallmatrix}\right) \in \Gamma'$, then from (\ref{modfunc}) we have $(-1)^kf(z)=f(z)$ for
all $f \in \mmer_k(\Gamma')$, from which it follows that
$\mmer_{2m+1}(\Gamma')=0$ for all integers $m$.  Thus, in
particular, $\mmer_{2m+1}=0$.
\end{remark}

For examples of modular forms on $M_k$ for even $k \geq 4$, we may
take the classical Eisenstein series of weight $k$:
\begin{equation} \label{Eis}
E_k(z):=1-\frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n)q^n
\end{equation}
where $B_k$ is the $k$th Bernoulli number and
$\sigma_{k-1}(n):=\sum_{d|n}d^{k-1}$.  We can formally define
$E_2$ using (\ref{Eis}), and though it is not a modular form, it
satisfies
the following transformation law for $\left(\begin{smallmatrix} a & b \\
c & d \end{smallmatrix} \right) \in \Gamma$:
\begin{equation} \label{e2}
 E_2\left( \frac{az+b}{cz+d}\right)(cz+d)^{-2}=E_2(z)+\frac{12c}{2 \pi i (cz+d)}.
\end{equation}
This transformation law turns out to play a role in many
arguments; a proof of it in this form is given in
\cite[p.~68]{sch}.

Other useful examples of modular forms are the discriminant
function
\begin{equation}
\Delta(z):=\frac{E_4(z)^3-E_6(z)^2}{1728}=q\prod_{n=1}^{\infty}(1-q^n)^{24}
\end{equation}
which is of weight $12$, and the $j$-function, which is a weakly
modular form of weight zero:
\begin{equation} \label{modjdef}
j(z):=\frac{E_4(z)^3}{\Delta(z)}=q^{-1}+744+196884q+21493760q^2+\cdots.
\end{equation}
We note that any element of $\minf_0$ is a polynomial in $j(z)$.
If we wish to emphasize for a proof that we are regarding $E_k$,
$\Delta$, $j$ as $q$-series (which can be either viewed formally
or as functions holomorphic in the punctured disc $0 <|q|<1$), we
write them as $E_k(q)$, $\Delta(q)$, and $J(q)$, respectively.

It is easy to see that $\minf_k(\Gamma')$ is a vector space over
$\CC$ for all congruence subgroups $\Gamma'$.  There exists an
important class of linear operators on these spaces, namely, the
Hecke operators $T_{k,n}$. These can be defined (in an admittedly
ad-hoc manner) by
\begin{equation} \label{heckedefo}
f(z)|T_{k,n}=n^{k-1}\mathop{\sum_{ad=n,\textrm{ }d>0}}_{0 \leq b
\leq d-1}f\left(\frac{az+b}{d}\right)
\end{equation}
or, equivalently,
\begin{equation} \label{heckedef}
f(z)|T_{k,n}:=\sum_{n \in \ZZ}\left(
\sum_{0<d|(m,n)}d^{k-1}a\left(\frac{mn}{d^2}\right)\right)q^n.
\end{equation}
If we define, for positive integers $d$, the $V$- and
$U$-operators $V(d)$ and $U(d)$ on formal $q$-series in $\CC[[q]]$
by
\begin{equation} \label{vdef}
 \left(\sum_{n \geq n_0} c(n)q^n \right)|V(d):=\sum_{n \geq
n_0}c(n)q^{dn}
\end{equation}
and
\begin{equation} \label{udef}
\left(\sum_{n \geq n_0} c(n)q^n \right)|U(d):=\sum_{n \geq
n_0}c(dn)q^{n}
\end{equation}
then we may write \begin{equation} \label{uvhecke}
T_{k,n}=\sum_{d|n}d^{k-1}V(d) \circ U(n/d).
\end{equation}
Note that if we identify a meromorphic modular form $f$ with its
$q$-expansion, we have
\begin{equation} \label{altvdef}
d^{k/2}f|V(d)=f|_k\left(\begin{smallmatrix} d & 0 \\ 0 &
1
\end{smallmatrix}\right).
\end{equation} For more natural definitions of these operators and a
discussion of their basic properties, see, for example, \cite[\S
III.5]{k} or \cite[\S VII]{S1}.

If we consider $\mathcal{M}^{\infty}:=\bigoplus_{k=0}^{\infty}
\mathcal{M}^{\infty}_k$ it is straightforward to see that we have
something better than a collection of vector spaces, we have a
graded algebra, where the grading is given by weight and the
multiplication operation is multiplication of functions (for proof
of this, see \cite[\S VII]{S1}). A question naturally suggests
itself: are there natural operators on this algebra?  As one
possible answer to this question, we define Ramanujan's theta
operator:
$$
\Theta:=\frac{1}{2 \pi i } \frac{d}{dz}=q \frac{d}{dq}.
$$
It is perhaps speaking loosely to call $\Theta$ an operator, but
$$
f(z) \mapsto \Theta f(z)-f(z)\frac{k}{12}E_2(z)
$$
is a derivation on $\mathcal{M}$.  In particular, we have the
following:

\begin{proposition} \label{thetprop}
If $f$ is in $\mmer _k(\Gamma')$
then
\begin{equation} \label{thetobs}
g(z)=\Theta f-f(z)\frac{k}{12}E_2  \in \mmer_{k+2}(\Gamma').
\end{equation}
The same statement is true with $\mmer_k(\Gamma')$ replaced by
$\minf_k(\Gamma')$ or $M_k(\Gamma')$ throughout.
\end{proposition}
\begin{proof}
By noting its affect on $q$-expansions, we see that applying the
$\Theta$ operator does not affect meromorphicity (resp.,
holomorphicity) at the cusps.  Thus we need only check the
functional equation. For $\gamma=\left(\begin{smallmatrix} a & b
\\c & d\end{smallmatrix}\right) \in \Gamma$, upon differentiating
the functional equation (\ref{modfunc}) we have
\begin{eqnarray*} \Theta
f(\gamma z)(cz+d)^{-k-2}&=&\Theta f(z)+\frac{ck}{2 \pi i}f(\gamma z)(cz+d)^{-k-1} \\
&=&\Theta f(z)+ \frac{ck}{2 \pi i}f(z)(cz+d)^{-1}.
\end{eqnarray*}
Using (\ref{e2}), for $\gamma \in \Gamma' \subset \Gamma$ we have
\begin{eqnarray*}
\Theta f (z) |_{k+2}\gamma &-&\frac{k}{12}\left(E_2(z)|_{2} \gamma
\right)\left(f(z)|_k \gamma \right)
\\ &=&\Theta f(z) +\frac{ck}{2 \pi i}f(\gamma
z)(cz+d)^{-k-1}-\frac{k}{12}E_2(z)f(z)-\left(\frac{k}{12}\right)\frac{12c}{2
\pi i (cz+d)}f(z) \\
&=& \Theta f(z)-\frac{k}{12}E_2(z)f(z).
\end{eqnarray*}
\end{proof}

\begin{remark} It is worth mentioning that there exists a family of ``Rankin-Cohen" brackets on
$\bigoplus_{k=0}^{\infty} M_k$ (defined using $\Theta$), one of
which gives this algebra the structure of a graded Lie algebra.
For their definition and basic properties see \cite{Z2}, and for
references to recent work, see \cite{BWO}.
\end{remark}

Now, given a modular form $f \in \mmer_k(\Gamma')$, normalized so
that its first nonzero $q$-expansion coefficient is $1$, we can
write
$$
f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}
$$
for some complex numbers $c(n)$, in some neighborhood of $\infty$.
Ignoring convergence issues for a moment (which will be dealt with
carefully in Lemma \ref{logdiv}), some easy manipulations with
$q$-series combined with Proposition \ref{thetprop} yield
\begin{equation} \label{ok}
\frac{\Theta f}{f}=h-\sum_{n \geq 1} \sum_{d|n}c(d)dq^n \in
\mmer_2(\Gamma')
\end{equation}
In the next section, we will prove the following characterization
of this logarithmic derivative:

\begin{theorem}[Bruinier, Kohnen, Ono, \cite{BKO}, \cite{O}] \label{bko} If $f(z)=\sum_{n=h}^{\infty}a(n)q^n
\in \mmer_k$ is normalized so that $a(h)=1$, then
$$
\frac{\Theta
f(z)}{f(z)}=\frac{k}{12}E_2(z)-\frac{E_4(z)^2E_6(z)}{\Delta(z)}\sum_{\tau_i
\in \mathfrak{F}} \frac{e_{\tau}\ord_{\tau}(f)}{j(z)-j(\tau)}.
$$
\end{theorem}

\begin{remark}  This formula has been generalized to several
genus zero congruence subgroups in \cite{Ahl1} (see \S \ref{Rth}
of this thesis) and Hecke subgroups of $\textrm{SL}_2(\RR)$ (see
\cite{CKo}). The author has also received a preprint \cite{DC}
giving a generalization to $\Gamma_0(N)$ for squarefree $N$.
\end{remark}

This formula alone is of interest in that it explicitly relates,
via equation (\ref{ok}), the product expansion exponents of $f$ to
special values of $j$, namely, $j(\tau)$ where $\tau$ is a zero or
pole of $f$. Further, it has been used to provide recursive
formulas for the coefficients of any modular form over $\Gamma$
(see \cite{BKO}), to provide infinite families of systems of
orthogonal polynomials divisible by the supersingular locus as
polynomials over $\FF_p$ (see \cite{BGNS}), (generalizing work of
Atkin described in \cite{kz}), and also to provide a
characterization of the characteristic polynomials of the Hecke
operators over $\Gamma$ (again in \cite{BKO}). We will not discuss
these applications in this thesis. We will, however, give one
additional application, which we defer for a moment in order to
introduce the concept of a $p$-adic modular form.

Following Serre, we define a $p$-adic modular form to be the
$p$-adic limit of a sequence of elements of
$\oplus_{k=0}^{\infty}M_k$ (a precise definition is given in \S
\ref{Spadic}). It turns out that in many cases of interest, the
logarithmic derivative of a modular form is a $p$-adic modular
form of weight $2$.  In particular, we have the following theorem
of Bruinier and Ono:

\begin{theorem}[\cite{BrO}] \label{modelth}
Let $f(z)=q^h(1+\sum_{n=1}^{\infty}a(n)q^n) \in q^h\OO_K[[q]] \cap
\mmer_k(\Gamma_0(1))$, where $\OO_K$ is the ring of integers of a
number field $K$. Moreover, let $c(n) \in K$ denote the algebraic
numbers defined by the formal infinite product
\begin{equation} \label{cexp}
f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}.
\end{equation}
If $f(z)$ is good at a prime $p$, then the formal power series
$$
\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n
$$
is a weight two $p$-adic modular form.
\end{theorem}

We offer a brief proof of this result, mainly as motivation for
the following generalization:

\begin{theorem} \label{gbro} Suppose $p \geq 5$ is prime.  Let $f(z)=q^h(1+\sum_{n=1}^{\infty}a(n)q^n) \in q^h\OO_K[[q]] \cap \mmer_k(\Gamma_0(p))$ where $\OO_K$ is the
ring of integers of a number field $K$.  Moreover, let $c(n) \in
K$ denote the algebraic numbers defined by the formal infinite
product (\ref{cexp}) for $f$. If $f$ is good at $p$, then the
formal power series
$$
\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n
$$
is a weight two $p$-adic modular form.
\end{theorem}
\noindent The proofs of both of these theorems appear in \S
\ref{pborc}.

\begin{remark} In theorems \ref{modelth} and \ref{gbro}, we allow
$h$ to be negative. The fact that the $c(n)$ are elements of $K$
(implicitly identified with an embedding $K \hookrightarrow \CC$)
will be obvious from the proof of Lemma \ref{logdiv}.
\end{remark}

The definition of ``good" in the preceding two theorems is given
in \S \ref{pborc} and discussed in some detail in \S \ref{good}.
As one example, the form $E_{p-1}$ is good at $p$. In general,
whether or not a form is good at $p$ is intimately related to the
question of whether or not the value of the $j$-function at the
zeros and poles of the form reduces to a supersingular
$j$-invariant in characteristic $p$ (which should come as no
surprise to those familiar with overconvergent $p$-adic modular
forms).  Through this connection we are able to relate these
$p$-adic modular forms to class numbers of imaginary quadratic
fields.  In particular, for small primes, we obtain $p$-adic class
number formulae involving sums of special values of the
$j$-function.

Before we can state this result, we must recall the notion of a
Heegner point.  A complex number $\tau$ of the form
$\tau=\frac{-b+\sqrt{b^2-4ac}}{2a}$ with $a,b,c \in \ZZ$, $\gcd
(a,b,c)=1$ and $b^2-4ac<0$ is known as a \emph{Heegner point} of
discriminant $d_{\tau}:=b^2-4ac$.  Heegner points are discussed at
some length in \S \ref{cm}.  Denote by $h_K$ the Hurwitz class
number of the number field $K$.  We have the following:

\begin{corollary}[Ono and Papanikolas, \cite{pj}] \label{classnumber}
Suppose that $d<-4$ is a fundamental discriminant of an imaginary
quadratic field and that $\tau$ is a Heegner point of discriminant
$d$.  If $K=\QQ(j(\tau))$, then the following are true:
\begin{enumerate}
\item If $d \equiv 5 \pmod{8}$, then as $2$-adic numbers we have
$$
h_{\QQ(\sqrt{d})}=-\frac{1}{720} \lim_{n \to \infty}
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{2^a-1} j\left(
\frac{2^{n-a} \tau+b}{2^a}\right)\right).
$$
\item If $d \equiv 2 \pmod{3}$, then as $3$-adic numbers we have
$$
h_{\QQ(\sqrt{d})}=-\frac{1}{360} \lim_{n \to \infty}
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{3^a-1} j\left(
\frac{3^{n-a} \tau+b}{3^a}\right)\right).
$$
\item If $d \equiv 2,3 \pmod{5}$, then as $5$-adic numbers we have
$$
h_{\QQ(\sqrt{d})}=-\frac{1}{180} \lim_{n \to \infty}
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{5^a-1} j\left(
\frac{5^{n-a} \tau+b}{5^a}\right)\right).
$$
\item If $d \equiv 3,5,6 \pmod{7}$, then as $7$-adic numbers we
have
$$
h_{\QQ(\sqrt{d})}=-\frac{1}{120} \lim_{n \to \infty}
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{7^a-1} j\left(
\frac{7^{n-a} \tau+b}{7^a}\right)\right).
$$
\end{enumerate}
\end{corollary}
\noindent In \S \ref{good}, we also use Theorem \ref{gbro} to
provide formulae of the same general form of those in Corollary
\ref{classnumber}, with a weight zero modular form in
$\Gamma_0(p)$ taking the place of the $j$-function (see Theorem
\ref{classnumber2}).

Before we begin the body of this work, we make a few remarks about
its structure. Sections \ref{Rth}, \ref{pborc}, and \ref{good}
contain results that have only been published recently, if at all,
and the primary purpose of this thesis is to collect their content
into one place.  Sections \ref{Spadic} and \ref{atle}, on the
other hand, are mostly derived from two well-known papers
(\cite{Sep} and \cite{AL}, respectively).  The author has provided
proofs of most of the results in these sections that are necessary
for the proof of theorems \ref{bko}, \ref{modelth}, and
\ref{gbro}. The notable exceptions are theorems \ref{sturm} and
\ref{newforms} which are proven in \cite{St} and \cite{L},
respectively.

In contrast, providing applications of theorems \ref{bko},
\ref{modelth} and \ref{gbro}, including Corollary
\ref{classnumber} and Theorem \ref{classnumber2}, requires results
for which we will not provide proofs; it would simply take us too
far afield. In particular, \S \ref{cm} is intended to give a brief
survey of the relevant definitions and theorems in the theory of
complex multiplication, but we omit the proofs of results usually
proven using class field theory and reduction theory (we refer the
reader to \cite[\S II]{Si2} or \cite{LaE} for a more complete
account). Ergo, \S \ref{cm} can be skipped without interrupting
the flow of ideas, especially if one is familiar with complex
multiplication and elementary calculations involving elliptic
curves.

