CoCalc Public Fileswww / projects / getz_senior_thesis.tex
Author: William A. Stein
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135\begin{document}
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141\title[Borcherds exponents]{Classical and $p$-adic modular forms arising from the Borcherds exponents of other modular forms}
142
143\author{Jayce Getz \\ Senior Thesis} \address{
1444404 South Ave. W \\ Missoula, MT
14559804}\email{getz@fas.harvard.edu}
146
147\date{\today}
148
149%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
150%          Abstract           %
151%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
152\begin{abstract}
153Let $f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}$ be a modular form
154on $\SL_2(\ZZ)$.  Formal logarithmic differentiation of $f$ yields
155a $q$-series $g(z):=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n$ whose
156coefficients are uniquely determined by the exponents of the
157original form.  We provide a formula, due to Bruinier, Kohnen, and
158Ono for $g(z)$ in terms of the values of the classical
159$j$-function at the zeros and poles of $f(z)$. Further, we give a
160variety of cases in which $g(z)$ is additionally a $p$-adic
161modular form in the classical sense of Serre.  As an application,
162we derive some $p$-adic formulae, due to Bruinier, Ono, and
163Papanikolas, in which the class numbers of a family of imaginary
164quadratic fields are written in terms of special values of the
165$j$-function at imaginary quadratic arguments.
166\end{abstract}
167\thanks {The author would like to thank his family for their constant personal and financial support,
168Particular thanks go to his little brother Joel, who is the
169coolest person in the world.  This thesis is dedicated to them.}
170
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175\maketitle
176
177%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
178%          Section:  Introduction           %
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180
181\section{Introduction}
182\label{intro}
183
184Suppose $f$ is a function on the upper half plane $\mathbb{H}$.
185For each positive integer $k$, define an action $|_k$ of
186$\textrm{GL}_2^+(\QQ)$ on the set of such $f$ by
187\begin{equation}
188f(z)|_k\gamma=\det(\gamma)^{k/2}(cz+d)^{-k}f\left(
189\frac{az+b}{cz+d} \right).
190\end{equation}
191Here $\gamma = \left( \begin{smallmatrix} a & b \\ c & d 192\end{smallmatrix} \right) \in \textrm{GL}_2^+(\QQ)$
193(with the exception of the proof of Theorem \ref{bko}, in this
194thesis we always use the symbol $\gamma$ in this sense).
195 Suppose $\Gamma' \subset \Gamma:=\textrm{SL}_2(\ZZ)$
196is a congruence subgroup. Let $\mathcal{M}_k^{\infty}(\Gamma')$
197(resp., $\mathcal{M}_k^{\textrm{mero}}(\Gamma')$) denote the space
198of holomorphic (resp., meromorphic) functions on the upper half
199plane $\mathbb{H}$ that satisfy the functional equation
200\begin{equation} \label{modfunc}
201f(z)|_k\gamma:=f(z)
202\end{equation}
203for all $\gamma \in \Gamma'$ and additionally are meromorphic at
204the cusps of $\Gamma'$ (for a precise description of this
205meromorphic at the cusps" condition, see \cite[\S III.3,
206p.~125]{k}). Such a function will be called a \emph{weakly modular
207form of weight $k$} (resp., \emph{meromorphic modular form of
208weight $k$}) following J-P. Serre's convention \cite[\S
209VII.2]{S1}. We further define $M_k(\Gamma') \subset 210\mathcal{M}_k^{\infty}(\Gamma')$ to be the space of weakly modular
211forms that, additionally, are holomorphic at the cusps of
212$\Gamma'$. Such a form will be called a \emph{holomorphic modular
213form}, or, simply, a \emph{modular form}.  For any congruence
214subgroup $\Gamma'$ containing the element
215$\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 216\end{smallmatrix}\right)$, meromorphicity of $f$ at the cusps of $\Gamma'$ implies that
217$f$ can be identified with a Fourier, or $q$-series, expansion
218\begin{equation} \label{qseries}
219f(z):=\sum_{n=n_0}^{\infty}a_nq^n
220\end{equation}
221where here, and throughout this thesis, $q:=e^{2 \pi i z}$.  In
222the case $\Gamma'=\Gamma$, this is in fact equivalent to
223meromorphicity at the cusps.  Holomorphicity at the cusps in the
224case of $\Gamma'=\Gamma$ (which are all in the same orbit as
225$\infty$ under the action of $\Gamma$) is equivalent to the
226statement that $n_0 \geq 0$.  Finally, a holomorphic modular form
227over $\Gamma'$ is said to be a \emph{cusp form} if it vanishes at
228the cusps of $\Gamma'$; we denote the space of cusp forms of
229weight $k$ over $\Gamma'$ by $S_k(\Gamma')$. In the case $f \in 230M_k(\Gamma)$, this is simply the assertion that in the expansion
231(\ref{qseries}) we have $n_0>0$. For convenience we define
232$\mmer_k:=\mmer_k(\Gamma)$,
233$\mathcal{M}_k^{\infty}:=\mathcal{M}_k^{\infty}(\Gamma)$ and
234$M_k:=M_k(\Gamma)$.
235
236We take the opportunity now to introduce the only congruence
237subgroup we will explicitly use in this thesis, namely the
238following level $N$ subgroup:
239$$240\Gamma_0(N):=\left\{ \begin{pmatrix} a & b \\ c & d 241\end{pmatrix} \in \Gamma : c \equiv 0 \pmod{N} \right\}. 242$$
243By convention, $\Gamma_0(1)=\Gamma$.
244\begin{remark}  If $\left( \begin{smallmatrix} -1 & 0 \\ 0 & -1 245\end{smallmatrix}\right) \in \Gamma'$, then from (\ref{modfunc}) we have $(-1)^kf(z)=f(z)$ for
246all $f \in \mmer_k(\Gamma')$, from which it follows that
247$\mmer_{2m+1}(\Gamma')=0$ for all integers $m$.  Thus, in
248particular, $\mmer_{2m+1}=0$.
249\end{remark}
250
251For examples of modular forms on $M_k$ for even $k \geq 4$, we may
252take the classical Eisenstein series of weight $k$:
253\begin{equation} \label{Eis}
254E_k(z):=1-\frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n)q^n
255\end{equation}
256where $B_k$ is the $k$th Bernoulli number and
257$\sigma_{k-1}(n):=\sum_{d|n}d^{k-1}$.  We can formally define
258$E_2$ using (\ref{Eis}), and though it is not a modular form, it
259satisfies
260the following transformation law for $\left(\begin{smallmatrix} a & b \\ 261c & d \end{smallmatrix} \right) \in \Gamma$:
262\begin{equation} \label{e2}
263 E_2\left( \frac{az+b}{cz+d}\right)(cz+d)^{-2}=E_2(z)+\frac{12c}{2 \pi i (cz+d)}.
264\end{equation}
265This transformation law turns out to play a role in many
266arguments; a proof of it in this form is given in
267\cite[p.~68]{sch}.
268
269Other useful examples of modular forms are the discriminant
270function
271\begin{equation}
272\Delta(z):=\frac{E_4(z)^3-E_6(z)^2}{1728}=q\prod_{n=1}^{\infty}(1-q^n)^{24}
273\end{equation}
274which is of weight $12$, and the $j$-function, which is a weakly
275modular form of weight zero:
276\begin{equation} \label{modjdef}
277j(z):=\frac{E_4(z)^3}{\Delta(z)}=q^{-1}+744+196884q+21493760q^2+\cdots.
278\end{equation}
279We note that any element of $\minf_0$ is a polynomial in $j(z)$.
280If we wish to emphasize for a proof that we are regarding $E_k$,
281$\Delta$, $j$ as $q$-series (which can be either viewed formally
282or as functions holomorphic in the punctured disc $0 <|q|<1$), we
283write them as $E_k(q)$, $\Delta(q)$, and $J(q)$, respectively.
284
285It is easy to see that $\minf_k(\Gamma')$ is a vector space over
286$\CC$ for all congruence subgroups $\Gamma'$.  There exists an
287important class of linear operators on these spaces, namely, the
288Hecke operators $T_{k,n}$. These can be defined (in an admittedly
290\begin{equation} \label{heckedefo}
292\leq d-1}f\left(\frac{az+b}{d}\right)
293\end{equation}
294or, equivalently,
295\begin{equation} \label{heckedef}
296f(z)|T_{k,n}:=\sum_{n \in \ZZ}\left(
297\sum_{0<d|(m,n)}d^{k-1}a\left(\frac{mn}{d^2}\right)\right)q^n.
298\end{equation}
299If we define, for positive integers $d$, the $V$- and
300$U$-operators $V(d)$ and $U(d)$ on formal $q$-series in $\CC[[q]]$
301by
302\begin{equation} \label{vdef}
303 \left(\sum_{n \geq n_0} c(n)q^n \right)|V(d):=\sum_{n \geq
304n_0}c(n)q^{dn}
305\end{equation}
306and
307\begin{equation} \label{udef}
308\left(\sum_{n \geq n_0} c(n)q^n \right)|U(d):=\sum_{n \geq
309n_0}c(dn)q^{n}
310\end{equation}
311then we may write \begin{equation} \label{uvhecke}
312T_{k,n}=\sum_{d|n}d^{k-1}V(d) \circ U(n/d).
313\end{equation}
314Note that if we identify a meromorphic modular form $f$ with its
315$q$-expansion, we have
316\begin{equation} \label{altvdef}
317d^{k/2}f|V(d)=f|_k\left(\begin{smallmatrix} d & 0 \\ 0 &
3181
319\end{smallmatrix}\right).
320\end{equation} For more natural definitions of these operators and a
321discussion of their basic properties, see, for example, \cite[\S
322III.5]{k} or \cite[\S VII]{S1}.
323
324If we consider $\mathcal{M}^{\infty}:=\bigoplus_{k=0}^{\infty} 325\mathcal{M}^{\infty}_k$ it is straightforward to see that we have
326something better than a collection of vector spaces, we have a
328multiplication operation is multiplication of functions (for proof
329of this, see \cite[\S VII]{S1}). A question naturally suggests
330itself: are there natural operators on this algebra?  As one
331possible answer to this question, we define Ramanujan's theta
332operator:
333$$334\Theta:=\frac{1}{2 \pi i } \frac{d}{dz}=q \frac{d}{dq}. 335$$
336It is perhaps speaking loosely to call $\Theta$ an operator, but
337$$338f(z) \mapsto \Theta f(z)-f(z)\frac{k}{12}E_2(z) 339$$
340is a derivation on $\mathcal{M}$.  In particular, we have the
341following:
342
343\begin{proposition} \label{thetprop}
344If $f$ is in $\mmer _k(\Gamma')$
345then
346\begin{equation} \label{thetobs}
347g(z)=\Theta f-f(z)\frac{k}{12}E_2  \in \mmer_{k+2}(\Gamma').
348\end{equation}
349The same statement is true with $\mmer_k(\Gamma')$ replaced by
350$\minf_k(\Gamma')$ or $M_k(\Gamma')$ throughout.
351\end{proposition}
352\begin{proof}
353By noting its affect on $q$-expansions, we see that applying the
354$\Theta$ operator does not affect meromorphicity (resp.,
355holomorphicity) at the cusps.  Thus we need only check the
356functional equation. For $\gamma=\left(\begin{smallmatrix} a & b 357\\c & d\end{smallmatrix}\right) \in \Gamma$, upon differentiating
358the functional equation (\ref{modfunc}) we have
359\begin{eqnarray*} \Theta
360f(\gamma z)(cz+d)^{-k-2}&=&\Theta f(z)+\frac{ck}{2 \pi i}f(\gamma z)(cz+d)^{-k-1} \\
361&=&\Theta f(z)+ \frac{ck}{2 \pi i}f(z)(cz+d)^{-1}.
362\end{eqnarray*}
363Using (\ref{e2}), for $\gamma \in \Gamma' \subset \Gamma$ we have
364\begin{eqnarray*}
365\Theta f (z) |_{k+2}\gamma &-&\frac{k}{12}\left(E_2(z)|_{2} \gamma
366\right)\left(f(z)|_k \gamma \right)
367\\ &=&\Theta f(z) +\frac{ck}{2 \pi i}f(\gamma
368z)(cz+d)^{-k-1}-\frac{k}{12}E_2(z)f(z)-\left(\frac{k}{12}\right)\frac{12c}{2
369\pi i (cz+d)}f(z) \\
370&=& \Theta f(z)-\frac{k}{12}E_2(z)f(z).
371\end{eqnarray*}
372\end{proof}
373
374\begin{remark} It is worth mentioning that there exists a family of Rankin-Cohen" brackets on
375$\bigoplus_{k=0}^{\infty} M_k$ (defined using $\Theta$), one of
376which gives this algebra the structure of a graded Lie algebra.
377For their definition and basic properties see \cite{Z2}, and for
378references to recent work, see \cite{BWO}.
