text:
\section{Introduction}
\label{section1}

Let $L$ be a lattice in $\C$, and consider elliptic functions with
respect to $L$.  A standard formula (see, e.g., equation IV.3.6
of~\cite{Chandrasekharan}), which we reprove in
Corollary~\ref{corollary3.6} of this article, states that if
$\alpha,\beta,\gamma \in \C - L$ and $\alpha + \beta + \gamma = 0$,
then
\begin{equation}
\label{equation1.1}
\frac{-1}{2} \cdot
\frac{\wp'(\alpha) - \wp'(\beta)}{\wp(\alpha) - \wp(\beta)}
= \zeta(\alpha) + \zeta(\beta) + \zeta(\gamma).
\end{equation}
Here $\wp$ and $\zeta$ are the Weierstrass $\wp$ and zeta functions
with respect to $L$; our notation for elliptic functions follows
\cite{Chandrasekharan}, unless otherwise specified.  Let us
temporarily call the above expression $\thelambda =
\thelambda_{\alpha, \beta, \gamma, L}$.  In terms of the
projective embedding of the elliptic curve $\E = \C/L$ as a plane
cubic using $\wp$ and $\wp'$, this essentially says that
$\thelambda$ is the slope of the line (in the affine part of the
plane) joining the images of $\alpha$ and $\beta$.  After a short
calculation, we obtain that $\thelambda$ can be written as the
absolutely convergent series
\begin{equation}
\label{equation1.2}
\thelambda
 =\zeta(\alpha) + \zeta(\beta) + \zeta(\gamma)
 = \sum_{\omega \in L}{}' \left(
           \frac{1}{\omega + \alpha}
         + \frac{1}{\omega + \beta}
         + \frac{1}{\omega + \gamma}
         - \frac{3}{\omega}
                        \right),
\end{equation}
where the notation $\sum'$ means that one omits the term $3/\omega$
from the summand when $\omega = 0$.  Note that the individual sums
such as $\sum_\omega 1/(\omega + \alpha)$ do not converge; however, if
$\alpha, \beta, \gamma \in \frac{1}{\myl}L$ for some integer $\myl$,
then the sums can be regularized by Hecke's method to obtain
Eisenstein series of weight~$1$ on the congruence subgroup $\Gammal$.
After overcoming some analytic hurdles, we indeed show in
Section~\ref{section2} below that $\thelambda$ is the value of a
suitable Eisenstein series of weight~$1$.  As for Eisenstein series in
weights $2$ and~$3$, these can be related to values of the $\wp$ and
$\wp'$ functions, in other words to the affine coordinates of the
torsion points of $\E$ corresponding to $\alpha$, $\beta$, and
$\gamma$.  This means that the values of Eisenstein series
of weights up to~$3$ can be computed from the Weierstrass model of the
varying elliptic curve $\E$ and its $\myl$-torsion, in other words
from the moduli problem that is parametrized by the modular curve
$X(\myl)$.  This is the ``moduli  interpretation'' referred to in our
title.

More generally, we can search for other ``moduli-friendly'' modular forms
on $\Gammal$ that have an agreeable modular interpretation; this allows
us to define such modular forms over more general base fields than
$\C$.
In truth, the first paragraph above reverses the order
in which this author came across the family of
moduli-friendly forms treated in this article.
The realization that $\thelambda$ was a modular form came
first, from different considerations.  Indeed, the expression of
$\thelambda$ as a slope quotient shows that it is the quotient of an
Eisenstein series of weight~$3$ by one of weight~$2$; thus viewing
$\thelambda$ as a function of the varying $L$  (as well as $\alpha,
\beta, \gamma$), we obtain that $\thelambda$ is a  meromorphic modular
form of weight~$1$.  Holomorphy of $\thelambda$ on the upper half
plane $\Half$ and at the cusps then follows from the addition formula
on the elliptic curve, namely, from the formula
$\thelambda^2 = \wp(\alpha) + \wp(\beta) + \wp(\gamma)$, which
expresses $\thelambda^2$ as a holomorphic modular form.  It was in
this way that we first collected a family of moduli-friendly forms
(Definition~\ref{definition3.1}, equation~\eqref{equation3.16} and
Theorem~\ref{theorem3.9} below), defined more generally than
in~\eqref{equation1.1} by the coefficients in the Laurent expansions
of certain elliptic functions, or rather the algebraic Laurent
expansions of certain elements of the function field of $\E$.  It was
only later in our investigations that we made the connection from
these forms to the Weierstrass $\zeta$ function and Eisenstein series
(Theorem~\ref{theorem2.8}, which now comes earlier in our treatment).