\section{A characterization of Ramanujan's theta operator}
\label{Rth}

As indicated above, in this section we will prove a useful
characterization of the derivative of a modular form.  First we
require some preparation.  Let $$ \fF:=\left\{z: -\frac{1}{2} \le
\mbox{Re}(z) \leq 0 \text{ and } |z| \geq 1 \right\} \cup
\left\{z:  0 < \mbox{Re}(z) < \frac{1}{2} \text{ and } |z| > 1
\right\}
$$
 be the standard fundamental domain for the action of $\mbox{SL}_2(\ZZ)$ on the upper half plane $\mathbb{H}$, and let
\begin{eqnarray} \label{edef}
e_{\tau} = \begin{cases} \frac{1}{2} & \mbox{if } \tau = i, \\
\frac{1}{3} & \mbox{if } \tau = e^{2\pi i/3}, \\ 1 &
\mbox{otherwise}. \end{cases}
\end{eqnarray}

The purpose of this section is to prove the characterization of
the logarithmic derivative of a modular form given by Theorem
\ref{bko}. The proof of the theorem requires two steps.  The first
is an identity due to Asai, Kaneko, and Ninomiya \cite{AKN}.  To
introduce this result, define $j_0(z):=1$, and, for $m>1$, define
$j_m(z)$ to be the unique weight zero meromorphic modular form
with $q$-expansion
\begin{equation} \label{jmdef}
j_m(z):=J_m(q):=q^{-m}+\sum_{n=1}^{\infty} a_m(n)q^n \in q^{-m}
\ZZ[[q]]
\end{equation}
We note that $j_m(z)$ is a polynomial in $j(z)$ for all $m$.  In
fact, it is a polynomial in $j$ with integral coefficients, for
$J_m(q)$ can be formed by subtracting suitable integer multiplies
of the $q$-series $J(q)^k \in q^{-k}\ZZ[[q]]$ from $J(q)^m$ (where
$0 \leq k < m$). The first few $j_m(z)$ follow:
\begin{eqnarray} \label{jmexamp}
j_0(z)=J_0(q)&=&1, \\
j_1(z)=J_1(q)&=&j(z)-744 = q^{-1}+196884q+\cdots, \\
j_2(z)=J_2(q)&=&j(z)^2-1488j(z)+159768=q^{-2}+42987520q+ \cdots, \\
j_3(z)=J_3(q)&=&j(z)^3-2232j(z)^2+1069956j(z)-36866976=q^{-3}+2592899910q+
\cdots.
\end{eqnarray}
We may equivalently define $J_0(q):=j_0(z):=1$,
$J_1(q):=j_1(z):=j(z)-744$, and
\begin{equation} \label{jmdefo}
J_m(q):=j_m(z):=m j_1(z)|T_{0,m}
\end{equation}
for $m>1$.  The equivalence of this definition to the $q$-series
definition (\ref{jmdef}) follows from (\ref{heckedef}) and the
fact that a weakly modular form, being a polynomial in $j$, is
uniquely determined by the coefficients of non-positive exponent
in its $q$-series expansion. Indeed, from this fact we see that
the $J_m(q)$ form a basis for $\minf_0$.

We have the following:

\begin{theorem} \label{aknf} As an identity of formal power series
in $\rho,q$, we have
\begin{equation} \label{aknfo}
\sum_{n=0}^{\infty} J_n(\rho)q^n=\frac{E_4(q)^2E_6(q)}{\Delta(q)}
\cdot \frac{1}{J(q)-J(\rho)}. \end{equation}
\end{theorem}

\begin{remark}
Asai, Kaneko, and Ninomiya show in \cite{AKN} how Theorem
\ref{aknf} implies the famous denominator formula for the Monster
Lie algebra, namely
$$
J(\rho)-J(q)=\rho^{-1} \prod_{m>0 \textrm{ and } n \in
\ZZ}(1-\rho^mq^n)^{\beta(mn)},
$$
where the coefficients $\beta(n)$ are defined by
$$
j_1(z)=\sum_{n=-1}^{\infty}\beta(n)q^n.
$$
\end{remark}

\begin{proof}[Proof of Theorem \ref{aknf}]

We require a companion set of functions $g_m(\rho)$ indexed by
positive integers $m$, the $m$th of which can be defined in
analogy with (\ref{jmdef}) as the unique weight $2$ weakly modular
form with $\rho$-expansion
\begin{equation} \label{gdef}
g_m(\rho):=\rho^{-m}+\sum_{n=1}^{\infty} b_m(n)\rho^n \in \minf_2.
\end{equation}
Alternately, we may define
$g_1(\rho):=\frac{E_{4}(\rho)^2E_6(\rho)}{\Delta(\rho)}$ and
$$
g_m(\rho):=m^{-1}g_1(\rho)|T_{2,m}.
$$
As before, the equivalence of these two definitions follows from
the definition of the $T_{2,m}$ and the fact that any weight $2$
weakly holomorphic form is uniquely determined by the coefficients
in its $q$-expansion of negative order.  We note that this fact
follows from the well-known ``$k/12$ valence formula" (see, for
example, \cite[\S III.2]{k}), as does the corresponding fact for
weight zero weakly holomorphic forms.  In fact, as in the weight
zero case, this implies that the $g_m(\rho)$ form a basis for the
space $\minf_2$.  Further, from (\ref{thetobs}), if $f \in
\minf_0$ then $\Theta f \in \minf_2$, and by simply looking at the
bases $\{J_m\}$,$\{g_m\}$ we have just written down we see that
every element of $\minf_2$ can be written as $\Theta f$ for some
$f \in \minf_0$.  In particular, it follows from this observation
and the definition of $\Theta$ that the constant term of any
element of $\minf_2$ is identically zero (which justifies the
indexing of (\ref{gdef})).

Now we note that
\begin{equation} \label{Jconst}
J_m(q):=mJ_1(q)|T_{0,m}=q^{-m}+m a_m(m)q+ \cdots
\end{equation}
and
\begin{equation} \label{gconst}
g_m(\rho):=m^{-1}g_1(\rho)|T_{2,m}=\rho^{-m}+b_1(m)q+\cdots
\end{equation}
for $m \geq 1$ simply by (\ref{heckedef}) and the fact that
$b_1(0)=0$. Further, by noting that the constant term of
$J_m(q)g_1(q) \in \minf_2$ must be zero by the comments in the
preceding paragraph and using (\ref{gdef}) and (\ref{Jconst}), we
have that
\begin{equation} \label{abconst}
b_1(m)=-ma_m(m)
\end{equation}
for $m \geq 1$.  Now $J(\rho)J_m(\rho) \in \mathcal{M}_0$ and
$J(q)g_m(q) \in \minf_2$ are uniquely determined by their $\rho$-
(resp., $q$-) expansion coefficients of non-positive exponent, as
we've remarked before. Define $\rho$-expansion coefficients $c(n)$
by
$$
J(\rho)=\rho^{-1}+\sum_{n=0}^{\infty}c(n)\rho^n
$$
By comparing coefficients using equalities (\ref{Jconst}),
(\ref{gconst}), (\ref{abconst}) and the observation that
$b_1(0)=0$, we obtain the recurrence relation
\begin{equation} \label{Jmrecur}
J(\rho)J_m(\rho)=J_{m+1}(\rho)+\sum_{i=0}^m c(m-i)J_i(\rho)-b_1(m)
\end{equation}
for all $m \geq 0$.  Thus, multiplying both sides of
(\ref{Jmrecur}) by $q^{m}$ and summing over $m \geq 0$ we obtain
\begin{equation} \label{messy}
J(\rho)\sum_{m=0}^{\infty}J_m(\rho)q^m=\frac{1}{q}(\sum_{m=0}^{\infty}J_m(\rho)q^m-1)+
(J(q)-\frac{1}{q})\sum_{m=0}^{\infty}J_m(\rho)q^m-g_1(q)+\frac{1}{q}.
\end{equation}
Noting that $g_1(q)=\frac{E_{4}(q)^2E_6(q)}{\Delta(q)}$, we see
that (\ref{messy}) is a rewriting of (\ref{aknfo}).
\end{proof}

\begin{corollary} \label{aknfu}
Fix $\tau \in \mathbb{H}$.  Then
$$
\frac{E_4(z)^2E_6(z)}{\Delta(z)}\frac{1}{j(z)-j(\tau)}=\sum_{n=0}^{\infty}j_m(\tau)q^n
$$
as meromorphic functions in $z$ on $\mathfrak{F}$.
\end{corollary}
\begin{proof} Compare Fourier ($q$-series) coefficients in a deleted
neighborhood of infinity using Theorem \ref{aknf}.
\end{proof}

\begin{remark} The main result of \cite{AKN} is the statement that the zeros of $j_m(z)$ in
$\mathfrak{F}$ are simple and are all contained in the
intersection of the unit circle with $\mathfrak{F}$.  The
technique they use is analogous to that used by Rankin and
Swinnerton-Dyer to prove that the ``nontrivial" zeros of $E_k(z)$
have the same property, see \cite{RSD}.  For yet another family of
modular forms whose zeros have the same property, see \cite{G}.
\end{remark}

We also require the following proposition, which follows from
basic complex analysis:
\begin{proposition}[\cite{BKO}] \label{logdiv}
Let $f=\sum_{n=h}^{\infty}a_f(n)q^n$ be a meromorphic function in
a neighborhood of $q=0$, normalized so that $a_f(h)=1$.  Then
there are complex numbers $c(n)$ such that
$$
f=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)},
$$
where the product converges in a sufficiently small neighborhood
of $q=0$.  Moreover,
\begin{equation} \label{logdive}
\frac{\Theta f}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n.
\end{equation}
\end{proposition}
\begin{remark} We will refer to the $c(n)$ associated to a given
meromorphic modular form $f$ by Proposition \ref{logdiv} as the
\emph{Borcherds exponents} of $f$.
\end{remark}
\begin{proof}

As usual, we understand that complex powers are defined by the
principle branch of the complex logarithm.  Write $F(q):=f(z)$,
and then note that $qF'(q)/F(q)$ is holomorphic at $q=0$.  We may
therefore write its Taylor expansion around $q=0$, valid in
$|q|<\epsilon$ for some $\epsilon>0$, as
\begin{equation} \label{ld1}
qF'(q)/F(q)=h-\sum_{n \geq 1} \alpha(n)q^n.
\end{equation}
For $n \geq 1$ define
$$
c(n):=\frac{1}{n}\sum_{d|n}\alpha(d)\mu(n/d)
$$
where $\mu$ is the M$\ddot{\textrm{o}}$bius function.  By
M$\ddot{\textrm{o}}$bius inversion we have
\begin{eqnarray} \label{moinv}
\alpha(n)=\sum_{d|n}c(d)d.
\end{eqnarray}

If we fix $q_0$ with $|q_0|<\epsilon$, then by absolute
convergence of (\ref{ld1}) we have
$\alpha(n)=\mathcal{O}(|q_0|^{-n})$ for all $n$.  Thus the double
sum
\begin{equation} \label{un1}
 \sum_{m,n \geq 1} c(n)nq^{mn}
\end{equation}
converges absolutely in $|q|<|q_0|$ and hence in $|q|<\epsilon$.

Suppose for the remainder of the proof that $|q|<\epsilon$.  From
(\ref{ld1}) and (\ref{moinv}) we have
\begin{eqnarray*}
\frac{d}{dq} \log(F(q)q^{-h})&=&\frac{F'(q)}{F(q)}-\frac{h}{q} \\
&=&-\sum_{n \geq 1}c(n) \frac{d}{dq}\left(\sum_{m \geq 1}
\frac{q^{mn}}{m} \right) \\
&=& \frac{d}{dq}\left(\sum_{n \geq 1} c(n) \log(1-q^n)\right).
\end{eqnarray*}
The interchange of summation and integration can be justified by
using local uniform convergence as we did in proving the absolute
convergence of (\ref{un1}).

Upon integrating, we obtain
$$
\log(F(q)q^{-h})=\sum_{n \geq 1} c(n)\log(1-q^n).
$$
Here we use the normalization $a_f(h)=1$.  Now $c(n)\log(1-q^n)$
and $\log(1-q^n)^{c(n)}$ differ by integer multiples of $2 \pi i$.
Since $c(n) \log(1-q^n) \to 0$ as $n \to \infty$, we have
$\log(1-q^n)^{c(n)} \to 0$ as well.  Thus, as $n \to \infty$,
these two quantities differ in value only finitely many times; it
follows that there exists an integer $N$ such that
$$
\log(F(q)q^{-h})=\sum_{n \geq 1} \log(1-q^n)^{c(n)}+2 \pi i N.
$$
Taking the exponential on both sides finishes the proof of the
proposition.
\end{proof}

We now prove Theorem \ref{bko}.

\begin{proof}[Proof of Theorem \ref{bko}]

Choose $C>0$ large enough so that all poles of $f$ in
$\mathfrak{F}$ (excluding any at the cusp at infinity) have
imaginary part less than $C$. Let $ L:=\{t+iC: -\frac{1}{2} \leq t
\leq \frac{1}{2} \}$ and consider the contour in $\mathbb{H}$
formed from the part of $\partial \mathfrak{F}$ of imaginary part
less than $C$ and $L$.  Modify this contour as in the proof of the
classical $k/12$ valence formula (see, for example, \cite[\S
III.2, p.~115]{k}), specifically, if there are poles of $f$ at $i$
or $\omega:=e^{2 \pi i/3}$ (which, by modularity, implies the
existence of a pole at $e^{\pi i/3}$), form half and ``sixth"
circles of radius $r>0$ around them, and if there are poles of $f$
on the boundary, form two half circles of radius $r>0$ around
them, one enclosing the pole on one side of the fundamental
domain, one not enclosing the pole which must exist on the other
side (given that $f$ is modular).  Call the left vertical side of
this contour $\gamma_1(r)$, the right vertical side $\gamma_2(r)$,
and the bottom $\gamma_3(r)$.  Take the modified contour
$\gamma_1(r) \cup L \cup \gamma_2(r) \cup \gamma_3(r)$ to have
positive (counterclockwise) orientation.

If we integrate
\begin{equation} \label{int1}
\frac{1}{2 \pi i} \frac{f'(z)}{f(z)} j_n(z)
\end{equation}
along this full contour and let $r \to 0$, by holomorphicity of
$j_n$ on $\mathbb{H}$ the integral will be equal to
\begin{equation} \label{int1way}
\sum_{\tau \in \mathfrak{F}-\{\omega,i\}} \ord_{\tau}(f)j_n(\tau).
\end{equation}
We can also integrate (\ref{int1}) in pieces, from which we see
that (\ref{int1way}) is equal to
\begin{eqnarray} \label{lotsoterms}
&&-\frac{1}{3}\ord_{\omega}(f)j_n(\omega)-\frac{1}{2}\ord_i(f)j_n(i)+\int_L
\frac{f'(z)}{f(z)}j_n(z)dz+\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)dz
\\\nonumber &=&-\frac{1}{3}\ord_{\omega}(f)j_n(\omega)-\frac{1}{2}\ord_i(f)j_n(i)+\frac{1}{2 \pi i}\int_{L'}\frac{F'(q)}{F(q)}J_n(q)
dq+\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)dz.
\end{eqnarray}
Here $L'$ is a simple loop around $q=0$.  By Proposition
\ref{logdiv} we have
$$
\frac{qF'(q)}{F(q)}=\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}
c(d)dq^n
$$
and thus, applying the residue theorem, we have
$$
\frac{1}{2 \pi i} \int_{L'}
\frac{F'(q)}{F(q)}J_n(q)dq=\sum_{d|n}c(d)d.
$$

We now deal with the last term in (\ref{lotsoterms}).  By
Proposition \ref{thetobs}, if the weight of $f$ is $k$, there
exists a weight $k+2$ modular form $g$ such that
\begin{eqnarray} \label{justamo}
\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)&=&2 \pi
i\int_{\gamma_3(r)}\frac{\Theta(f)}{f}j_n(z)dz\\\nonumber&=&2 \pi
i\int_{\gamma_3(r)}\frac{g(z)}{f(z)}j_n(z)dz+ 2 \pi i
\int_{\gamma_3(r)}\frac{k}{12}j_n(z)E_2(z)dz
\end{eqnarray}
Now let $\beta$ denote the path along the unit circle from $i$ to
$\omega$, taken with positive orientation, and $S$ the fractional
linear transformation defined by $S(z)=-1/z$. Then
$\gamma_3=-\beta+S\beta$, and thus the right hand side of equation
(\ref{justamo}) is equal to
\begin{eqnarray*}
&&\left(\int_{-\beta}\frac{g(z)}{f(z)}j_n(z)dz+
\int_{S\beta}\frac{g(z)}{f(z)}j_n(z)dz
\right)+\frac{k}{12}\left(\int_{-\beta}j_n(z)E_2(z)dz+\int_{S\beta}j_n(z)E_2(z)dz\right)\\
&=&\frac{k}{12}\left(\int_{-\beta}j_n(z)E_2(z)dz+\int_{S\beta}j_n(z)E_2(z)dz\right)\\
&=&\frac{k}{12} \left( \int_{-\beta}j_n(z)E_2(z)dz+\int_{\beta}j_n(z)E_2(z)dz+\int_{\beta}\frac{12}{2 \pi i}\frac{j_n(z)}{z}dz \right)\\
&=&\frac{k}{2 \pi i}\int_{\beta}\frac{j_n(z)}{z}dz.
\end{eqnarray*}
To obtain the first equality we used the functional equation for
elements of $\minf_2$ along with a standard change of variables
(which introduces a factor of $1/z^2$).  To move from the second
line to the third we used the functional equation for elements of
$\minf_2$, a change of variables, and the functional equation
(\ref{e2}) for $E_2(z)$.