379\end{remark}
380
381Now, given a modular form $f \in \mmer_k(\Gamma')$, normalized so
382that its first nonzero $q$-expansion coefficient is $1$, we can
383write
384$$385f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)} 386$$
387for some complex numbers $c(n)$, in some neighborhood of $\infty$.
388Ignoring convergence issues for a moment (which will be dealt with
389carefully in Lemma \ref{logdiv}), some easy manipulations with
390$q$-series combined with Proposition \ref{thetprop} yield
391\begin{equation} \label{ok}
392\frac{\Theta f}{f}=h-\sum_{n \geq 1} \sum_{d|n}c(d)dq^n \in
393\mmer_2(\Gamma')
394\end{equation}
395In the next section, we will prove the following characterization
396of this logarithmic derivative:
397
398\begin{theorem}[Bruinier, Kohnen, Ono, \cite{BKO}, \cite{O}] \label{bko} If $f(z)=\sum_{n=h}^{\infty}a(n)q^n 399\in \mmer_k$ is normalized so that $a(h)=1$, then
400$$401\frac{\Theta 402f(z)}{f(z)}=\frac{k}{12}E_2(z)-\frac{E_4(z)^2E_6(z)}{\Delta(z)}\sum_{\tau_i 403\in \mathfrak{F}} \frac{e_{\tau}\ord_{\tau}(f)}{j(z)-j(\tau)}. 404$$
405\end{theorem}
406
407\begin{remark}  This formula has been generalized to several
408genus zero congruence subgroups in \cite{Ahl1} (see \S \ref{Rth}
409of this thesis) and Hecke subgroups of $\textrm{SL}_2(\RR)$ (see
410\cite{CKo}). The author has also received a preprint \cite{DC}
411giving a generalization to $\Gamma_0(N)$ for squarefree $N$.
412\end{remark}
413
414This formula alone is of interest in that it explicitly relates,
415via equation (\ref{ok}), the product expansion exponents of $f$ to
416special values of $j$, namely, $j(\tau)$ where $\tau$ is a zero or
417pole of $f$. Further, it has been used to provide recursive
418formulas for the coefficients of any modular form over $\Gamma$
419(see \cite{BKO}), to provide infinite families of systems of
420orthogonal polynomials divisible by the supersingular locus as
421polynomials over $\FF_p$ (see \cite{BGNS}), (generalizing work of
422Atkin described in \cite{kz}), and also to provide a
423characterization of the characteristic polynomials of the Hecke
424operators over $\Gamma$ (again in \cite{BKO}). We will not discuss
425these applications in this thesis. We will, however, give one
426additional application, which we defer for a moment in order to
427introduce the concept of a $p$-adic modular form.
428
429Following Serre, we define a $p$-adic modular form to be the
430$p$-adic limit of a sequence of elements of
431$\oplus_{k=0}^{\infty}M_k$ (a precise definition is given in \S
432\ref{Spadic}). It turns out that in many cases of interest, the
433logarithmic derivative of a modular form is a $p$-adic modular
434form of weight $2$.  In particular, we have the following theorem
435of Bruinier and Ono:
436
437\begin{theorem}[\cite{BrO}] \label{modelth}
438Let $f(z)=q^h(1+\sum_{n=1}^{\infty}a(n)q^n) \in q^h\OO_K[[q]] \cap 439\mmer_k(\Gamma_0(1))$, where $\OO_K$ is the ring of integers of a
440number field $K$. Moreover, let $c(n) \in K$ denote the algebraic
441numbers defined by the formal infinite product
442\begin{equation} \label{cexp}
443f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}.
444\end{equation}
445If $f(z)$ is good at a prime $p$, then the formal power series
446$$447\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n 448$$
449is a weight two $p$-adic modular form.
450\end{theorem}
451
452We offer a brief proof of this result, mainly as motivation for
453the following generalization:
454
455\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
456ring of integers of a number field $K$.  Moreover, let $c(n) \in 457K$ denote the algebraic numbers defined by the formal infinite
458product (\ref{cexp}) for $f$. If $f$ is good at $p$, then the
459formal power series
460$$461\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n 462$$
463is a weight two $p$-adic modular form.
464\end{theorem}
465\noindent The proofs of both of these theorems appear in \S
466\ref{pborc}.
467
468\begin{remark} In theorems \ref{modelth} and \ref{gbro}, we allow
469$h$ to be negative. The fact that the $c(n)$ are elements of $K$
470(implicitly identified with an embedding $K \hookrightarrow \CC$)
471will be obvious from the proof of Lemma \ref{logdiv}.
472\end{remark}
473
474The definition of good" in the preceding two theorems is given
475in \S \ref{pborc} and discussed in some detail in \S \ref{good}.
476As one example, the form $E_{p-1}$ is good at $p$. In general,
477whether or not a form is good at $p$ is intimately related to the
478question of whether or not the value of the $j$-function at the
479zeros and poles of the form reduces to a supersingular
480$j$-invariant in characteristic $p$ (which should come as no
481surprise to those familiar with overconvergent $p$-adic modular
482forms).  Through this connection we are able to relate these
483$p$-adic modular forms to class numbers of imaginary quadratic
484fields.  In particular, for small primes, we obtain $p$-adic class
485number formulae involving sums of special values of the
486$j$-function.
487
488Before we can state this result, we must recall the notion of a
489Heegner point.  A complex number $\tau$ of the form
490$\tau=\frac{-b+\sqrt{b^2-4ac}}{2a}$ with $a,b,c \in \ZZ$, $\gcd 491(a,b,c)=1$ and $b^2-4ac<0$ is known as a \emph{Heegner point} of
492discriminant $d_{\tau}:=b^2-4ac$.  Heegner points are discussed at
493some length in \S \ref{cm}.  Denote by $h_K$ the Hurwitz class
494number of the number field $K$.  We have the following:
495
496\begin{corollary}[Ono and Papanikolas, \cite{pj}] \label{classnumber}
497Suppose that $d<-4$ is a fundamental discriminant of an imaginary
498quadratic field and that $\tau$ is a Heegner point of discriminant
499$d$.  If $K=\QQ(j(\tau))$, then the following are true:
500\begin{enumerate}
501\item If $d \equiv 5 \pmod{8}$, then as $2$-adic numbers we have
502$$503h_{\QQ(\sqrt{d})}=-\frac{1}{720} \lim_{n \to \infty} 504\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{2^a-1} j\left( 505\frac{2^{n-a} \tau+b}{2^a}\right)\right). 506$$
507\item If $d \equiv 2 \pmod{3}$, then as $3$-adic numbers we have
508$$509h_{\QQ(\sqrt{d})}=-\frac{1}{360} \lim_{n \to \infty} 510\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{3^a-1} j\left( 511\frac{3^{n-a} \tau+b}{3^a}\right)\right). 512$$
513\item If $d \equiv 2,3 \pmod{5}$, then as $5$-adic numbers we have
514$$515h_{\QQ(\sqrt{d})}=-\frac{1}{180} \lim_{n \to \infty} 516\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{5^a-1} j\left( 517\frac{5^{n-a} \tau+b}{5^a}\right)\right). 518$$
519\item If $d \equiv 3,5,6 \pmod{7}$, then as $7$-adic numbers we
520have
521$$522h_{\QQ(\sqrt{d})}=-\frac{1}{120} \lim_{n \to \infty} 523\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{7^a-1} j\left( 524\frac{7^{n-a} \tau+b}{7^a}\right)\right). 525$$
526\end{enumerate}
527\end{corollary}
528\noindent In \S \ref{good}, we also use Theorem \ref{gbro} to
529provide formulae of the same general form of those in Corollary
530\ref{classnumber}, with a weight zero modular form in
531$\Gamma_0(p)$ taking the place of the $j$-function (see Theorem
532\ref{classnumber2}).
533
534Before we begin the body of this work, we make a few remarks about
535its structure. Sections \ref{Rth}, \ref{pborc}, and \ref{good}
536contain results that have only been published recently, if at all,
537and the primary purpose of this thesis is to collect their content
538into one place.  Sections \ref{Spadic} and \ref{atle}, on the
539other hand, are mostly derived from two well-known papers
540(\cite{Sep} and \cite{AL}, respectively).  The author has provided
541proofs of most of the results in these sections that are necessary
542for the proof of theorems \ref{bko}, \ref{modelth}, and
543\ref{gbro}. The notable exceptions are theorems \ref{sturm} and
544\ref{newforms} which are proven in \cite{St} and \cite{L},
545respectively.
546
547In contrast, providing applications of theorems \ref{bko},
548\ref{modelth} and \ref{gbro}, including Corollary
549\ref{classnumber} and Theorem \ref{classnumber2}, requires results
550for which we will not provide proofs; it would simply take us too
551far afield. In particular, \S \ref{cm} is intended to give a brief
552survey of the relevant definitions and theorems in the theory of
553complex multiplication, but we omit the proofs of results usually
554proven using class field theory and reduction theory (we refer the
555reader to \cite[\S II]{Si2} or \cite{LaE} for a more complete
556account). Ergo, \S \ref{cm} can be skipped without interrupting
557the flow of ideas, especially if one is familiar with complex
558multiplication and elementary calculations involving elliptic
559curves.
560
561\section{A characterization of Ramanujan's theta operator}
562\label{Rth}
563
564As indicated above, in this section we will prove a useful
565characterization of the derivative of a modular form.  First we
566require some preparation.  Let $$\fF:=\left\{z: -\frac{1}{2} \le 567\mbox{Re}(z) \leq 0 \text{ and } |z| \geq 1 \right\} \cup 568\left\{z: 0 < \mbox{Re}(z) < \frac{1}{2} \text{ and } |z| > 1 569\right\} 570$$
571 be the standard fundamental domain for the action of $\mbox{SL}_2(\ZZ)$ on the upper half plane $\mathbb{H}$, and let
572\begin{eqnarray} \label{edef}
573e_{\tau} = \begin{cases} \frac{1}{2} & \mbox{if } \tau = i, \\
574\frac{1}{3} & \mbox{if } \tau = e^{2\pi i/3}, \\ 1 &
575\mbox{otherwise}. \end{cases}
576\end{eqnarray}
577
578The purpose of this section is to prove the characterization of
579the logarithmic derivative of a modular form given by Theorem
580\ref{bko}. The proof of the theorem requires two steps.  The first
581is an identity due to Asai, Kaneko, and Ninomiya \cite{AKN}.  To
582introduce this result, define $j_0(z):=1$, and, for $m>1$, define
583$j_m(z)$ to be the unique weight zero meromorphic modular form
584with $q$-expansion
585\begin{equation} \label{jmdef}
586j_m(z):=J_m(q):=q^{-m}+\sum_{n=1}^{\infty} a_m(n)q^n \in q^{-m}
587\ZZ[[q]]
588\end{equation}
589We note that $j_m(z)$ is a polynomial in $j(z)$ for all $m$.  In
590fact, it is a polynomial in $j$ with integral coefficients, for
591$J_m(q)$ can be formed by subtracting suitable integer multiplies
592of the $q$-series $J(q)^k \in q^{-k}\ZZ[[q]]$ from $J(q)^m$ (where
593$0 \leq k < m$). The first few $j_m(z)$ follow:
594\begin{eqnarray} \label{jmexamp}
595j_0(z)=J_0(q)&=&1, \\
596j_1(z)=J_1(q)&=&j(z)-744 = q^{-1}+196884q+\cdots, \\
597j_2(z)=J_2(q)&=&j(z)^2-1488j(z)+159768=q^{-2}+42987520q+ \cdots, \\
598j_3(z)=J_3(q)&=&j(z)^3-2232j(z)^2+1069956j(z)-36866976=q^{-3}+2592899910q+
599\cdots.
600\end{eqnarray}
601We may equivalently define $J_0(q):=j_0(z):=1$,
602$J_1(q):=j_1(z):=j(z)-744$, and
603\begin{equation} \label{jmdefo}
604J_m(q):=j_m(z):=m j_1(z)|T_{0,m}
605\end{equation}
606for $m>1$.  The equivalence of this definition to the $q$-series
607definition (\ref{jmdef}) follows from (\ref{heckedef}) and the
608fact that a weakly modular form, being a polynomial in $j$, is
609uniquely determined by the coefficients of non-positive exponent
610in its $q$-series expansion. Indeed, from this fact we see that
611the $J_m(q)$ form a basis for $\minf_0$.