In particular, this article provides a moduli-friendly, algebraic
treatment of all holomorphic Eisenstein series of arbitrary weight
$\wtj$ on $\Gammal$, and gives a natural way to express these
Eisenstein series as polynomials in the Eisenstein series of
weight~$1$.  This occupies Sections \ref{section2} and~\ref{section3}
of this article.  In fact, all the modular forms that we obtain belong
to a ring $\ringRl$, which turns out to be generated by the algebraic
version of the Eisenstein series in weight~$1$ for $\myl \geq 3$
(Theorem~\ref{theorem3.12}).  This result is similar to the
results proved in~\cite{BorisovGunnells}, where Borisov and Gunnells
define and study ``toric modular forms'' on $\Gamma_1(\myl)$, and
prove that the ring of toric modular forms is generated by certain
Eisenstein series in weight~$1$, and that it is stable under the
Hecke operators $T_n$ for $\Gamma_1(\myl)$; their proofs rely
on $q$-expansions of modular forms.  Thus the results in this article
include a generalization to $\Gammal$ of the ring of toric modular
forms introduced in~\cite{BorisovGunnells}.
(See also~\cite{Cornelissen} for a study of the ring generated by
weight~$1$ Eisenstein series in the Drinfeld modular case.)  The above
article~\cite{BorisovGunnells}, as well as the subsequent
articles~\cite{BorisovGunnellsNonvanishing,BorisovGunnellsHigherWeight,
BorisovGunnellsPopescuEquations}, were a definite inspiration for
several of the results in this article, even though our proofs tend
to proceed along different lines (most notably, without any
$q$-expansions).

Sections \ref{section4} and~\ref{section5} contain the main technical
results of this article.
Continuing the analogy with~\cite{BorisovGunnells}, we also prove
invariance of our ring $\ringRl$ under the Hecke algebra.
We first use an algebraic method to prove this in weights $2$ and~$3$
(Propositions \ref{proposition4.6}, \ref{proposition4.8},
and~\ref{proposition4.11}, with whose proofs we are rather pleased).
We then combine the Hecke invariance in these low weights with analytic
techniques (Rankin-Selberg and nonvanishing of $L$-functions), along
with some algebraic geometry of sufficiently positive line bundles on
curves, to prove that over $\C$, the ring $\ringRl$ contains \emph{all}
modular forms of weights $\wtj \geq 2$ (Theorem~\ref{theorem5.1}).  In
weight~$1$, our ring contains precisely the Eisenstein series.  Thus
our ring is of necessity stable under Hecke operators, this time for
the full (noncommutative) Hecke algebra of all double cosets for
$\Gammal$.  Our result that $\ringRl$ contains all modular forms in
higher weights is analogous to the results
in~\cite{BorisovGunnellsNonvanishing,BorisovGunnellsHigherWeight}
for toric modular forms on $\Gamma_1(\myl)$.  Borisov and Gunnells
prove there that the cuspidal part of the toric modular forms in
weight~$2$ consists of all cusp forms with nonvanishing central
$L$-value, while in weight $\wtj \geq 3$, the cuspidal part is all of
$\cuspforms_\wtj(\Gamma_1(\myl))$.  Their approach also uses
nonvanishing of $L$-functions, but is otherwise somewhat different;
see the introduction to Section~\ref{section5} below.  In that
section, we use our results so far to study models of the modular
curve $X(\myl)$.  We use our moduli-friendly interpretation of the
elements of $\ringRl$ to show the final result of this article
(Theorem~\ref{theorem5.5}), which can be stated in the following
striking manner: for $\myl \geq 3$, the slopes of lines joining the
$\myl$-torsion points of any \emph{one} elliptic curve over $\Q$
with $j \neq 0, 1728$ (for example, $\E_0: y^2 = x^3 + 3141x + 5926$)
contain enough information to deduce equations for $X(\myl)$, which
parametrizes the $\myl$-torsion of \emph{all} elliptic curves.
Moreover, the computations involved to find the equations for
$X(\myl)$ are all exact computations in the number field
$\Q(\E_0[\myl])$, and yield a model for $X(\myl)$ over the cyclotomic
field $\Q(\boldmu_\myl)$.  In particular, no infinite series or other
approximations in $\C$ are necessary.

