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\begin{document}
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% Title, Etc. %
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\title[Borcherds exponents]{Classical and $p$-adic modular forms arising from the Borcherds exponents of other modular forms}
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\author{Jayce Getz \\ Senior Thesis} \address{
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4404 South Ave. W \\ Missoula, MT
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59804}\email{getz@fas.harvard.edu}
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\date{\today}
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% Abstract %
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\begin{abstract}
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Let $f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}$ be a modular form
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on $\SL_2(\ZZ)$. Formal logarithmic differentiation of $f$ yields
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a $q$-series $g(z):=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n$ whose
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coefficients are uniquely determined by the exponents of the
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original form. We provide a formula, due to Bruinier, Kohnen, and
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Ono for $g(z)$ in terms of the values of the classical
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$j$-function at the zeros and poles of $f(z)$. Further, we give a
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variety of cases in which $g(z)$ is additionally a $p$-adic
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modular form in the classical sense of Serre. As an application,
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we derive some $p$-adic formulae, due to Bruinier, Ono, and
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Papanikolas, in which the class numbers of a family of imaginary
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quadratic fields are written in terms of special values of the
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$j$-function at imaginary quadratic arguments.
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\end{abstract}
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\thanks {The author would like to thank his family for their constant personal and financial support,
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Particular thanks go to his little brother Joel, who is the
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coolest person in the world. This thesis is dedicated to them.}
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\maketitle
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% Section: Introduction %
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\section{Introduction}
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\label{intro}
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Suppose $f$ is a function on the upper half plane $\mathbb{H}$.
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For each positive integer $k$, define an action $|_k$ of
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$\textrm{GL}_2^+(\QQ)$ on the set of such $f$ by
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\begin{equation}
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f(z)|_k\gamma=\det(\gamma)^{k/2}(cz+d)^{-k}f\left(
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\frac{az+b}{cz+d} \right).
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\end{equation}
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Here $\gamma = \left( \begin{smallmatrix} a & b \\ c & d
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\end{smallmatrix} \right) \in \textrm{GL}_2^+(\QQ)$
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(with the exception of the proof of Theorem \ref{bko}, in this
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thesis we always use the symbol $\gamma$ in this sense).
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Suppose $\Gamma' \subset \Gamma:=\textrm{SL}_2(\ZZ)$
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is a congruence subgroup. Let $\mathcal{M}_k^{\infty}(\Gamma')$
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(resp., $\mathcal{M}_k^{\textrm{mero}}(\Gamma')$) denote the space
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of holomorphic (resp., meromorphic) functions on the upper half
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plane $\mathbb{H}$ that satisfy the functional equation
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\begin{equation} \label{modfunc}
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f(z)|_k\gamma:=f(z)
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\end{equation}
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for all $\gamma \in \Gamma'$ and additionally are meromorphic at
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the cusps of $\Gamma'$ (for a precise description of this
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``meromorphic at the cusps" condition, see \cite[\S III.3,
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p.~125]{k}). Such a function will be called a \emph{weakly modular
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form of weight $k$} (resp., \emph{meromorphic modular form of
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weight $k$}) following J-P. Serre's convention \cite[\S
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VII.2]{S1}. We further define $M_k(\Gamma') \subset
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\mathcal{M}_k^{\infty}(\Gamma')$ to be the space of weakly modular
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forms that, additionally, are holomorphic at the cusps of
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$\Gamma'$. Such a form will be called a \emph{holomorphic modular
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form}, or, simply, a \emph{modular form}. For any congruence
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subgroup $\Gamma'$ containing the element
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$\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1
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\end{smallmatrix}\right)$, meromorphicity of $f$ at the cusps of $\Gamma'$ implies that
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$f$ can be identified with a Fourier, or $q$-series, expansion
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\begin{equation} \label{qseries}
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f(z):=\sum_{n=n_0}^{\infty}a_nq^n
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\end{equation}
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where here, and throughout this thesis, $q:=e^{2 \pi i z}$. In
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the case $\Gamma'=\Gamma$, this is in fact equivalent to
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meromorphicity at the cusps. Holomorphicity at the cusps in the
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case of $\Gamma'=\Gamma$ (which are all in the same orbit as
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$\infty$ under the action of $\Gamma$) is equivalent to the
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statement that $n_0 \geq 0$. Finally, a holomorphic modular form
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over $\Gamma'$ is said to be a \emph{cusp form} if it vanishes at
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the cusps of $\Gamma'$; we denote the space of cusp forms of
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weight $k$ over $\Gamma'$ by $S_k(\Gamma')$. In the case $f \in
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M_k(\Gamma)$, this is simply the assertion that in the expansion
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(\ref{qseries}) we have $n_0>0$. For convenience we define
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$\mmer_k:=\mmer_k(\Gamma)$,
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$\mathcal{M}_k^{\infty}:=\mathcal{M}_k^{\infty}(\Gamma)$ and
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$M_k:=M_k(\Gamma)$.
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We take the opportunity now to introduce the only congruence
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subgroup we will explicitly use in this thesis, namely the
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following level $N$ subgroup:
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$$
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\Gamma_0(N):=\left\{ \begin{pmatrix} a & b \\ c & d
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\end{pmatrix} \in \Gamma : c \equiv 0 \pmod{N} \right\}.
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$$
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By convention, $\Gamma_0(1)=\Gamma$.
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\begin{remark} If $\left( \begin{smallmatrix} -1 & 0 \\ 0 & -1
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\end{smallmatrix}\right) \in \Gamma'$, then from (\ref{modfunc}) we have $(-1)^kf(z)=f(z)$ for
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all $f \in \mmer_k(\Gamma')$, from which it follows that
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$\mmer_{2m+1}(\Gamma')=0$ for all integers $m$. Thus, in
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particular, $\mmer_{2m+1}=0$.
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\end{remark}
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For examples of modular forms on $M_k$ for even $k \geq 4$, we may
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take the classical Eisenstein series of weight $k$:
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\begin{equation} \label{Eis}
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E_k(z):=1-\frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n)q^n
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\end{equation}
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where $B_k$ is the $k$th Bernoulli number and
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$\sigma_{k-1}(n):=\sum_{d|n}d^{k-1}$. We can formally define
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$E_2$ using (\ref{Eis}), and though it is not a modular form, it
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satisfies
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the following transformation law for $\left(\begin{smallmatrix} a & b \\
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c & d \end{smallmatrix} \right) \in \Gamma$:
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\begin{equation} \label{e2}
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E_2\left( \frac{az+b}{cz+d}\right)(cz+d)^{-2}=E_2(z)+\frac{12c}{2 \pi i (cz+d)}.
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\end{equation}
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This transformation law turns out to play a role in many
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arguments; a proof of it in this form is given in
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\cite[p.~68]{sch}.
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Other useful examples of modular forms are the discriminant
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function
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\begin{equation}
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\Delta(z):=\frac{E_4(z)^3-E_6(z)^2}{1728}=q\prod_{n=1}^{\infty}(1-q^n)^{24}
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\end{equation}
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which is of weight $12$, and the $j$-function, which is a weakly
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modular form of weight zero:
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\begin{equation} \label{modjdef}
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j(z):=\frac{E_4(z)^3}{\Delta(z)}=q^{-1}+744+196884q+21493760q^2+\cdots.
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\end{equation}
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We note that any element of $\minf_0$ is a polynomial in $j(z)$.
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If we wish to emphasize for a proof that we are regarding $E_k$,
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$\Delta$, $j$ as $q$-series (which can be either viewed formally
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or as functions holomorphic in the punctured disc $0 <|q|<1$), we
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write them as $E_k(q)$, $\Delta(q)$, and $J(q)$, respectively.
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It is easy to see that $\minf_k(\Gamma')$ is a vector space over
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$\CC$ for all congruence subgroups $\Gamma'$. There exists an
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important class of linear operators on these spaces, namely, the
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Hecke operators $T_{k,n}$. These can be defined (in an admittedly
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ad-hoc manner) by
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\begin{equation} \label{heckedefo}
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f(z)|T_{k,n}=n^{k-1}\mathop{\sum_{ad=n,\textrm{ }d>0}}_{0 \leq b
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\leq d-1}f\left(\frac{az+b}{d}\right)
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\end{equation}
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or, equivalently,
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\begin{equation} \label{heckedef}
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f(z)|T_{k,n}:=\sum_{n \in \ZZ}\left(
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\sum_{0<d|(m,n)}d^{k-1}a\left(\frac{mn}{d^2}\right)\right)q^n.
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\end{equation}
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If we define, for positive integers $d$, the $V$- and
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$U$-operators $V(d)$ and $U(d)$ on formal $q$-series in $\CC[[q]]$
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by
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\begin{equation} \label{vdef}
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\left(\sum_{n \geq n_0} c(n)q^n \right)|V(d):=\sum_{n \geq
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n_0}c(n)q^{dn}
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\end{equation}
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and
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\begin{equation} \label{udef}
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\left(\sum_{n \geq n_0} c(n)q^n \right)|U(d):=\sum_{n \geq
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n_0}c(dn)q^{n}
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\end{equation}
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then we may write \begin{equation} \label{uvhecke}
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T_{k,n}=\sum_{d|n}d^{k-1}V(d) \circ U(n/d).
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\end{equation}
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Note that if we identify a meromorphic modular form $f$ with its
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$q$-expansion, we have
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\begin{equation} \label{altvdef}
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d^{k/2}f|V(d)=f|_k\left(\begin{smallmatrix} d & 0 \\ 0 &
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1
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\end{smallmatrix}\right).
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\end{equation} For more natural definitions of these operators and a
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discussion of their basic properties, see, for example, \cite[\S
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III.5]{k} or \cite[\S VII]{S1}.