612
613We have the following:
614
615\begin{theorem} \label{aknf} As an identity of formal power series
616in $\rho,q$, we have
617\begin{equation} \label{aknfo}
618\sum_{n=0}^{\infty} J_n(\rho)q^n=\frac{E_4(q)^2E_6(q)}{\Delta(q)}
619\cdot \frac{1}{J(q)-J(\rho)}. \end{equation}
620\end{theorem}
621
622\begin{remark}
623Asai, Kaneko, and Ninomiya show in \cite{AKN} how Theorem
624\ref{aknf} implies the famous denominator formula for the Monster
625Lie algebra, namely
626$$627J(\rho)-J(q)=\rho^{-1} \prod_{m>0 \textrm{ and } n \in 628\ZZ}(1-\rho^mq^n)^{\beta(mn)}, 629$$
630where the coefficients $\beta(n)$ are defined by
631$$632j_1(z)=\sum_{n=-1}^{\infty}\beta(n)q^n. 633$$
634\end{remark}
635
636\begin{proof}[Proof of Theorem \ref{aknf}]
637
638We require a companion set of functions $g_m(\rho)$ indexed by
639positive integers $m$, the $m$th of which can be defined in
640analogy with (\ref{jmdef}) as the unique weight $2$ weakly modular
641form with $\rho$-expansion
642\begin{equation} \label{gdef}
643g_m(\rho):=\rho^{-m}+\sum_{n=1}^{\infty} b_m(n)\rho^n \in \minf_2.
644\end{equation}
645Alternately, we may define
646$g_1(\rho):=\frac{E_{4}(\rho)^2E_6(\rho)}{\Delta(\rho)}$ and
647$$648g_m(\rho):=m^{-1}g_1(\rho)|T_{2,m}. 649$$
650As before, the equivalence of these two definitions follows from
651the definition of the $T_{2,m}$ and the fact that any weight $2$
652weakly holomorphic form is uniquely determined by the coefficients
653in its $q$-expansion of negative order.  We note that this fact
654follows from the well-known $k/12$ valence formula" (see, for
655example, \cite[\S III.2]{k}), as does the corresponding fact for
656weight zero weakly holomorphic forms.  In fact, as in the weight
657zero case, this implies that the $g_m(\rho)$ form a basis for the
658space $\minf_2$.  Further, from (\ref{thetobs}), if $f \in 659\minf_0$ then $\Theta f \in \minf_2$, and by simply looking at the
660bases $\{J_m\}$,$\{g_m\}$ we have just written down we see that
661every element of $\minf_2$ can be written as $\Theta f$ for some
662$f \in \minf_0$.  In particular, it follows from this observation
663and the definition of $\Theta$ that the constant term of any
664element of $\minf_2$ is identically zero (which justifies the
665indexing of (\ref{gdef})).
666
667Now we note that
668\begin{equation} \label{Jconst}
669J_m(q):=mJ_1(q)|T_{0,m}=q^{-m}+m a_m(m)q+ \cdots
670\end{equation}
671and
672\begin{equation} \label{gconst}
673g_m(\rho):=m^{-1}g_1(\rho)|T_{2,m}=\rho^{-m}+b_1(m)q+\cdots
674\end{equation}
675for $m \geq 1$ simply by (\ref{heckedef}) and the fact that
676$b_1(0)=0$. Further, by noting that the constant term of
677$J_m(q)g_1(q) \in \minf_2$ must be zero by the comments in the
678preceding paragraph and using (\ref{gdef}) and (\ref{Jconst}), we
679have that
680\begin{equation} \label{abconst}
681b_1(m)=-ma_m(m)
682\end{equation}
683for $m \geq 1$.  Now $J(\rho)J_m(\rho) \in \mathcal{M}_0$ and
684$J(q)g_m(q) \in \minf_2$ are uniquely determined by their $\rho$-
685(resp., $q$-) expansion coefficients of non-positive exponent, as
686we've remarked before. Define $\rho$-expansion coefficients $c(n)$
687by
688$$689J(\rho)=\rho^{-1}+\sum_{n=0}^{\infty}c(n)\rho^n 690$$
691By comparing coefficients using equalities (\ref{Jconst}),
692(\ref{gconst}), (\ref{abconst}) and the observation that
693$b_1(0)=0$, we obtain the recurrence relation
694\begin{equation} \label{Jmrecur}
695J(\rho)J_m(\rho)=J_{m+1}(\rho)+\sum_{i=0}^m c(m-i)J_i(\rho)-b_1(m)
696\end{equation}
697for all $m \geq 0$.  Thus, multiplying both sides of
698(\ref{Jmrecur}) by $q^{m}$ and summing over $m \geq 0$ we obtain
699\begin{equation} \label{messy}
700J(\rho)\sum_{m=0}^{\infty}J_m(\rho)q^m=\frac{1}{q}(\sum_{m=0}^{\infty}J_m(\rho)q^m-1)+
701(J(q)-\frac{1}{q})\sum_{m=0}^{\infty}J_m(\rho)q^m-g_1(q)+\frac{1}{q}.
702\end{equation}
703Noting that $g_1(q)=\frac{E_{4}(q)^2E_6(q)}{\Delta(q)}$, we see
704that (\ref{messy}) is a rewriting of (\ref{aknfo}).
705\end{proof}
706
707\begin{corollary} \label{aknfu}
708Fix $\tau \in \mathbb{H}$.  Then
709$$710\frac{E_4(z)^2E_6(z)}{\Delta(z)}\frac{1}{j(z)-j(\tau)}=\sum_{n=0}^{\infty}j_m(\tau)q^n 711$$
712as meromorphic functions in $z$ on $\mathfrak{F}$.
713\end{corollary}
714\begin{proof} Compare Fourier ($q$-series) coefficients in a deleted
715neighborhood of infinity using Theorem \ref{aknf}.
716\end{proof}
717
718\begin{remark} The main result of \cite{AKN} is the statement that the zeros of $j_m(z)$ in
719$\mathfrak{F}$ are simple and are all contained in the
720intersection of the unit circle with $\mathfrak{F}$.  The
721technique they use is analogous to that used by Rankin and
722Swinnerton-Dyer to prove that the nontrivial" zeros of $E_k(z)$
723have the same property, see \cite{RSD}.  For yet another family of
724modular forms whose zeros have the same property, see \cite{G}.
725\end{remark}
726
727We also require the following proposition, which follows from
728basic complex analysis:
729\begin{proposition}[\cite{BKO}] \label{logdiv}
730Let $f=\sum_{n=h}^{\infty}a_f(n)q^n$ be a meromorphic function in
731a neighborhood of $q=0$, normalized so that $a_f(h)=1$.  Then
732there are complex numbers $c(n)$ such that
733$$734f=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}, 735$$
736where the product converges in a sufficiently small neighborhood
737of $q=0$.  Moreover,
738\begin{equation} \label{logdive}
739\frac{\Theta f}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n.
740\end{equation}
741\end{proposition}
742\begin{remark} We will refer to the $c(n)$ associated to a given
743meromorphic modular form $f$ by Proposition \ref{logdiv} as the
744\emph{Borcherds exponents} of $f$.
745\end{remark}
746\begin{proof}
747
748As usual, we understand that complex powers are defined by the
749principle branch of the complex logarithm.  Write $F(q):=f(z)$,
750and then note that $qF'(q)/F(q)$ is holomorphic at $q=0$.  We may
751therefore write its Taylor expansion around $q=0$, valid in
752$|q|<\epsilon$ for some $\epsilon>0$, as
753\begin{equation} \label{ld1}
754qF'(q)/F(q)=h-\sum_{n \geq 1} \alpha(n)q^n.
755\end{equation}
756For $n \geq 1$ define
757$$758c(n):=\frac{1}{n}\sum_{d|n}\alpha(d)\mu(n/d) 759$$
760where $\mu$ is the M$\ddot{\textrm{o}}$bius function.  By
761M$\ddot{\textrm{o}}$bius inversion we have
762\begin{eqnarray} \label{moinv}
763\alpha(n)=\sum_{d|n}c(d)d.
764\end{eqnarray}
765
766If we fix $q_0$ with $|q_0|<\epsilon$, then by absolute
767convergence of (\ref{ld1}) we have
768$\alpha(n)=\mathcal{O}(|q_0|^{-n})$ for all $n$.  Thus the double
769sum
770\begin{equation} \label{un1}
771 \sum_{m,n \geq 1} c(n)nq^{mn}
772\end{equation}
773converges absolutely in $|q|<|q_0|$ and hence in $|q|<\epsilon$.
774
775Suppose for the remainder of the proof that $|q|<\epsilon$.  From
776(\ref{ld1}) and (\ref{moinv}) we have
777\begin{eqnarray*}
778\frac{d}{dq} \log(F(q)q^{-h})&=&\frac{F'(q)}{F(q)}-\frac{h}{q} \\
779&=&-\sum_{n \geq 1}c(n) \frac{d}{dq}\left(\sum_{m \geq 1}
780\frac{q^{mn}}{m} \right) \\
781&=& \frac{d}{dq}\left(\sum_{n \geq 1} c(n) \log(1-q^n)\right).
782\end{eqnarray*}
783The interchange of summation and integration can be justified by
784using local uniform convergence as we did in proving the absolute
785convergence of (\ref{un1}).
786
787Upon integrating, we obtain
788$$789\log(F(q)q^{-h})=\sum_{n \geq 1} c(n)\log(1-q^n). 790$$
791Here we use the normalization $a_f(h)=1$.  Now $c(n)\log(1-q^n)$
792and $\log(1-q^n)^{c(n)}$ differ by integer multiples of $2 \pi i$.
793Since $c(n) \log(1-q^n) \to 0$ as $n \to \infty$, we have
794$\log(1-q^n)^{c(n)} \to 0$ as well.  Thus, as $n \to \infty$,
795these two quantities differ in value only finitely many times; it
796follows that there exists an integer $N$ such that
797$$798\log(F(q)q^{-h})=\sum_{n \geq 1} \log(1-q^n)^{c(n)}+2 \pi i N. 799$$
800Taking the exponential on both sides finishes the proof of the
801proposition.
802\end{proof}
803
804We now prove Theorem \ref{bko}.
805
806\begin{proof}[Proof of Theorem \ref{bko}]
807
808Choose $C>0$ large enough so that all poles of $f$ in
809$\mathfrak{F}$ (excluding any at the cusp at infinity) have
810imaginary part less than $C$. Let $L:=\{t+iC: -\frac{1}{2} \leq t 811\leq \frac{1}{2} \}$ and consider the contour in $\mathbb{H}$
812formed from the part of $\partial \mathfrak{F}$ of imaginary part
813less than $C$ and $L$.  Modify this contour as in the proof of the
814classical $k/12$ valence formula (see, for example, \cite[\S
815III.2, p.~115]{k}), specifically, if there are poles of $f$ at $i$
816or $\omega:=e^{2 \pi i/3}$ (which, by modularity, implies the
817existence of a pole at $e^{\pi i/3}$), form half and sixth"
818circles of radius $r>0$ around them, and if there are poles of $f$
819on the boundary, form two half circles of radius $r>0$ around
820them, one enclosing the pole on one side of the fundamental
821domain, one not enclosing the pole which must exist on the other
822side (given that $f$ is modular).  Call the left vertical side of
823this contour $\gamma_1(r)$, the right vertical side $\gamma_2(r)$,
824and the bottom $\gamma_3(r)$.  Take the modified contour
825$\gamma_1(r) \cup L \cup \gamma_2(r) \cup \gamma_3(r)$ to have
826positive (counterclockwise) orientation.
827
828If we integrate
829\begin{equation} \label{int1}
830\frac{1}{2 \pi i} \frac{f'(z)}{f(z)} j_n(z)
831\end{equation}
832along this full contour and let $r \to 0$, by holomorphicity of
833$j_n$ on $\mathbb{H}$ the integral will be equal to
834\begin{equation} \label{int1way}
835\sum_{\tau \in \mathfrak{F}-\{\omega,i\}} \ord_{\tau}(f)j_n(\tau).
836\end{equation}
837We can also integrate (\ref{int1}) in pieces, from which we see
838that (\ref{int1way}) is equal to
839\begin{eqnarray} \label{lotsoterms}
840&&-\frac{1}{3}\ord_{\omega}(f)j_n(\omega)-\frac{1}{2}\ord_i(f)j_n(i)+\int_L
841\frac{f'(z)}{f(z)}j_n(z)dz+\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)dz
842\\\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)
843dq+\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)dz.