Since our results are moduli-friendly and largely algebraic as opposed
to analytic (except for Theorem~\ref{theorem5.1}), the approach in
this article has the advantage that large parts of the theory work
over more general fields $\fieldk$ than $\C$, provided that $6\myl$ is
invertible in $\fieldk$, and that the $\myl$th roots of unity are
contained in $\fieldk$.  Our approach also has the benefit of yielding
a more direct connection to Eisenstein series and to moduli of
elliptic curves without using $q$-expansions at any stage.  We thus
hope that the techniques we have developed can be of use in the study
of modular forms over indefinite quaternion algebras and of Shimura
curves.

To summarize, here are the main results in this article:
\begin{itemize}
\item
A purely algebraic way to evaluate any Eisenstein series at a
noncuspidal point $\p \in X(\myl)$, in terms of a Weierstrass equation
for the elliptic curve $\E_\p$ corresponding to $\p$ in the moduli
interpretation, along with the coordinates of the $\myl$-torsion
$\E_\p[\myl]$ (this is in Sections \ref{section2} and~\ref{section3},
which also include effectively computable expressions for Eisenstein
series of any weight as polynomials in Eisenstein series of
weight~$1$)
\item
An expression for Eisenstein series of weights $1$ and~$2$ as
absolutely convergent sums, without the need for Hecke's method of
analytically continuing
$\sum_{c,d} (c\tau + d)^{-\wtj} \abs{c\tau + d}^{-2s}$ in the
parameter $s \in \C$ (Section~\ref{section2})
\item
Several relations between the moduli-friendly forms, proved
algebraically by a consideration of the moduli of elliptic curves
(simpler relations in Section~\ref{section3}, and deeper relations
in Section~\ref{section4}, which include the action of Hecke operators
in weights $2$ and~$3$)
\item
A proof that $\ringRl$ contains all modular forms of weights $\geq 2$;
thus the only modular forms that are missed by $\ringRl$ are
the cusp forms of weight~$1$ (Theorem~\ref{theorem5.1}), which is
in some sense not surprising, since these correspond to Galois
representations of Artin type, and are the most intractable from an
arithmetic viewpoint
\item
A systematic method to produce models for the curve $X(\myl)$
(Theorem~\ref{theorem5.5}; this model of a curve was called
``Representation~B'' in~\cite{KKMasymptotic}, and allows for
efficient computation in the Jacobian of $X(\myl)$).  The idea is
to use one fixed elliptic curve $\E_0$ to produce sufficiently many points
on a projective model for $X(\myl)$, so that only one curve $X(\myl)$ can
reasonably interpolate through all these points.  This involves lengthy but
purely algebraic computations in the field $\Q(\E_0[\myl])$, and
generalizes directly to all modular curves.
\end{itemize}

\textbf{Acknowledgements.}
This research was partially supported by the University Research Board at the
American University of Beirut, and the Lebanese National Council for
Scientific Research, through the grants ``Equations for modular and
Shimura curves''.  The author is grateful to L. Merel for helpful
discussions about the Hecke action, and to R. Ramakrishna for useful
comments on the manuscript.

\section{Eisenstein series and Laurent expansions of elliptic functions}
\label{section2}

Our first goal in this section is to describe a rearrangement of the sum in
Eisenstein series that converges absolutely for all weights $\wtj$, not
just for $\wtj \geq 3$.  Let $\tau \in \Half$, where $\Half$ is the
complex upper half plane, and consider the lattice
$L = L_\tau = \Z + \Z\tau$.  Recall the definition of Eisenstein series on
the principal congruence subgroup $\Gammal$, with $\myl \geq 1$.