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If we consider $\mathcal{M}^{\infty}:=\bigoplus_{k=0}^{\infty}
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\mathcal{M}^{\infty}_k$ it is straightforward to see that we have
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something better than a collection of vector spaces, we have a
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graded algebra, where the grading is given by weight and the
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multiplication operation is multiplication of functions (for proof
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of this, see \cite[\S VII]{S1}). A question naturally suggests
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itself: are there natural operators on this algebra? As one
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possible answer to this question, we define Ramanujan's theta
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operator:
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$$
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\Theta:=\frac{1}{2 \pi i } \frac{d}{dz}=q \frac{d}{dq}.
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$$
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It is perhaps speaking loosely to call $\Theta$ an operator, but
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$$
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f(z) \mapsto \Theta f(z)-f(z)\frac{k}{12}E_2(z)
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$$
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is a derivation on $\mathcal{M}$. In particular, we have the
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following:
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\begin{proposition} \label{thetprop}
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If $f$ is in $\mmer _k(\Gamma')$
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then
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\begin{equation} \label{thetobs}
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g(z)=\Theta f-f(z)\frac{k}{12}E_2 \in \mmer_{k+2}(\Gamma').
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\end{equation}
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The same statement is true with $\mmer_k(\Gamma')$ replaced by
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$\minf_k(\Gamma')$ or $M_k(\Gamma')$ throughout.
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\end{proposition}
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\begin{proof}
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By noting its affect on $q$-expansions, we see that applying the
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$\Theta$ operator does not affect meromorphicity (resp.,
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holomorphicity) at the cusps. Thus we need only check the
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functional equation. For $\gamma=\left(\begin{smallmatrix} a & b
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\\c & d\end{smallmatrix}\right) \in \Gamma$, upon differentiating
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the functional equation (\ref{modfunc}) we have
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\begin{eqnarray*} \Theta
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f(\gamma z)(cz+d)^{-k-2}&=&\Theta f(z)+\frac{ck}{2 \pi i}f(\gamma z)(cz+d)^{-k-1} \\
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&=&\Theta f(z)+ \frac{ck}{2 \pi i}f(z)(cz+d)^{-1}.
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\end{eqnarray*}
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Using (\ref{e2}), for $\gamma \in \Gamma' \subset \Gamma$ we have
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\begin{eqnarray*}
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\Theta f (z) |_{k+2}\gamma &-&\frac{k}{12}\left(E_2(z)|_{2} \gamma
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\right)\left(f(z)|_k \gamma \right)
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\\ &=&\Theta f(z) +\frac{ck}{2 \pi i}f(\gamma
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z)(cz+d)^{-k-1}-\frac{k}{12}E_2(z)f(z)-\left(\frac{k}{12}\right)\frac{12c}{2
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\pi i (cz+d)}f(z) \\
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&=& \Theta f(z)-\frac{k}{12}E_2(z)f(z).
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\end{eqnarray*}
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\end{proof}
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\begin{remark} It is worth mentioning that there exists a family of ``Rankin-Cohen" brackets on
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$\bigoplus_{k=0}^{\infty} M_k$ (defined using $\Theta$), one of
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which gives this algebra the structure of a graded Lie algebra.
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For their definition and basic properties see \cite{Z2}, and for
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references to recent work, see \cite{BWO}.
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\end{remark}
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Now, given a modular form $f \in \mmer_k(\Gamma')$, normalized so
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that its first nonzero $q$-expansion coefficient is $1$, we can
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write
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$$
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f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}
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$$
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for some complex numbers $c(n)$, in some neighborhood of $\infty$.
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Ignoring convergence issues for a moment (which will be dealt with
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carefully in Lemma \ref{logdiv}), some easy manipulations with
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$q$-series combined with Proposition \ref{thetprop} yield
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\begin{equation} \label{ok}
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\frac{\Theta f}{f}=h-\sum_{n \geq 1} \sum_{d|n}c(d)dq^n \in
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\mmer_2(\Gamma')
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\end{equation}
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In the next section, we will prove the following characterization
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of this logarithmic derivative:
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\begin{theorem}[Bruinier, Kohnen, Ono, \cite{BKO}, \cite{O}] \label{bko} If $f(z)=\sum_{n=h}^{\infty}a(n)q^n
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\in \mmer_k$ is normalized so that $a(h)=1$, then
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$$
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\frac{\Theta
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f(z)}{f(z)}=\frac{k}{12}E_2(z)-\frac{E_4(z)^2E_6(z)}{\Delta(z)}\sum_{\tau_i
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\in \mathfrak{F}} \frac{e_{\tau}\ord_{\tau}(f)}{j(z)-j(\tau)}.
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$$
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\end{theorem}
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\begin{remark} This formula has been generalized to several
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genus zero congruence subgroups in \cite{Ahl1} (see \S \ref{Rth}
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of this thesis) and Hecke subgroups of $\textrm{SL}_2(\RR)$ (see
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\cite{CKo}). The author has also received a preprint \cite{DC}
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giving a generalization to $\Gamma_0(N)$ for squarefree $N$.
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\end{remark}
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This formula alone is of interest in that it explicitly relates,
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via equation (\ref{ok}), the product expansion exponents of $f$ to
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special values of $j$, namely, $j(\tau)$ where $\tau$ is a zero or
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pole of $f$. Further, it has been used to provide recursive
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formulas for the coefficients of any modular form over $\Gamma$
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(see \cite{BKO}), to provide infinite families of systems of
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orthogonal polynomials divisible by the supersingular locus as
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polynomials over $\FF_p$ (see \cite{BGNS}), (generalizing work of
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Atkin described in \cite{kz}), and also to provide a
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characterization of the characteristic polynomials of the Hecke
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operators over $\Gamma$ (again in \cite{BKO}). We will not discuss
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these applications in this thesis. We will, however, give one
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additional application, which we defer for a moment in order to
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introduce the concept of a $p$-adic modular form.
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Following Serre, we define a $p$-adic modular form to be the
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$p$-adic limit of a sequence of elements of
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$\oplus_{k=0}^{\infty}M_k$ (a precise definition is given in \S
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\ref{Spadic}). It turns out that in many cases of interest, the
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logarithmic derivative of a modular form is a $p$-adic modular
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form of weight $2$. In particular, we have the following theorem
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of Bruinier and Ono:
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\begin{theorem}[\cite{BrO}] \label{modelth}
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Let $f(z)=q^h(1+\sum_{n=1}^{\infty}a(n)q^n) \in q^h\OO_K[[q]] \cap
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\mmer_k(\Gamma_0(1))$, where $\OO_K$ is the ring of integers of a
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number field $K$. Moreover, let $c(n) \in K$ denote the algebraic
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numbers defined by the formal infinite product
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\begin{equation} \label{cexp}
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f(z)=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)}.
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\end{equation}
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If $f(z)$ is good at a prime $p$, then the formal power series
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$$
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\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n
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$$
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is a weight two $p$-adic modular form.
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\end{theorem}
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We offer a brief proof of this result, mainly as motivation for
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the following generalization:
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\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
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ring of integers of a number field $K$. Moreover, let $c(n) \in
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K$ denote the algebraic numbers defined by the formal infinite
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product (\ref{cexp}) for $f$. If $f$ is good at $p$, then the
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formal power series
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$$
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\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}c(d)dq^n
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$$
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is a weight two $p$-adic modular form.
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\end{theorem}
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\noindent The proofs of both of these theorems appear in \S
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\ref{pborc}.
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\begin{remark} In theorems \ref{modelth} and \ref{gbro}, we allow
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$h$ to be negative. The fact that the $c(n)$ are elements of $K$
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(implicitly identified with an embedding $K \hookrightarrow \CC$)
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will be obvious from the proof of Lemma \ref{logdiv}.
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\end{remark}
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The definition of ``good" in the preceding two theorems is given
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in \S \ref{pborc} and discussed in some detail in \S \ref{good}.
476
As one example, the form $E_{p-1}$ is good at $p$. In general,
477
whether or not a form is good at $p$ is intimately related to the
478
question of whether or not the value of the $j$-function at the
479
zeros and poles of the form reduces to a supersingular
480
$j$-invariant in characteristic $p$ (which should come as no
481
surprise to those familiar with overconvergent $p$-adic modular
482
forms). Through this connection we are able to relate these
483
$p$-adic modular forms to class numbers of imaginary quadratic
484
fields. In particular, for small primes, we obtain $p$-adic class
485
number formulae involving sums of special values of the
486
$j$-function.
487
488
Before we can state this result, we must recall the notion of a
489
Heegner 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
492
discriminant $d_{\tau}:=b^2-4ac$. Heegner points are discussed at
493
some length in \S \ref{cm}. Denote by $h_K$ the Hurwitz class
494
number of the number field $K$. We have the following:
495
496
\begin{corollary}[Ono and Papanikolas, \cite{pj}] \label{classnumber}
497
Suppose that $d<-4$ is a fundamental discriminant of an imaginary
498
quadratic 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
$$
503
h_{\QQ(\sqrt{d})}=-\frac{1}{720} \lim_{n \to \infty}
504
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{2^a-1} j\left(
505
\frac{2^{n-a} \tau+b}{2^a}\right)\right).
506
$$
507
\item If $d \equiv 2 \pmod{3}$, then as $3$-adic numbers we have
508
$$
509
h_{\QQ(\sqrt{d})}=-\frac{1}{360} \lim_{n \to \infty}
510
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{3^a-1} j\left(
511
\frac{3^{n-a} \tau+b}{3^a}\right)\right).
512
$$
513
\item If $d \equiv 2,3 \pmod{5}$, then as $5$-adic numbers we have
514
$$
515
h_{\QQ(\sqrt{d})}=-\frac{1}{180} \lim_{n \to \infty}
516
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{5^a-1} j\left(
517
\frac{5^{n-a} \tau+b}{5^a}\right)\right).