844\end{eqnarray}
845Here $L'$ is a simple loop around $q=0$.  By Proposition
846\ref{logdiv} we have
847$$848\frac{qF'(q)}{F(q)}=\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n} 849c(d)dq^n 850$$
851and thus, applying the residue theorem, we have
852$$853\frac{1}{2 \pi i} \int_{L'} 854\frac{F'(q)}{F(q)}J_n(q)dq=\sum_{d|n}c(d)d. 855$$
856
857We now deal with the last term in (\ref{lotsoterms}).  By
858Proposition \ref{thetobs}, if the weight of $f$ is $k$, there
859exists a weight $k+2$ modular form $g$ such that
860\begin{eqnarray} \label{justamo}
861\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)&=&2 \pi
862i\int_{\gamma_3(r)}\frac{\Theta(f)}{f}j_n(z)dz\\\nonumber&=&2 \pi
863i\int_{\gamma_3(r)}\frac{g(z)}{f(z)}j_n(z)dz+ 2 \pi i
864\int_{\gamma_3(r)}\frac{k}{12}j_n(z)E_2(z)dz
865\end{eqnarray}
866Now let $\beta$ denote the path along the unit circle from $i$ to
867$\omega$, taken with positive orientation, and $S$ the fractional
868linear transformation defined by $S(z)=-1/z$. Then
869$\gamma_3=-\beta+S\beta$, and thus the right hand side of equation
870(\ref{justamo}) is equal to
871\begin{eqnarray*}
872&&\left(\int_{-\beta}\frac{g(z)}{f(z)}j_n(z)dz+
873\int_{S\beta}\frac{g(z)}{f(z)}j_n(z)dz
874\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)\\
875&=&\frac{k}{12}\left(\int_{-\beta}j_n(z)E_2(z)dz+\int_{S\beta}j_n(z)E_2(z)dz\right)\\
876&=&\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)\\
877&=&\frac{k}{2 \pi i}\int_{\beta}\frac{j_n(z)}{z}dz.
878\end{eqnarray*}
879To obtain the first equality we used the functional equation for
880elements of $\minf_2$ along with a standard change of variables
881(which introduces a factor of $1/z^2$).  To move from the second
882line to the third we used the functional equation for elements of
883$\minf_2$, a change of variables, and the functional equation
884(\ref{e2}) for $E_2(z)$.
885
886Now, instead of trying to evaluate $\frac{k}{2 \pi i} 887\int_{\beta}\frac{j_n(z)}{z}dz$ directly, we plug $f=\Delta$ into
888(\ref{lotsoterms}), notice that $\sum_{\tau \in 889\mathfrak{F}}\ord_{\tau}(f)j_n(\tau)=0$, and thereby obtain
890\begin{eqnarray*}
891\int_{\beta} \frac{j_n(z)}{z}dz&=&-\frac{1}{12}\int_{\beta}\frac{\Delta'(q)}{\Delta(q)}J_n(q)dq\\
892&=&-\frac{1}{12}\sum_{d|n}c(d)d
893\\
894&=&-2 \sigma_1(n)
895\end{eqnarray*}
896where $c(d) \equiv 24$ are (just for the purposes of the preceding
897equation) the product expansion exponents of
898$\Delta(z)=q\prod_{n=1}^{\infty}(1-q^n)^{24}$.
899
900Thus, collecting all of this, equation (\ref{lotsoterms}) implies
901that
902$$903\sum_{\tau \in \mathfrak{F}} 904e_{\tau}\ord_{\tau}(f)j_n(\tau)=\sum_{d|n}c(d)d-2k \sigma_1(n) 905$$
906
907Now we recall that by Theorem \ref{aknf}, it is sufficient to show
908that
909$$910\frac{\Theta(f)}{f}=\frac{kE_2}{12}- 911\sum_{n=1}^{\infty}\left(\sum_{\tau \in \mathfrak{F}} 912e_{\tau}\ord_{\tau}(f)j_n(\tau)\right)q^n. 913$$
914To prove this identity, we apply Proposition \ref{logdiv}, note
915$$916\frac{k}{12}E_2(z)=\frac{k}{12}-2k\sum_{n=1}^{\infty}\sigma_1(n)q^n, 917$$
918and argue coefficient by coefficient.  The only coefficient that
919might be unclear is the constant $n=0$ term.  In this case, on the
920left we have $h,$ which is the order of $f$ at infinity, and on
921the right we have $\frac{k}{12}-\sum_{\tau \in 922\mathfrak{F}}e_{\tau}\ord_{\tau}(f)$, which is precisely
923$\ord_{\infty}(f)=h$ by the $k/12$ valence formula (for example,
924see \cite[\S III.II, p.~115]{k}).
925
926\end{proof}
927
928We remark here that the derivative formula of Theorem \ref{bko}
929explicitly relates, via Proposition \ref{logdive}, the
930coefficients $c(n)$ of the product expansion of a modular form to
931a specific weight $2$ meromorphic modular form.  This relationship
932is in the spirit of the work of Borcherds on the product expansion
933exponents of Jacobi forms with Heegner divisors. See \cite{Borc}
934for the details of this theory.
935
936As we mentioned in the introduction, Ahlgren, in \cite{Ahl1}, has
937proven a generalization of Theorem \ref{gbro} to certain genus
938zero congruence subgroups.  We will state his theorem after fixing
939some notation. Define Dedekind's eta-function
940$$941\eta(z):=q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^n) 942$$
943as usual.  For $p=2,3,5,7$ or $13$, let
944$$945j^{(p)}(z):=\left(\frac{\eta(z)}{\eta 946(pz)}\right)^{\frac{24}{p-1}} \in \minf_0(\Gamma_0(p)). 947$$
948This $j^{(p)}(z)$ is a modular form with a simple pole at $\infty$
949and a simple zero (with respect to local coordinates) at $0$.
950Additionally, its restriction to a fundamental domain for the
951action of $\Gamma_0(p)$ on $\HH$ forms a bijection from that
952fundamental domain to $\CC$. In analogy with (\ref{jmdef}), we now
953define a sequence of modular functions
954$\{j^{(p)}_m(z)\}_{m=0}^{\infty}$. Let $j_0^{(p)}(z):=1$ and for
955$m>0$ let $j_m^{(p)}(z) \in \minf_0(\Gamma_0(p))$ be the unique
956modular function which is holomorphic on $\HH$, vanishes at the
957cusp $0$ and whose Fourier expansion at infinity has the form
958\begin{eqnarray}
959\label{jmpdef}
960 j_m^{(p)}(z)=q^{-m}+c(0)+c(1)q+c(2)q^2+ \cdots.
961\end{eqnarray}
962Because $\Gamma_0(p)$ is genus zero, each of these functions can
963be written as monic polynomials in $j_1^{(p)}(z)=j^{(p)}(z)$ with
964constant term equal to zero.  For example, we have
965\begin{eqnarray*}
966j_0^{(5)}(z)&=&1, \\
967j_1^{(5)}(z)&=&j^{(5)}(z)=q^{-1}-6+9q+10q^2-30q^3+ \cdots \\
968j_2^{(5)}(z)&=&j^{(5)}(z)^2+12j^{(5)}(z)=q^{-2}-18+20q+21q^2+192q^3+
969\cdots \\
970j_3^{(5)}(z)&=&j^{(5)}(z)^3+18j^{(5)}(z)^2+81j^{(5)}(z)=q^{-3}-24-90q+288q^2+144q^3
971+ \cdots
972\end{eqnarray*}
973In analogy with our definition of $\mathfrak{F}$, we define
974$\mathfrak{F}_p$ to be a fundamental domain for the action of
975$\Gamma_0(p)$ on $\HH$, taking the convention that
976$\mathfrak{F}_p$ does not include the two cusps $\infty$ and $0$.
977If $\tau \in \HH$, then (in analogy with (\ref{edef})) we define
978$e^{(p)}_{\tau} \in \left\{1,\frac{1}{2},\frac{1}{3}\right\}$ by
979$$980e_{\tau}^{(p)}:=( \textrm{the order of the isotropy subgroup of } 981\tau \textrm{ in } \Gamma_0(p)/\{ \pm I\} )^{-1}. 982$$
983We can now state the following theorem:
984
985\begin{theorem}[\cite{Ahl1}] \label{scott} Suppose that $p \in 986\{2,3,5,7,13\}$ and that $f(z)=\sum_{n=h}^{\infty} a(n)q^n \in 987\mmer_k(\Gamma_0(p))$, normalized so that $a(h)=1$.  Then
988\begin{eqnarray*}
989\frac{\theta f}{f} =-\sum_{\tau \in \mathfrak{F}_p}
990\left(e^{(p)}_{\tau}\sum_{n=1}^{\infty}
991j_n^{(p)}(\tau)q^n\right)+\frac{h-k/12}{p-1} \cdot
992pE_2|V(p)+\frac{pk/12-h}{p-1} \cdot E_2.
993\end{eqnarray*}
994\end{theorem}
995
996We will not provide a proof of this theorem; it is entirely
997analogous to the proof of Theorem \ref{bko} except for some
998difficulties which naturally arise when dealing with congruence
999subgroups.  We note that a formula analogous to Corollary
1000\ref{aknfu} holds in the $\Gamma_0(p)$ case for $p \in 1001\{2,3,5,7,13\}$ as well (see \cite{Ahl1}).
1002
1003\section{Serre's $p$-adic modular forms} \label{Spadic}
1004
1005We begin with the notion of congruent $q$-series.  Two $q$-series
1006$f(z)=\sum_{n=n_0}^{\infty}a(n)q^n \in q^{n_0}\ZZ[[q]]$ and $g(z)= 1007\sum_{m=m_0}^{\infty}b(m)q^m\in q^{m_0}\ZZ[[q]]$ are said to be
1008\emph{congruent modulo $N$} if
1009$$1010a(k) \equiv b(k) \pmod{N} 1011$$
1012for all $k$.  For primes $p$, we say that a $q$-series $f(z)$ with
1013integral coefficients is a \emph{weakly modular form modulo $p^n$}
1014if it is congruent modulo $p^n$ to a modular form $g(z) \in \minf 1015\cap q^{-m_0}\ZZ[[q]]$. This is written as
1016$$1017f(z) \equiv g(z) \pmod{p^n} 1018$$
1019We note here that the theory of modular forms modulo prime powers
1020is quite well developed; for a basic introduction, see \cite[\S
1021IV.X]{La}, and for a variety of interesting number-theoretic
1022applications, see \cite{O}.
1023
1024We begin by establishing some well-known congruences involving the
1025Eisenstein series $E_k(z)$.  First we recall two classical
1026Bernoulli number congruences (see \cite[p.~233-238]{IR}).  Let
1027$D_n$ be the denominator of the $n$th Bernoulli number, written in
1028lowest terms.  The von Staudt-Clausen congruences state
1029\begin{equation}
1030\label{VS} D_n=6\prod_{(p_i-1) |n}p_i
1031\end{equation}
1032where the $p_i$'s are prime. Let $p\ge5$ be prime. Now suppose $m 1033\geq 2$ is even and $m' \equiv m \pmod{\phi(p^r)}$ where $\phi$ is
1034the Euler $\phi$-function.  Then the Kummer congruences state
1035\begin{equation}
1036\label{Kummer} \frac{(1-p^{m'-1})B_{m'}}{m'} \equiv
1037\frac{(1-p^{m-1})B_{m}}{m} \pmod{p^r}.
1038\end{equation}
1039
1040Using these congruences, we prove the following lemma:
1041\begin{lemma}
1042\label{pmoddy} For $r \geq 1$ and odd primes $p$, the following
1043$q$-series congruences hold:
1044\begin{equation}
1045\label{1wrclev} (E_{p-1}(z))^{p^{r-1}} \equiv 1 \pmod{p^r}
1046\end{equation}
1047and
1048\begin{equation}
1049\label{E2} E_{\phi(p^r)+2}(z) \equiv E_2(z) \pmod{p^r}.
1050\end{equation}
1051\end{lemma}
1052\begin{proof}
1053For (\ref{1wrclev}), we have
1054$$1055\left(E_{p-1}(z)\right)^{p^{r-1}}=\left( 10561-\frac{2(p-1)}{B_{p-1}}\sum_{n=1}^{\infty}\sigma_{p-1}(n)q^n 1057\right)^{p^{r-1}}=\left( 10581-\frac{2(p-1)D_{p-1}}{U_{p-1}}\sum_{n=1}^{\infty}\sigma_{p-1}(n)q^n 1059\right)^{p^{r-1}} 1060$$
1061where $U_{p-1}$ is an integer coprime to $D_{p-1}$.  From
1062(\ref{VS}) we have $p|D_{p-1}$ which implies (\ref{1wrclev}) after
1063an application of the binomial theorem.
1064
1065To prove (\ref{E2}), if we let $m'=2$, $m=\phi(p^r)+2$ in
1066(\ref{Kummer}) and note that $p^{\phi(p^r)+1} \equiv p 1067\pmod{p^{r}}$ we obtain
1068$$1069\frac{B_2}{2} \equiv \frac{B_{\phi(p^r)+2}}{\phi(p^r)+2} 1070\pmod{p^r}. 1071$$
1072Also by Euler's theorem, $$\sigma_1(n) \equiv 1073\sigma_{\phi(p^r)+1}(n) \pmod{p^r}.$$  With these two observations
1074we have
1075$$1076E_2(z)=1-\frac{2(2)}{B_2} \sum_{n=1}^{\infty}\sigma_1(n)q^n \equiv 10771-\frac{2(\phi(p^r)+2)}{B_{\phi(p^r)+2}} 1078\sum_{n=1}^{\infty}\sigma_{\phi(p^r)+1}(n)q^n \pmod{p^r}. 1079$$
1080\end{proof}
1081
1082\noindent We also record here the following congruences, which
1083will be useful in \S \ref{good}:
1084
1085\begin{lemma} \label{24k} Suppose $k \geq 4$ is even.  Then
1086\begin{equation*}
1087E_k(z) \equiv 1 \pmod{24},
1088\end{equation*}
1089and, if $p \geq 5$ is a prime such that $(p-1) \mid k$,
1090\begin{equation*}
1091E_k(z) \equiv 1 \pmod{p}.