\begin{definition}
\label{definition2.1}
For $\torsi, \torsj \in \Z$, let $\alpha = \alpha_\tau
  = (\torsi \tau + \torsj)/\myl \in \frac{1}{\myl}L_\tau$.
For an integer $\wtj \geq 1$ and for $s \in \C$, we define,
following~\cite{HeckeEisenstein}, the Eisenstein series of weight~$\wtj$ on
$\Gammal$:
\begin{equation}
\label{equation2.1}
\begin{split}
\Ej&(\tau, \alpha; s)
   = \sum_{\omega \in L_\tau}{}'
       \frac{1}{(\alpha+\omega)^\wtj \abs{\alpha+\omega}^{2s}}
\\
   &= \sum_{(\intm, \intn) \in \Z^2}{}'
       \bigl[(\intm + \torsi/\myl)\tau + \intn + \torsj/\myl\bigr]^{-\wtj}
       \bigl|(\intm + \torsi/\myl)\tau + \intn + \torsj/\myl\bigr|^{-2s},
\\
\end{split}
\end{equation}
\begin{equation}
\label{equation2.2}
\Ej(\tau, \alpha) = \Ej(\tau, \alpha; 0),
                           \quad \text{by analytic continuation}.
\end{equation}
Here the notation $\sum_\omega'$ omits $\omega = -\alpha$ in case we have
$\alpha \in L_\tau$; similarly for $\sum_{(\intm, \intn)}'$.
The sum for $\Ej(\tau, \alpha; s)$ converges for $\Real s > 1 - \wtj/2$,
and hence when $\wtj \geq 3$ we have the absolutely convergent series
$\Ej(\tau, \alpha) = \sum_\omega' (\alpha + \omega)^{-\wtj}$.
For $\wtj \geq 1$, Hecke showed that $\Ej(\tau,\alpha;s)$ can be
analytically continued to all $s \in \C$, and that
$\Eone(\tau,\alpha)$ is a holomorphic function of $\tau$, while
$\Etwo(\tau, \alpha)$ is the sum of
$-2\pi i / ( \tau - \overline{\tau})$ and a holomorphic function of $\tau$.
Since
$\Ej(\tau, \alpha; s)$ depends only on the class of $\alpha$ modulo
$L_\tau$, we can view $\alpha$ as an $\myl$-torsion point on the
elliptic curve $\E = \E_\tau = \C/L_\tau$.  We shall nonetheless take
care to distinguish between $\alpha \in \C$ and its image
$\pointP_\alpha \in \E$.
\end{definition}

We reformulate our Eisenstein series in terms of divisors on $\C$ and on
$\E$.  We establish the following notation to distinguish the notation
for the formal sums of points appearing in divisors from sums in $\C$ and
from the group operation on $\E$:
\begin{itemize}
\item
A divisor on $\C$ will be written $\tilde{\divD} = \sum_{\alpha} m_\alpha
(\alpha)$, and its image in $\E$ is $\divD = \sum_{\alpha} m_\alpha
(\pointP_\alpha)$.  Note that the $\alpha$ need not be distinct modulo $L$,
so some cancellation can occur in the formal sum for $\divD$.  We call
$\tilde{\divD}$ a \emph{lift} of $\divD$.
\item
The group operations of addition, inversion, and multiplication by an
integer $n \in \Z$ on points $\pointP, \pointQ \in \E$ are given by
\begin{equation}
\label{equation2.3}
\pointP, \pointQ \mapsto \pointP \oplus \pointQ,
\qquad
\pointP \mapsto \ominus \pointP = [-1]\pointP,
\qquad
\pointP \mapsto [n] \pointP = \pointP \oplus \cdots \oplus \pointP,
\text{ if } n \geq 1.
\end{equation}
\end{itemize}
We denote by $\Pzero \in \E$ the additive identity in that group.

\begin{definition}
\label{definition2.2}
Let $\divD$ be a divisor on $\E$ that is supported on the $\ell$-torsion
points $\El$, and choose any lift
$\tilde{\divD} = \sum_\alpha m_\alpha (\alpha)$ of $\divD$ to $\C$.
We then define the following Eisenstein series on $\Gammal$:
\begin{equation}
\label{equation2.4}
\Ej(\tau, \divD; s) = \sum_\alpha m_\alpha \Ej(\tau, \alpha; s),
\qquad\qquad
\Ej(\tau, \divD) = \Ej(\tau, \divD; 0).
\end{equation}
It is immediate that the definition does not depend on the choice of lift
$\tilde{\divD}$.
We remind the reader that the values
$\alpha \in \frac{1}{\myl} L_\tau$
(and corresponding points $\pointP_\alpha \in \El$)
vary with $\tau$, as in Definition~\ref{definition2.1}.
\end{definition}

Our observation is that suitable choices of the lift $\tilde{\divD}$ lead
to series for $\Ej(\tau, \divD; s)$ with good convergence for all
$\wtj \geq 1$.  We motivate our discussion with the classical fact
that a divisor
$\divD = \sum_\alpha m_\alpha (\pointP_\alpha)$ on $\E$ is principal
if and only if
\begin{equation}
\label{equation2.5}
\deg \divD := \sum_\alpha m_\alpha = 0, \qquad\qquad
\bigoplus \divD := \bigoplus_\alpha [m_\alpha] \pointP_\alpha = \Pzero.
\end{equation}
The latter sum above is evaluated in $\E$.