518
$$
519
\item If $d \equiv 3,5,6 \pmod{7}$, then as $7$-adic numbers we
520
have
521
$$
522
h_{\QQ(\sqrt{d})}=-\frac{1}{120} \lim_{n \to \infty}
523
\Trace_{K/\QQ}\left( \sum_{a=0}^n \sum_{b=0}^{7^a-1} j\left(
524
\frac{7^{n-a} \tau+b}{7^a}\right)\right).
525
$$
526
\end{enumerate}
527
\end{corollary}
528
\noindent In \S \ref{good}, we also use Theorem \ref{gbro} to
529
provide 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
534
Before we begin the body of this work, we make a few remarks about
535
its structure. Sections \ref{Rth}, \ref{pborc}, and \ref{good}
536
contain results that have only been published recently, if at all,
537
and the primary purpose of this thesis is to collect their content
538
into one place. Sections \ref{Spadic} and \ref{atle}, on the
539
other hand, are mostly derived from two well-known papers
540
(\cite{Sep} and \cite{AL}, respectively). The author has provided
541
proofs of most of the results in these sections that are necessary
542
for 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},
545
respectively.
546
547
In contrast, providing applications of theorems \ref{bko},
548
\ref{modelth} and \ref{gbro}, including Corollary
549
\ref{classnumber} and Theorem \ref{classnumber2}, requires results
550
for which we will not provide proofs; it would simply take us too
551
far afield. In particular, \S \ref{cm} is intended to give a brief
552
survey of the relevant definitions and theorems in the theory of
553
complex multiplication, but we omit the proofs of results usually
554
proven using class field theory and reduction theory (we refer the
555
reader to \cite[\S II]{Si2} or \cite{LaE} for a more complete
556
account). Ergo, \S \ref{cm} can be skipped without interrupting
557
the flow of ideas, especially if one is familiar with complex
558
multiplication and elementary calculations involving elliptic
559
curves.
560
561
\section{A characterization of Ramanujan's theta operator}
562
\label{Rth}
563
564
As indicated above, in this section we will prove a useful
565
characterization of the derivative of a modular form. First we
566
require 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}
573
e_{\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
578
The purpose of this section is to prove the characterization of
579
the logarithmic derivative of a modular form given by Theorem
580
\ref{bko}. The proof of the theorem requires two steps. The first
581
is an identity due to Asai, Kaneko, and Ninomiya \cite{AKN}. To
582
introduce 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
584
with $q$-expansion
585
\begin{equation} \label{jmdef}
586
j_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}
589
We note that $j_m(z)$ is a polynomial in $j(z)$ for all $m$. In
590
fact, it is a polynomial in $j$ with integral coefficients, for
591
$J_m(q)$ can be formed by subtracting suitable integer multiplies
592
of 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}
595
j_0(z)=J_0(q)&=&1, \\
596
j_1(z)=J_1(q)&=&j(z)-744 = q^{-1}+196884q+\cdots, \\
597
j_2(z)=J_2(q)&=&j(z)^2-1488j(z)+159768=q^{-2}+42987520q+ \cdots, \\
598
j_3(z)=J_3(q)&=&j(z)^3-2232j(z)^2+1069956j(z)-36866976=q^{-3}+2592899910q+
599
\cdots.
600
\end{eqnarray}
601
We 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}
604
J_m(q):=j_m(z):=m j_1(z)|T_{0,m}
605
\end{equation}
606
for $m>1$. The equivalence of this definition to the $q$-series
607
definition (\ref{jmdef}) follows from (\ref{heckedef}) and the
608
fact that a weakly modular form, being a polynomial in $j$, is
609
uniquely determined by the coefficients of non-positive exponent
610
in its $q$-series expansion. Indeed, from this fact we see that
611
the $J_m(q)$ form a basis for $\minf_0$.
612
613
We have the following:
614
615
\begin{theorem} \label{aknf} As an identity of formal power series
616
in $\rho,q$, 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}
623
Asai, Kaneko, and Ninomiya show in \cite{AKN} how Theorem
624
\ref{aknf} implies the famous denominator formula for the Monster
625
Lie algebra, namely
626
$$
627
J(\rho)-J(q)=\rho^{-1} \prod_{m>0 \textrm{ and } n \in
628
\ZZ}(1-\rho^mq^n)^{\beta(mn)},
629
$$
630
where the coefficients $\beta(n)$ are defined by
631
$$
632
j_1(z)=\sum_{n=-1}^{\infty}\beta(n)q^n.
633
$$
634
\end{remark}
635
636
\begin{proof}[Proof of Theorem \ref{aknf}]
637
638
We require a companion set of functions $g_m(\rho)$ indexed by
639
positive integers $m$, the $m$th of which can be defined in
640
analogy with (\ref{jmdef}) as the unique weight $2$ weakly modular
641
form with $\rho$-expansion
642
\begin{equation} \label{gdef}
643
g_m(\rho):=\rho^{-m}+\sum_{n=1}^{\infty} b_m(n)\rho^n \in \minf_2.
644
\end{equation}
645
Alternately, we may define
646
$g_1(\rho):=\frac{E_{4}(\rho)^2E_6(\rho)}{\Delta(\rho)}$ and
647
$$
648
g_m(\rho):=m^{-1}g_1(\rho)|T_{2,m}.
649
$$
650
As before, the equivalence of these two definitions follows from
651
the definition of the $T_{2,m}$ and the fact that any weight $2$
652
weakly holomorphic form is uniquely determined by the coefficients
653
in its $q$-expansion of negative order. We note that this fact
654
follows from the well-known ``$k/12$ valence formula" (see, for
655
example, \cite[\S III.2]{k}), as does the corresponding fact for
656
weight zero weakly holomorphic forms. In fact, as in the weight
657
zero case, this implies that the $g_m(\rho)$ form a basis for the
658
space $\minf_2$. Further, from (\ref{thetobs}), if $f \in
659
\minf_0$ then $\Theta f \in \minf_2$, and by simply looking at the
660
bases $\{J_m\}$,$\{g_m\}$ we have just written down we see that
661
every element of $\minf_2$ can be written as $\Theta f$ for some
662
$f \in \minf_0$. In particular, it follows from this observation
663
and the definition of $\Theta$ that the constant term of any
664
element of $\minf_2$ is identically zero (which justifies the
665
indexing of (\ref{gdef})).
666
667
Now we note that
668
\begin{equation} \label{Jconst}
669
J_m(q):=mJ_1(q)|T_{0,m}=q^{-m}+m a_m(m)q+ \cdots
670
\end{equation}
671
and
672
\begin{equation} \label{gconst}
673
g_m(\rho):=m^{-1}g_1(\rho)|T_{2,m}=\rho^{-m}+b_1(m)q+\cdots
674
\end{equation}
675
for $m \geq 1$ 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
678
preceding paragraph and using (\ref{gdef}) and (\ref{Jconst}), we
679
have that
680
\begin{equation} \label{abconst}
681
b_1(m)=-ma_m(m)
682
\end{equation}
683
for $m \geq 1$. 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
686
we've remarked before. Define $\rho$-expansion coefficients $c(n)$
687
by
688
$$
689
J(\rho)=\rho^{-1}+\sum_{n=0}^{\infty}c(n)\rho^n
690
$$
691
By 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}
695
J(\rho)J_m(\rho)=J_{m+1}(\rho)+\sum_{i=0}^m c(m-i)J_i(\rho)-b_1(m)
696
\end{equation}
697
for 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}
700
J(\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}
703
Noting that $g_1(q)=\frac{E_{4}(q)^2E_6(q)}{\Delta(q)}$, we see
704
that (\ref{messy}) is a rewriting of (\ref{aknfo}).
705
\end{proof}
706
707
\begin{corollary} \label{aknfu}
708
Fix $\tau \in \mathbb{H}$. Then
709
$$
710
\frac{E_4(z)^2E_6(z)}{\Delta(z)}\frac{1}{j(z)-j(\tau)}=\sum_{n=0}^{\infty}j_m(\tau)q^n
711
$$
712
as meromorphic functions in $z$ on $\mathfrak{F}$.
713
\end{corollary}
714
\begin{proof} Compare Fourier ($q$-series) coefficients in a deleted
715
neighborhood 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
720
intersection of the unit circle with $\mathfrak{F}$. The
721
technique they use is analogous to that used by Rankin and
722
Swinnerton-Dyer to prove that the ``nontrivial" zeros of $E_k(z)$
723
have the same property, see \cite{RSD}. For yet another family of
724
modular forms whose zeros have the same property, see \cite{G}.
725
\end{remark}
726
727
We also require the following proposition, which follows from
728
basic complex analysis:
729
\begin{proposition}[\cite{BKO}] \label{logdiv}
730
Let $f=\sum_{n=h}^{\infty}a_f(n)q^n$ be a meromorphic function in
731
a neighborhood of $q=0$, normalized so that $a_f(h)=1$. Then
732
there are complex numbers $c(n)$ such that
733
$$
734
f=q^h\prod_{n=1}^{\infty}(1-q^n)^{c(n)},
735
$$
736
where the product converges in a sufficiently small neighborhood
737
of $q=0$. 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
743
meromorphic modular form $f$ by Proposition \ref{logdiv} as the
744
\emph{Borcherds exponents} of $f$.
745
\end{remark}
746
\begin{proof}
747
748
As usual, we understand that complex powers are defined by the
749
principle branch of the complex logarithm. Write $F(q):=f(z)$,
750
and then note that $qF'(q)/F(q)$ is holomorphic at $q=0$. We may
751
therefore write its Taylor expansion around $q=0$, valid in
752
$|q|<\epsilon$ for some $\epsilon>0$, as
753
\begin{equation} \label{ld1}
754
qF'(q)/F(q)=h-\sum_{n \geq 1} \alpha(n)q^n.