1092\end{equation*}
1093\end{lemma}
1094\begin{proof}
1095These both follow immediately from the von Staudt-Clausen equation
1096(\ref{VS}).
1097\end{proof}
1098
1099
1100Before we can proceed any farther, we must generalize the notion
1101of congruent modular forms introduced above.  Let $K$ be a number
1102field with ring of integers $\mathcal{O}_K$, and $\mathfrak{m} 1103\subset \mathcal{O}_K$ an ideal.  We define the \emph{order of $f$
1104modulo $\mathfrak{m}$} by
1105$$1106\ord_{\mathfrak{m}}(f):=\min\{n:a(n) \not \in \mathfrak{m} \} 1107$$
1108with the convention that $\ord_{\mathfrak{m}}(f):=+\infty$ if
1109$a(n) \in \mathfrak{m}$ for all $n$.  Though this is certainly not
1110obvious a priori, given a modular form with coefficients in
1111$\mathcal{O}_K$, one need only check finitely many $q$-series
1112coefficients to calculate $\ord_{\mathfrak{m}}(f)$. The following
1113theorem of Sturm (see \cite[\S 2.9]{O} or \cite{St}) makes this
1114precise:
1115\begin{theorem} \label{sturm}
1116Suppose $k \geq 0$ is an integer and $K$ is a number field with
1117ring of integers $O_K$.  Moreover let $f=\sum_{n=0}^{\infty} 1118a(n)q^n \in M_k(\Gamma_0(N)) \cap \OO_K[[q]]$.  If $\mathfrak{m} 1119\subset \mathcal{O}_K$ is an ideal for which
1120$$1121\ord_{\mathfrak{m}}(f)>\frac{k}{12}[\Gamma_0(1):\Gamma_0(N)] 1122$$
1123then $\ord_{\mathfrak{m}}(f)= + \infty$.
1124\end{theorem}
1125\begin{remark}We will not prove this theorem in this thesis.  We will only
1126require it for the proofs of theorems \ref{modelth} and
1127\ref{gbro}, and there we only invoke it briefly to prove that we
1128can normalize certain forms so that they have coefficients in a
1129ring of integers.  To see how this works, consider some form $f 1130\in M_k(\Gamma_0(N))$ with $p$-integral algebraic coefficients.
1131Then we can pick an integer $M \equiv 1 \pmod{p}$ such that the
1132first $\frac{k}{12}[\Gamma_0(1):\Gamma_0(N)]$ coefficients of $Mf$
1133are contained in the ring of integers of some number field
1134$\OO_K$. Applying Theorem \ref{sturm}, it follows that all of the
1135coefficients of $Mf$ are in $\OO_K$, in other words, we have
1136produced a form $Mf \equiv f \pmod{p}$ with algebraic integer
1137coefficients.
1138\end{remark}
1139
1140Elements of $M_k(\Gamma')$ have the extremely useful property that
1141they determined by their first few $q$-series coefficients.
1142Though, as noted above, we will not need Theorem \ref{sturm} until
1143\S \ref{pborc}, we included it at this point to call the reader's
1144attention to the fact that a similar statement is true when
1145working with modular forms congruent modulo ideals in a number
1146field.
1147
1148We are now in a position to justify the title of this section. Let
1149$K$ be a number field and let $\mathcal{O}_v$ be the completion of
1150its ring of integers at a finite place $v$ with residue
1151characteristic $p$.  Moreover, let $\lambda$ be a uniformizer for
1152$\mathcal{O}_v$.  We make the following:
1153\begin{definition} A formal power
1154series
1155$$1156f:=\sum_{n=0}^{\infty}a(n)q^n \in \mathcal{O}_v[[q]] 1157$$
1158is a \textbf{$p$-adic modular form of weight $k \in 1159\mathcal{O}_v$} if there is a sequence $f_i \in 1160\mathcal{O}_v[[q]]$ of holomorphic modular forms on $\Gamma$ with
1161weights $k_i$ for which $\ord_{\lambda}(f_i-f) \to +\infty$ and
1162$\ord_{\lambda}(k-k_i) \to + \infty$.
1163\end{definition}
1164\begin{remark} This is Serre's original definition  of a
1165$p$-adic modular form \cite{Sep}.  The notion of a $p$-adic
1166modular form has been substantially generalized by Katz; for an
1167introduction and an explanation of how the two definitions relate,
1168see \cite[\S I]{Go}.
1169\end{remark}
1170Thus we observe, with the help of Lemma \ref{pmoddy}, that $E_2$
1171and $1$ are both $p$-adic modular forms, or, more precisely, are
1172$p$-adic modular forms when identified with their $q$-expansions
1173considered as elements of $\OO_v[[q]]$ (the $q$-expansion of $1$
1174is just $1+0q+0q^2+\cdots$). Further, any element of $M_k \cap 1175\OO_v[[q]]$ is trivially a $p$-adic modular form.
1176
1177The only nontrivial result we will require from the theory of
1178$p$-adic modular forms is a theorem, due to Serre, which allows us
1179to compute the constant term of a $p$-adic modular form in terms
1180of a $p$-adic limit of its other coefficients for small primes
1181$p$. Let $\zeta_p^*(s)$ be the Kubota-Leopoldt $p$-adic zeta
1182function. We have
1183
1184\begin{theorem}[Theorem 7, \cite{Sep}] \label{Ser} If $p \leq 7$ is prime and
1185$$1186f=\sum_{n=0}^{\infty} a(n)q^n 1187$$
1188is a $p$-adic modular form of weight $k \neq 0$, then
1189$$1190a(0)=\frac{\zeta_p^*(1-k)}{2} \cdot \lim_{n \to + \infty} a(p^n). 1191$$
1192\end{theorem}
1193
1194This theorem is proven by decomposing the vector space $M$ of
1195$p$-adic modular forms into $M = E \oplus N$, where $N$ is a space
1196on which the $U$ operator (defined exactly as in (\ref{udef}))
1197acts nilpotently and $E$ is a space on which $U$ acts bijectively.
1198It turns out that for $2 \leq p \leq 7$ prime, $E$ is spanned by
1199the reductions of Eisenstein series, and $N$ is spanned by the
1200reductions of cusp forms.  By analyzing each subspace, the theorem
1201follows.  For a complete proof, see \cite[\S 2.3]{Sep}.
1202Incidentally, \cite{Sep} is a beautiful paper, and provides an
1203interesting counterpoint to Katz's geometric approach to $p$-adic
1204modular forms.
1205
1206Also mentioned in \cite[\S 1.6]{Sep} is the fact that
1207$\zeta_p^*(1-k)=(1-p^{k-1})\zeta(1-k)$ for even integers $k \geq 12082$, where $\zeta(s)$ is the usual characteristic zero Riemann zeta
1209function. In the sequel we will only be interested in the special
1210case $k=2$, in which we have:
1211$$1212\zeta_p^*(1-2)=(1-p)\zeta(-1)=\frac{p-1}{12}. 1213$$
1214Thus we immediately have the following corollary of Theorem
1215\ref{Ser}:
1216\begin{corollary}
1217\label{Serre} If $p \leq 7$ is prime and
1218$$1219f=\sum_{n=0}^{\infty} a(n)q^n 1220$$
1221is a $p$-adic modular form of weight $k \neq 0$, then
1222$$1223a(0)=\frac{p-1}{24} \cdot \lim_{n \to + \infty} a(p^n). 1224$$
1225\end{corollary}
1226
1227\section{Varying the level} \label{atle}
1228
1229Given a modular form $f \in M_k(\Gamma_0(M))$ (resp., $f \in 1230S_k(\Gamma_0(M))$) and recalling (\ref{vdef}) and (\ref{altvdef}),
1231it is not hard to verify using the functional equation
1232(\ref{modfunc}) that $f|V(d) \in M_k(\Gamma_0(dM))$ (resp.,
1233$f|V(d) \in S_k(\Gamma_0(dM))$).  These forms are holdovers from
1234lower levels; they're nothing new, which justifies the notation
1235$$1236S_k(\Gamma_0(N)) \supset 1237S_k^{\textrm{old}}(\Gamma_0(N)):=\bigoplus_{dM | 1238N}S_k(\Gamma_0(M))|V(d). 1239$$
1240We define \emph{the space of newforms $\Sn_k(\Gamma_0(N))$} to be
1241the orthogonal complement to $\So_k(\Gamma_0(N)$ with respect to a
1242certain inner product, called the \emph{Petersson inner product}
1243(see \cite[\S III.4]{La} or \cite[\S III.3]{k}).  As a first
1244example, for $p \geq 3$ prime, we have
1245\begin{eqnarray} \label{decomp1}
1246 M_2(\Gamma_0(p))=\langle
1247E_2(z)-pE_2(pz) \rangle \oplus \Sn_2(\Gamma_0(p))
1248\end{eqnarray}
1249because $M_2(\Gamma)=0$.  One can check that $E_2(z)-pE_2(pz)$
1250satisfies the requisite functional equation using (\ref{e2}). For
1251arbitrary weights, the space of newforms has the useful property
1252that it is preserved under the action of the Hecke operators.  It
1253is also invariant under another operator, the Atkin-Lehner
1254involution, which we now define.
1255\begin{definition} For a prime divisor $p$ of $N$ with
1256$\ord_p(N)=\ell$, let $Q_p:=p^{\ell}$.  We define the
1257\textbf{Atkin-Lehner operator $|_k W(Q_p)$} on $M_k(\Gamma_0(N))$
1258by any matrix
1259$$1260W(Q_p):=\left( \begin{smallmatrix} Q_pa & b \\ N c & Q_pd 1261\end{smallmatrix} \right) \in M_{2 \times 2}(\ZZ) 1262$$
1263with determinant $Q_p$, where $a,b,c,d \in \ZZ$.  Further, define
1264the \textbf{Fricke involution $|_k W(N)$} on $M_k(\Gamma_0(N))$ by
1265the matrix
1266$$1267W(N):=\begin{pmatrix} 0 & -1 \\ N & 0 \end{pmatrix}. 1268$$
1269\end{definition}
1270\noindent Well-definition of $|_k W(Q_p)$ follows from the
1271functional equation of $f \in M_k(\Gamma_0(N))$ and the fact that
1272$W(Q_p)$ is unique up to left multiplication by elements of
1273$\Gamma_0(N)$.  We note here that for $f \in M_k(\Gamma_0(p))$ we
1274have $f|_kW(Q_p)=f|_kW(p)$.  By abuse of language, we will call
1275$W(p)$ an Atkin-Lehner operator in this setting.
1276
1277We now are in a position to make the following:
1278\begin{definition} A \textbf{newform} in $\Sn_k(\Gamma_0(N))$ is a
1279normalized cusp form that is an eigenform for all the Hecke
1280operators, all of the Atkin-Lehner involutions $|_k W(Q_p)$ for
1281$p|N$, and the Fricke involution $|_k W(N)$.
1282\end{definition}
1283
1284Newforms enjoy remarkable properties.  We recall a few such
1285properties on the more utilitarian side of things:
1286\begin{theorem} \label{newforms}
1287Suppose that $k$ is a positive even integer.  Then
1288\begin{enumerate}
1289\item  The space $\Sn_k(\Gamma_0(N))$ has a basis of newforms.
1290
1291\item  If $f(z)=\sum_{n=1}^{\infty} a(n)q^n \in 1292\Sn_k(\Gamma_0(N))$ is a newform, then there is a number field $K$
1293with the property that for every integer $n$ we have $a(n) \in 1294\OO_K$, the ring of algebraic integers of $K$.
1295
1296\item If $f \in \Sn_k(\Gamma_0(N))$ is a newform then there is an
1297integer $\lambda_f \in \{\pm1\}$ for which
1298$$1299f|_kW(Q_p)=\lambda_pf. 1300$$
1301\end{enumerate}
1302\end{theorem}
1303\noindent For the statements of a collection of results, including
1304the above, on newforms, see \cite[\S 2.4,\S 2.5]{O}.  For proofs,
1305see \cite{AL}, and for generalizations, see \cite{L} and \cite{M}.