\begin{definition}
\label{definition2.3}
Let $\divD$ be a principal divisor on $\E$.  A \emph{principal lift} of
$\divD$ is a divisor $\tilde{\divD} = \sum_\alpha m_\alpha (\alpha)$ on
$\C$ satisfying
\begin{equation}
\label{equation2.6}
\sum_\alpha m_\alpha = 0, \qquad\qquad
\sum_\alpha m_\alpha \alpha = 0
        \quad \text{ (both sums evaluated in } \C).
\end{equation}
An arbitrary lift $\tilde{\divD}$ would \textit{a priori} merely satisfy
$\sum_\alpha m_\alpha \alpha \in L$.
\end{definition}

It is easy to see that principal lifts always exist.  For example, if
$\alpha = (\torsi\tau + \torsj)/\myl$, then the divisor $\divD =
\myl(\pointP_\alpha) - \myl(\Pzero)$ is principal, and all of the following
are principal lifts of $\divD$:
\begin{equation}
\label{equation2.7}
\begin{split}
\tilde{\divD_1} &= \myl(\alpha) - (\myl - 1)(0) - (\torsi \tau + \torsj),
  \\
\tilde{\divD_2} &= (\myl - 1)(\alpha) + (\alpha - \torsi \tau - \torsj)
   - \myl(0), \\
\tilde{\divD_3} &= (\myl + 1)(\alpha) - (\alpha + \torsi \tau + \torsj)
   - \myl(0). \\
\end{split}
\end{equation}

\begin{proposition}
\label{proposition2.4}
Given a principal divisor $\divD$ supported on $\El$, choose a principal
lift $\tilde{\divD}$ satisfying~\eqref{equation2.6}.  Then
\begin{equation}
\label{equation2.8}
\sum_{\alpha}
    \frac{m_\alpha}{(\alpha + \omega)^\wtj \abs{\alpha + \omega}^{2s}}
 = O \left( \frac{1}{\abs{\omega}^{2s+j+2}} \right),
   \text{ for large } \abs{\omega}.
\end{equation}
We hence obtain for all $\wtj \geq 1$ the following convergent double
series (where the notation $\sum_\alpha'$ means that we omit $\alpha =
-\omega$ if it appears in the inner sum):
\begin{equation}
\label{equation2.9}
\Ej(\tau, \divD)
  = \left.
      \sum_{\omega \in L} \sum_{\alpha}{}'
        \frac{m_\alpha}{(\alpha + \omega)^\wtj \abs{\alpha + \omega}^{2s}}
    \right|_{s=0}
 =  \sum_{\omega \in L}
       \left( \sum_{\alpha}{}' \frac{m_\alpha}{(\alpha + \omega)^\wtj}
       \right).
\end{equation}
Note that the outer sum over $\omega$ is absolutely convergent for
$\Real s > -\wtj/2$, even though the double sum converges only
conditionally.
\end{proposition}
\begin{proof}
Define the $\mathcal{C}^\infty$ function
$F(u) = \frac{1}{(u+\omega)^\wtj \abs{u+\omega}^{2s}}
  = (u + \omega)^{-\wtj - s} (\overline{u} + \overline{\omega})^{-s}$,
upon taking suitable branches for the powers.  By Taylor's formula with
respect to $u$ and $\overline{u}$, we have
\begin{equation}
\label{equation2.10}
\begin{split}
F(\alpha) &= F(0)
+ \left.\frac{\partial F}{\partial u}\right|_{u = 0} \alpha
+ \left.\frac{\partial F}{\partial \overline{u}}\right|_{u=0}
                                                     \overline{\alpha} \\
&+ \int_{u=0}^{\alpha} \left[
       (\alpha - u) \frac{\partial^2 F}{\partial u^2} \, du
      + (\overline{\alpha} - \overline{u})
           \frac{\partial^2 F}{\partial u \partial\overline{u}}\, du
      + (\alpha - u)
           \frac{\partial^2 F}{\partial u \partial\overline{u}}\, d\overline{u}
      + (\overline{\alpha} - \overline{u})
           \frac{\partial^2 F}{\partial\overline{u}^2}\, d\overline{u}
       \right].\\
\end{split}
\end{equation}
The integral is along any suitable path in the complex plane from $0$ to
$\alpha$, say for example a line segment.  We hence obtain an
expansion valid for $\abs{\omega} > 2\abs{\alpha}$:
\begin{equation}
\label{equation2.11}
\frac{1}{(\alpha+\omega)^\wtj \abs{\alpha+\omega}^{2s}}
 = \frac{1}{\omega^\wtj \abs{\omega}^{2s}}
   - \frac{(s+\wtj)\alpha}{\omega^{\wtj+1}\abs{\omega}^{2s}}
   - \frac{s\>\overline{\alpha}}{\omega^{\wtj-1}\abs{\omega}^{2s+2}}
   + O\left( \frac{1}{\abs{\omega}^{2s+\wtj+2}} \right).
\end{equation}
Here the implied constant depends on $\alpha$, $\wtj$, and $s$, and is
uniform in $\tau$ when $\tau$ is restricted to a compact subset of
$\Half$.  Now multiply~\eqref{equation2.11} by $m_\alpha$, and sum
over $\alpha$ to obtain \eqref{equation2.8} and~\eqref{equation2.9}.
(Note that the sum $\sum_\alpha'$ does not omit any $\alpha$ once
$\abs{\omega}$ is sufficiently large).
\end{proof}