755
\end{equation}
756
For $n \geq 1$ define
757
$$
758
c(n):=\frac{1}{n}\sum_{d|n}\alpha(d)\mu(n/d)
759
$$
760
where $\mu$ is the M$\ddot{\textrm{o}}$bius function. By
761
M$\ddot{\textrm{o}}$bius inversion we have
762
\begin{eqnarray} \label{moinv}
763
\alpha(n)=\sum_{d|n}c(d)d.
764
\end{eqnarray}
765
766
If we fix $q_0$ with $|q_0|<\epsilon$, then by absolute
767
convergence of (\ref{ld1}) we have
768
$\alpha(n)=\mathcal{O}(|q_0|^{-n})$ for all $n$. Thus the double
769
sum
770
\begin{equation} \label{un1}
771
\sum_{m,n \geq 1} c(n)nq^{mn}
772
\end{equation}
773
converges absolutely in $|q|<|q_0|$ and hence in $|q|<\epsilon$.
774
775
Suppose 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*}
783
The interchange of summation and integration can be justified by
784
using local uniform convergence as we did in proving the absolute
785
convergence of (\ref{un1}).
786
787
Upon integrating, we obtain
788
$$
789
\log(F(q)q^{-h})=\sum_{n \geq 1} c(n)\log(1-q^n).
790
$$
791
Here we use the normalization $a_f(h)=1$. Now $c(n)\log(1-q^n)$
792
and $\log(1-q^n)^{c(n)}$ differ by integer multiples of $2 \pi i$.
793
Since $c(n) \log(1-q^n) \to 0$ as $n \to \infty$, we have
794
$\log(1-q^n)^{c(n)} \to 0$ as well. Thus, as $n \to \infty$,
795
these two quantities differ in value only finitely many times; it
796
follows that there exists an integer $N$ such that
797
$$
798
\log(F(q)q^{-h})=\sum_{n \geq 1} \log(1-q^n)^{c(n)}+2 \pi i N.
799
$$
800
Taking the exponential on both sides finishes the proof of the
801
proposition.
802
\end{proof}
803
804
We now prove Theorem \ref{bko}.
805
806
\begin{proof}[Proof of Theorem \ref{bko}]
807
808
Choose $C>0$ large enough so that all poles of $f$ in
809
$\mathfrak{F}$ (excluding any at the cusp at infinity) have
810
imaginary part less than $C$. Let $ L:=\{t+iC: -\frac{1}{2} \leq t
811
\leq \frac{1}{2} \}$ and consider the contour in $\mathbb{H}$
812
formed from the part of $\partial \mathfrak{F}$ of imaginary part
813
less than $C$ and $L$. Modify this contour as in the proof of the
814
classical $k/12$ valence formula (see, for example, \cite[\S
815
III.2, p.~115]{k}), specifically, if there are poles of $f$ at $i$
816
or $\omega:=e^{2 \pi i/3}$ (which, by modularity, implies the
817
existence of a pole at $e^{\pi i/3}$), form half and ``sixth"
818
circles of radius $r>0$ around them, and if there are poles of $f$
819
on the boundary, form two half circles of radius $r>0$ around
820
them, one enclosing the pole on one side of the fundamental
821
domain, one not enclosing the pole which must exist on the other
822
side (given that $f$ is modular). Call the left vertical side of
823
this contour $\gamma_1(r)$, the right vertical side $\gamma_2(r)$,
824
and 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
826
positive (counterclockwise) orientation.
827
828
If we integrate
829
\begin{equation} \label{int1}
830
\frac{1}{2 \pi i} \frac{f'(z)}{f(z)} j_n(z)
831
\end{equation}
832
along 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}
837
We can also integrate (\ref{int1}) in pieces, from which we see
838
that (\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)
843
dq+\int_{\gamma_3(r)}\frac{f'(z)}{f(z)}j_n(z)dz.
844
\end{eqnarray}
845
Here $L'$ is a simple loop around $q=0$. By Proposition
846
\ref{logdiv} we have
847
$$
848
\frac{qF'(q)}{F(q)}=\frac{\Theta(f)}{f}=h-\sum_{n=1}^{\infty}\sum_{d|n}
849
c(d)dq^n
850
$$
851
and 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
857
We now deal with the last term in (\ref{lotsoterms}). By
858
Proposition \ref{thetobs}, if the weight of $f$ is $k$, there
859
exists 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
862
i\int_{\gamma_3(r)}\frac{\Theta(f)}{f}j_n(z)dz\\\nonumber&=&2 \pi
863
i\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}
866
Now let $\beta$ denote the path along the unit circle from $i$ to
867
$\omega$, taken with positive orientation, and $S$ the fractional
868
linear 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*}
879
To obtain the first equality we used the functional equation for
880
elements of $\minf_2$ along with a standard change of variables
881
(which introduces a factor of $1/z^2$). To move from the second
882
line 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
886
Now, 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*}
896
where $c(d) \equiv 24$ are (just for the purposes of the preceding
897
equation) the product expansion exponents of
898
$\Delta(z)=q\prod_{n=1}^{\infty}(1-q^n)^{24}$.
899
900
Thus, collecting all of this, equation (\ref{lotsoterms}) implies
901
that
902
$$
903
\sum_{\tau \in \mathfrak{F}}
904
e_{\tau}\ord_{\tau}(f)j_n(\tau)=\sum_{d|n}c(d)d-2k \sigma_1(n)
905
$$
906
907
Now we recall that by Theorem \ref{aknf}, it is sufficient to show
908
that
909
$$
910
\frac{\Theta(f)}{f}=\frac{kE_2}{12}-
911
\sum_{n=1}^{\infty}\left(\sum_{\tau \in \mathfrak{F}}
912
e_{\tau}\ord_{\tau}(f)j_n(\tau)\right)q^n.
913
$$
914
To prove this identity, we apply Proposition \ref{logdiv}, note
915
$$
916
\frac{k}{12}E_2(z)=\frac{k}{12}-2k\sum_{n=1}^{\infty}\sigma_1(n)q^n,
917
$$
918
and argue coefficient by coefficient. The only coefficient that
919
might be unclear is the constant $n=0$ term. In this case, on the
920
left we have $h,$ which is the order of $f$ at infinity, and on
921
the 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,
924
see \cite[\S III.II, p.~115]{k}).
925
926
\end{proof}
927
928
We remark here that the derivative formula of Theorem \ref{bko}
929
explicitly relates, via Proposition \ref{logdive}, the
930
coefficients $c(n)$ of the product expansion of a modular form to
931
a specific weight $2$ meromorphic modular form. This relationship
932
is in the spirit of the work of Borcherds on the product expansion
933
exponents of Jacobi forms with Heegner divisors. See \cite{Borc}
934
for the details of this theory.
935
936
As we mentioned in the introduction, Ahlgren, in \cite{Ahl1}, has
937
proven a generalization of Theorem \ref{gbro} to certain genus
938
zero congruence subgroups. We will state his theorem after fixing
939
some notation. Define Dedekind's eta-function
940
$$
941
\eta(z):=q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^n)
942
$$
943
as usual. For $p=2,3,5,7$ or $13$, let
944
$$
945
j^{(p)}(z):=\left(\frac{\eta(z)}{\eta
946
(pz)}\right)^{\frac{24}{p-1}} \in \minf_0(\Gamma_0(p)).
947
$$
948
This $j^{(p)}(z)$ is a modular form with a simple pole at $\infty$
949
and a simple zero (with respect to local coordinates) at $0$.
950
Additionally, its restriction to a fundamental domain for the
951
action of $\Gamma_0(p)$ on $\HH$ forms a bijection from that
952
fundamental domain to $\CC$. In analogy with (\ref{jmdef}), we now
953
define 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
956
modular function which is holomorphic on $\HH$, vanishes at the
957
cusp $0$ 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}
962
Because $\Gamma_0(p)$ is genus zero, each of these functions can
963
be written as monic polynomials in $j_1^{(p)}(z)=j^{(p)}(z)$ with
964
constant term equal to zero. For example, we have
965
\begin{eqnarray*}
966
j_0^{(5)}(z)&=&1, \\
967
j_1^{(5)}(z)&=&j^{(5)}(z)=q^{-1}-6+9q+10q^2-30q^3+ \cdots \\
968
j_2^{(5)}(z)&=&j^{(5)}(z)^2+12j^{(5)}(z)=q^{-2}-18+20q+21q^2+192q^3+
969
\cdots \\
970
j_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*}
973
In 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 $0$.
977
If $\tau \in \HH$, then (in analogy with (\ref{edef})) we define
978
$e^{(p)}_{\tau} \in \left\{1,\frac{1}{2},\frac{1}{3}\right\}$ by
979
$$
980
e_{\tau}^{(p)}:=( \textrm{the order of the isotropy subgroup of }
981
\tau \textrm{ in } \Gamma_0(p)/\{ \pm I\} )^{-1}.
982
$$
983
We 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}
991
j_n^{(p)}(\tau)q^n\right)+\frac{h-k/12}{p-1} \cdot
992
pE_2|V(p)+\frac{pk/12-h}{p-1} \cdot E_2.
993
\end{eqnarray*}
994
\end{theorem}
995
996
We will not provide a proof of this theorem; it is entirely
997
analogous to the proof of Theorem \ref{bko} except for some
998
difficulties which naturally arise when dealing with congruence
999
subgroups. 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
1005
We 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
$$
1010
a(k) \equiv b(k) \pmod{N}
1011
$$
1012
for all $k$. For primes $p$, we say that a $q$-series $f(z)$ with
1013
integral coefficients is a \emph{weakly modular form modulo $p^n$}
1014
if 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
$$
1017
f(z) \equiv g(z) \pmod{p^n}
1018
$$
1019
We note here that the theory of modular forms modulo prime powers
1020
is quite well developed; for a basic introduction, see \cite[\S
1021
IV.X]{La}, and for a variety of interesting number-theoretic
1022
applications, see \cite{O}.