Now, instead of trying to evaluate $\frac{k}{2 \pi i}
\int_{\beta}\frac{j_n(z)}{z}dz$ directly, we plug $f=\Delta$ into
(\ref{lotsoterms}), notice that $\sum_{\tau \in
\mathfrak{F}}\ord_{\tau}(f)j_n(\tau)=0$, and thereby obtain
\begin{eqnarray*}
\int_{\beta} \frac{j_n(z)}{z}dz&=&-\frac{1}{12}\int_{\beta}\frac{\Delta'(q)}{\Delta(q)}J_n(q)dq\\
&=&-\frac{1}{12}\sum_{d|n}c(d)d
\\
&=&-2 \sigma_1(n)
\end{eqnarray*}
where $c(d) \equiv 24$ are (just for the purposes of the preceding
equation) the product expansion exponents of
$\Delta(z)=q\prod_{n=1}^{\infty}(1-q^n)^{24}$.

Thus, collecting all of this, equation (\ref{lotsoterms}) implies
that
$$
\sum_{\tau \in \mathfrak{F}}
e_{\tau}\ord_{\tau}(f)j_n(\tau)=\sum_{d|n}c(d)d-2k \sigma_1(n)
$$

Now we recall that by Theorem \ref{aknf}, it is sufficient to show
that
$$
\frac{\Theta(f)}{f}=\frac{kE_2}{12}-
\sum_{n=1}^{\infty}\left(\sum_{\tau \in \mathfrak{F}}
e_{\tau}\ord_{\tau}(f)j_n(\tau)\right)q^n.
$$
To prove this identity, we apply Proposition \ref{logdiv}, note
$$
\frac{k}{12}E_2(z)=\frac{k}{12}-2k\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
and argue coefficient by coefficient.  The only coefficient that
might be unclear is the constant $n=0$ term.  In this case, on the
left we have $h,$ which is the order of $f$ at infinity, and on
the right we have $\frac{k}{12}-\sum_{\tau \in
\mathfrak{F}}e_{\tau}\ord_{\tau}(f)$, which is precisely
$\ord_{\infty}(f)=h$ by the $k/12$ valence formula (for example,
see \cite[\S III.II, p.~115]{k}).

\end{proof}

We remark here that the derivative formula of Theorem \ref{bko}
explicitly relates, via Proposition \ref{logdive}, the
coefficients $c(n)$ of the product expansion of a modular form to
a specific weight $2$ meromorphic modular form.  This relationship
is in the spirit of the work of Borcherds on the product expansion
exponents of Jacobi forms with Heegner divisors. See \cite{Borc}
for the details of this theory.

As we mentioned in the introduction, Ahlgren, in \cite{Ahl1}, has
proven a generalization of Theorem \ref{gbro} to certain genus
zero congruence subgroups.  We will state his theorem after fixing
some notation. Define Dedekind's eta-function
$$
\eta(z):=q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^n)
$$
as usual.  For $p=2,3,5,7$ or $13$, let
$$
j^{(p)}(z):=\left(\frac{\eta(z)}{\eta
(pz)}\right)^{\frac{24}{p-1}} \in \minf_0(\Gamma_0(p)).
$$
This $j^{(p)}(z)$ is a modular form with a simple pole at $\infty$
and a simple zero (with respect to local coordinates) at $0$.
Additionally, its restriction to a fundamental domain for the
action of $\Gamma_0(p)$ on $\HH$ forms a bijection from that
fundamental domain to $\CC$. In analogy with (\ref{jmdef}), we now
define a sequence of modular functions
$\{j^{(p)}_m(z)\}_{m=0}^{\infty}$. Let $j_0^{(p)}(z):=1$ and for
$m>0$ let $j_m^{(p)}(z) \in \minf_0(\Gamma_0(p))$ be the unique
modular function which is holomorphic on $\HH$, vanishes at the
cusp $0$ and whose Fourier expansion at infinity has the form
\begin{eqnarray}
\label{jmpdef}
 j_m^{(p)}(z)=q^{-m}+c(0)+c(1)q+c(2)q^2+ \cdots.
\end{eqnarray}
Because $\Gamma_0(p)$ is genus zero, each of these functions can
be written as monic polynomials in $j_1^{(p)}(z)=j^{(p)}(z)$ with
constant term equal to zero.  For example, we have
\begin{eqnarray*}
j_0^{(5)}(z)&=&1, \\
j_1^{(5)}(z)&=&j^{(5)}(z)=q^{-1}-6+9q+10q^2-30q^3+ \cdots \\
j_2^{(5)}(z)&=&j^{(5)}(z)^2+12j^{(5)}(z)=q^{-2}-18+20q+21q^2+192q^3+
\cdots \\
j_3^{(5)}(z)&=&j^{(5)}(z)^3+18j^{(5)}(z)^2+81j^{(5)}(z)=q^{-3}-24-90q+288q^2+144q^3
+ \cdots
\end{eqnarray*}
In analogy with our definition of $\mathfrak{F}$, we define
$\mathfrak{F}_p$ to be a fundamental domain for the action of
$\Gamma_0(p)$ on $\HH$, taking the convention that
$\mathfrak{F}_p$ does not include the two cusps $\infty$ and $0$.
If $\tau \in \HH$, then (in analogy with (\ref{edef})) we define
$e^{(p)}_{\tau} \in \left\{1,\frac{1}{2},\frac{1}{3}\right\}$ by
$$
e_{\tau}^{(p)}:=( \textrm{the order of the isotropy subgroup of }
\tau \textrm{ in } \Gamma_0(p)/\{ \pm I\} )^{-1}.
$$
We can now state the following theorem:

\begin{theorem}[\cite{Ahl1}] \label{scott} Suppose that $p \in
\{2,3,5,7,13\}$ and that $f(z)=\sum_{n=h}^{\infty} a(n)q^n \in
\mmer_k(\Gamma_0(p))$, normalized so that $a(h)=1$.  Then
\begin{eqnarray*}
\frac{\theta f}{f} =-\sum_{\tau \in \mathfrak{F}_p}
\left(e^{(p)}_{\tau}\sum_{n=1}^{\infty}
j_n^{(p)}(\tau)q^n\right)+\frac{h-k/12}{p-1} \cdot
pE_2|V(p)+\frac{pk/12-h}{p-1} \cdot E_2.
\end{eqnarray*}
\end{theorem}

We will not provide a proof of this theorem; it is entirely
analogous to the proof of Theorem \ref{bko} except for some
difficulties which naturally arise when dealing with congruence
subgroups.  We note that a formula analogous to Corollary
\ref{aknfu} holds in the $\Gamma_0(p)$ case for $p \in
\{2,3,5,7,13\}$ as well (see \cite{Ahl1}).

\section{Serre's $p$-adic modular forms} \label{Spadic}

We begin with the notion of congruent $q$-series.  Two $q$-series
$f(z)=\sum_{n=n_0}^{\infty}a(n)q^n \in q^{n_0}\ZZ[[q]]$ and $g(z)=
\sum_{m=m_0}^{\infty}b(m)q^m\in q^{m_0}\ZZ[[q]]$ are said to be
\emph{congruent modulo $N$} if
$$
a(k) \equiv b(k) \pmod{N}
$$
for all $k$.  For primes $p$, we say that a $q$-series $f(z)$ with
integral coefficients is a \emph{weakly modular form modulo $p^n$}
if it is congruent modulo $p^n$ to a modular form $g(z) \in \minf
\cap q^{-m_0}\ZZ[[q]]$. This is written as
$$
f(z) \equiv g(z) \pmod{p^n}
$$
We note here that the theory of modular forms modulo prime powers
is quite well developed; for a basic introduction, see \cite[\S
IV.X]{La}, and for a variety of interesting number-theoretic
applications, see \cite{O}.

We begin by establishing some well-known congruences involving the
Eisenstein series $E_k(z)$.  First we recall two classical
Bernoulli number congruences (see \cite[p.~233-238]{IR}).  Let
$D_n$ be the denominator of the $n$th Bernoulli number, written in
lowest terms.  The von Staudt-Clausen congruences state
\begin{equation}
\label{VS} D_n=6\prod_{(p_i-1) |n}p_i
\end{equation}
where the $p_i$'s are prime. Let $p\ge5$ be prime. Now suppose $m
\geq 2$ is even and $m' \equiv m \pmod{\phi(p^r)}$ where $\phi$ is
the Euler $\phi$-function.  Then the Kummer congruences state
\begin{equation}
\label{Kummer} \frac{(1-p^{m'-1})B_{m'}}{m'} \equiv
\frac{(1-p^{m-1})B_{m}}{m} \pmod{p^r}.
\end{equation}

Using these congruences, we prove the following lemma:
\begin{lemma}
\label{pmoddy} For $r \geq 1$ and odd primes $p$, the following
$q$-series congruences hold:
\begin{equation}
\label{1wrclev} (E_{p-1}(z))^{p^{r-1}} \equiv 1 \pmod{p^r}
\end{equation}
and
\begin{equation}
\label{E2} E_{\phi(p^r)+2}(z) \equiv E_2(z) \pmod{p^r}.
\end{equation}
\end{lemma}
\begin{proof}
For (\ref{1wrclev}), we have
$$
\left(E_{p-1}(z)\right)^{p^{r-1}}=\left(
1-\frac{2(p-1)}{B_{p-1}}\sum_{n=1}^{\infty}\sigma_{p-1}(n)q^n
\right)^{p^{r-1}}=\left(
1-\frac{2(p-1)D_{p-1}}{U_{p-1}}\sum_{n=1}^{\infty}\sigma_{p-1}(n)q^n
\right)^{p^{r-1}}
$$
where $U_{p-1}$ is an integer coprime to $D_{p-1}$.  From
(\ref{VS}) we have $p|D_{p-1}$ which implies (\ref{1wrclev}) after
an application of the binomial theorem.

To prove (\ref{E2}), if we let $m'=2$, $m=\phi(p^r)+2$ in
(\ref{Kummer}) and note that $p^{\phi(p^r)+1} \equiv p
\pmod{p^{r}}$ we obtain
$$
\frac{B_2}{2} \equiv \frac{B_{\phi(p^r)+2}}{\phi(p^r)+2}
\pmod{p^r}.
$$
Also by Euler's theorem, $$\sigma_1(n) \equiv
\sigma_{\phi(p^r)+1}(n) \pmod{p^r}.$$  With these two observations
we have
$$
E_2(z)=1-\frac{2(2)}{B_2} \sum_{n=1}^{\infty}\sigma_1(n)q^n \equiv
1-\frac{2(\phi(p^r)+2)}{B_{\phi(p^r)+2}}
\sum_{n=1}^{\infty}\sigma_{\phi(p^r)+1}(n)q^n \pmod{p^r}.
$$
\end{proof}

\noindent We also record here the following congruences, which
will be useful in \S \ref{good}:

\begin{lemma} \label{24k} Suppose $k \geq 4$ is even.  Then
\begin{equation*}
E_k(z) \equiv 1 \pmod{24},
\end{equation*}
and, if $p \geq 5$ is a prime such that $(p-1) \mid k$,
\begin{equation*}
E_k(z) \equiv 1 \pmod{p}.
\end{equation*}
\end{lemma}
\begin{proof}
These both follow immediately from the von Staudt-Clausen equation
(\ref{VS}).
\end{proof}


Before we can proceed any farther, we must generalize the notion
of congruent modular forms introduced above.  Let $K$ be a number
field with ring of integers $\mathcal{O}_K$, and $\mathfrak{m}
\subset \mathcal{O}_K$ an ideal.  We define the \emph{order of $f$
modulo $\mathfrak{m}$} by
$$
\ord_{\mathfrak{m}}(f):=\min\{n:a(n) \not \in \mathfrak{m} \}
$$
with the convention that $\ord_{\mathfrak{m}}(f):=+\infty$ if
$a(n) \in \mathfrak{m}$ for all $n$.  Though this is certainly not
obvious a priori, given a modular form with coefficients in
$\mathcal{O}_K$, one need only check finitely many $q$-series
coefficients to calculate $\ord_{\mathfrak{m}}(f)$. The following
theorem of Sturm (see \cite[\S 2.9]{O} or \cite{St}) makes this
precise:
\begin{theorem} \label{sturm}
Suppose $k \geq 0$ is an integer and $K$ is a number field with
ring of integers $O_K$.  Moreover let $f=\sum_{n=0}^{\infty}
a(n)q^n \in M_k(\Gamma_0(N)) \cap \OO_K[[q]]$.  If $\mathfrak{m}
\subset \mathcal{O}_K$ is an ideal for which
$$
\ord_{\mathfrak{m}}(f)>\frac{k}{12}[\Gamma_0(1):\Gamma_0(N)]
$$
then $\ord_{\mathfrak{m}}(f)= + \infty$.
\end{theorem}
\begin{remark}We will not prove this theorem in this thesis.  We will only
require it for the proofs of theorems \ref{modelth} and
\ref{gbro}, and there we only invoke it briefly to prove that we
can normalize certain forms so that they have coefficients in a
ring of integers.  To see how this works, consider some form $f
\in M_k(\Gamma_0(N))$ with $p$-integral algebraic coefficients.
Then we can pick an integer $M \equiv 1 \pmod{p}$ such that the
first $\frac{k}{12}[\Gamma_0(1):\Gamma_0(N)]$ coefficients of $Mf$
are contained in the ring of integers of some number field
$\OO_K$. Applying Theorem \ref{sturm}, it follows that all of the
coefficients of $Mf$ are in $\OO_K$, in other words, we have
produced a form $Mf \equiv f \pmod{p}$ with algebraic integer
coefficients.
\end{remark}

Elements of $M_k(\Gamma')$ have the extremely useful property that
they determined by their first few $q$-series coefficients.
Though, as noted above, we will not need Theorem \ref{sturm} until
\S \ref{pborc}, we included it at this point to call the reader's
attention to the fact that a similar statement is true when
working with modular forms congruent modulo ideals in a number
field.

We are now in a position to justify the title of this section. Let
$K$ be a number field and let $\mathcal{O}_v$ be the completion of
its ring of integers at a finite place $v$ with residue
characteristic $p$.  Moreover, let $\lambda$ be a uniformizer for
$\mathcal{O}_v$.  We make the following:
\begin{definition} A formal power
series
$$
f:=\sum_{n=0}^{\infty}a(n)q^n \in \mathcal{O}_v[[q]]
$$
is a \textbf{$p$-adic modular form of weight $k \in
\mathcal{O}_v$} if there is a sequence $f_i \in
\mathcal{O}_v[[q]]$ of holomorphic modular forms on $\Gamma$ with
weights $k_i$ for which $\ord_{\lambda}(f_i-f) \to +\infty$ and
$\ord_{\lambda}(k-k_i) \to + \infty$.
\end{definition}
\begin{remark} This is Serre's original definition  of a
$p$-adic modular form \cite{Sep}.  The notion of a $p$-adic
modular form has been substantially generalized by Katz; for an
introduction and an explanation of how the two definitions relate,
see \cite[\S I]{Go}.
\end{remark}
Thus we observe, with the help of Lemma \ref{pmoddy}, that $E_2$
and $1$ are both $p$-adic modular forms, or, more precisely, are
$p$-adic modular forms when identified with their $q$-expansions
considered as elements of $\OO_v[[q]]$ (the $q$-expansion of $1$
is just $1+0q+0q^2+\cdots$). Further, any element of $M_k \cap
\OO_v[[q]]$ is trivially a $p$-adic modular form.