\begin{remark}
\label{remark2.5}
Note that we always obtain holomorphic functions of $\tau$ above.  In
the setting of weight~$\wtj=2$, this arises because we have always
taken $\deg \divD = 0$, so the nonholomorphic terms cancel.
\end{remark}

Proposition~\ref{proposition2.4} allows us to rederive Hecke's second
definition of weight~$1$ Eisenstein series as ``division values'' of the
Weierstrass $\zeta$ function in Section~6 of~\cite{HeckeZurTheorie},
as well as Corollary~3.4.24  of~\cite{Katz}; we reprove those results
in~\eqref{equation2.14} below.  Recall the absolutely and uniformly
convergent series for $\zeta(z)$ for $z$ in a compact subset of
$\C-L_\tau$:
\begin{equation}
\label{equation2.12}
\zeta(z)
  = \frac{1}{z} +
    \sum_{0 \neq \omega \in L} \left[
         \frac{1}{z - \omega} + \frac{1}{\omega} + \frac{z}{\omega^2}
         \right]
  = \frac{1}{z} +
    \sum_{0 \neq \omega \in L} \frac{z^2}{(z-\omega)\omega^2}.
\end{equation}
It is a standard fact that
$\zeta(z + \intm \tau + \intn) = \zeta(z) + 2\intm \eta_2 + 2\intn \eta_1$
for $\intm, \intn \in \Z$ (with ``constants'' $\eta_i = \eta_i(L)$
satisfying $2\eta_1\tau - 2\eta_2 = 2\pi i$).  Here
we follow the notation of Chapter~IV of \cite{Chandrasekharan}; note that
Hecke and other authors use $\eta_i$ for what we have called $2\eta_i$.
Moreover, $\zeta$ is an odd function of $z$, and in fact its
Laurent expansion near $0$ is $\zeta(z) = z^{-1} + O(z^3)$.

\begin{corollary}
\label{corollary2.6}
Let $\divD$ be a principal divisor supported on $\El$, and take a
principal lift $\tilde{\divD} = \sum_\alpha m_\alpha (\alpha)$ for
which every instance of $\Pzero$ in $\divD$ is lifted to $\alpha =
0$.  Then
\begin{equation}
\label{equation2.13}
\Eone(\tau,\divD) = \sum_{\alpha \neq 0} m_\alpha \zeta(\alpha).
\end{equation}
Moreover, let $\pointP_\alpha \in \El - \{\Pzero\}$, with any choice
of lift
$\alpha = (\torsi \tau + \torsj)/\myl$ with $\torsi, \torsj \in \Z$.
Then
\begin{equation}
\label{equation2.14}
\begin{split}
\Eone(\tau,\pointP_\alpha)
         &=
     \zeta(\alpha)
   + \frac{1}{\myl} \left[
         \zeta(\alpha) - \zeta(\alpha + \torsi \tau + \torsj)
     \right] \\
         &=
     \zeta\left(\frac{\torsi \tau + \torsj}{\myl}\right)
   - \frac{\torsi}{\myl} \cdot 2\eta_2
   - \frac{\torsj}{\myl} \cdot 2\eta_1.\\
\end{split}
\end{equation}
\end{corollary}
\begin{proof}
Write
$\tilde{\divD} = m_0 (0) + \sum_{\alpha \neq 0} m_\alpha (\alpha)$,
with $\alpha \neq 0 \implies \alpha \notin L$ by our assumption on
$\tilde{\divD}$.  Changing the sign of $\omega$
in~\eqref{equation2.12}, we obtain
\begin{equation}
\label{equation2.15}
\sum_{\alpha \neq 0} m_\alpha \zeta(\alpha)
 = \sum_{\alpha \neq 0} \frac{m_\alpha}{\alpha}
 + \sum_{\omega \neq 0}
   \sum_{\alpha \neq 0}
      \left[
         \frac{m_\alpha}{\alpha + \omega}
        - \frac{m_\alpha}{\omega}
        + \frac{m_\alpha \alpha}{\omega^2}
      \right].
\end{equation}
The change of order of summation is justified by the good convergence
of the series for $\zeta$ and because the sum over $\alpha$ is finite.
Since $\tilde{\divD}$ satisfies~\eqref{equation2.6}, we have
$\sum_{\alpha \neq 0} m_\alpha = - m_0$ and $\sum_{\alpha \neq 0}
m_\alpha \alpha = 0$, which allows us to rewrite the above sum in the
form of~\eqref{equation2.9} (at the cost of replacing absolute
convergence with conditional convergence), and hence to
obtain~\eqref{equation2.13}.  Now apply this result in the case
$\divD = \myl (\alpha) - \myl(0)$, using the principal lift
$\tilde{\divD_3}$ from~\eqref{equation2.7}.  This
yields~\eqref{equation2.14}, because
$\Eone(\tau, \myl (\alpha) - \myl(0))
= \myl \Eone(\tau, \alpha) - \myl \Eone(\tau, 0)$ and
$\Eone(\tau,0) = 0$ (more generally, $\Ej(\tau, -\beta; s) = (-1)^\wtj
\Ej(\tau, \beta; s)$).
\end{proof}