1023
1024
We begin by establishing some well-known congruences involving the
1025
Eisenstein series $E_k(z)$. First we recall two classical
1026
Bernoulli number congruences (see \cite[p.~233-238]{IR}). Let
1027
$D_n$ be the denominator of the $n$th Bernoulli number, written in
1028
lowest 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}
1032
where 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
1034
the 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
1040
Using 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}
1047
and
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}
1053
For (\ref{1wrclev}), we have
1054
$$
1055
\left(E_{p-1}(z)\right)^{p^{r-1}}=\left(
1056
1-\frac{2(p-1)}{B_{p-1}}\sum_{n=1}^{\infty}\sigma_{p-1}(n)q^n
1057
\right)^{p^{r-1}}=\left(
1058
1-\frac{2(p-1)D_{p-1}}{U_{p-1}}\sum_{n=1}^{\infty}\sigma_{p-1}(n)q^n
1059
\right)^{p^{r-1}}
1060
$$
1061
where $U_{p-1}$ is an integer coprime to $D_{p-1}$. From
1062
(\ref{VS}) we have $p|D_{p-1}$ which implies (\ref{1wrclev}) after
1063
an application of the binomial theorem.
1064
1065
To 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
$$
1069
\frac{B_2}{2} \equiv \frac{B_{\phi(p^r)+2}}{\phi(p^r)+2}
1070
\pmod{p^r}.
1071
$$
1072
Also by Euler's theorem, $$\sigma_1(n) \equiv
1073
\sigma_{\phi(p^r)+1}(n) \pmod{p^r}.$$ With these two observations
1074
we have
1075
$$
1076
E_2(z)=1-\frac{2(2)}{B_2} \sum_{n=1}^{\infty}\sigma_1(n)q^n \equiv
1077
1-\frac{2(\phi(p^r)+2)}{B_{\phi(p^r)+2}}
1078
\sum_{n=1}^{\infty}\sigma_{\phi(p^r)+1}(n)q^n \pmod{p^r}.
1079
$$
1080
\end{proof}
1081
1082
\noindent We also record here the following congruences, which
1083
will be useful in \S \ref{good}:
1084
1085
\begin{lemma} \label{24k} Suppose $k \geq 4$ is even. Then
1086
\begin{equation*}
1087
E_k(z) \equiv 1 \pmod{24},
1088
\end{equation*}
1089
and, if $p \geq 5$ is a prime such that $(p-1) \mid k$,
1090
\begin{equation*}
1091
E_k(z) \equiv 1 \pmod{p}.
1092
\end{equation*}
1093
\end{lemma}
1094
\begin{proof}
1095
These both follow immediately from the von Staudt-Clausen equation
1096
(\ref{VS}).
1097
\end{proof}
1098
1099
1100
Before we can proceed any farther, we must generalize the notion
1101
of congruent modular forms introduced above. Let $K$ be a number
1102
field with ring of integers $\mathcal{O}_K$, and $\mathfrak{m}
1103
\subset \mathcal{O}_K$ an ideal. We define the \emph{order of $f$
1104
modulo $\mathfrak{m}$} by
1105
$$
1106
\ord_{\mathfrak{m}}(f):=\min\{n:a(n) \not \in \mathfrak{m} \}
1107
$$
1108
with the convention that $\ord_{\mathfrak{m}}(f):=+\infty$ if
1109
$a(n) \in \mathfrak{m}$ for all $n$. Though this is certainly not
1110
obvious a priori, given a modular form with coefficients in
1111
$\mathcal{O}_K$, one need only check finitely many $q$-series
1112
coefficients to calculate $\ord_{\mathfrak{m}}(f)$. The following
1113
theorem of Sturm (see \cite[\S 2.9]{O} or \cite{St}) makes this
1114
precise:
1115
\begin{theorem} \label{sturm}
1116
Suppose $k \geq 0$ is an integer and $K$ is a number field with
1117
ring of integers $O_K$. Moreover let $f=\sum_{n=0}^{\infty}
1118
a(n)q^n \in M_k(\Gamma_0(N)) \cap \OO_K[[q]]$. If $\mathfrak{m}
1119
\subset \mathcal{O}_K$ is an ideal for which
1120
$$
1121
\ord_{\mathfrak{m}}(f)>\frac{k}{12}[\Gamma_0(1):\Gamma_0(N)]
1122
$$
1123
then $\ord_{\mathfrak{m}}(f)= + \infty$.
1124
\end{theorem}
1125
\begin{remark}We will not prove this theorem in this thesis. We will only
1126
require it for the proofs of theorems \ref{modelth} and
1127
\ref{gbro}, and there we only invoke it briefly to prove that we
1128
can normalize certain forms so that they have coefficients in a
1129
ring of integers. To see how this works, consider some form $f
1130
\in M_k(\Gamma_0(N))$ with $p$-integral algebraic coefficients.
1131
Then we can pick an integer $M \equiv 1 \pmod{p}$ such that the
1132
first $\frac{k}{12}[\Gamma_0(1):\Gamma_0(N)]$ coefficients of $Mf$
1133
are contained in the ring of integers of some number field
1134
$\OO_K$. Applying Theorem \ref{sturm}, it follows that all of the
1135
coefficients of $Mf$ are in $\OO_K$, in other words, we have
1136
produced a form $Mf \equiv f \pmod{p}$ with algebraic integer
1137
coefficients.
1138
\end{remark}
1139
1140
Elements of $M_k(\Gamma')$ have the extremely useful property that
1141
they determined by their first few $q$-series coefficients.
1142
Though, 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
1144
attention to the fact that a similar statement is true when
1145
working with modular forms congruent modulo ideals in a number
1146
field.
1147
1148
We 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
1150
its ring of integers at a finite place $v$ with residue
1151
characteristic $p$. Moreover, let $\lambda$ be a uniformizer for
1152
$\mathcal{O}_v$. We make the following:
1153
\begin{definition} A formal power
1154
series
1155
$$
1156
f:=\sum_{n=0}^{\infty}a(n)q^n \in \mathcal{O}_v[[q]]
1157
$$
1158
is 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
1161
weights $k_i$ 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
1166
modular form has been substantially generalized by Katz; for an
1167
introduction and an explanation of how the two definitions relate,
1168
see \cite[\S I]{Go}.
1169
\end{remark}
1170
Thus we observe, with the help of Lemma \ref{pmoddy}, that $E_2$
1171
and $1$ are both $p$-adic modular forms, or, more precisely, are
1172
$p$-adic modular forms when identified with their $q$-expansions
1173
considered as elements of $\OO_v[[q]]$ (the $q$-expansion of $1$
1174
is 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
1177
The only nontrivial result we will require from the theory of
1178
$p$-adic modular forms is a theorem, due to Serre, which allows us
1179
to compute the constant term of a $p$-adic modular form in terms
1180
of a $p$-adic limit of its other coefficients for small primes
1181
$p$. Let $\zeta_p^*(s)$ be the Kubota-Leopoldt $p$-adic zeta
1182
function. We have
1183
1184
\begin{theorem}[Theorem 7, \cite{Sep}] \label{Ser} If $p \leq 7$ is prime and
1185
$$
1186
f=\sum_{n=0}^{\infty} a(n)q^n
1187
$$
1188
is a $p$-adic modular form of weight $k \neq 0$, then
1189
$$
1190
a(0)=\frac{\zeta_p^*(1-k)}{2} \cdot \lim_{n \to + \infty} a(p^n).
1191
$$
1192
\end{theorem}
1193
1194
This 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
1196
on which the $U$ operator (defined exactly as in (\ref{udef}))
1197
acts nilpotently and $E$ is a space on which $U$ acts bijectively.
1198
It turns out that for $2 \leq p \leq 7$ prime, $E$ is spanned by
1199
the reductions of Eisenstein series, and $N$ is spanned by the
1200
reductions of cusp forms. By analyzing each subspace, the theorem
1201
follows. For a complete proof, see \cite[\S 2.3]{Sep}.
1202
Incidentally, \cite{Sep} is a beautiful paper, and provides an
1203
interesting counterpoint to Katz's geometric approach to $p$-adic
1204
modular forms.
1205
1206
Also 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 $k \geq
1208
2$, where $\zeta(s)$ is the usual characteristic zero Riemann zeta
1209
function. In the sequel we will only be interested in the special
1210
case $k=2$, in which we have:
1211
$$
1212
\zeta_p^*(1-2)=(1-p)\zeta(-1)=\frac{p-1}{12}.
1213
$$
1214
Thus 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
$$
1219
f=\sum_{n=0}^{\infty} a(n)q^n
1220
$$
1221
is a $p$-adic modular form of weight $k \neq 0$, then
1222
$$
1223
a(0)=\frac{p-1}{24} \cdot \lim_{n \to + \infty} a(p^n).
1224
$$
1225
\end{corollary}
1226
1227
\section{Varying the level} \label{atle}
1228
1229
Given a modular form $f \in M_k(\Gamma_0(M))$ (resp., $f \in
1230
S_k(\Gamma_0(M))$) and recalling (\ref{vdef}) and (\ref{altvdef}),
1231
it 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
1234
lower levels; they're nothing new, which justifies the notation
1235
$$
1236
S_k(\Gamma_0(N)) \supset
1237
S_k^{\textrm{old}}(\Gamma_0(N)):=\bigoplus_{dM |
1238
N}S_k(\Gamma_0(M))|V(d).