1306
1307We began this section by discussing how one can raise the level of
1308an element of $M_k(\Gamma_0(N))$ to obtain an element of
1309$M_k(\Gamma_0(MN))$.  We now discuss the \emph{trace operator
1310$\textrm{Tr}^{MN}_N$}, which lowers the level.  For coprime $M,N$,
1311define
1312$$1313\Tr^{MN}_N:M_k(\Gamma_0(MN)) \to M_k(\Gamma_0(N)) 1314$$
1315by
1316$$1317\Tr^{MN}_N(f) =\sum_{i=1}^rf|_k \gamma_i 1318$$
1319where $\{\gamma_1,...,\gamma_r\}$ is a complete set of coset
1320representatives for $\Gamma_0(NM) \backslash \Gamma_0(N)$.  The
1321fact that $\Tr^{MN}_N(f) \in M_k(\Gamma_0(N))$ is immediate;
1322acting on $\Tr^{MN}_N(f)$ by an element of $\Gamma_0(N)$ simply
1323permutes the $\gamma_i$ by the invariance of $f$ under the action
1324of $\Gamma_0(NM)$.  We have the following explicit formula for
1325$\Tr^{Np}_p$:
1326\begin{lemma}[\cite{MO}] \label{tr} Suppose that $p$ is an odd prime and that $p 1327\nmid N$.  If $f \in M_k(\Gamma_0(Np))$ then
1328$$1329\Tr_N^{Np}(f)=f+p^{1-k/2}f|_kW(p)U(p) 1330$$
1331\end{lemma}
1332\begin{proof}
1333A complete set of coset representatives for $\Gamma_0(Np)$ in
1334$\Gamma_0(N)$ is given by
1335$$1336\left\{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\right\} \cup 1337\left\{ 1338\begin{pmatrix} 1 & 0 \\ N & 1 \end{pmatrix}\begin{pmatrix} 1 & j \\ 0 & 13391\end{pmatrix}\right\}_{j=0}^{p-1}. 1340$$
1341We also have
1342$$1343\begin{pmatrix} 1 & 0 \\ N & 1\end{pmatrix} \begin{pmatrix} 1 & j 1344\\ 0 & 1 1345\end{pmatrix}= \begin{pmatrix} 1/p & 0 \\ 0 & 1/p \end{pmatrix} 1346\begin{pmatrix} p & a \\ Np & pb \end{pmatrix} \begin{pmatrix} 1 & 1347j-a \\ 0 & p 1348\end{pmatrix} 1349$$
1350where
1351$$1352\begin{pmatrix} p & a \\ Np & pb \end{pmatrix} 1353$$
1354is a matrix for $W(p)$.  Since scalar matrices act trivially on
1355$M_k(\Gamma_0(Np))$,
1356$$1357\Tr^{MN}_N(f) = f + \sum_{j=1}^{p-1} f|_k W(p) \begin{pmatrix} 1 & 1358j \\ 0 & p 1359\end{pmatrix}. 1360$$
1361By considering $q$-expansions, we have
1362$$1363\sum_{j=0}^{p-1} g\left( \frac{z+j}{p}\right)=p(g|U(p))(z), 1364$$
1365which completes the proof of the lemma.
1366\end{proof}
1367It is well-known that if $p$ is prime with $p \nmid N$, then
1368$\Tr_N^{Np}(f)=0$ for $f \in \Sn_k(\Gamma_0(Np))$ (see \cite{L}).
1369Combining this observation with Lemma \ref{tr} yields the
1370following:
1371\begin{proposition}[\cite{AL}] \label{wphecke} If $f \in 1372\Sn_k(\Gamma_0(p))$, then
1373$$1374f|_kW(p)=-p^{1-k/2}f|U(p). 1375$$
1376\end{proposition}
1377\begin{proof}
1378First suppose that $f$ is a newform.  From Lemma \ref{tr}, we have
1379$$13800=\Tr_1^{p}(f)=f+p^{1-k/2}f|_kW(p)U(p). 1381$$
1382Thus
1383\begin{equation} \label{needitnow}
1384f=-p^{1-k/2}f|_kW(p)U(p).
1385\end{equation}
1386 Note that $U(p)=T_{k,p}$ because the
1387level is $p$ (see (\ref{uvhecke})). Now note that $f$, being a
1388newform, is an eigenform both for the Hecke operators and $W(p)$
1389(by Theorem \ref{newforms}).  Thus the actions of $W(p)$ and
1390$U(p)$ on $f$ commute.  With all this in mind, applying $W(p)$ to
1391both sides of (\ref{needitnow}), we have
1392\begin{eqnarray*}
1393f|_kW(p)&=&-p^{1-k/2}f|_kW(p)U(p)W(p) \\
1394&=&-p^{1-k/2}f|_kW(p)^2U(p) \\
1395&=&-p^{1-k/2}f|_kU(p).
1396\end{eqnarray*}
1397To derive the last equality, we used the fact that the action of
1398$W(p)^2$ is trivial, which can be seen from directly from a matrix
1399representation of $W(p)$: $\left(\begin{smallmatrix} 0 & -1 \\ p & 14000 1401\end{smallmatrix}\right) \left(\begin{smallmatrix} 0 & -1 \\ p & 0 1402\end{smallmatrix}\right)=\left(\begin{smallmatrix} -p & 0 \\ 0 & 1403-p 1404\end{smallmatrix}\right)$.  Since $U(p)$ and $W(p)$ are both
1405linear operators, the proposition now follows for all $f \in 1406\Sn_k(\Gamma_0(p))$.
1407\end{proof}
1408
1409
1410\section{$p$-adic properties of Borcherds exponents} \label{pborc}
1411We begin with the following:
1412\begin{definition} Let $f$ be a meromorphic modular form of weight $k$ over $\Gamma$ or
1413$\Gamma_0(p)$
1414 whose poles and zeros, away from $z=\infty$,
1415are at the points $z_1,...,z_s \in \mathbb{H}$. We say that $f(z)$
1416is \emph{\textbf{good at $p$}} if there is a holomorphic modular
1417form $\mathcal{E}_f(z) \in M_b(\Gamma)$ with $p$-integral
1418algebraic coefficients for which the following are true:
1419\begin{enumerate}
1420\item As $q$-series, $\mathcal{E}_f(z) \equiv 1 \pmod{p}$. \item
1421For each $1 \leq i \leq s$ we have $\mathcal{E}_f(z_i)=0$.
1422\end{enumerate}
1423\end{definition}
1424
1425\begin{remark}[1] It follows immediately that if $f$ and $g$ are
1426good, then $fg$ is good.
1427\end{remark}
1428
1429\begin{remark}[2] As
1430mentioned in the introduction, we will provide several families of
1431good forms in \S \ref{good}; other families are provided in
1432\cite{BrO}. Unfortunately, the author has not thought carefully
1433about interesting examples of forms which are not good.
1434\end{remark}
1435
1436In view of the observations we made in sections \ref{intro} and
1437\ref{Rth}, it is now straightforward to prove Theorem
1438\ref{modelth}:
1439
1440\begin{proof}[Proof of Theorem \ref{modelth}]
1441By examining the proof of Proposition \ref{thetobs}, we see that
1442if $f$ is a meromorphic modular form of weight $k$ over $\Gamma$,
1443then
1444\begin{equation} \label{fonow}
1445\widetilde{f}:=12 \Theta f(z)-kE_2(z) f(z)
1446\end{equation}
1447is a meromorphic modular form of weight $k+2$ over $\Gamma$.
1448Further, from (\ref{fonow}) we see that the poles of
1449$\widetilde{f}(z)$ are supported at the poles of $f(z)$.
1450
1451Now consider
1452$$1453\frac{\theta f}{f} 1454=\frac{1}{12}\left(\frac{\widetilde{f}(z)}{f(z)}+kE_2(z) \right). 1455$$
1456By \ref{pmoddy}, $E_2$ is a $p$-adic modular form of weight $2$
1457with integer coefficients.  Thus it suffices to show that
1458$\widetilde{f}/f$ is as well.  If $b$ is the weight of
1459$\mathcal{E}_f(z)$, then note
1460$\mathcal{E}_f(z)^{p^j}\widetilde{f}/f \in M_{p^jb+2}$. If
1461$\mathcal{E}_f(z)^{p^j}\widetilde{f}/f$ does not have algebraic
1462integer coefficients, then multiply it by a suitable integer
1463$t_{j+1} \equiv 1 \pmod{p^{j+1}}$ so that the resulting series
1464does.  Thus we have
1465$$1466t_{j+1}\mathcal{E}_f(z)^{p^j}\frac{\widetilde{f}}{f} \equiv 1467\frac{\widetilde{f}}{f} \pmod{p^{j+1}}. 1468$$
1469If we define
1470$F_{j+1}(z):=t_{j+1}\mathcal{E}(z)^{p^j}\widetilde{f}(z)/f(z)$,
1471then we have that $\{F_{j+1}\}$ is a sequence of holomorphic
1472modular forms whose coefficients $p$-adically converge to
1473$\widetilde{F}(z)/F(z)$ and whose weights $p$-adically converge to
1474$2$.
1475\end{proof}
1476We will devote the rest this section to proving Theorem
1477\ref{gbro}, a generalization of Bruinier and Ono's result to forms
1478of prime level $p \geq 5$. We require two lemmas before we start
1479on the main body of the proof.  The first is most naturally proven
1480using the notion of the divisor polynomial of a modular form,
1481which we now recall. If $k\geq 4$ is even, then define
1482$\widetilde{E}_k(z)$ by
1483\begin{equation}\label{Wtag2.5}
1484\widetilde{E}_k(z):=\begin{cases} 1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 0\pmod{12},\\
1485E_{4}(z)^2E_6(z) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12},\\
1486E_4(z)\ \ \ \ \ &{\text {\rm if}}\ k\equiv 4\pmod{12},\\
1487E_6(z)\ \ \ \ \ &{\text {\rm if}}\ k\equiv 6\pmod{12},\\
1488E_4(z)^2\ \ \ \ \ &{\text {\rm if}}\ k\equiv 8\pmod{12},\\
1489E_{4}(z)E_6(z) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 10\pmod{12},
1490\end{cases}
1491\end{equation}
1492and polynomials $h_k$ by
1493\begin{equation}
1494h_k(x):=\begin{cases}   1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 0\pmod{12},\\
1495            x^2(x-1728) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12},\\
1496               x \ \ \ \ \ &{\text {\rm if}}\ k\equiv 4\pmod{12},\\
1497               x-1728 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 6\pmod{12},\\
1498               x^2 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 8\pmod{12},\\
1499               x(x-1728) \ \ \ \ \ &{\text {\rm if}}\ k\equiv
1500               10\pmod{12}.
1501 \end{cases}
1502\end{equation}
1503
1504Further, define $m(k)$ by
1505$$1506m(k):=\begin{cases} 1507 \lfloor k/12\rfloor \ \ \ \ \ &{\text {\rm if}}\ k\not \equiv 2\pmod{12},\\ 1508 \lfloor k/12\rfloor -1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12}. 1509\end{cases} 1510$$
1511With this notation, if $f(z) \in M_k$ and $\widetilde{F}(f,x)$ is
1512the unique rational function in $x$ for which
1513\begin{equation}
1514\label{divyopoly}
1515f(z)=\Delta(z)^{m(k)}\widetilde{E}_k(z)\widetilde{F}(f,j(z)),
1516\end{equation}
1517then $\widetilde{F}(f, x)$ is a polynomial; this follows from the
1518familiar fact that any element of $\minf_0$ is a polynomial in
1519$j$.  We will refer to
1520\begin{equation} \label{divisorpoly}
1521F(f,x):=h_k(x)\widetilde{F}(f,x)
1522\end{equation}
1523as the \emph{divisor polynomial} for $f$. From (\ref{Wtag2.5}),
1524(\ref{divyopoly}) and the classical $k/12$ valence formula (again,
1525see \cite[\S III.2]{k}) the polynomial $F(f,x)$ will have a zero
1526of order $n_k$ precisely at $j(z_k)$ for all zeros $z_k$ of $f$,
1527where $n_k:=\ord_{z_k}(f)$. For a discussion of divisor
1528polynomials, see \cite[\S 2.6]{O}.
1529\begin{lemma} \label{pint}
1530Suppose $f =q^h\prod_{n=1}^\infty(1-q^n)^{c(n)} \in 1531\mmer_k(\Gamma_0(p)) \cap q^h\OO_K[[q]]$ for some number field $K$
1532 and some prime $p \geq 5$, and further that $f$ is good at $p$.  Then
1533$$1534\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p) \in 1535\mmer_2(\Gamma_0(p)) 1536$$
1537is $p$-integral.