We now turn to the second goal of this section, which is to relate
Eisenstein series on $\Gammal$ to Laurent expansions of elliptic
functions.

\begin{definition}
\label{definition2.7}
Let $\divD$ be a principal divisor on $\E$, and let $m_0$ be the
multiplicity of $\Pzero$ in $\divD$.  We define an element
$\fD$ of the function field of $\E$, which we also view as an
elliptic function on $\C$ with respect to $L$, by the requirements
\begin{equation}
\label{equation2.16}
\Divisor(\fD) = \divD, \qquad\qquad
\fD = z^{m_0}
      ( 1 + O(z) ),\quad \text{ near } z=0.
\end{equation}
Here the first requirement determines $\fD$ up to a nonzero constant
factor, and the second requirement normalizes the constant so as to
fix our choice of $\fD$.  Our normalization ensures that for principal
divisors $\divD$ and $\divE$,
\begin{equation}
\label{equation2.16.5}
  \thef_{\divD + \divE} = \fD \cdot \fE.
\end{equation}
\end{definition}
We remark that the precise normalization of the constant factor in
$\fD$ will be needed in later sections of this article; it is not
significant in this section, since we will mainly consider the
logarithmic differential $d\fD/\fD$.

\begin{theorem}
\label{theorem2.8}
Let $\divD$ be a principal divisor, and take a principal lift
$\tilde{\divD} = \sum_\alpha m_\alpha (\alpha)$.   Make the same
assumption on $\tilde{\divD}$ as in Corollary~\ref{corollary2.6}.
Then
\begin{equation}
\label{equation2.17}
\frac{d\fD}{\fD} = \sum_{\alpha}  m_\alpha \zeta(z-\alpha) \, dz
 = \sum_{\omega \in L} \left[
      \sum_{\alpha} \frac{m_\alpha}{z - \alpha - \omega}
   \right] dz,
\end{equation}
where the last series has similar convergence properties to the series
of~\eqref{equation2.9}.  Furthermore, if $\divD$ is supported on
$\El$, then the Laurent series expansion of $d\fD/\fD$ near $z=0$ is
\begin{equation}
\label{equation2.18}
\frac{d\fD}{\fD} = \left(
        \frac{m_0}{z} - \sum_{\wtj \geq 1} \Ej(\tau,\divD) z^{\wtj-1}
        \right) dz.
\end{equation}
\end{theorem}
\begin{proof}
It is classical (see, for example, Section IV.3 of \cite{Chandrasekharan})
that we can express $\fD$ up to a nonzero constant $C = C_\tau$ in
terms of the Weierstrass $\sigma$ function, provided that we have
taken a principal lift $\tilde{\divD}$:
\begin{equation}
\label{equation2.19}
\fD(z) = C \prod_{\alpha} \Bigl[\sigma(z - \alpha)^{m_\alpha}\Bigr].
\end{equation}
Taking logarithmic differentials yields the first equality
in~\eqref{equation2.17}, since $\sigma'/\sigma = \zeta$.  The second
equality now follows from substituting the series for $\zeta$ and
using the fact that
$\sum_\alpha [m_\alpha/\omega + m_\alpha(z-\alpha)/\omega^2] = 0$.