The only nontrivial result we will require from the theory of
$p$-adic modular forms is a theorem, due to Serre, which allows us
to compute the constant term of a $p$-adic modular form in terms
of a $p$-adic limit of its other coefficients for small primes
$p$. Let $\zeta_p^*(s)$ be the Kubota-Leopoldt $p$-adic zeta
function. We have

\begin{theorem}[Theorem 7, \cite{Sep}] \label{Ser} If $p \leq 7$ is prime and
$$
f=\sum_{n=0}^{\infty} a(n)q^n
$$
is a $p$-adic modular form of weight $k \neq 0$, then
$$
a(0)=\frac{\zeta_p^*(1-k)}{2} \cdot \lim_{n \to + \infty} a(p^n).
$$
\end{theorem}

This theorem is proven by decomposing the vector space $M$ of
$p$-adic modular forms into $M = E \oplus N$, where $N$ is a space
on which the $U$ operator (defined exactly as in (\ref{udef}))
acts nilpotently and $E$ is a space on which $U$ acts bijectively.
It turns out that for $2 \leq p \leq 7$ prime, $E$ is spanned by
the reductions of Eisenstein series, and $N$ is spanned by the
reductions of cusp forms.  By analyzing each subspace, the theorem
follows.  For a complete proof, see \cite[\S 2.3]{Sep}.
Incidentally, \cite{Sep} is a beautiful paper, and provides an
interesting counterpoint to Katz's geometric approach to $p$-adic
modular forms.

Also mentioned in \cite[\S 1.6]{Sep} is the fact that
$\zeta_p^*(1-k)=(1-p^{k-1})\zeta(1-k)$ for even integers $k \geq
2$, where $\zeta(s)$ is the usual characteristic zero Riemann zeta
function. In the sequel we will only be interested in the special
case $k=2$, in which we have:
$$
\zeta_p^*(1-2)=(1-p)\zeta(-1)=\frac{p-1}{12}.
$$
Thus we immediately have the following corollary of Theorem
\ref{Ser}:
\begin{corollary}
\label{Serre} If $p \leq 7$ is prime and
$$
f=\sum_{n=0}^{\infty} a(n)q^n
$$
is a $p$-adic modular form of weight $k \neq 0$, then
$$
a(0)=\frac{p-1}{24} \cdot \lim_{n \to + \infty} a(p^n).
$$
\end{corollary}

\section{Varying the level} \label{atle}

Given a modular form $f \in M_k(\Gamma_0(M))$ (resp., $f \in
S_k(\Gamma_0(M))$) and recalling (\ref{vdef}) and (\ref{altvdef}),
it is not hard to verify using the functional equation
(\ref{modfunc}) that $f|V(d) \in M_k(\Gamma_0(dM))$ (resp.,
$f|V(d) \in S_k(\Gamma_0(dM))$).  These forms are holdovers from
lower levels; they're nothing new, which justifies the notation
$$
S_k(\Gamma_0(N)) \supset
S_k^{\textrm{old}}(\Gamma_0(N)):=\bigoplus_{dM |
N}S_k(\Gamma_0(M))|V(d).
$$
We define \emph{the space of newforms $\Sn_k(\Gamma_0(N))$} to be
the orthogonal complement to $\So_k(\Gamma_0(N)$ with respect to a
certain inner product, called the \emph{Petersson inner product}
(see \cite[\S III.4]{La} or \cite[\S III.3]{k}).  As a first
example, for $p \geq 3$ prime, we have
\begin{eqnarray} \label{decomp1}
 M_2(\Gamma_0(p))=\langle
E_2(z)-pE_2(pz) \rangle \oplus \Sn_2(\Gamma_0(p))
\end{eqnarray}
because $M_2(\Gamma)=0$.  One can check that $E_2(z)-pE_2(pz)$
satisfies the requisite functional equation using (\ref{e2}). For
arbitrary weights, the space of newforms has the useful property
that it is preserved under the action of the Hecke operators.  It
is also invariant under another operator, the Atkin-Lehner
involution, which we now define.
\begin{definition} For a prime divisor $p$ of $N$ with
$\ord_p(N)=\ell$, let $Q_p:=p^{\ell}$.  We define the
\textbf{Atkin-Lehner operator $|_k W(Q_p)$} on $M_k(\Gamma_0(N))$
by any matrix
$$
W(Q_p):=\left( \begin{smallmatrix} Q_pa & b \\ N c & Q_pd
\end{smallmatrix} \right) \in M_{2 \times 2}(\ZZ)
$$
with determinant $Q_p$, where $a,b,c,d \in \ZZ$.  Further, define
the \textbf{Fricke involution $|_k W(N)$} on $M_k(\Gamma_0(N))$ by
the matrix
$$
W(N):=\begin{pmatrix} 0 & -1 \\ N & 0 \end{pmatrix}.
$$
\end{definition}
\noindent Well-definition of $|_k W(Q_p)$ follows from the
functional equation of $f \in M_k(\Gamma_0(N))$ and the fact that
$W(Q_p)$ is unique up to left multiplication by elements of
$\Gamma_0(N)$.  We note here that for $f \in M_k(\Gamma_0(p))$ we
have $f|_kW(Q_p)=f|_kW(p)$.  By abuse of language, we will call
$W(p)$ an Atkin-Lehner operator in this setting.

We now are in a position to make the following:
\begin{definition} A \textbf{newform} in $\Sn_k(\Gamma_0(N))$ is a
normalized cusp form that is an eigenform for all the Hecke
operators, all of the Atkin-Lehner involutions $|_k W(Q_p)$ for
$p|N$, and the Fricke involution $|_k W(N)$.
\end{definition}

Newforms enjoy remarkable properties.  We recall a few such
properties on the more utilitarian side of things:
\begin{theorem} \label{newforms}
Suppose that $k$ is a positive even integer.  Then
\begin{enumerate}
\item  The space $\Sn_k(\Gamma_0(N))$ has a basis of newforms.

\item  If $f(z)=\sum_{n=1}^{\infty} a(n)q^n \in
\Sn_k(\Gamma_0(N))$ is a newform, then there is a number field $K$
with the property that for every integer $n$ we have $a(n) \in
\OO_K$, the ring of algebraic integers of $K$.

\item If $f \in \Sn_k(\Gamma_0(N))$ is a newform then there is an
integer $\lambda_f \in \{\pm1\}$ for which
$$
f|_kW(Q_p)=\lambda_pf.
$$
\end{enumerate}
\end{theorem}
\noindent For the statements of a collection of results, including
the above, on newforms, see \cite[\S 2.4,\S 2.5]{O}.  For proofs,
see \cite{AL}, and for generalizations, see \cite{L} and \cite{M}.

We began this section by discussing how one can raise the level of
an element of $M_k(\Gamma_0(N))$ to obtain an element of
$M_k(\Gamma_0(MN))$.  We now discuss the \emph{trace operator
$\textrm{Tr}^{MN}_N$}, which lowers the level.  For coprime $M,N$,
define
$$
\Tr^{MN}_N:M_k(\Gamma_0(MN)) \to M_k(\Gamma_0(N))
$$
by
$$
\Tr^{MN}_N(f) =\sum_{i=1}^rf|_k \gamma_i
$$
where $\{\gamma_1,...,\gamma_r\}$ is a complete set of coset
representatives for $\Gamma_0(NM) \backslash \Gamma_0(N)$.  The
fact that $\Tr^{MN}_N(f) \in M_k(\Gamma_0(N))$ is immediate;
acting on $\Tr^{MN}_N(f)$ by an element of $\Gamma_0(N)$ simply
permutes the $\gamma_i$ by the invariance of $f$ under the action
of $\Gamma_0(NM)$.  We have the following explicit formula for
$\Tr^{Np}_p$:
\begin{lemma}[\cite{MO}] \label{tr} Suppose that $p$ is an odd prime and that $p
\nmid N$.  If $f \in M_k(\Gamma_0(Np))$ then
$$
\Tr_N^{Np}(f)=f+p^{1-k/2}f|_kW(p)U(p)
$$
\end{lemma}
\begin{proof}
A complete set of coset representatives for $\Gamma_0(Np)$ in
$\Gamma_0(N)$ is given by
$$
\left\{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\right\} \cup
\left\{
\begin{pmatrix} 1 & 0 \\ N & 1 \end{pmatrix}\begin{pmatrix} 1 & j \\ 0 &
1\end{pmatrix}\right\}_{j=0}^{p-1}.
$$
We also have
$$
\begin{pmatrix} 1 & 0 \\ N & 1\end{pmatrix} \begin{pmatrix} 1 & j
\\ 0 & 1
\end{pmatrix}= \begin{pmatrix} 1/p & 0 \\ 0 & 1/p \end{pmatrix}
\begin{pmatrix} p & a \\ Np & pb \end{pmatrix} \begin{pmatrix} 1 &
j-a \\ 0 & p
\end{pmatrix}
$$
where
$$
\begin{pmatrix} p & a \\ Np & pb \end{pmatrix}
$$
is a matrix for $W(p)$.  Since scalar matrices act trivially on
$M_k(\Gamma_0(Np))$,
$$
\Tr^{MN}_N(f) = f + \sum_{j=1}^{p-1} f|_k W(p) \begin{pmatrix} 1 &
j \\ 0 & p
\end{pmatrix}.
$$
By considering $q$-expansions, we have
$$
\sum_{j=0}^{p-1} g\left( \frac{z+j}{p}\right)=p(g|U(p))(z),
$$
which completes the proof of the lemma.
\end{proof}
It is well-known that if $p$ is prime with $p \nmid N$, then
$\Tr_N^{Np}(f)=0$ for $f \in \Sn_k(\Gamma_0(Np))$ (see \cite{L}).
Combining this observation with Lemma \ref{tr} yields the
following:
\begin{proposition}[\cite{AL}] \label{wphecke} If $f \in
\Sn_k(\Gamma_0(p))$, then
$$
f|_kW(p)=-p^{1-k/2}f|U(p).
$$
\end{proposition}
\begin{proof}
First suppose that $f$ is a newform.  From Lemma \ref{tr}, we have
$$
0=\Tr_1^{p}(f)=f+p^{1-k/2}f|_kW(p)U(p).
$$
Thus
\begin{equation} \label{needitnow}
f=-p^{1-k/2}f|_kW(p)U(p).
\end{equation}
 Note that $U(p)=T_{k,p}$ because the
level is $p$ (see (\ref{uvhecke})). Now note that $f$, being a
newform, is an eigenform both for the Hecke operators and $W(p)$
(by Theorem \ref{newforms}).  Thus the actions of $W(p)$ and
$U(p)$ on $f$ commute.  With all this in mind, applying $W(p)$ to
both sides of (\ref{needitnow}), we have
\begin{eqnarray*}
f|_kW(p)&=&-p^{1-k/2}f|_kW(p)U(p)W(p) \\
&=&-p^{1-k/2}f|_kW(p)^2U(p) \\
&=&-p^{1-k/2}f|_kU(p).
\end{eqnarray*}
To derive the last equality, we used the fact that the action of
$W(p)^2$ is trivial, which can be seen from directly from a matrix
representation of $W(p)$: $\left(\begin{smallmatrix} 0 & -1 \\ p &
0
\end{smallmatrix}\right) \left(\begin{smallmatrix} 0 & -1 \\ p & 0
\end{smallmatrix}\right)=\left(\begin{smallmatrix} -p & 0 \\ 0 &
-p
\end{smallmatrix}\right)$.  Since $U(p)$ and $W(p)$ are both
linear operators, the proposition now follows for all $f \in
\Sn_k(\Gamma_0(p))$.
\end{proof}


\section{$p$-adic properties of Borcherds exponents} \label{pborc}
We begin with the following:
\begin{definition} Let $f$ be a meromorphic modular form of weight $k$ over $\Gamma$ or
$\Gamma_0(p)$
 whose poles and zeros, away from $z=\infty$,
are at the points $z_1,...,z_s \in \mathbb{H}$. We say that $f(z)$
is \emph{\textbf{good at $p$}} if there is a holomorphic modular
form $\mathcal{E}_f(z) \in M_b(\Gamma)$ with $p$-integral
algebraic coefficients for which the following are true:
\begin{enumerate}
\item As $q$-series, $\mathcal{E}_f(z) \equiv 1 \pmod{p}$. \item
For each $1 \leq i \leq s$ we have $\mathcal{E}_f(z_i)=0$.
\end{enumerate}
\end{definition}

\begin{remark}[1] It follows immediately that if $f$ and $g$ are
good, then $fg$ is good.
\end{remark}

\begin{remark}[2] As
mentioned in the introduction, we will provide several families of
good forms in \S \ref{good}; other families are provided in
\cite{BrO}. Unfortunately, the author has not thought carefully
about interesting examples of forms which are not good.
\end{remark}

In view of the observations we made in sections \ref{intro} and
\ref{Rth}, it is now straightforward to prove Theorem
\ref{modelth}:

\begin{proof}[Proof of Theorem \ref{modelth}]
By examining the proof of Proposition \ref{thetobs}, we see that
if $f$ is a meromorphic modular form of weight $k$ over $\Gamma$,
then
\begin{equation} \label{fonow}
\widetilde{f}:=12 \Theta f(z)-kE_2(z) f(z)
\end{equation}
is a meromorphic modular form of weight $k+2$ over $\Gamma$.
Further, from (\ref{fonow}) we see that the poles of
$\widetilde{f}(z)$ are supported at the poles of $f(z)$.

Now consider
$$
\frac{\theta f}{f}
=\frac{1}{12}\left(\frac{\widetilde{f}(z)}{f(z)}+kE_2(z) \right).
$$
By \ref{pmoddy}, $E_2$ is a $p$-adic modular form of weight $2$
with integer coefficients.  Thus it suffices to show that
$\widetilde{f}/f$ is as well.  If $b$ is the weight of
$\mathcal{E}_f(z)$, then note
$\mathcal{E}_f(z)^{p^j}\widetilde{f}/f \in M_{p^jb+2}$. If
$\mathcal{E}_f(z)^{p^j}\widetilde{f}/f$ does not have algebraic
integer coefficients, then multiply it by a suitable integer
$t_{j+1} \equiv 1 \pmod{p^{j+1}}$ so that the resulting series
does.  Thus we have
$$
t_{j+1}\mathcal{E}_f(z)^{p^j}\frac{\widetilde{f}}{f} \equiv
\frac{\widetilde{f}}{f} \pmod{p^{j+1}}.
$$
If we define
$F_{j+1}(z):=t_{j+1}\mathcal{E}(z)^{p^j}\widetilde{f}(z)/f(z)$,
then we have that $\{F_{j+1}\}$ is a sequence of holomorphic
modular forms whose coefficients $p$-adically converge to
$\widetilde{F}(z)/F(z)$ and whose weights $p$-adically converge to
$2$.
\end{proof}
We will devote the rest this section to proving Theorem
\ref{gbro}, a generalization of Bruinier and Ono's result to forms
of prime level $p \geq 5$. We require two lemmas before we start
on the main body of the proof.  The first is most naturally proven
using the notion of the divisor polynomial of a modular form,
which we now recall. If $k\geq 4$ is even, then define
$\widetilde{E}_k(z)$ by
\begin{equation}\label{Wtag2.5}
\widetilde{E}_k(z):=\begin{cases} 1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 0\pmod{12},\\
E_{4}(z)^2E_6(z) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12},\\
E_4(z)\ \ \ \ \ &{\text {\rm if}}\ k\equiv 4\pmod{12},\\
E_6(z)\ \ \ \ \ &{\text {\rm if}}\ k\equiv 6\pmod{12},\\
E_4(z)^2\ \ \ \ \ &{\text {\rm if}}\ k\equiv 8\pmod{12},\\
E_{4}(z)E_6(z) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 10\pmod{12},
\end{cases}
\end{equation}
and polynomials $h_k$ by
\begin{equation}
h_k(x):=\begin{cases}   1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 0\pmod{12},\\
            x^2(x-1728) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12},\\
               x \ \ \ \ \ &{\text {\rm if}}\ k\equiv 4\pmod{12},\\
               x-1728 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 6\pmod{12},\\
               x^2 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 8\pmod{12},\\
               x(x-1728) \ \ \ \ \ &{\text {\rm if}}\ k\equiv
               10\pmod{12}.
 \end{cases}
\end{equation}