1538\end{lemma}
1539\begin{proof}  Note that $F(\mathcal{E}_f,j)$ has $p$-integral algebraic
1540coefficients as a $q$-series and as a polynomial because
1541$\mathcal{E}_f$ has $p$-integral algebraic coefficients.  Thus, if
1542$z_1,...,z_n$ are the zeros and poles of $f$ as before (written
1543without multiplicity),
1544$$1545G(j(z)):=(j(z)-j(z_1))\cdots(j(z)-j(z_n)) 1546$$
1547has $p$-integral algebraic $q$-series coefficients.  Because no
1548prime above $p$ divides the $q$-expansion coefficient of lowest
1549exponent in $G(j(z))$, we also have that $(G(j))^{-1}$ is
1550$p$-integral (again as a $q$-series). Thus we may write
1551$$1552\frac{\Theta(f)-k(12)^{-1}E_2}{f}=\frac{g}{G(j)} 1553$$
1554where $g \in M_2(\Gamma_0(p)) \cap \overline{\QQ} [[q]]$ has
1555$p$-integral algebraic coefficients.  We have
1556$$1557\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p)=\left(\frac{1}{G(j)}\right)|_0W(p)g|_2W(p). 1558$$
1559We will prove that each of the factors on the right hand side is
1560$p$-integral.  First,
1561\begin{eqnarray*}
1562\left(\frac{1}{G(j(z))}\right)|_0W(p)&=&\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
15630 & -1 \\ p & 0
1564\end{matrix}\right)=\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
15650 & -1 \\ 1 & 0 \end{matrix}\right)\left(\begin{matrix} p & 0
1566\\ 0 & 1
1567\end{matrix}\right)\\ &=&\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
1568p & 0 \\ 0 & 1
1569\end{matrix}\right)=\frac{1}{G(j(pz))},
1570\end{eqnarray*}
1571which is evidently $p$-integral.  Now note that we can write
1572$g=c_1(E_2(z)-pE_2(pz))+h(z)$, where $h(z) \in \Sn_2(\Gamma_0(p))$
1573has $p$-integral algebraic coefficients and $c_1$ is a
1574$p$-integral algebraic number.  From Proposition \ref{wphecke} we
1575have $h(z)|_2W(p)=-h(z)|U(p)$, which is $p$-integral by the
1576$q$-series definition (\ref{udef}) of the $U(p)$ operator.  Using
1577(\ref{e2}), we also have
1578\begin{eqnarray*}
1579(E_2(z)-pE_2(pz))|_2W(p)&=&E_2(z)|_2\begin{pmatrix}0 & -1 \\ 1 & 0
1580\end{pmatrix}\begin{pmatrix}p & 0 \\ 0 & 1
1581\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
1582\end{pmatrix}-\frac{12}{2 \pi i z}
1583-E_2(z) \\&=&pE_2(pz)-E_2(z).
1584\end{eqnarray*}
1585which is also $p$-integral.  Since we have dealt with both
1586factors, the lemma follows.
1587\end{proof}
1588\begin{remark} If restrict to the case $k=0$, this lemma is also
1589true for $p=3$; the proof is the same.
1590\end{remark}
1591
1592Define
1593\begin{equation}
1594\widetilde{E}_3(z):=E_2(z)-3E_2(3z) \in M_2(\Gamma_0(3))
1595\end{equation}
1596(see \ref{decomp1}) and
1597\begin{equation}
1598\widetilde{E}_p:=E_{p-1}(z)-p^{(p-1)/2}(E_{p-1}(z)|_{p-1}W(p)) \in
1599M_{p-1}(\Gamma_0(p))
1600\end{equation}
1601for primes $p \geq 5$.  We have following:
1602\begin{lemma}
1603If $p$ is an odd prime, then
1604\begin{eqnarray} \label{twidE1}
1605\widetilde{E}_p(z) &\equiv& 1 \pmod{p} \\
1606\label{twidE2} (\widetilde{E}_p(z)|_{p-1}W(p)) &\equiv& 0
1607\pmod{p^{(p-1)/2+1}}
1608\end{eqnarray}
1609\end{lemma}
1610\begin{proof}  For $p=3$, the first claim is obvious, and the
1611second follows from the end of the proof of Lemma \ref{pint}. For
1612$p \geq 5$ we compute
1613\begin{eqnarray*}
1614E_{p-1}|_{p-1}W(p) &=&E_{p-1}|_{p-1}\begin{pmatrix} 0 & -1
1615\\ p & 0 \end{pmatrix} \\
1616&=& E_{p-1}|_{p-1}\begin{pmatrix} 0 & -1
1617\\ 1 & 0 \end{pmatrix} \begin{pmatrix} p & 0
1618\\ 0 & 1 \end{pmatrix} \\
1619&=& p^{(p-1)/2}E_{p-1} | V(p)
1620\end{eqnarray*}
1621>From Lemma \ref{pmoddy}, we know that $E_{p-1}$ is $p$-integral.
1622Thus we have the congruence $\widetilde{E}_p \equiv E_{p-1} 1623\pmod{p}$, which yields $\widetilde{E}_p \equiv 1 \pmod{p}$ for
1624all odd primes $p$ after an application of Lemma \ref{pmoddy}.
1625
1626For the second claim, we have
1627\begin{eqnarray*}
1628\widetilde{E}_p|_{p-1}W(p) &=&
1629E_{p-1}|_{p-1}W(p)-p^{(p-1)/2}E_{p-1}|_{p-1}W(p)W(p) \\
1630&=& E_{p-1}|_{p-1}W(p)-p^{(p-1)/2}E_{p-1} \\
1631&=& p^{(p-1)/2}E_{p-1} | V(p)-p^{(p-1)/2}E_{p-1}.
1632\end{eqnarray*}
1633We note the $p$-integrality of $E_{p-1}$ and $E_{p-1}|V(p)$ and
1634again apply Lemma \ref{pmoddy} to finish the proof of the lemma.
1635\end{proof}
1636
1637We now prove Theorem \ref{gbro}.  The two main inputs into this
1638proof are the ideas behind the proof of Theorem \ref{modelth} and
1639Serre's proof that a newform in $\Sn_k(\Gamma_0(p))$ is a $p$-adic
1640modular form (see \cite{MO} and \cite{Sep}).
1641\begin{proof}[Proof of Theorem \ref{gbro}]
1642By (\ref{thetobs}), there exists a meromorphic modular form $g$ on
1643$\Gamma_0(p)$ so that
1644$$1645\frac{\Theta f}{f}=\frac{g}{f}+\frac{k}{12}E_2. 1646$$
1647Because $E_2$ is a $p$-adic modular form of weight two, it
1648suffices to show that the same is true of $\frac{\Theta 1649f-k(12)^{-1} E_2 f}{f}=\frac{g}{f}$.
1650
1651Fix a positive integer $r$.  Then (using the fact that $f$ is good
1652at $p$), we have
1653$$1654(\mathcal{E}_f)^{p^{r-1}} \frac{\Theta f-k(12)^{-1} E_2 f}{f} \in 1655M_{2+p^{r-1}b}(\Gamma_0(p)) 1656$$
1657where $b$ is the weight of $\mathcal{E}_f$.  Further, this form is
1658congruent modulo $p^r$ to $g/f$. Now consider
1659$$1660f_r(z):=(\widetilde{E}_p)^{p^{r-1}}(\mathcal{E}_f)^{p^{r-1}} 1661\frac{\Theta f-k(12)^{-1} E_2 f}{f} \equiv \frac{g}{f} \pmod{p^r}. 1662$$
1663We clearly have $f_r \in M_{2+p^{r-1}b+p^r-p^{r-1}}(\Gamma_0(p))$.
1664We now take the trace of these $f_r$ to lower their level. We
1665certainly have $\Tr_1^p(f_r) \in M_{2+p^{r-1}b+p^r-p^{r-1}}$, and
1666we will prove shortly that $\Tr_1^p(f_r) \equiv f_r \equiv 1667\frac{g}{f} \pmod{p^r}$.  Now, as in the proof of Theorem
1668\ref{modelth}, choose a suitable integer $t_r \equiv 1 1669\pmod{p^{r}}$ such that $t_r\Tr_1^p(f_r)$ has coefficients in the
1670ring of integers $\OO_{K_r}$ of some number field
1671$\mathcal{O}_{K_r}$ (this normalization may or may not be
1672necessary depending on $\mathcal{E}_f$). Then
1673$\{t_r\Tr_1^p(f_r)\}$ forms a sequence of holomorphic modular
1674forms over $\Gamma$ whose coefficients converge $p$-adically to
1675$g/f$ and whose weights converge to $2$, thus $g/f$ is a $p$-adic
1676modular form of weight $2$.
1677
1678We now prove that $\Trace_1^p(f_r) \equiv f_r \pmod{p^{r}}$. By
1679Lemma \ref{tr}, we have
1680\begin{eqnarray*}
1681\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)
1682\\ &=&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)
1683(\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)
1684\end{eqnarray*}
1685Because $f$ is good, applying Lemma \ref{pint} implies that
1686$\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p)$ is
1687$p$-integral, which, together with the definition of $U(p)$,
1688implies that
1689\begin{eqnarray} \label{1ord}
1690\left(\frac{\Theta f-k(12)^{-1} E_2 f}{f}\right)|_2W(p)U(p)
1691\end{eqnarray}
1692is $p$-integral.  Using (\ref{twidE2}), we also compute
1693\begin{eqnarray} \label{2ord}
1694\nonumber &&\widetilde{E}^{p^{r-1}}|_{p^r-p^{r-1}}W(p)U(p)
1695\\ &=&(\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}}},
1696\end{eqnarray}
1697and, just from the definitions,
1698\begin{eqnarray} \label{3ord}
1699&&
1700(\mathcal{E}_f)^{p^{r-1}}|_{p^{r-1}}W(p)U(p)\\
1701\nonumber &=&
1702p^{p^{r-1}b/2}(\mathcal{E}_f)^{p^{r-1}}|_{p^{r-1}b}\left(\begin{smallmatrix}
1703p & 0 \\ 0 & 1
1704\end{smallmatrix}\right)U(p) \equiv 0 \pmod{p^{p^{r-1}b/2}}.
1705\end{eqnarray}
1706The inequalities (\ref{1ord}), (\ref{2ord}) and (\ref{3ord})
1707together imply (as claimed) that $\Tr_1^p(f_r) \equiv f_r 1708\pmod{p^{r}}$.
1709\end{proof}
1710\begin{remark} As with Lemma \ref{pint}, Theorem \ref{gbro} is
1711true in the case $k=0$ for $p=3$ as well.  We will use this fact
1712without further comment in the proof of Theorem
1713\ref{classnumber2}.
1714\end{remark}
1715
1716\section{CM elliptic curves and supersingularity}
1717\label{cm}
1718
1719As indicated in the introduction, the construction of explicit
1720families of good forms will require a discussion of complex
1721multiplication and supersingularity, which we now begin. Recall
1722that for an elliptic curve $E/\CC$, there exists a lattice $L 1723\subset \CC$ such that
1724\begin{eqnarray} \label{Eisom}
1725\CC/L &\widetilde{\longrightarrow}& E \\ \nonumber z \not \in L
1726&\mapsto& (\wp(z,L),\wp'(z,L),1) \\ \nonumber z \in L &\mapsto&
1727(0:1:0)
1728\end{eqnarray}
1729is an analytic isomorphism.  Here $\wp$ is the classical
1730Weierstrass $\wp$-function.  Conversely, given any lattice $L 1731\subset \CC$, one can show that there exists an elliptic curve $E$
1732for which an analytic isomorphism of the form (\ref{Eisom}) holds.
1733Under this correspondence between lattices and elliptic curves,
1734isomorphism classes of elliptic curves over $\CC$ correspond to
1735equivalence classes of lattices, where the equivalence is given by
1736$L \sim L'$ if $L=cL'$ for some $c \in \CC^*$.  By way of
1737terminology, the map $L' \to L$ given by multiplication by $c \in 1738\CC^*$ is called a \emph{homothety}, and two lattices related in
1739such a way are called \emph{homothetic}.
1740 Note that we may choose a lattice $L_{\tau}$ with basis $\{ \tau, 1741 1\}$ with $\tau \in \HH$ in each homothety class. Different bases of
1742 $L_{\tau}$ are given by applying elements of
1743 $\Gamma$ to the basis $\{ \tau, 1744 1\}$; it follows that we may take $\tau \in 1745 \mathfrak{F}$.  With this stipulation, the basis $\{\tau,1\}$
1746 is uniquely determined. We will denote
1747 by $E_{\tau}$ the corresponding elliptic curve under the map
1748$$1749\CC/L_{\tau} \to E_{\tau}. 1750$$
1751We call this map (which is induced by (\ref{Eisom})) an
1752\emph{analytic representation} of $E_{\tau}$.