1239
$$
1240
We define \emph{the space of newforms $\Sn_k(\Gamma_0(N))$} to be
1241
the orthogonal complement to $\So_k(\Gamma_0(N)$ with respect to a
1242
certain inner product, called the \emph{Petersson inner product}
1243
(see \cite[\S III.4]{La} or \cite[\S III.3]{k}). As a first
1244
example, for $p \geq 3$ prime, we have
1245
\begin{eqnarray} \label{decomp1}
1246
M_2(\Gamma_0(p))=\langle
1247
E_2(z)-pE_2(pz) \rangle \oplus \Sn_2(\Gamma_0(p))
1248
\end{eqnarray}
1249
because $M_2(\Gamma)=0$. One can check that $E_2(z)-pE_2(pz)$
1250
satisfies the requisite functional equation using (\ref{e2}). For
1251
arbitrary weights, the space of newforms has the useful property
1252
that it is preserved under the action of the Hecke operators. It
1253
is also invariant under another operator, the Atkin-Lehner
1254
involution, 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 $M_k(\Gamma_0(N))$
1258
by any matrix
1259
$$
1260
W(Q_p):=\left( \begin{smallmatrix} Q_pa & b \\ N c & Q_pd
1261
\end{smallmatrix} \right) \in M_{2 \times 2}(\ZZ)
1262
$$
1263
with determinant $Q_p$, where $a,b,c,d \in \ZZ$. Further, define
1264
the \textbf{Fricke involution $|_k W(N)$} on $M_k(\Gamma_0(N))$ by
1265
the matrix
1266
$$
1267
W(N):=\begin{pmatrix} 0 & -1 \\ N & 0 \end{pmatrix}.
1268
$$
1269
\end{definition}
1270
\noindent Well-definition of $|_k W(Q_p)$ follows from the
1271
functional 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
1274
have $f|_kW(Q_p)=f|_kW(p)$. By abuse of language, we will call
1275
$W(p)$ an Atkin-Lehner operator in this setting.
1276
1277
We now are in a position to make the following:
1278
\begin{definition} A \textbf{newform} in $\Sn_k(\Gamma_0(N))$ is a
1279
normalized cusp form that is an eigenform for all the Hecke
1280
operators, 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
1284
Newforms enjoy remarkable properties. We recall a few such
1285
properties on the more utilitarian side of things:
1286
\begin{theorem} \label{newforms}
1287
Suppose 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 $K$
1293
with 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
1297
integer $\lambda_f \in \{\pm1\}$ for which
1298
$$
1299
f|_kW(Q_p)=\lambda_pf.
1300
$$
1301
\end{enumerate}
1302
\end{theorem}
1303
\noindent For the statements of a collection of results, including
1304
the above, on newforms, see \cite[\S 2.4,\S 2.5]{O}. For proofs,
1305
see \cite{AL}, and for generalizations, see \cite{L} and \cite{M}.
1306
1307
We began this section by discussing how one can raise the level of
1308
an 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 $M,N$,
1311
define
1312
$$
1313
\Tr^{MN}_N:M_k(\Gamma_0(MN)) \to M_k(\Gamma_0(N))
1314
$$
1315
by
1316
$$
1317
\Tr^{MN}_N(f) =\sum_{i=1}^rf|_k \gamma_i
1318
$$
1319
where $\{\gamma_1,...,\gamma_r\}$ is a complete set of coset
1320
representatives for $\Gamma_0(NM) \backslash \Gamma_0(N)$. The
1321
fact that $\Tr^{MN}_N(f) \in M_k(\Gamma_0(N))$ is immediate;
1322
acting on $\Tr^{MN}_N(f)$ by an element of $\Gamma_0(N)$ simply
1323
permutes the $\gamma_i$ by the invariance of $f$ under the action
1324
of $\Gamma_0(NM)$. 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}
1333
A complete set of coset representatives for $\Gamma_0(Np)$ in
1334
$\Gamma_0(N)$ is given by
1335
$$
1336
\left\{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\right\} \cup
1337
\left\{
1338
\begin{pmatrix} 1 & 0 \\ N & 1 \end{pmatrix}\begin{pmatrix} 1 & j \\ 0 &
1339
1\end{pmatrix}\right\}_{j=0}^{p-1}.
1340
$$
1341
We also have
1342
$$
1343
\begin{pmatrix} 1 & 0 \\ N & 1\end{pmatrix} \begin{pmatrix} 1 & j
1344
\\ 0 & 1
1345
\end{pmatrix}= \begin{pmatrix} 1/p & 0 \\ 0 & 1/p \end{pmatrix}
1346
\begin{pmatrix} p & a \\ Np & pb \end{pmatrix} \begin{pmatrix} 1 &
1347
j-a \\ 0 & p
1348
\end{pmatrix}
1349
$$
1350
where
1351
$$
1352
\begin{pmatrix} p & a \\ Np & pb \end{pmatrix}
1353
$$
1354
is a matrix for $W(p)$. Since scalar matrices act trivially on
1355
$M_k(\Gamma_0(Np))$,
1356
$$
1357
\Tr^{MN}_N(f) = f + \sum_{j=1}^{p-1} f|_k W(p) \begin{pmatrix} 1 &
1358
j \\ 0 & p
1359
\end{pmatrix}.
1360
$$
1361
By considering $q$-expansions, we have
1362
$$
1363
\sum_{j=0}^{p-1} g\left( \frac{z+j}{p}\right)=p(g|U(p))(z),
1364
$$
1365
which completes the proof of the lemma.
1366
\end{proof}
1367
It 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}).
1369
Combining this observation with Lemma \ref{tr} yields the
1370
following:
1371
\begin{proposition}[\cite{AL}] \label{wphecke} If $f \in
1372
\Sn_k(\Gamma_0(p))$, then
1373
$$
1374
f|_kW(p)=-p^{1-k/2}f|U(p).
1375
$$
1376
\end{proposition}
1377
\begin{proof}
1378
First suppose that $f$ is a newform. From Lemma \ref{tr}, we have
1379
$$
1380
0=\Tr_1^{p}(f)=f+p^{1-k/2}f|_kW(p)U(p).
1381
$$
1382
Thus
1383
\begin{equation} \label{needitnow}
1384
f=-p^{1-k/2}f|_kW(p)U(p).
1385
\end{equation}
1386
Note that $U(p)=T_{k,p}$ because the
1387
level is $p$ (see (\ref{uvhecke})). Now note that $f$, being a
1388
newform, 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
1391
both sides of (\ref{needitnow}), we have
1392
\begin{eqnarray*}
1393
f|_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*}
1397
To 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
1399
representation of $W(p)$: $\left(\begin{smallmatrix} 0 & -1 \\ p &
1400
0
1401
\end{smallmatrix}\right) \left(\begin{smallmatrix} 0 & -1 \\ p & 0
1402
\end{smallmatrix}\right)=\left(\begin{smallmatrix} -p & 0 \\ 0 &
1403
-p
1404
\end{smallmatrix}\right)$. Since $U(p)$ and $W(p)$ are both
1405
linear 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}
1411
We 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 $z=\infty$,
1415
are at the points $z_1,...,z_s \in \mathbb{H}$. We say that $f(z)$
1416
is \emph{\textbf{good at $p$}} if there is a holomorphic modular
1417
form $\mathcal{E}_f(z) \in M_b(\Gamma)$ with $p$-integral
1418
algebraic coefficients for which the following are true:
1419
\begin{enumerate}
1420
\item As $q$-series, $\mathcal{E}_f(z) \equiv 1 \pmod{p}$. \item
1421
For 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
1426
good, then $fg$ is good.
1427
\end{remark}
1428
1429
\begin{remark}[2] As
1430
mentioned in the introduction, we will provide several families of
1431
good forms in \S \ref{good}; other families are provided in
1432
\cite{BrO}. Unfortunately, the author has not thought carefully
1433
about interesting examples of forms which are not good.
1434
\end{remark}
1435
1436
In 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}]
1441
By examining the proof of Proposition \ref{thetobs}, we see that
1442
if $f$ is a meromorphic modular form of weight $k$ over $\Gamma$,
1443
then
1444
\begin{equation} \label{fonow}
1445
\widetilde{f}:=12 \Theta f(z)-kE_2(z) f(z)
1446
\end{equation}
1447
is a meromorphic modular form of weight $k+2$ over $\Gamma$.
1448
Further, from (\ref{fonow}) we see that the poles of
1449
$\widetilde{f}(z)$ are supported at the poles of $f(z)$.
1450
1451
Now consider
1452
$$
1453
\frac{\theta f}{f}
1454
=\frac{1}{12}\left(\frac{\widetilde{f}(z)}{f(z)}+kE_2(z) \right).
1455
$$
1456
By \ref{pmoddy}, $E_2$ is a $p$-adic modular form of weight $2$
1457
with 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
1462
integer coefficients, then multiply it by a suitable integer
1463
$t_{j+1} \equiv 1 \pmod{p^{j+1}}$ so that the resulting series
1464
does. Thus we have
1465
$$
1466
t_{j+1}\mathcal{E}_f(z)^{p^j}\frac{\widetilde{f}}{f} \equiv
1467
\frac{\widetilde{f}}{f} \pmod{p^{j+1}}.