Further, define $m(k)$ by
$$
m(k):=\begin{cases}
 \lfloor k/12\rfloor \ \ \ \ \ &{\text {\rm if}}\ k\not \equiv 2\pmod{12},\\
           \lfloor k/12\rfloor -1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12}.
\end{cases}
$$
With this notation, if $f(z) \in M_k$ and $\widetilde{F}(f,x)$ is
the unique rational function in $x$ for which
\begin{equation}
\label{divyopoly}
f(z)=\Delta(z)^{m(k)}\widetilde{E}_k(z)\widetilde{F}(f,j(z)),
\end{equation}
then $\widetilde{F}(f, x)$ is a polynomial; this follows from the
familiar fact that any element of $\minf_0$ is a polynomial in
$j$.  We will refer to
\begin{equation} \label{divisorpoly}
F(f,x):=h_k(x)\widetilde{F}(f,x)
\end{equation}
as the \emph{divisor polynomial} for $f$. From (\ref{Wtag2.5}),
(\ref{divyopoly}) and the classical $k/12$ valence formula (again,
see \cite[\S III.2]{k}) the polynomial $F(f,x)$ will have a zero
of order $n_k$ precisely at $j(z_k)$ for all zeros $z_k$ of $f$,
where $n_k:=\ord_{z_k}(f)$. For a discussion of divisor
polynomials, see \cite[\S 2.6]{O}.
\begin{lemma} \label{pint}
Suppose $f =q^h\prod_{n=1}^\infty(1-q^n)^{c(n)} \in
\mmer_k(\Gamma_0(p)) \cap q^h\OO_K[[q]]$ for some number field $K$
 and some prime $p \geq 5$, and further that $f$ is good at $p$.  Then
$$
\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p) \in
\mmer_2(\Gamma_0(p))
$$
is $p$-integral.
\end{lemma}
\begin{proof}  Note that $F(\mathcal{E}_f,j)$ has $p$-integral algebraic
coefficients as a $q$-series and as a polynomial because
$\mathcal{E}_f$ has $p$-integral algebraic coefficients.  Thus, if
$z_1,...,z_n$ are the zeros and poles of $f$ as before (written
without multiplicity),
$$
G(j(z)):=(j(z)-j(z_1))\cdots(j(z)-j(z_n))
$$
has $p$-integral algebraic $q$-series coefficients.  Because no
prime above $p$ divides the $q$-expansion coefficient of lowest
exponent in $G(j(z))$, we also have that $(G(j))^{-1}$ is
$p$-integral (again as a $q$-series). Thus we may write
$$
\frac{\Theta(f)-k(12)^{-1}E_2}{f}=\frac{g}{G(j)}
$$
where $g \in M_2(\Gamma_0(p)) \cap \overline{\QQ} [[q]]$ has
$p$-integral algebraic coefficients.  We have
$$
\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p)=\left(\frac{1}{G(j)}\right)|_0W(p)g|_2W(p).
$$
We will prove that each of the factors on the right hand side is
$p$-integral.  First,
\begin{eqnarray*}
\left(\frac{1}{G(j(z))}\right)|_0W(p)&=&\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
0 & -1 \\ p & 0
\end{matrix}\right)=\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
0 & -1 \\ 1 & 0 \end{matrix}\right)\left(\begin{matrix} p & 0
\\ 0 & 1
\end{matrix}\right)\\ &=&\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
p & 0 \\ 0 & 1
\end{matrix}\right)=\frac{1}{G(j(pz))},
\end{eqnarray*}
which is evidently $p$-integral.  Now note that we can write
$g=c_1(E_2(z)-pE_2(pz))+h(z)$, where $h(z) \in \Sn_2(\Gamma_0(p))$
has $p$-integral algebraic coefficients and $c_1$ is a
$p$-integral algebraic number.  From Proposition \ref{wphecke} we
have $h(z)|_2W(p)=-h(z)|U(p)$, which is $p$-integral by the
$q$-series definition (\ref{udef}) of the $U(p)$ operator.  Using
(\ref{e2}), we also have
\begin{eqnarray*}
(E_2(z)-pE_2(pz))|_2W(p)&=&E_2(z)|_2\begin{pmatrix}0 & -1 \\ 1 & 0
\end{pmatrix}\begin{pmatrix}p & 0 \\ 0 & 1
\end{pmatrix}-p^2(pz)^{-2}E_2(-1/z) \\ &=& \left(\frac{12}{2 \pi i z} + E_2(z)\right)|_2\begin{pmatrix}p & 0 \\ 0 & 1
\end{pmatrix}-\frac{12}{2 \pi i z}
-E_2(z) \\&=&pE_2(pz)-E_2(z).
\end{eqnarray*}
which is also $p$-integral.  Since we have dealt with both
factors, the lemma follows.
\end{proof}
\begin{remark} If restrict to the case $k=0$, this lemma is also
true for $p=3$; the proof is the same.
\end{remark}

Define
\begin{equation}
\widetilde{E}_3(z):=E_2(z)-3E_2(3z) \in M_2(\Gamma_0(3))
\end{equation}
(see \ref{decomp1}) and
\begin{equation}
\widetilde{E}_p:=E_{p-1}(z)-p^{(p-1)/2}(E_{p-1}(z)|_{p-1}W(p)) \in
M_{p-1}(\Gamma_0(p))
\end{equation}
for primes $p \geq 5$.  We have following:
\begin{lemma}
If $p$ is an odd prime, then
\begin{eqnarray} \label{twidE1}
\widetilde{E}_p(z) &\equiv& 1 \pmod{p} \\
\label{twidE2} (\widetilde{E}_p(z)|_{p-1}W(p)) &\equiv& 0
\pmod{p^{(p-1)/2+1}}
\end{eqnarray}
\end{lemma}
\begin{proof}  For $p=3$, the first claim is obvious, and the
second follows from the end of the proof of Lemma \ref{pint}. For
$p \geq 5$ we compute
\begin{eqnarray*}
E_{p-1}|_{p-1}W(p) &=&E_{p-1}|_{p-1}\begin{pmatrix} 0 & -1
\\ p & 0 \end{pmatrix} \\
&=& E_{p-1}|_{p-1}\begin{pmatrix} 0 & -1
\\ 1 & 0 \end{pmatrix} \begin{pmatrix} p & 0
\\ 0 & 1 \end{pmatrix} \\
&=& p^{(p-1)/2}E_{p-1} | V(p)
\end{eqnarray*}
>From Lemma \ref{pmoddy}, we know that $E_{p-1}$ is $p$-integral.
Thus we have the congruence $\widetilde{E}_p \equiv E_{p-1}
\pmod{p}$, which yields $\widetilde{E}_p \equiv 1 \pmod{p}$ for
all odd primes $p$ after an application of Lemma \ref{pmoddy}.

For the second claim, we have
\begin{eqnarray*}
\widetilde{E}_p|_{p-1}W(p) &=&
E_{p-1}|_{p-1}W(p)-p^{(p-1)/2}E_{p-1}|_{p-1}W(p)W(p) \\
&=& E_{p-1}|_{p-1}W(p)-p^{(p-1)/2}E_{p-1} \\
&=& p^{(p-1)/2}E_{p-1} | V(p)-p^{(p-1)/2}E_{p-1}.
\end{eqnarray*}
We note the $p$-integrality of $E_{p-1}$ and $E_{p-1}|V(p)$ and
again apply Lemma \ref{pmoddy} to finish the proof of the lemma.
\end{proof}

We now prove Theorem \ref{gbro}.  The two main inputs into this
proof are the ideas behind the proof of Theorem \ref{modelth} and
Serre's proof that a newform in $\Sn_k(\Gamma_0(p))$ is a $p$-adic
modular form (see \cite{MO} and \cite{Sep}).
\begin{proof}[Proof of Theorem \ref{gbro}]
By (\ref{thetobs}), there exists a meromorphic modular form $g$ on
$\Gamma_0(p)$ so that
$$
\frac{\Theta f}{f}=\frac{g}{f}+\frac{k}{12}E_2.
$$
Because $E_2$ is a $p$-adic modular form of weight two, it
suffices to show that the same is true of $\frac{\Theta
f-k(12)^{-1} E_2 f}{f}=\frac{g}{f}$.

Fix a positive integer $r$.  Then (using the fact that $f$ is good
at $p$), we have
$$
(\mathcal{E}_f)^{p^{r-1}} \frac{\Theta f-k(12)^{-1} E_2 f}{f} \in
M_{2+p^{r-1}b}(\Gamma_0(p))
$$
where $b$ is the weight of $\mathcal{E}_f$.  Further, this form is
congruent modulo $p^r$ to $g/f$. Now consider
$$
f_r(z):=(\widetilde{E}_p)^{p^{r-1}}(\mathcal{E}_f)^{p^{r-1}}
\frac{\Theta f-k(12)^{-1} E_2 f}{f} \equiv \frac{g}{f} \pmod{p^r}.
$$
We clearly have $f_r \in M_{2+p^{r-1}b+p^r-p^{r-1}}(\Gamma_0(p))$.
We now take the trace of these $f_r$ to lower their level. We
certainly have $\Tr_1^p(f_r) \in M_{2+p^{r-1}b+p^r-p^{r-1}}$, and
we will prove shortly that $\Tr_1^p(f_r) \equiv f_r \equiv
\frac{g}{f} \pmod{p^r}$.  Now, as in the proof of Theorem
\ref{modelth}, choose a suitable integer $t_r \equiv 1
\pmod{p^{r}}$ such that $t_r\Tr_1^p(f_r)$ has coefficients in the
ring of integers $\OO_{K_r}$ of some number field
$\mathcal{O}_{K_r}$ (this normalization may or may not be
necessary depending on $\mathcal{E}_f$). Then
$\{t_r\Tr_1^p(f_r)\}$ forms a sequence of holomorphic modular
forms over $\Gamma$ whose coefficients converge $p$-adically to
$g/f$ and whose weights converge to $2$, thus $g/f$ is a $p$-adic
modular form of weight $2$.

We now prove that $\Trace_1^p(f_r) \equiv f_r \pmod{p^{r}}$. By
Lemma \ref{tr}, we have
\begin{eqnarray*}
\Tr_1^p(f_r)&=&f_r+p^{1-(2+p^{r-1}b+p^r-p^{r-1})/2}f_r|_{(2+p^{r-1}b+p^r-p^{r-1})/2}W(p)U(p)
\\ &=&f_r+p^{1-(2+p^{r-1}b+p^r-p^{r-1})/2} \\ &&\times \left(\left(\frac{\Theta f-k(12)^{-1} E_2 f}{f}\right)|_2W(p)
(\widetilde{E}_p)^{p^{r-1}}|_{p^r-p^{r-1}}W(p)(\mathcal{E}_f)^{p^{r-1}}|_{p^{r-1}b}W(p)\right)U(p)
\end{eqnarray*}
Because $f$ is good, applying Lemma \ref{pint} implies that
$\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p)$ is
$p$-integral, which, together with the definition of $U(p)$,
implies that
\begin{eqnarray} \label{1ord}
\left(\frac{\Theta f-k(12)^{-1} E_2 f}{f}\right)|_2W(p)U(p)
\end{eqnarray}
is $p$-integral.  Using (\ref{twidE2}), we also compute
\begin{eqnarray} \label{2ord}
\nonumber &&\widetilde{E}^{p^{r-1}}|_{p^r-p^{r-1}}W(p)U(p)
\\ &=&(\widetilde{E}_p|_{p-1}W(p))^{p^{r-1}}|U(p) \equiv 0 \pmod{p^{(p-1)p^{r-1}/2+p^{r-1}}},
\end{eqnarray}
and, just from the definitions,
\begin{eqnarray} \label{3ord}
&&
(\mathcal{E}_f)^{p^{r-1}}|_{p^{r-1}}W(p)U(p)\\
\nonumber &=&
p^{p^{r-1}b/2}(\mathcal{E}_f)^{p^{r-1}}|_{p^{r-1}b}\left(\begin{smallmatrix}
p & 0 \\ 0 & 1
\end{smallmatrix}\right)U(p) \equiv 0 \pmod{p^{p^{r-1}b/2}}.
\end{eqnarray}
The inequalities (\ref{1ord}), (\ref{2ord}) and (\ref{3ord})
together imply (as claimed) that $\Tr_1^p(f_r) \equiv f_r
\pmod{p^{r}}$.
\end{proof}
\begin{remark} As with Lemma \ref{pint}, Theorem \ref{gbro} is
true in the case $k=0$ for $p=3$ as well.  We will use this fact
without further comment in the proof of Theorem
\ref{classnumber2}.
\end{remark}

\section{CM elliptic curves and supersingularity}
\label{cm}

As indicated in the introduction, the construction of explicit
families of good forms will require a discussion of complex
multiplication and supersingularity, which we now begin. Recall
that for an elliptic curve $E/\CC$, there exists a lattice $L
\subset \CC$ such that
\begin{eqnarray} \label{Eisom}
\CC/L &\widetilde{\longrightarrow}& E \\ \nonumber z \not \in L
&\mapsto& (\wp(z,L),\wp'(z,L),1) \\ \nonumber z \in L &\mapsto&
(0:1:0)
\end{eqnarray}
is an analytic isomorphism.  Here $\wp$ is the classical
Weierstrass $\wp$-function.  Conversely, given any lattice $L
\subset \CC$, one can show that there exists an elliptic curve $E$
for which an analytic isomorphism of the form (\ref{Eisom}) holds.
Under this correspondence between lattices and elliptic curves,
isomorphism classes of elliptic curves over $\CC$ correspond to
equivalence classes of lattices, where the equivalence is given by
$L \sim L'$ if $L=cL'$ for some $c \in \CC^*$.  By way of
terminology, the map $L' \to L$ given by multiplication by $c \in
\CC^*$ is called a \emph{homothety}, and two lattices related in
such a way are called \emph{homothetic}.
 Note that we may choose a lattice $L_{\tau}$ with basis $\{ \tau,
 1\}$ with $\tau \in \HH$ in each homothety class. Different bases of
 $L_{\tau}$ are given by applying elements of
 $\Gamma$ to the basis $\{ \tau,
 1\}$; it follows that we may take $\tau \in
 \mathfrak{F}$.  With this stipulation, the basis $\{\tau,1\}$
 is uniquely determined. We will denote
 by $E_{\tau}$ the corresponding elliptic curve under the map
$$
\CC/L_{\tau} \to E_{\tau}.
$$
We call this map (which is induced by (\ref{Eisom})) an
\emph{analytic representation} of $E_{\tau}$.

We now wish to make this analytic representation more explicit;
additionally, because it will be useful later, we work in a
slightly more general context. Let $E/K$ be an elliptic curve over
a field $K$ of characteristic not equal to $2$ or $3$. Up to
isomorphism,
 we can assume that $E$ is given in affine coordinates by
\begin{equation} \label{affequ}
E:y^2=4x^3-g_4x-g_6
\end{equation}
(see, for example, \cite[\S III.2]{kn}).  If we restrict to the
case $K=\CC$, with the normalizations given above, the map
(\ref{Eisom}) just formalizes the parametrization
$$
E:(\wp'(z,L))^2=4(\wp(z,L))^3-g_4\wp(z,L)-g_6.
$$
that exists for some lattice $L \subset \CC$.

We now wish define the $j$-invariant of $E_{\tau}$, and show how
it relates to $j(\tau)$. First, the \emph{discriminant
$\Delta(E)$} of the elliptic curve $E/k$ is defined as
\begin{equation} \label{deltaE}
\Delta(E)=(2 \pi)^{-12}(g_4^3-27g_6^2).
\end{equation}
\begin{remark}
It is important to observe that the discriminant function
$\Delta(E)$ is \emph{not} equal to the discriminant of the cubic
polynomial defining the curve.  Since the discriminant of the
polynomial defining an elliptic curve $E$ is \emph{not} an
isomorphism invariant of $E$, there are a variety of essentially
equivalent ways to define the discriminant; the reason for our
particular definition will soon be apparent.
\end{remark}
We define the \emph{$j$-invariant of $E$} to be the quantity
\begin{equation} \label{je}
 j(E):=\frac{1728g_4^3}{(2 \pi)^{12}\Delta(E)}.
\end{equation}
One can show by elementary means over any field $K$ of
characteristic not equal to $2$ or $3$ that $j(E)$ is indeed an
invariant of the isomorphism class of $E$, and, further, given any
$j(E) \in K$, there exists a curve of $j$-invariant $j(E)$ (see
\cite[\S III.2]{kn}).