1753
1754We now wish to make this analytic representation more explicit;
1755additionally, because it will be useful later, we work in a
1756slightly more general context. Let $E/K$ be an elliptic curve over
1757a field $K$ of characteristic not equal to $2$ or $3$. Up to
1758isomorphism,
1759 we can assume that $E$ is given in affine coordinates by
1760\begin{equation} \label{affequ}
1761E:y^2=4x^3-g_4x-g_6
1762\end{equation}
1763(see, for example, \cite[\S III.2]{kn}).  If we restrict to the
1764case $K=\CC$, with the normalizations given above, the map
1765(\ref{Eisom}) just formalizes the parametrization
1766$$1767E:(\wp'(z,L))^2=4(\wp(z,L))^3-g_4\wp(z,L)-g_6. 1768$$
1769that exists for some lattice $L \subset \CC$.
1770
1771We now wish define the $j$-invariant of $E_{\tau}$, and show how
1772it relates to $j(\tau)$. First, the \emph{discriminant
1773$\Delta(E)$} of the elliptic curve $E/k$ is defined as
1774\begin{equation} \label{deltaE}
1775\Delta(E)=(2 \pi)^{-12}(g_4^3-27g_6^2).
1776\end{equation}
1777\begin{remark}
1778It is important to observe that the discriminant function
1779$\Delta(E)$ is \emph{not} equal to the discriminant of the cubic
1780polynomial defining the curve.  Since the discriminant of the
1781polynomial defining an elliptic curve $E$ is \emph{not} an
1782isomorphism invariant of $E$, there are a variety of essentially
1783equivalent ways to define the discriminant; the reason for our
1784particular definition will soon be apparent.
1785\end{remark}
1786We define the \emph{$j$-invariant of $E$} to be the quantity
1787\begin{equation} \label{je}
1788 j(E):=\frac{1728g_4^3}{(2 \pi)^{12}\Delta(E)}.
1789\end{equation}
1790One can show by elementary means over any field $K$ of
1791characteristic not equal to $2$ or $3$ that $j(E)$ is indeed an
1792invariant of the isomorphism class of $E$, and, further, given any
1793$j(E) \in K$, there exists a curve of $j$-invariant $j(E)$ (see
1794\cite[\S III.2]{kn}).
1795
1796Note the similarity of (\ref{je}) and (\ref{modjdef}). This is no
1797accident.  Let $\CC/L_{\tau} \to E_{\tau}$ be an analytic
1798representation. It turns out that, with the normalizations given
1799above, we have $g_2=\frac{4}{3}\pi^4E_4(\tau)$,
1800$g_3=\frac{8}{27}\pi^6E_6(\tau)$. Hence, we have
1801$$1802\Delta(E)=\frac{(E_4(\tau)^3-E_6(\tau)^2)}{1728}=\Delta(\tau) 1803$$
1804and
1805\begin{eqnarray} \label{jcoincid}
1806j(E_{\tau})=j(\tau).
1807\end{eqnarray}
1808Thus the coincidence of the $j$" in $j$-function and
1809$j$-invariant is really no coincidence. Indeed, noting the fact
1810that as the $j$-invariant varies over $K$ it parameterizes
1811isomorphism classes of elliptic curves over $K$ (at least if we
1812continue to assume that the characteristic of $K$ is not $2$ or
1813$3$), and recalling that the $j$-function is a bijection between
1814$\mathfrak{F}$ and $\CC$, we have a bijective map
1815$$1816\mathfrak{F} \longleftrightarrow \left\{ \textrm{isomorphism 1817classes of } E/ \CC \right\}. 1818$$
1819\noindent For proofs of the statements we just made on the
1820equality of the various definitions of $j$ and $\Delta$, see
1821\cite[\S I and p. 112]{k}.  For a basic introduction to the theory
1822of elliptic curves, see \cite{kn}.
1823
1824Later we will be giving examples of elliptic curves in the form
1825$E:y^2=x^3+ax+b$ for some $a,b \in k$.  It is easy to see that
1826given any elliptic curve over $k$ with defining affine equation
1827$y^2=4x^3+cx+d$, if the characteristic of $k$ is not $2$, then
1828this curve is isomorphic to a curve with defining affine equation
1829$y^2=x^3+\widetilde{c}x+\widetilde{d}$ for some
1830$\widetilde{c},\widetilde{d} \in K$. We shall call an elliptic
1831curve written in this form an elliptic curve in \emph{Weierstrass
1832form}. We now write formally as a proposition some elementary
1833properties of curves written in Weierstrass form; for a proof, see
1834\cite[\S III.2]{kn}
1835
1836\begin{proposition} \label{explicitj}
1837Suppose $a,b \in K$, where $K$ is a field with characteristic not
1838equal to $2$ or $3$.  Then the discriminant of $x^3+ax+b$ is
1839$-4a^3-27b^2$. If the discriminant is nonzero, then
1840$E:y^2=x^3+ax+b$ is nonsingular. Further, the $j$-invariant of
1841$E:y^2=x^3+ax+b$ is $1728\frac{4a^3}{4a^3+27b^2}$.
1842\end{proposition}
1843
1844Now that we have (\ref{Eisom}) and the isomorphism invariant
1845$j(E)$ in hand, we completely understand isomorphism classes of
1846elliptic curves over $\CC$ considered as analytic objects; they
1847are explicitly parameterized by $\wp(z,L_{\tau})$ (considered as a
1848function of $\tau \in \mathfrak{F}$). For example, define
1849\emph{$E[N]$, the $N$-division points of $E$}, to be the points of
1850$E$ of order dividing $N$. Viewing $E/\CC$ as $\CC/L_{\tau}$, it
1851is evident that $E[N]$ is simply the group
1852$\frac{1}{N}L_{\tau}/L_{\tau}$, that is,
1853$$1854E[N] \approx \ZZ/N\ZZ \times \ZZ/N \ZZ. 1855$$
1856The ring of endomorphisms of $E$, or $\eo (E)$, can also be
1857understood in a relatively straightforward manner using analytic
1858representations.  To begin, we have the following:
1859\begin{lemma} \label{multilam} Let $L,M$ be two lattices in $\CC$, and let
1860$$1861\lambda: \CC/L \to \CC/M 1862$$
1863be a complex analytic homomorphism.  Then there exists a complex
1864number $\alpha$ so that the following diagram commutes:
1865\begin{eqnarray*}
1866\begin{matrix}
1867\alpha: & \CC & \rightarrow & \CC \\
1868& \downarrow & & \downarrow \\
1869\lambda: & \CC/L& \rightarrow & \CC/M.
1870\end{matrix}
1871\end{eqnarray*}
1872Here the top map is multiplication by $\alpha$ and the bottom is
1873the homomorphism $\lambda$.
1874\end{lemma}
1875
1876\begin{proof}[Proof (compare \cite{L})] In a neighborhood of
1877zero, $\lambda$ can be expressed by a power series
1878$$1879\lambda(z) = a_0 + a_1 z +a_2z^2 + \cdots, 1880$$
1881On the other hand, $\lambda$ is a homomorphism, so $a_0=0$ and
1883$$1884\lambda(z+z') \equiv \lambda(z) + \lambda(z') \pmod{M}. 1885$$
1886If we choose a small enough neighborhood $U$ of zero, we must have
1887that this congruence is an equality in $U$; thus
1888$$1889\lambda(z)=a_1z 1890$$
1891for $z \in U$.  But for any $z \in \CC$, $z/n$ is in $U$ for
1892sufficiently large integers $n$, and from this we conclude that,
1893identifying $z$ with its reduction modulo $L$,
1894$$1895\lambda(z)=\lambda 1896\left(n\left(\frac{z}{n}\right)\right)=n\lambda\left(\frac{z}{n}\right)=na_1\left(\frac{z}{n}\right)=a_1z. 1897$$
1898\end{proof}
1899\begin{remark} Abusing notation, we will often denote the
1900complex number $\alpha$ and the homomorphism $\lambda$ by the same
1901symbol $\lambda$.  We will also only be considering the special
1902case $L=M$ of Lemma \ref{multilam}.\end{remark}
1903
1904It is clear that any $\lambda \in \ZZ$ will induce an endomorphism
1905of $\CC/L_{\tau}$, which we can then identify with an element of
1906$\eo (E_{\tau})$.  We will call these endomorphisms the
1907\emph{trivial endomorphisms of $E_{\tau}$}.  We have the
1908following:
1909\begin{definition}
1910  If $E/\CC$ is an elliptic curve with nontrivial elements in its
1911  endomorphism ring $\eo(E/\CC)$, then we say \textbf{$E$ is a curve
1912  with complex multiplication}, or, briefly, \textbf{$E$ has CM}.
1913\end{definition}
1914The complex numbers $\lambda$ inducing a nontrivial endomorphism
1915of a lattice $L$ turn out to be algebraic numbers; more
1916specifically, they are quadratic over $\QQ$.  Before we formalize
1917and prove this as a proposition, we offer another definition,
1918which will also be useful in \S \ref{good}:
1919\begin{definition}
1920Suppose $\tau \in \HH$ is the root of a quadratic equation with
1921integer coefficients; that is, $\tau=\frac{-b+\sqrt{b^2-4ac}}{2a}$
1922with $a,b,c \in \ZZ$ and $\gcd(a,b,c)=1$.  We say that $\tau$ is a
1923\textbf{Heegner point} and that $d_{\tau}=b^2-4ac$ is the
1924\textbf{discriminant} of $\tau$.
1925\end{definition}
1926
1927\begin{proposition} \label{cmequiv}
1928Suppose $E/\CC$ is an elliptic curve.  Then
1929\begin{enumerate}
1930\item Every nontrivial endomorphism of $E/\CC$ is induced (in the
1931sense of Theorem \ref{multilam}) either by a Heegner point
1932$\lambda \in \HH$ or by $-\lambda$ for a Heegner point $\lambda 1933\in \HH$.
1934
1935\item The curve $E/\CC$ has CM if and only if $j(E)=j(\tau)$ for
1936some Heegner point $\tau \in \mathfrak{F}$.
1937
1938\item The curve $E/\CC$ has CM if and only if $\eo (E) \cong 1939\mathcal{O}$, where $\mathcal{O}$ is an order in an imaginary
1940quadratic number field $K$.
1941\end{enumerate}
1942\end{proposition}
1943\begin{proof}
1944The endomorphism ring of $E$ is unchanged if we replace it with
1945another elliptic curve isomorphic to it, so we assume without loss
1946of generality that $E=E_{\tau}$, $\tau \in \mathfrak{F}$. Thus we
1947have an analytic representation
1948$$1949\CC/L_{\tau} \to E_{\tau}. 1950$$
1951As we proved in Lemma \ref{multilam}, a nontrivial automorphism of
1952$E_{\tau}$ can now be realized as a $\lambda \in \CC^*-\ZZ$ such
1953that
1954$$1955\lambda L_{\tau} \subset L_{\tau} 1956$$
1957or, equivalently, for some $\left(\begin{smallmatrix} a & b \\ 1958c& d 1959\end{smallmatrix} \right) \in \textrm{GL}_2(\QQ) \cap M_{2 \times 2}(\ZZ)$,
1960\begin{eqnarray*}
1961\lambda \tau &=& a \tau+b \\
1962\lambda  &=& c \tau +d.
1963\end{eqnarray*}
1964This implies that $\lambda$ is a root of the quadratic equation
1965\begin{equation*}
1966\left|\begin{matrix} x-a & -b \\ -c & x-d \end{matrix}\right|=0.
1967\end{equation*}
1968
1969Thus $\lambda$ is a quadratic irrational algebraic integer. Now
1970note that $\tau$ cannot be real; otherwise $L_{\tau}$ would not be
1971a lattice, and $c \neq 0$, for then $\lambda$ would be an integer.
1972Thus $\QQ(\tau)=\QQ(\lambda)$, and, further, both $\lambda$ and
1973$\tau$ are imaginary quadratic numbers.  This proves (1).
1974
1975We've also proven the only if" implication of (2), just by
1976recalling that $j$ is an isomorphism invariant. The other
1977direction follows similarly: note that if $j(E)=j(\tau)$ with
1978$\tau$ a Heegner point, then $E_{\tau} \approx E$, and $E_{\tau}$
1979is evidently CM.
1980
1981Finally, for (3), note that if $E$ is CM, as proven above, there
1982is an isomorphic curve $E_{\tau}$ where $\tau$ is a Heegner point.
1983Thus $\eo (E) \approx \eo (L_{\tau})$, and, again as proven above,
1984any complex number inducing a nontrivial endomorphism of
1985$L_{\tau}$ is an element of $\mathcal{O}_{\QQ(\tau)}$, the ring of
1986integers of $\QQ(\tau)$, but not an element of $\ZZ$.  With this
1987observation in mind it is easy to see that the evident map $\eo 1988(L_{\tau}) \to \mathcal{O}_{\QQ(\tau)}$ is a homomorphism of rings
1989with identity, and, further, the image of this homomorphism is not
1990contained in $\ZZ \subset \mathcal{O}_{\QQ(\tau)}$.  Thus
1991$\eo(L_{\tau}) \approx \eo (E_{\tau})$ is isomorphic to an order
1992in \$\mathcal{O}_{