1468
$$
1469
If we define
1470
$F_{j+1}(z):=t_{j+1}\mathcal{E}(z)^{p^j}\widetilde{f}(z)/f(z)$,
1471
then we have that $\{F_{j+1}\}$ is a sequence of holomorphic
1472
modular 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}
1476
We will devote the rest this section to proving Theorem
1477
\ref{gbro}, a generalization of Bruinier and Ono's result to forms
1478
of prime level $p \geq 5$. We require two lemmas before we start
1479
on the main body of the proof. The first is most naturally proven
1480
using the notion of the divisor polynomial of a modular form,
1481
which 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},\\
1485
E_{4}(z)^2E_6(z) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12},\\
1486
E_4(z)\ \ \ \ \ &{\text {\rm if}}\ k\equiv 4\pmod{12},\\
1487
E_6(z)\ \ \ \ \ &{\text {\rm if}}\ k\equiv 6\pmod{12},\\
1488
E_4(z)^2\ \ \ \ \ &{\text {\rm if}}\ k\equiv 8\pmod{12},\\
1489
E_{4}(z)E_6(z) \ \ \ \ \ &{\text {\rm if}}\ k\equiv 10\pmod{12},
1490
\end{cases}
1491
\end{equation}
1492
and polynomials $h_k$ by
1493
\begin{equation}
1494
h_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
1504
Further, define $m(k)$ by
1505
$$
1506
m(k):=\begin{cases}
1507
\lfloor k/12\rfloor \ \ \ \ \ &{\text {\rm if}}\ k\not \equiv 2\pmod{12},\\
1508
\lfloor k/12\rfloor -1 \ \ \ \ \ &{\text {\rm if}}\ k\equiv 2\pmod{12}.
1509
\end{cases}
1510
$$
1511
With this notation, if $f(z) \in M_k$ and $\widetilde{F}(f,x)$ is
1512
the unique rational function in $x$ for which
1513
\begin{equation}
1514
\label{divyopoly}
1515
f(z)=\Delta(z)^{m(k)}\widetilde{E}_k(z)\widetilde{F}(f,j(z)),
1516
\end{equation}
1517
then $\widetilde{F}(f, x)$ is a polynomial; this follows from the
1518
familiar fact that any element of $\minf_0$ is a polynomial in
1519
$j$. We will refer to
1520
\begin{equation} \label{divisorpoly}
1521
F(f,x):=h_k(x)\widetilde{F}(f,x)
1522
\end{equation}
1523
as the \emph{divisor polynomial} for $f$. From (\ref{Wtag2.5}),
1524
(\ref{divyopoly}) and the classical $k/12$ valence formula (again,
1525
see \cite[\S III.2]{k}) the polynomial $F(f,x)$ will have a zero
1526
of order $n_k$ precisely at $j(z_k)$ for all zeros $z_k$ of $f$,
1527
where $n_k:=\ord_{z_k}(f)$. For a discussion of divisor
1528
polynomials, see \cite[\S 2.6]{O}.
1529
\begin{lemma} \label{pint}
1530
Suppose $f =q^h\prod_{n=1}^\infty(1-q^n)^{c(n)} \in
1531
\mmer_k(\Gamma_0(p)) \cap q^h\OO_K[[q]]$ for some number field $K$
1532
and some prime $p \geq 5$, and further that $f$ is good at $p$. Then
1533
$$
1534
\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p) \in
1535
\mmer_2(\Gamma_0(p))
1536
$$
1537
is $p$-integral.
1538
\end{lemma}
1539
\begin{proof} Note that $F(\mathcal{E}_f,j)$ has $p$-integral algebraic
1540
coefficients 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
1543
without multiplicity),
1544
$$
1545
G(j(z)):=(j(z)-j(z_1))\cdots(j(z)-j(z_n))
1546
$$
1547
has $p$-integral algebraic $q$-series coefficients. Because no
1548
prime above $p$ divides the $q$-expansion coefficient of lowest
1549
exponent 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
$$
1552
\frac{\Theta(f)-k(12)^{-1}E_2}{f}=\frac{g}{G(j)}
1553
$$
1554
where $g \in M_2(\Gamma_0(p)) \cap \overline{\QQ} [[q]]$ has
1555
$p$-integral algebraic coefficients. We have
1556
$$
1557
\left(\frac{\Theta(f)-k(12)^{-1}E_2}{f}\right)|_2W(p)=\left(\frac{1}{G(j)}\right)|_0W(p)g|_2W(p).
1558
$$
1559
We 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}
1563
0 & -1 \\ p & 0
1564
\end{matrix}\right)=\left(\frac{1}{G(j(z))}\right)|_0\left(\begin{matrix}
1565
0 & -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}
1568
p & 0 \\ 0 & 1
1569
\end{matrix}\right)=\frac{1}{G(j(pz))},
1570
\end{eqnarray*}
1571
which is evidently $p$-integral. Now note that we can write
1572
$g=c_1(E_2(z)-pE_2(pz))+h(z)$, where $h(z) \in \Sn_2(\Gamma_0(p))$
1573
has $p$-integral algebraic coefficients and $c_1$ is a
1574
$p$-integral algebraic number. From Proposition \ref{wphecke} we
1575
have $h(z)|_2W(p)=-h(z)|U(p)$, 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*}
1585
which is also $p$-integral. Since we have dealt with both
1586
factors, the lemma follows.
1587
\end{proof}
1588
\begin{remark} If restrict to the case $k=0$, this lemma is also
1589
true for $p=3$; the proof is the same.
1590
\end{remark}
1591
1592
Define
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
1599
M_{p-1}(\Gamma_0(p))
1600
\end{equation}
1601
for primes $p \geq 5$. We have following:
1602
\begin{lemma}
1603
If $p$ 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
1611
second follows from the end of the proof of Lemma \ref{pint}. For
1612
$p \geq 5$ we compute
1613
\begin{eqnarray*}
1614
E_{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.
1622
Thus we have the congruence $\widetilde{E}_p \equiv E_{p-1}
1623
\pmod{p}$, which yields $\widetilde{E}_p \equiv 1 \pmod{p}$ for
1624
all odd primes $p$ after an application of Lemma \ref{pmoddy}.
1625
1626
For the second claim, we have
1627
\begin{eqnarray*}
1628
\widetilde{E}_p|_{p-1}W(p) &=&
1629
E_{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*}
1633
We note the $p$-integrality of $E_{p-1}$ and $E_{p-1}|V(p)$ and
1634
again apply Lemma \ref{pmoddy} to finish the proof of the lemma.
1635
\end{proof}
1636
1637
We now prove Theorem \ref{gbro}. The two main inputs into this
1638
proof are the ideas behind the proof of Theorem \ref{modelth} and
1639
Serre's proof that a newform in $\Sn_k(\Gamma_0(p))$ is a $p$-adic
1640
modular form (see \cite{MO} and \cite{Sep}).
1641
\begin{proof}[Proof of Theorem \ref{gbro}]
1642
By (\ref{thetobs}), there exists a meromorphic modular form $g$ on
1643
$\Gamma_0(p)$ so that
1644
$$
1645
\frac{\Theta f}{f}=\frac{g}{f}+\frac{k}{12}E_2.
1646
$$
1647
Because $E_2$ is a $p$-adic modular form of weight two, it
1648
suffices to show that the same is true of $\frac{\Theta
1649
f-k(12)^{-1} E_2 f}{f}=\frac{g}{f}$.
1650
1651
Fix a positive integer $r$. Then (using the fact that $f$ is good
1652
at $p$), we have
1653
$$
1654
(\mathcal{E}_f)^{p^{r-1}} \frac{\Theta f-k(12)^{-1} E_2 f}{f} \in
1655
M_{2+p^{r-1}b}(\Gamma_0(p))
1656
$$
1657
where $b$ is the weight of $\mathcal{E}_f$. Further, this form is
1658
congruent modulo $p^r$ to $g/f$. Now consider
1659
$$
1660
f_r(z):=(\widetilde{E}_p)^{p^{r-1}}(\mathcal{E}_f)^{p^{r-1}}
1661
\frac{\Theta f-k(12)^{-1} E_2 f}{f} \equiv \frac{g}{f} \pmod{p^r}.
1662
$$
1663
We clearly have $f_r \in M_{2+p^{r-1}b+p^r-p^{r-1}}(\Gamma_0(p))$.
1664
We now take the trace of these $f_r$ to lower their level. We
1665
certainly have $\Tr_1^p(f_r) \in M_{2+p^{r-1}b+p^r-p^{r-1}}$, and
1666
we 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
1670
ring of integers $\OO_{K_r}$ of some number field
1671
$\mathcal{O}_{K_r}$ (this normalization may or may not be
1672
necessary depending on $\mathcal{E}_f$). Then
1673
$\{t_r\Tr_1^p(f_r)\}$ forms a sequence of holomorphic modular
1674
forms over $\Gamma$ whose coefficients converge $p$-adically to
1675
$g/f$ and whose weights converge to $2$, thus $g/f$ is a $p$-adic
1676
modular form of weight $2$.
1677
1678
We now prove that $\Trace_1^p(f_r) \equiv f_r \pmod{p^{r}}$. By
1679
Lemma \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*}
1685
Because $f$ 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 $U(p)$,
1688
implies 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}
1692
is $p$-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}
1697
and, 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 &=&
1702
p^{p^{r-1}b/2}(\mathcal{E}_f)^{p^{r-1}}|_{p^{r-1}b}\left(\begin{smallmatrix}
1703
p & 0 \\ 0 & 1
1704
\end{smallmatrix}\right)U(p) \equiv 0 \pmod{p^{p^{r-1}b/2}}.
1705
\end{eqnarray}
1706
The inequalities (\ref{1ord}), (\ref{2ord}) and (\ref{3ord})
1707
together 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
1711
true in the case $k=0$ for $p=3$ as well. We will use this fact
1712
without 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
1719
As indicated in the introduction, the construction of explicit
1720
families of good forms will require a discussion of complex
1721
multiplication and supersingularity, which we now begin. Recall
1722
that 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}
1729
is an analytic isomorphism. Here $\wp$ is the classical
1730
Weierstrass $\wp$-function. Conversely, given any lattice $L
1731
\subset \CC$, one can show that there exists an elliptic curve $E$
1732
for which an analytic isomorphism of the form (\ref{Eisom}) holds.