Note the similarity of (\ref{je}) and (\ref{modjdef}). This is no
accident.  Let $\CC/L_{\tau} \to E_{\tau}$ be an analytic
representation. It turns out that, with the normalizations given
above, we have $g_2=\frac{4}{3}\pi^4E_4(\tau)$,
$g_3=\frac{8}{27}\pi^6E_6(\tau)$. Hence, we have
$$
\Delta(E)=\frac{(E_4(\tau)^3-E_6(\tau)^2)}{1728}=\Delta(\tau)
$$
and
\begin{eqnarray} \label{jcoincid}
j(E_{\tau})=j(\tau).
\end{eqnarray}
Thus the coincidence of the ``$j$" in $j$-function and
$j$-invariant is really no coincidence. Indeed, noting the fact
that as the $j$-invariant varies over $K$ it parameterizes
isomorphism classes of elliptic curves over $K$ (at least if we
continue to assume that the characteristic of $K$ is not $2$ or
$3$), and recalling that the $j$-function is a bijection between
$\mathfrak{F}$ and $\CC$, we have a bijective map
$$
\mathfrak{F} \longleftrightarrow \left\{ \textrm{isomorphism
classes of } E/ \CC  \right\}.
$$
\noindent For proofs of the statements we just made on the
equality of the various definitions of $j$ and $\Delta$, see
\cite[\S I and p. 112]{k}.  For a basic introduction to the theory
of elliptic curves, see \cite{kn}.

Later we will be giving examples of elliptic curves in the form
$E:y^2=x^3+ax+b$ for some $a,b \in k$.  It is easy to see that
given any elliptic curve over $k$ with defining affine equation
$y^2=4x^3+cx+d$, if the characteristic of $k$ is not $2$, then
this curve is isomorphic to a curve with defining affine equation
$y^2=x^3+\widetilde{c}x+\widetilde{d}$ for some
$\widetilde{c},\widetilde{d} \in K$. We shall call an elliptic
curve written in this form an elliptic curve in \emph{Weierstrass
form}. We now write formally as a proposition some elementary
properties of curves written in Weierstrass form; for a proof, see
\cite[\S III.2]{kn}

\begin{proposition} \label{explicitj}
Suppose $a,b \in K$, where $K$ is a field with characteristic not
equal to $2$ or $3$.  Then the discriminant of $x^3+ax+b$ is
$-4a^3-27b^2$. If the discriminant is nonzero, then
$E:y^2=x^3+ax+b$ is nonsingular. Further, the $j$-invariant of
$E:y^2=x^3+ax+b$ is $1728\frac{4a^3}{4a^3+27b^2}$.
\end{proposition}

Now that we have (\ref{Eisom}) and the isomorphism invariant
$j(E)$ in hand, we completely understand isomorphism classes of
elliptic curves over $\CC$ considered as analytic objects; they
are explicitly parameterized by $\wp(z,L_{\tau})$ (considered as a
function of $\tau \in \mathfrak{F}$). For example, define
\emph{$E[N]$, the $N$-division points of $E$}, to be the points of
$E$ of order dividing $N$. Viewing $E/\CC$ as $\CC/L_{\tau}$, it
is evident that $E[N]$ is simply the group
$\frac{1}{N}L_{\tau}/L_{\tau}$, that is,
$$
E[N] \approx \ZZ/N\ZZ \times \ZZ/N \ZZ.
$$
The ring of endomorphisms of $E$, or $\eo (E)$, can also be
understood in a relatively straightforward manner using analytic
representations.  To begin, we have the following:
\begin{lemma} \label{multilam} Let $L,M$ be two lattices in $\CC$, and let
$$
\lambda: \CC/L \to \CC/M
$$
be a complex analytic homomorphism.  Then there exists a complex
number $\alpha$ so that the following diagram commutes:
\begin{eqnarray*}
\begin{matrix}
\alpha: & \CC & \rightarrow & \CC \\
& \downarrow & & \downarrow \\
\lambda: & \CC/L& \rightarrow & \CC/M.
\end{matrix}
\end{eqnarray*}
Here the top map is multiplication by $\alpha$ and the bottom is
the homomorphism $\lambda$.
\end{lemma}

\begin{proof}[Proof (compare \cite{L})] In a neighborhood of
zero, $\lambda$ can be expressed by a power series
$$
\lambda(z) = a_0 + a_1 z +a_2z^2 + \cdots,
$$
On the other hand, $\lambda$ is a homomorphism, so $a_0=0$ and
additionally we have
$$
\lambda(z+z') \equiv \lambda(z) + \lambda(z') \pmod{M}.
$$
If we choose a small enough neighborhood $U$ of zero, we must have
that this congruence is an equality in $U$; thus
$$
\lambda(z)=a_1z
$$
for $z \in U$.  But for any $z \in \CC$, $z/n$ is in $U$ for
sufficiently large integers $n$, and from this we conclude that,
identifying $z$ with its reduction modulo $L$,
$$
\lambda(z)=\lambda
\left(n\left(\frac{z}{n}\right)\right)=n\lambda\left(\frac{z}{n}\right)=na_1\left(\frac{z}{n}\right)=a_1z.
$$
\end{proof}
\begin{remark} Abusing notation, we will often denote the
complex number $\alpha$ and the homomorphism $\lambda$ by the same
symbol $\lambda$.  We will also only be considering the special
case $L=M$ of Lemma \ref{multilam}.\end{remark}

It is clear that any $\lambda \in \ZZ$ will induce an endomorphism
of $\CC/L_{\tau}$, which we can then identify with an element of
$\eo (E_{\tau})$.  We will call these endomorphisms the
\emph{trivial endomorphisms of $E_{\tau}$}.  We have the
following:
\begin{definition}
  If $E/\CC$ is an elliptic curve with nontrivial elements in its
  endomorphism ring $\eo(E/\CC)$, then we say \textbf{$E$ is a curve
  with complex multiplication}, or, briefly, \textbf{$E$ has CM}.
\end{definition}
The complex numbers $\lambda$ inducing a nontrivial endomorphism
of a lattice $L$ turn out to be algebraic numbers; more
specifically, they are quadratic over $\QQ$.  Before we formalize
and prove this as a proposition, we offer another definition,
which will also be useful in \S \ref{good}:
\begin{definition}
Suppose $\tau \in \HH$ is the root of a quadratic equation with
integer coefficients; that is, $\tau=\frac{-b+\sqrt{b^2-4ac}}{2a}$
with $a,b,c \in \ZZ$ and $\gcd(a,b,c)=1$.  We say that $\tau$ is a
\textbf{Heegner point} and that $d_{\tau}=b^2-4ac$ is the
\textbf{discriminant} of $\tau$.
\end{definition}

\begin{proposition} \label{cmequiv}
Suppose $E/\CC$ is an elliptic curve.  Then
\begin{enumerate}
\item Every nontrivial endomorphism of $E/\CC$ is induced (in the
sense of Theorem \ref{multilam}) either by a Heegner point
$\lambda \in \HH$ or by $-\lambda$ for a Heegner point $\lambda
\in \HH$.

\item The curve $E/\CC$ has CM if and only if $j(E)=j(\tau)$ for
some Heegner point $\tau \in \mathfrak{F}$.

\item The curve $E/\CC$ has CM if and only if $\eo (E) \cong
\mathcal{O}$, where $\mathcal{O}$ is an order in an imaginary
quadratic number field $K$.
\end{enumerate}
\end{proposition}
\begin{proof}
The endomorphism ring of $E$ is unchanged if we replace it with
another elliptic curve isomorphic to it, so we assume without loss
of generality that $E=E_{\tau}$, $\tau \in \mathfrak{F}$. Thus we
have an analytic representation
$$
\CC/L_{\tau} \to E_{\tau}.
$$
As we proved in Lemma \ref{multilam}, a nontrivial automorphism of
$E_{\tau}$ can now be realized as a $\lambda \in \CC^*-\ZZ$ such
that
$$
\lambda L_{\tau} \subset L_{\tau}
$$
or, equivalently, for some $\left(\begin{smallmatrix} a & b \\
c& d
\end{smallmatrix} \right) \in \textrm{GL}_2(\QQ) \cap M_{2 \times 2}(\ZZ)$,
\begin{eqnarray*}
\lambda \tau &=& a \tau+b \\
\lambda  &=& c \tau +d.
\end{eqnarray*}
This implies that $\lambda$ is a root of the quadratic equation
\begin{equation*}
\left|\begin{matrix} x-a & -b \\ -c & x-d \end{matrix}\right|=0.
\end{equation*}

Thus $\lambda$ is a quadratic irrational algebraic integer. Now
note that $\tau$ cannot be real; otherwise $L_{\tau}$ would not be
a lattice, and $c \neq 0$, for then $\lambda$ would be an integer.
Thus $\QQ(\tau)=\QQ(\lambda)$, and, further, both $\lambda$ and
$\tau$ are imaginary quadratic numbers.  This proves (1).

We've also proven the ``only if" implication of (2), just by
recalling that $j$ is an isomorphism invariant. The other
direction follows similarly: note that if $j(E)=j(\tau)$ with
$\tau$ a Heegner point, then $E_{\tau} \approx E$, and $E_{\tau}$
is evidently CM.

Finally, for (3), note that if $E$ is CM, as proven above, there
is an isomorphic curve $E_{\tau}$ where $\tau$ is a Heegner point.
Thus $\eo (E) \approx \eo (L_{\tau})$, and, again as proven above,
any complex number inducing a nontrivial endomorphism of
$L_{\tau}$ is an element of $\mathcal{O}_{\QQ(\tau)}$, the ring of
integers of $\QQ(\tau)$, but not an element of $\ZZ$.  With this
observation in mind it is easy to see that the evident map $\eo
(L_{\tau}) \to \mathcal{O}_{\QQ(\tau)}$ is a homomorphism of rings
with identity, and, further, the image of this homomorphism is not
contained in $\ZZ \subset \mathcal{O}_{\QQ(\tau)}$.  Thus
$\eo(L_{\tau}) \approx \eo (E_{\tau})$ is isomorphic to an order
in $\mathcal{O}_{\QQ(\tau)}$. Conversely, note that if $\eo (E)
\approx \mathcal{O}$, with $\mathcal{O}$ an order in a quadratic
imaginary field, then $\eo (E)$ is not isomorphic to $\ZZ$, so $E$
must be CM.
\end{proof}

By way of terminology, if $\tau \in \HH$ is a Heegner point, then
$j(\tau) \in \CC$ is called a \emph{singular modulus}.  In view of
parts (3) and (4) of the last proposition, one might guess that
these singular moduli would be of interest in the study of the
arithmetic of imaginary quadratic number fields.  This is indeed
the case, but before we can explain anything in any more detail we
must briefly explore the connection between the CM elliptic curves
and quadratic imaginary fields. The content of our discussion is
derived mostly from \cite[\S II]{Si2}.

We've seen in Proposition \ref{cmequiv} that every CM elliptic
curve has endomorphism ring isomorphic to an order in a quadratic
number field.  We now work in an opposite direction.  Fix an
imaginary quadratic number field $K$, and let $\mathcal{O}_K$ be
its ring of integers.  We wish to study the following sets:
\begin{eqnarray} \label{elldef} \mathcal{ELL}(\mathcal{O}_K):&=&
\frac{\{\textrm{elliptic curves } E/\CC \textrm{ with } \eo(E)
\cong \mathcal{O}_K\}}{\textrm{isomorphism over } \CC}
\\\nonumber &\cong& \frac{ \{ \textrm{lattices } L \textrm{ with
} \eo(L) \cong \mathcal{O}_K \}}{\textrm{homothety}}.
\end{eqnarray}

We now show that these sets are nonempty for any imaginary
quadratic number field $K$. Fix an embedding $K \hookrightarrow
\CC$. Given any nonzero fractional ideal $\mathfrak{a} \subset K$,
we know from elementary algebraic number theory that the image of
$\mathfrak{a}$ under our chosen embedding (which we will also
denote by $\mathfrak{a}$) is a lattice in $\CC$.  Denote by
$E_{\mathfrak{a}}$
 the elliptic curve associated to this lattice.  We have
 \begin{eqnarray*}
 \eo (E_{\mathfrak{a}}) &\cong& \{ \alpha \in \CC : \alpha
 \mathfrak{a} \subset \mathfrak{a} \} \\
 &=& \{\alpha \in K : \alpha \mathfrak{a} \subset
 \mathfrak{a} \} \textrm{ since } \mathfrak{a} \subset K, \\
 &=& \OO_K \textrm{ since } \mathfrak{a} \textrm{ is a
 fractional ideal.}
 \end{eqnarray*}
Thus given $\OO_K$, we can find an elliptic curve $E$ with $\eo
(E) \approx \OO_K$.  Further, since homothetic lattices give rise
to isomorphic elliptic curves, if $c \in K$, then $E_{(c)
\mathfrak{a}} \approx E_{\mathfrak{a}}$.  In other words,
multiplying a fractional ideal by a principal ideal in $\OO_K$
does not change the elliptic curve that arises from that ideal. In
particular, if we denote by $\mathcal{CL}(K)$ the ideal class
group of $K$, that is,
$$
\mathcal{CL}(K) :=\frac{\{\textrm{nonzero fractional ideals of }
K\}}{\{\textrm{nonzero principal ideals of } K\}}.
$$
then we have a map
\begin{eqnarray*}
\mathcal{CL}(K) &\longrightarrow& \mathcal{ELL}(\OO_{K}) \\
\overline{\mathfrak{a}} &\longmapsto& E_{\mathfrak{a}}
\end{eqnarray*}
where $\overline{\mathfrak{a}}$ is the ideal class of
$\mathfrak{a} \in \mathcal{CL}(k)$.  More generally, if $L$ is any
lattice and $\mathfrak{a}$ any nonzero fractional ideal of $K$,
then define the product
$$
\mathfrak{a}L:= \{ \alpha_1 \lambda_1 + \cdots + \alpha_r
\lambda_r: \alpha_i \in \mathfrak{a}, \lambda_i \in L\}.
$$
Now fix a lattice $L$ with $E_{L} \in \mathcal{ELL}(\OO_K)$ its
associated elliptic curve.  One can show in an elementary manner
that the map
\begin{eqnarray} \label{claction}
\mathcal{CL}(k) & \longrightarrow & \mathcal{ELL}(\OO_K) \\
\nonumber \overline{\mathfrak{a}} & \longmapsto &
E_{\mathfrak{a}^{-1}L}
\end{eqnarray}
defines a simply transitive action of $\mathcal{CL}(K)$ on
$\mathcal{ELL}(\OO_K)$ (see Proposition 1.2, \cite[\S II.2]{Si2}).
In particular, because $\mathcal{CL}(K)$ is finite,
$\mathcal{ELL}(\OO_K)$ is as well. This observation is the main
input into Proposition \ref{galconj} below.  Fix the notation
$$
h_K=|\mathcal{CL}(K)|.
$$

For $\sigma \in \textrm{Aut}(\CC)$, let $c^{\sigma}$ denote
$\sigma(c)$ for all $c \in \CC$, and let $E^{\sigma}$ denote the
elliptic curve formed by letting $\sigma$ act on the coefficients
of the defining affine equation of $E$. Further, if $\phi :E \to
E$ is an endomorphism of $E$, then denote by
$\phi^{\sigma}:E^{\sigma} \to E^{\sigma}$ the induced endomorphism
(i.e. isogeny from $E^{\sigma}$ to itself) of $E^{\sigma}$.

\begin{remark} We are implicitly identifying the ring of analytic endomorphisms of $E$, thought of
as a lattice, with the ring of algebraic endomorphisms of $E$,
thought of as group.  It is a fact that these two rings are indeed
isomorphic; see \cite[\S VI.4, Theorem 4.1]{Si}.
\end{remark}

Then we have the following:
\begin{proposition} \label{galconj} Let $E/\CC$ be an representative of a class of elliptic
curves in $\mathcal{ELL}(\OO_K)$ for $\OO_K$ the ring of integers
of an imaginary quadratic field $K$.  We have
\begin{enumerate}
\item $j(E) \in \overline{\QQ}$.

\item Let $E_1,...,E_{h_K}$ be a complete set of representatives
for $\mathcal{ELL}(\OO_K)$.  Then $j(E_1),...,j(E_{h_K})$ are the
$\textrm{Gal}(\overline{K}/K)$ conjugates for $j(E)$.
\end{enumerate}
\end{proposition}
\begin{proof}

Let $\sigma: \CC \to \CC$ be a field automorphism of $\CC$.  First
note that $\eo(E^{\sigma}) \simeq \eo(E)$, simply because if
$\phi:E \to E$ is any endomorphism of $E$, then
$\phi^{\sigma}:E^{\sigma} \to E^{\sigma}$ is an endomorphism of
$E^{\sigma}$.  Thus $\eo(E^{\sigma}) \approx \eo(E)$. In
particular, as $\sigma$ varies, $E^{\sigma}$ varies over only
finitely many $\CC$-automorphism classes of elliptic curves with
endomorphism ring isomorphic to $\mathcal{O}_K$ because the action
(\ref{claction}) is simply transitive and the class group is
finite.

Now $E^{\sigma}$ is obtained from $E$ by letting $\sigma$ act on
the coefficients of the affine equation defining $E$.  The
invariant $j(E)$ is just a rational combination of those
coefficients, so we have
$$
j(E^{\sigma})=j(E)^{\sigma}.
$$
Since the isomorphism class of an elliptic curve is determined by
its $j$-invariant, and, as we've noted above, there are only
finitely many $\CC$-isomorphism classes in $\{E^{\sigma}\}_{\sigma
\in \textrm{Aut}(\CC)}$, it follows that $j(E)^{\sigma}$ takes on
only finitely many values as $\sigma$ ranges over
$\textrm{Aut}(\CC)$.  Therefore $[\QQ(j(E)):\QQ]$ is finite, so
$j(E)$ is an algebraic number.  This completes the proof of (1).

To prove (2), first note that the action of $\mathcal{CL}(K)$ on
$\mathcal{ELL}(\OO_K)$ induces a simply transitive action
$\Psi:\mathcal{CL}(K) \to \{ j(E_1),...,j(E_{h_K}) \}$ if we
identify an isomorphism class of elliptic curves with its
$j$-invariant.  One then defines a surjective homomorphism $\Phi:
\textrm{Gal}(\overline{K}/K) \to \mathcal{CL}(K)$ such that the
canonical action of $\textrm{Gal}(\overline{K}/K)$ on the set $\{
j(E_1),...,j(E_{h_K}) \}$ is just $\Psi \circ \Phi$.  See \cite[\S
II.2]{Si2} for the construction of this homomorphism; the proof of
Theorem 4.3, \cite[\S II.4]{Si2} shows that it has the desired
property. The fact that (\ref{claction}) is simply transitive then
immediately yields the desired result.
\end{proof}

Actually, the $j(E)$ for $E$ as in the previous proposition are
integral, which can be proven by constructing certain polynomials
related to $n$-isogenies of elliptic curves. This proof requires
no more machinery than that which we have already developed (and
in fact many of the ideas behind it will be used in the proof of
Theorem \ref{classnumber2}), but it is rather long, and the reader
would do just as well to read it in \cite[\S II.6]{Si2}. We state
this fact as a theorem:

\begin{theorem} \label{jint} Let $E/\CC$ be an elliptic curve with
complex multiplication.  Then $j(E)$ is an algebraic integer.
\end{theorem}

We now wish to restate Proposition \ref{galconj} using the
language of Heegner points.  Recall that an integer $d \neq 1$ is
a \emph{fundamental discriminant} if it is not divisible by the
square of any odd prime and satisfies either $d \equiv 1 \pmod{4}$
or $d \equiv 8,12 \pmod{16}$. We prove the following lemma:

\begin{lemma} \label{hkcount} Let $d<0$ be a fundamental discriminant, and $K=\QQ(\sqrt{d})$.
Then there are precisely $h_K$ Heegner points of discriminant $d$
in $\mathfrak{F}$.
\end{lemma}
\begin{proof}
Notice that if $\tau=\frac{-b+\sqrt{b^2-4ac}}{2a} \in
\mathfrak{F}$ with $a,b,c \in \ZZ$, $\gcd(a,b,c)=1$, and
$b^2-4ac=d$, then $ax^2+bxy+cy^2$ is a primitive positive definite
quadratic form of discriminant $d$. Since $d$ is fundamental, the
number of such forms is precisely $|\mathcal{CL}(K)|=h_K$.
\end{proof}
\noindent We now have the following corollary of Proposition
\ref{galconj}:

\begin{corollary} \label{jconj}
Let $d<0$ be a fundamental discriminant and
$\tau_1,...,\tau_{h_K}$ be the Heegner points of discriminant $d$
in $\mathfrak{F}$.  Then $j(\tau_i) \in \overline{\ZZ}$ for all
$i$, and $j(\tau_1),...,j(\tau_{h_K})$ is a complete set of Galois
conjugates under the action of $\textrm{Gal}(\overline{K}/K)$.
\end{corollary}
\begin{proof}
First note that the map
\begin{eqnarray} \label{firstbij} \nonumber
\mathfrak{F} &\longleftrightarrow& \frac{ \{ \textrm{lattices } L
\subset \CC \} }{ \textrm{homothety} } \\ z &\longmapsto& [L_z]
\end{eqnarray}
is a bijection. Suppose $\tau \in \mathfrak{F}$ is a Heegner point
of discriminant $d$.  Using (\ref{jcoincid}), we have that
$j(E_{\tau})=j(\tau)$, which implies by part (2) of Proposition
\ref{cmequiv} that $E_{\tau}$ has endomorphism ring isomorphic to
an order in an imaginary quadratic number field.  This implies
that the same is true of $L_{\tau}$.  By the proof of Proposition
\ref{cmequiv}, we may take this imaginary quadratic number field
to be $K$. In fact, $\eo(L_\tau) \cong \OO_K$, the full ring of
integers. To see this, we observe that
$\mathcal{O}_K=\ZZ[\frac{1+\sqrt{d}}{2}]$ if $d$ is odd (resp.,
$\mathcal{O}_K=\ZZ[\frac{\sqrt{d}}{2}]$ if $d$ is even), hence one
need only check that $\frac{1+\sqrt{d}}{2} \cdot \tau \in
L_{\tau}$ (resp., $\frac{\sqrt{d}}{2} \cdot \tau \in L_{\tau}$).
We omit this calculation.

With this claim in hand, (\ref{firstbij}) yields an injection
\begin{eqnarray} \label{heegbij}
\nonumber \{\textrm{Heegner points in } \mathfrak{F} \textrm{ of
discriminant } d\} &\hookrightarrow& \frac{ \{ \textrm{lattices }
L \subset \CC \textrm{ with } \eo(L) \cong \OO_K \} }{ \textrm{homothety} } \\
\tau &\longmapsto& [L_{\tau}].
\end{eqnarray}
The set on the left hand side of (\ref{heegbij}) has cardinality
$h_K$ by Lemma \ref{hkcount} as does the set on the right hand
side by (\ref{elldef}) and the fact that the action
(\ref{claction}) is simply transitive.  Thus (\ref{heegbij}) is a
bijection.

We now identify the set on the right of (\ref{heegbij}) with
(\ref{elldef}).  Applying Proposition \ref{galconj} then yields
the corollary.
\end{proof}

We close our discussion of the connection between the j-invariants
of CM elliptic curves over $\CC$ by noting that the propositions
and lemmas we have proven above are really the elementary
preliminaries to a discussion of the class field theory of
imaginary quadratic fields.  This is one of two cases  in which
the class field theory of an extension of $\QQ$ has been
explicitly worked out (the other is the class field theory of
$\QQ$ itself). It would take us too far afield to prove the
following theorem, but it seems important to state for the reader
the main result in the class field theory of imaginary quadratic
extensions of $\QQ$, namely a characterization of the Hilbert
class field of such an extension. The connection with the CM
theory of elliptic curves will be evident.

\begin{theorem} Let $E$ be an elliptic curve representing an
isomorphism class in $\mathcal{ELL}(\OO_K)$.  Then $K(j(E))$ is
the maximal unramified abelian extension of $K$, that is,
$K(j(E))$ is the Hilbert class field of $K$.
\end{theorem}
\noindent For a proof of this result, see either \cite[\S
II.4]{Si2} or \cite[\S 10.1]{LaE}.

It should come as no surprise that in order to apply our
discussion of CM to the construction of $p$-adic modular forms, we
will have to understand at some level what happens when we move
from elliptic curves defined over $\overline{\QQ}$ to those
defined over a field of prime characteristic greater than or equal
to $5$. We begin with the notion of good reduction:

\begin{definition}
Let $K$ be a number field, and let $\mathfrak{P} \subset \OO_K$ be
a prime ideal.  An elliptic curve $E/\overline{\QQ}:y^2=x^3+ax+b$
with $\mathfrak{P}$-integral coefficients $a,b$ is said to have
\textbf{good reduction at $\mathfrak{P}$} if the reduced elliptic
curve $\widetilde{E}:y^2=x^3+\widetilde{a}x+\widetilde{b}$ is
nonsingular.  Here $\widetilde{a}$ denotes the reduction of $a$ in
$\OO_K/\mathfrak{P}$.
\end{definition}

\begin{remark}[1] For ease of exposition, we have restricted our definition of
good reduction so as only to include elliptic curves written in
Weierstrass form.  With this definition, whether or not an
elliptic curve $E$ has good reduction at a particular prime is not
an invariant of the isomorphism class of the curve. For a more
general (and natural) discussion of good reduction, see \cite[\S
VII.5]{Si}.
\end{remark}

\begin{remark}[2] Suppose $\mathfrak{P}$ is a prime ideal above a prime integer $p \not \in \{ 2,3\}$.
>From Proposition \ref{explicitj}, we have an easy way to determine
whether or not the elliptic curve $E/\overline{\QQ}:y^2=x^3+ax+b$
has good reduction at $\mathfrak{P}$; this is the case if and only
if $\ord_{\mathfrak{P}}(-4a^3-27b^2)=0$.
\end{remark}

We discussed two algebraic objects attached to an elliptic curve
defined in characteristic zero, namely, its group of $N$-division
points and its endomorphism ring.  The corresponding objects for
elliptic curves defined over fields of positive characteristic are
a good deal more subtle; a proper treatment of them would be a
thesis in itself.  In the remainder of this section we indicate
some of what is true about them as motivation for the concept of
supersingularity.

First, suppose $E$ is the reduction of an elliptic curve or an
elliptic curve defined over a field $K$ of prime characteristic $p
\geq 5$.  It is natural to ask for a description of the groups
$E[N]$. For $N$ coprime to $p$, we have the same answer as before,
namely
$$
E[N] \approx \ZZ/N\ZZ \times \ZZ/N\ZZ
$$
(see, for example, Corollary 6.4 of \cite[\S III.6]{Si}).  The
behavior is quite different at $p$; we have
$$
E[p^e] \approx \{0\} \textrm{ for all } e \in \NN
$$
or
$$
E[p^e] \approx \ZZ/p^e \ZZ \textrm{ for all } e \in \NN.
$$
Again, see \cite[\S III.6]{Si}.  In the first case, we say that
$E$ is \emph{supersingular}.

The endomorphism rings, which we completely described for elliptic
curves over $\CC$ using analytic parameterizations, also behave in
a much less straightforward manner upon reduction of the curve. In
particular, let $K$ be a number field, and let $E/\overline{\QQ}$
have good reduction at $\mathfrak{P} \subset \OO_K$. Then one can
show that the natural reduction map
$$
\eo(E) \to \eo(\widetilde{E})
$$
is injective (see \cite[\S VII.3]{Si} and Proposition 4.4,
\cite[\S II.4]{Si2}). If $E$ has CM, and this map is \emph{not}
surjective, then it turns out that $\eo(\widetilde{E})$, instead
of being an order in a quadratic imaginary field, is an order in a
quaternion algebra (see Corollary 9.4, \cite[\S III.9 ]{Si}).
Whether or not $E$ has CM, the condition that $\eo(\widetilde{E})$
is an order in a quaternion algebra is in fact equivalent to
supersingularity of the curve as defined above; sometimes it is
said that a supersingular elliptic curve has ``extra"
endomorphisms.  We write the two definitions of supersingularity
we have encountered as a displayed definition so they are not lost
in the text:

\begin{definition} Let $K$ be a number field, $\mathfrak{P}\subset \OO_K$ an prime ideal above $p \geq 3$.
An elliptic curve $E/\overline{\QQ}$ with good reduction at
$\mathfrak{P}$ is said to be \textbf{supersingular at
$\mathfrak{P}$} if one of the following equivalent conditions
holds:
\begin{enumerate}
\item $E[p^e]=0$ for all $e>0$.

\item $\eo(E)$ is an order in a quaternion algebra.
\end{enumerate}
\end{definition}
\begin{remark} In fact, if $E[p^k]=0$ for some $k > 0$, then $E[p^e]=0$ for all $e
>0$.
\end{remark}
\noindent For a discussion of the equivalence of these two
definitions, see \cite[\S V.3]{Si}. Essentially everything proven
therein is derived from the classical results of Deuring in
\cite{D}.

Recalling that $j(E)$ is an isomorphism invariant of $E$, and
given the central role that the study of $j$-invariants plays in
CM theory, it should come as no surprise that it enters the
discussion here as well.  We note first that if $E/\overline{\QQ}$
has good reduction at a prime ideal $\mathfrak{P}\subset \OO_K$
above $p \geq 5$, then $j(E)$ is $p$-integral.  This follows
trivially for any curve written in Weierstrass form by Proposition
\ref{explicitj} (and we are restricting our discussion of good
reduction to curves in Weierstrass form).  Thus we can make the
following:

\begin{definition} Let $K$ be a number field, and suppose $E/\overline{\QQ}$ is supersingular at a
prime ideal $\mathfrak{P}\subset \OO_K$ above a prime $p \geq 5$.
Then the reduction of $j(E)$ in $\overline{\mathbb{F}}_p$ is said
to be a \textbf{supersingular $j$-invariant} over
$\overline{\mathbb{F}}_p$.
\end{definition}

Using the dictionary between Heegner points and CM elliptic curves
we have developed, we can state the following theorem, which
yields a particularly nice method of deciding whether or not a CM
curve is supersingular at a particular prime:

\begin{theorem} \label{supsing}
Let $\tau$ be a Heegner point of discriminant $d_{\tau}$, and
$E_{\tau}$ be an elliptic curve with $j$-invariant $j(\tau)$.  Fix
a prime $p \geq 5$, and suppose that $p$ is inert or ramified in
$Q(\sqrt{d_{\tau}})$.  If $E_{\tau}$ has good reduction at
$\mathfrak{p}$ for all primes $\mathfrak{p}$ above $p$ in
$\QQ(j(\tau))$, then $j(\tau)$ reduces to a supersingular
$j$-invariant in $\overline{\FF}_p$.
\end{theorem}
\noindent See \cite[\S 13.4, p. 182]{LaE} for a proof of this
result.

We are now almost ready to state the theorem which will be the
main tool used in the construction of a family of good forms. We
fix the notation
\begin{equation}
\Omega_{p}:= \left\{j_E : j_E \text{ is a supersingular }
j\text{-invariant over } \overline{\FF}_p\right\}
\end{equation}
and  \begin{equation}g_p:=|\Omega_{p}|.\end{equation} Further,
define the supersingular locus $\sS (x)$ as
\begin{equation}
\label{superloci} \sS (x) = \prod_{j_E \in \Omega_{p}} (x-j_E) \in
\FF_p[x].
\end{equation}
We have swept under the rug the assertion that $|\Omega_p|$ is
finite and $\Ss(x) \in \FF_p[x]$.  For proofs of these assertions,
see either \cite{kz} or \cite[\S V.4]{Si}.  We have the following
result of Deligne:
\begin{theorem}[Deligne]
\label{deligne} If $p \ge 5$ is prime, then
\begin{equation*}
\sS (x) \equiv F(E_{p-1},x) \pmod p.
\end{equation*}
\end{theorem}
\noindent Note that the ``$F$" in Theorem \ref{deligne} is a
divisor polynomial (see \S \ref{pborc}).  An elementary proof of
Theorem \ref{deligne} using only basic properties of Hasse
invariants and complex analysis can be found in \cite{kz}.