1733
Under this correspondence between lattices and elliptic curves,
1734
isomorphism classes of elliptic curves over $\CC$ correspond to
1735
equivalence classes of lattices, where the equivalence is given by
1736
$L \sim L'$ if $L=cL'$ for some $c \in \CC^*$. By way of
1737
terminology, the map $L' \to L$ given by multiplication by $c \in
1738
\CC^*$ is called a \emph{homothety}, and two lattices related in
1739
such 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
$$
1749
\CC/L_{\tau} \to E_{\tau}.
1750
$$
1751
We call this map (which is induced by (\ref{Eisom})) an
1752
\emph{analytic representation} of $E_{\tau}$.
1753
1754
We now wish to make this analytic representation more explicit;
1755
additionally, because it will be useful later, we work in a
1756
slightly more general context. Let $E/K$ be an elliptic curve over
1757
a field $K$ of characteristic not equal to $2$ or $3$. Up to
1758
isomorphism,
1759
we can assume that $E$ is given in affine coordinates by
1760
\begin{equation} \label{affequ}
1761
E: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
1764
case $K=\CC$, with the normalizations given above, the map
1765
(\ref{Eisom}) just formalizes the parametrization
1766
$$
1767
E:(\wp'(z,L))^2=4(\wp(z,L))^3-g_4\wp(z,L)-g_6.
1768
$$
1769
that exists for some lattice $L \subset \CC$.
1770
1771
We now wish define the $j$-invariant of $E_{\tau}$, and show how
1772
it 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}
1778
It is important to observe that the discriminant function
1779
$\Delta(E)$ is \emph{not} equal to the discriminant of the cubic
1780
polynomial defining the curve. Since the discriminant of the
1781
polynomial defining an elliptic curve $E$ is \emph{not} an
1782
isomorphism invariant of $E$, there are a variety of essentially
1783
equivalent ways to define the discriminant; the reason for our
1784
particular definition will soon be apparent.
1785
\end{remark}
1786
We 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}
1790
One can show by elementary means over any field $K$ of
1791
characteristic not equal to $2$ or $3$ that $j(E)$ is indeed an
1792
invariant 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
1796
Note the similarity of (\ref{je}) and (\ref{modjdef}). This is no
1797
accident. Let $\CC/L_{\tau} \to E_{\tau}$ be an analytic
1798
representation. It turns out that, with the normalizations given
1799
above, 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
$$
1802
\Delta(E)=\frac{(E_4(\tau)^3-E_6(\tau)^2)}{1728}=\Delta(\tau)
1803
$$
1804
and
1805
\begin{eqnarray} \label{jcoincid}
1806
j(E_{\tau})=j(\tau).
1807
\end{eqnarray}
1808
Thus the coincidence of the ``$j$" in $j$-function and
1809
$j$-invariant is really no coincidence. Indeed, noting the fact
1810
that as the $j$-invariant varies over $K$ it parameterizes
1811
isomorphism classes of elliptic curves over $K$ (at least if we
1812
continue 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
$$
1816
\mathfrak{F} \longleftrightarrow \left\{ \textrm{isomorphism
1817
classes of } E/ \CC \right\}.
1818
$$
1819
\noindent For proofs of the statements we just made on the
1820
equality of the various definitions of $j$ and $\Delta$, see
1821
\cite[\S I and p. 112]{k}. For a basic introduction to the theory
1822
of elliptic curves, see \cite{kn}.
1823
1824
Later 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
1826
given 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
1828
this 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
1831
curve written in this form an elliptic curve in \emph{Weierstrass
1832
form}. We now write formally as a proposition some elementary
1833
properties of curves written in Weierstrass form; for a proof, see
1834
\cite[\S III.2]{kn}
1835
1836
\begin{proposition} \label{explicitj}
1837
Suppose $a,b \in K$, where $K$ is a field with characteristic not
1838
equal 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
1844
Now that we have (\ref{Eisom}) and the isomorphism invariant
1845
$j(E)$ in hand, we completely understand isomorphism classes of
1846
elliptic curves over $\CC$ considered as analytic objects; they
1847
are explicitly parameterized by $\wp(z,L_{\tau})$ (considered as a
1848
function 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
1851
is evident that $E[N]$ is simply the group
1852
$\frac{1}{N}L_{\tau}/L_{\tau}$, that is,
1853
$$
1854
E[N] \approx \ZZ/N\ZZ \times \ZZ/N \ZZ.
1855
$$
1856
The ring of endomorphisms of $E$, or $\eo (E)$, can also be
1857
understood in a relatively straightforward manner using analytic
1858
representations. To begin, we have the following:
1859
\begin{lemma} \label{multilam} Let $L,M$ be two lattices in $\CC$, and let
1860
$$
1861
\lambda: \CC/L \to \CC/M
1862
$$
1863
be a complex analytic homomorphism. Then there exists a complex
1864
number $\alpha$ 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*}
1872
Here the top map is multiplication by $\alpha$ and the bottom is
1873
the homomorphism $\lambda$.
1874
\end{lemma}
1875
1876
\begin{proof}[Proof (compare \cite{L})] In a neighborhood of
1877
zero, $\lambda$ can be expressed by a power series
1878
$$
1879
\lambda(z) = a_0 + a_1 z +a_2z^2 + \cdots,
1880
$$
1881
On the other hand, $\lambda$ is a homomorphism, so $a_0=0$ and
1882
additionally we have
1883
$$
1884
\lambda(z+z') \equiv \lambda(z) + \lambda(z') \pmod{M}.
1885
$$
1886
If we choose a small enough neighborhood $U$ of zero, we must have
1887
that this congruence is an equality in $U$; thus
1888
$$
1889
\lambda(z)=a_1z
1890
$$
1891
for $z \in U$. But for any $z \in \CC$, $z/n$ is in $U$ for
1892
sufficiently large integers $n$, and from this we conclude that,
1893
identifying $z$ 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
1900
complex number $\alpha$ and the homomorphism $\lambda$ by the same
1901
symbol $\lambda$. We will also only be considering the special
1902
case $L=M$ of Lemma \ref{multilam}.\end{remark}
1903
1904
It is clear that any $\lambda \in \ZZ$ will induce an endomorphism
1905
of $\CC/L_{\tau}$, 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
1908
following:
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}
1914
The complex numbers $\lambda$ inducing a nontrivial endomorphism
1915
of a lattice $L$ turn out to be algebraic numbers; more
1916
specifically, they are quadratic over $\QQ$. Before we formalize
1917
and prove this as a proposition, we offer another definition,
1918
which will also be useful in \S \ref{good}:
1919
\begin{definition}
1920
Suppose $\tau \in \HH$ is the root of a quadratic equation with
1921
integer coefficients; that is, $\tau=\frac{-b+\sqrt{b^2-4ac}}{2a}$
1922
with $a,b,c \in \ZZ$ 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}
1928
Suppose $E/\CC$ is an elliptic curve. Then
1929
\begin{enumerate}
1930
\item Every nontrivial endomorphism of $E/\CC$ is induced (in the
1931
sense 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
1936
some 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
1940
quadratic number field $K$.
1941
\end{enumerate}
1942
\end{proposition}
1943
\begin{proof}
1944
The endomorphism ring of $E$ is unchanged if we replace it with
1945
another elliptic curve isomorphic to it, so we assume without loss
1946
of generality that $E=E_{\tau}$, $\tau \in \mathfrak{F}$. Thus we
1947
have an analytic representation
1948
$$
1949
\CC/L_{\tau} \to E_{\tau}.
1950
$$
1951
As we proved in Lemma \ref{multilam}, a nontrivial automorphism of
1952
$E_{\tau}$ can now be realized as a $\lambda \in \CC^*-\ZZ$ such
1953
that
1954
$$
1955
\lambda L_{\tau} \subset L_{\tau}
1956
$$
1957
or, equivalently, for some $\left(\begin{smallmatrix} a & b \\
1958
c& d
1959
\end{smallmatrix} \right) \in \textrm{GL}_2(\QQ) \cap M_{2 \times 2}(\ZZ)$,
1960
\begin{eqnarray*}
1961
\lambda \tau &=& a \tau+b \\
1962
\lambda &=& c \tau +d.
1963
\end{eqnarray*}
1964
This 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
1969
Thus $\lambda$ is a quadratic irrational algebraic integer. Now
1970
note that $\tau$ cannot be real; otherwise $L_{\tau}$ would not be
1971
a lattice, and $c \neq 0$, for then $\lambda$ would be an integer.
1972
Thus $\QQ(\tau)=\QQ(\lambda)$, and, further, both $\lambda$ and
1973
$\tau$ are imaginary quadratic numbers. This proves (1).
1974
1975
We've also proven the ``only if" implication of (2), just by
1976
recalling that $j$ is an isomorphism invariant. The other
1977
direction follows similarly: note that if $j(E)=j(\tau)$ with
1978
$\tau$ a Heegner point, then $E_{\tau} \approx E$, and $E_{\tau}$
1979
is evidently CM.
1980
1981
Finally, for (3), note that if $E$ is CM, as proven above, there
1982
is an isomorphic curve $E_{\tau}$ where $\tau$ is a Heegner point.
1983
Thus $\eo (E) \approx \eo (L_{\tau})$, and, again as proven above,
1984
any complex number inducing a nontrivial endomorphism of
1985
$L_{\tau}$ is an element of $\mathcal{O}_{\QQ(\tau)}$, the ring of
1986
integers of $\QQ(\tau)$, but not an element of $\ZZ$. With this
1987
observation 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
1989
with identity, and, further, the image of this homomorphism is not
1990
contained in $\ZZ \subset \mathcal{O}_{\QQ(\tau)}$. Thus
1991
$\eo(L_{\tau}) \approx \eo (E_{\tau})$ is isomorphic to an order
1992
in $\mathcal