In this section we have asked the reader to accept many results
without proof.  An explicit example could be enlightening:

\begin{example} Consider the elliptic curve $E:y^2=x^3+x$.  Using Proposition
\ref{explicitj}, we calculate that the discriminant of this curve
is $-4$ and conclude that it has good reduction at any prime $p
\geq 3$. We then use Proposition \ref{explicitj} to calculate that
the $j$-invariant is $1728$.  It is a standard fact from the
theory of modular functions that $j$ maps the arc from $e^{2 \pi i
/3}=-\frac{1}{2}+\frac{\sqrt{-3}}{2}$ to $i$ along the unit disc
$\{ z:|z|=1 \}$ bijectively onto the interval $[0,1728]$; in
particular, $j(i)=1728$.  It follows that from Proposition
\ref{cmequiv} that $E$ has CM with endomorphism ring an order in
$K=\QQ(i)$.

It is a fact from elementary number theory that a prime $p \geq 5$
splits in $K$ if and only if $p \equiv 1 \pmod 4$. Therefore, by
Theorem \ref{supsing}, we should expect that $E:y^2=x^3+x$ is
supersingular when considered as a curve in $\FF_p$ for primes $p
\geq 5$ with $p \equiv 3 \pmod 4$.  We prove this in an elementary
manner by demonstrating that $|E/\FF_p|=p+1$.  This implies
$E/\FF_p$ has no $p$-torsion, which is one of our definitions of
supersingularity.

Let $\left(\frac{\cdot}{\cdot}\right)$ denote the typical Legendre
symbol.  We have
$$
\left( \frac{x^3+x}{p}\right)=\left(\frac{x}{p}\right)\left(
\frac{x^2+1}{p}\right).
$$  Because $p \equiv 3 \pmod{4}$, $\left( \frac{-1}{p}\right)=-1$,
so
$$
\left(
\frac{(-x)^3+(-x)}{p}\right)=\left(\frac{-x}{p}\right)\left(
\frac{(-x)^2+1}{p}\right)=\left(
\frac{-1}{p}\right)\left(\frac{x}{p}\right)\left(
\frac{x^2+1}{p}\right)=-\left(\frac{x}{p}\right)\left(
\frac{x^2+1}{p}\right).
$$
By pairing $x$ and $-x$ for $x \in \FF_p^*$, we can conclude that
exactly half the $p-1$ elements $x \in \FF_p^*$ have the property
that $x^3+x$ is a quadratic residue.  Each such $x$ yields two
solutions to the equation $y^2=x^3+x$ over $\FF_p^*$,
corresponding to $\pm y$.  Adding in the point $(x,y)=(0,0)$ and
the point at infinity, we have $|E/\FF_p|=p+1$.
\end{example}

\section{Good forms} \label{good}

As promised, in this section we now provide several families of
good forms.  Before we begin, however, we recall the following
congruences involving singular moduli:
\begin{proposition}[Gross and Zagier] \label{gz} Let $p<12$ be a prime and $\tau$ be a Heegner
point of discriminant $d<0$.  If $\left(\frac{d}{p}\right)=-1$ we
have
\begin{eqnarray*}
j(\tau) &\equiv  0 \pmod{2^{15}} &\textrm{ if } p=2 \\
j(\tau) &\equiv  12^3 \pmod{3^6} & \textrm{ if } p=3 \\
j(\tau) &\equiv  0 \pmod{5^3} &\textrm{ if } p=5 \\
j(\tau) &\equiv  12^3 \pmod{7^2} &\textrm{ if } p=7 \\
\end{eqnarray*}
\end{proposition}
\begin{proof}If $p \neq 2$, then this proposition follows from elementary
manipulations of elliptic curves and Theorem \ref{supsing}.  The
case $p=2$ is more complicated.  For a proof, see \cite[Corollary
2.5]{GZ}.
\end{proof}

We can now state and prove the following:

\begin{theorem} \label{goodexamp}
Let $K$ be a number field and $\mathcal{O}_K$ be its ring of
integers.  Suppose that $f(z)=q^h
\prod_{n=1}^{\infty}(1-q^n)^{c(n)} \in \mmer_k(\Gamma_0(p)) \cap
q^h\mathcal{O}_K[[q]]$ has poles and zeros at the Heegner points
$\tau_1,...,\tau_s \in \mathfrak{F}$, all of fixed discriminant
$d<0$. Then the following are true:
\begin{enumerate}
\item If $p \geq 5$ is a prime for which $\left(
\frac{d}{p}\right) \in \{0,-1\}$ and
$$
\prod_{i=1}^s j(\tau_i)(j(\tau_i)-1728) \not \equiv 0 \pmod p
$$
then $f$ is good at $p$.

\item If $p \in \{2,3,5,7\}$ and $\left( \frac{d}{p}\right)=-1$
then $f$ is good at $p$.
\end{enumerate}
\end{theorem}
\begin{remark}[1] The work of Gross and Zagier on
differences of singular moduli \cite{GZ} provides a simple
description of those primes $p$ which do not satisfy the
congruence condition given in part (1) of Theorem \ref{goodexamp}.
\end{remark}
\begin{remark}[2] In \cite{BrO}, Bruinier and Ono provide several
additional families of good modular forms.  The proof that these
forms are good requires the classical work of Deuring on singular
moduli (see \cite{D}) as well as further results from \cite{GZ},
but the method of proof is essentially the same.
\end{remark}
\begin{remark}[3] The proof of (2) was given by Ono and
Papanikolas in \cite{pj}.
\end{remark}
\begin{proof}
By the definition of good, we must produce a holomorphic modular
form $\mathcal{E}_f$ on $\Gamma$ with algebraic $p$-integral
coefficients for which $\mathcal{E}_f(\tau_i)=0$ for $1 \leq i
\leq s$ that additionally satisfies the congruence
$$
\mathcal{E}_f(z) \equiv 1 \pmod{p}.
$$
We first prove (1).  For each $1 \leq i \leq s$ let $A_i$ be the
curve
\begin{equation} \label{specurve}
A_i: y^2=x^3-108j(\tau_i)(j(\tau_i)-1728)
x-432j(\tau_i)(j(\tau_i)-1728)^2.
\end{equation}
This curve is defined over the number field $\QQ(j(\tau_i))$.  Let
$\mathfrak{p}$ be a prime ideal of $\mathcal{O}_{\QQ(j(\tau_i))}$
lying above a prime $p \geq 5$.  By assumption,
$$
j(\tau_i)(j(\tau_i)-1728) \not \equiv 0 \pmod{\mathfrak{p}}.
$$
We check that the discriminant of the cubic defining $A_i$ is
nonzero modulo $\mathfrak{p}$. Applying Proposition
\ref{explicitj} and the definition of good reduction, this will
simultaneously show us that $A_i$ is an elliptic curve (i.e. it is
nonsingular) and that it has good reduction at $\mathfrak{p}$. The
discriminant of the cubic defining $A_i$ is
\begin{eqnarray} \label{deltacalc}
\nonumber
-4(-108j(\tau_i)(j(\tau_i)-1728))^3-27(-432j(\tau_i)(j(\tau_i)-1728)^2)^2&\\
&=&2^{14} 3^{12}j(\tau_i)^2(j(\tau_i)-1728)^3
\end{eqnarray}
By assumption, $j(\tau_i)(j(\tau_i)-1728) \not \equiv 0 \pmod{p}$,
so (\ref{deltacalc}) is nonzero modulo $\mathfrak{p}$; in other
words, $A_i$ is an elliptic curve that has good reduction at
$\mathfrak{p}$.

Using Proposition \ref{explicitj} and (\ref{deltacalc}) (the
discriminant of the cubic defining $A_i$), we compute that
$$
j(A_i)=1728\frac{4(-108j(\tau_i)(j(\tau_i)-1728))^3}{-2^{14}3^{12}j(\tau_i)^2(j(\tau_i)-1728)^3}=j(\tau_i).
$$
Thus we have a CM curve $A_i$ with good reduction at
$\mathfrak{p}$ for all primes $\mathfrak{p} \subset
\OO_{\QQ(j(\tau_i))}$ above an integer prime $p \geq 5$.  The
condition $\left(\frac{d}{p}\right) \in \{0,-1\}$ implies that $p$
does not split in $\QQ(\sqrt{d})$, ergo, applying Theorem
\ref{supsing}, we have that $j(\tau_i)$ reduces to a supersingular
$j$-invariant in $\overline{\FF}_p$.

Since $j(\tau_i)$ is supersingular, Theorem \ref{deligne} implies
that there exists some $Q_i \in \mathfrak{F}$ such that
$E_{p-1}(Q_i)=0$ and $j(Q_i) \equiv j(\tau_i) \pmod{p}$, which
implies
$$
(j(z)-j(Q_i)) \equiv (j(z)-j(\tau_i)) \pmod{p}.
$$
Recalling from Lemma \ref{pmoddy} that $E_{p-1}(z) \equiv 1
\pmod{p}$, we may take
$$
\mathcal{E}_f(z):=\prod_{i=1}^s\left(E_{p-1}(z)\frac{j(z)-j(\tau_i)}{j(z)-j(Q_i)}\right).
$$
For (2), we claim that we may take
\begin{equation*}
\mathcal{E}_f(z)=\prod_{i=1}^s\Delta(z)(j(z)-j(\tau_i)) \in
M_{12s}(\Gamma)
\end{equation*}
Because $j(\tau_i)$ is an algebraic integer, the weight $12s$
holomorphic modular form $\mathcal{E}_f$ has coefficients in the
ring of integers of some fixed number field $K$.  It is evident
that $\mathcal{E}_f(\tau_i)=0$ for all $1 \leq i \leq s$.  In view
of the fact that
$$
\Delta(z)=\frac{E_4(z)^3}{j(z)}=\frac{E_6(z)^2}{j(z)-1728}
$$
the congruences in Lemma \ref{24k} and Proposition \ref{gz} yield
the desired result.
\end{proof}

With the machinery we have now developed, it is straightforward to
prove Corollary \ref{classnumber}:

\begin{proof}[Proof of Corollary \ref{classnumber}]

Let $\tau_1(:=\tau),\tau_2,...,\tau_{h_{\QQ(\sqrt{d})}} \in
\mathfrak{F}$ be the Heegner points of discriminant $d$ (see Lemma
\ref{hkcount}). Define
$$
f_d(z):=\prod_{s=1}^{h_{\QQ(\sqrt{d})}}(j(z)-j(\tau_s))
$$
Our assumptions on $d$ guarantee, via part (2) of Theorem
\ref{goodexamp}, that $f_d$ is good at $p$ for all relevant pairs
of $p$ and $d$.  Therefore,
$$
\frac{\Theta f_d(z)}{f_d(z)}
$$
is a weight two $p$-adic modular form by Theorem \ref{modelth}. On
the other hand, applying Theorem \ref{bko}, we have
$$
\frac{\Theta f_d(z)}{f_d(z)}=-\frac{E_4(z)^2E_6(z)}{\Delta(z)}
\sum_{\tau \in \mathfrak{F}} \frac{e_{\tau}
\ord_{\tau}(f_d)}{j(z)-j(\tau)}
$$
which, by Corollary \ref{aknfu}, yields
$$
\frac{\Theta f_d(z)}{f_d(z)}=-\sum_{s=1}^{h_{\QQ(\sqrt{d})}}
\left(\sum_{n=0}^{\infty}j_n(\tau_s)q^n\right)=-h_{\QQ(\sqrt{d})}-
\sum_{n=1}^{\infty}\left(\sum_{s=1}^{h_{\QQ(\sqrt{d})}}
j_n(\tau_s)\right)q^n.
$$
Recall that from (\ref{jmdefo}) that $j_1(z):=j(z)-744$, and
$j_m(z):=mj_1(z)|T_{0,m}$.  From (\ref{heckedefo}) (the definition
of $T_{0,m}$) we have the first of the two following equalities:
\begin{eqnarray} \label{mthcoef}
\nonumber \sum_{s=1}^{h_{\QQ(\sqrt{d})}}
j_{p^m}(\tau_s)&=&\sum_{s=1}^{h_{\QQ(\sqrt{d})}}\left(\sum_{a=0}^m\sum_{b=0}^{p^a-1}
\left(j\left( \frac{p^{m-a} \tau_s+b}{p^a}\right) -744 \right) \right) \\
&=&
-\frac{744h_{\QQ(\sqrt{d})}(1-p^{m+1})}{1-p}+\Trace_{K/\QQ}\left(
\sum_{a=0}^m \sum_{b=0}^{p^a-1} j \left( \frac{p^{m-a}
\tau_1+b}{p^a} \right)\right).
\end{eqnarray}
The second equality follows from Corollary \ref{galconj} and the
observation that $j_m(\tau)$ is a polynomial in $j(\tau)$ with
integer coefficients.

The computation in (\ref{mthcoef}), when restricted to $m=p^n$,
gives us the $p^n$th coefficient of the $p$-adic modular form
$\Theta f_d(z)/f_d(z)$.  The constant term of this $p$-adic
modular form is precisely $-h_{\QQ(\sqrt{d})}$, so, by Corollary
\ref{Serre}, we have
$$
-h_{\QQ(\sqrt{d})}=\frac{p-1}{24} \cdot \lim_{n \to infty}
\left(-\frac{744h_{\QQ(\sqrt{d})}(1-p^{n+1})}{1-p}+\textrm{Tr}_{K/\QQ}\left(
\sum_{a=0}^{n} \sum_{b=0}^{p^a-1} j \left( \frac{p^{n-a}
\tau_1+b}{p^a} \right)\right) \right)
$$
as $p$-adic numbers.  Simplifying this expression yields the
corollary.
\end{proof}

Recall the weight zero modular forms $j^{(p)}(z) \in
\minf_0(\Gamma_0(p))$ introduced in \S \ref{Rth}.  We now provide
formulae involving $j^{(p)}(z)$ similar to those in Corollary
\ref{classnumber}:

\begin{theorem} \label{classnumber2}
Suppose that $d<-4$ is a fundamental discriminant of an imaginary
quadratic field and that $\tau$ is a Heegner point of discriminant
$d$. Then the following are true
\begin{enumerate}
\item Let $K=\QQ(j^{(3)}(\tau),j(\tau))$.  If $d \equiv 2
\pmod{3}$, then
$$
\lim_{n \to \infty} \left( \Trace_{K/\QQ}\left( \sum_{a=0}^n
\sum_{b=0}^{3^a-1} j^{(3)}\left( \frac{3^{n-a}
\tau+b}{3^a}\right)\right) \right)=0
$$
$3$-adically. \item Let $K=\QQ(j^{5}(\tau),j(\tau))$.  If $d
\equiv 2,3 \pmod{5}$, then
$$
\lim_{n \to \infty} \left(\Trace_{K/\QQ}\left( \sum_{a=0}^n
\sum_{b=0}^{5^a-1} j^{(5)}\left( \frac{5^{n-a}
\tau+b}{5^a}\right)\right)\right)=0
$$
$5$-adically. \item Let $K=\QQ(j^{(7)}(\tau),j(\tau))$.  If $d
\equiv 3,5,6 \pmod{7}$, then
$$
\lim_{n \to \infty} \left(\Trace_{K/\QQ}\left( \sum_{a=0}^n
\sum_{b=0}^{7^a-1} j^{(7)}\left( \frac{7^{n-a}
\tau+b}{7^a}\right)\right) \right)=0
$$
$7$-adically.
\end{enumerate}
\end{theorem}
We will see in the course of the proof of Theorem
\ref{classnumber2} that $j^{(p)}(\tau)$ is an algebraic integer of
degree $(p+1) \cdot h_{\QQ(\sqrt{d})}$ for $p$ and $\tau$ as in
the statement of the theorem. Before we begin the main body of the
proof, we require the following lemma:
\begin{lemma} \label{snp}
Suppose $p \in \{3,5,7,13\}$ and $0 \leq n \leq p$. Let
$\gamma_1,...,\gamma_{p+1}$ be a complete set of coset
representatives for $\Gamma_0(p)$ in $\Gamma$, and further let
$s_{n,p}(z)$ be the $n$th symmetric polynomial in $\{
j^{(p)}(\gamma_i \cdot z)\}_{i=1}^{p+1}$.  Then the $q$-series
expansion of $s_{n,p}(z)$ has integer coefficients, that is,
$$
s_{n,p}(z) \in q^{-n}\ZZ[[q]].
$$
\end{lemma}
\begin{proof}As
we recalled in Lemma \ref{tr}, a complete set of coset
representatives for $\Gamma_0(p)$ in $\Gamma$ is given by
$$
\left\{ \begin{pmatrix} 1 & 0 \\