We can now prove~\eqref{equation2.18}.  The first term in the Laurent
expansion is easy, and the other terms are equivalent to showing
that $\Res_{z=0} \left[ z^{-\wtj} \frac{d\fD}{\fD} \right] =
-\Ej(\tau,\divD)$ for $\wtj \geq 1$.  This residue can be computed by a
contour integral on a small circle enclosing $z=0$.   Since the sum over
$\omega$ in~\eqref{equation2.17} converges well, we are justified in
computing the residue term-by-term, using the expansion
$\frac{1}{z-\beta}
 = - \frac{1}{\beta} - \frac{z}{\beta^2} - \frac{z^2}{\beta^3}
      - \cdots$
for $\beta \neq 0$ to compute residues for each inner sum over $\alpha$
that occurs as a term in the sum over $\omega$.  Comparing
with~\eqref{equation2.9} yields the desired result.
\end{proof}

\begin{remark}
\label{remark2.9}
Note that~\eqref{equation2.17} nicely confirms the fact (a simple
consequence of~\eqref{equation2.16}) that the
differential form $d\fD/\fD$ is periodic with respect to $L$, has only
simple poles, and has residue $m_\alpha$ at all points
$\alpha + \omega$.  We would have liked to use this fact to give a
different proof of~\eqref{equation2.18}, by taking the contour
integral of $z^{-\wtj}\frac{d\fD}{\fD}$ around a large circle with
center at $0$ and radius $R$ (or perhaps using a large
parallelogram).  At least for $\wtj \geq 2$, this approach works,
since the contour integral tends to zero as $R \to \infty$.  This
explains the minus sign in our results, as well as the summation
$\sum'$ for Eisenstein series, which gives special treatment to the
pole at $z=0$ due to the presence of $z^{-\wtj}$.  However, we were
not able to push through this argument for the important case
$\wtj = 1$.  This is because an argument based only on the locations
and residues of the poles of the differential form $d\fD/\fD$
cannot distinguish it from any other differential form
$d\fD/\fD + C\,dz$ where $C$ is a constant.
\end{remark}

The above theorem appears to relate Laurent expansions of elliptic
functions only to those Eisenstein series $\Ej(\tau,\divD)$ where
$\divD$ is principal.  On the other hand, $\Ej(\tau,\divD)$ depends
$\Z$-linearly on $\divD$ (in fact, so does $d\fD/\fD$,
by~\eqref{equation2.16.5}), so we are led to consider linear
combinations of Eisenstein series.

\begin{proposition}
\label{proposition2.10}
Let $\myl \geq 2$.  Then for all $\wtj \geq 1$, the $\C$-span of the
Eisenstein series
$\{ \Ej(\tau,\divD) \mid \divD \text{ principal, supported on } \El\}$
consists of all holomorphic Eisenstein series of weight~$\wtj$ on
$\Gammal$.
\end{proposition}
\begin{proof}
For all $\pointP \in \El$, the divisor $\myl (\pointP) - \myl(\Pzero)$
is principal, so the $\C$-span of our Eisenstein series contains
all Eisenstein series of the form
$\Ej(\tau,\pointP) - \Ej(\tau, \Pzero)$.  (If $\wtj$ is odd, then
$\Ej(\tau, \Pzero) = 0$ as we have already noted in the proof of
Corollary~\ref{corollary2.6}, so we are done.  But we will not use
this fact).  We conclude that our $\C$-span contains all combinations
$\sum_{\pointP \in \El} c_\pointP \Ej(\tau, \pointP)$ for which
$\sum_\pointP c_\pointP = 0$.  If $\wtj = 2$, then this is the space
of all holomorphic Eisenstein series, since we want the nonholomorphic
terms  $2\pi i/(\tau - \overline{\tau})$ in $\Etwo$ to cancel.
If $\wtj \neq 2$, then it suffices to show that we can obtain
$\Ej(\tau, \Pzero)$ (which is of course an Eisenstein series on
$\Gamma(1)$).  To this end, consider the principal divisor
$\divD = \bigl[\sum_{\pointP \in \El} (\pointP)\bigr]
           - \myl^2 (\Pzero)$.
We obtain $\Ej(\tau, \divD) = (\myl^\wtj - \myl^2)\Ej(\tau, \Pzero)$,
so we are done, since $\myl^\wtj - \myl^2 \neq 0$ by our assumptions on
$\myl$ and $\wtj$.
\end{proof}

\begin{remark}
\label{remark2.11}
The insistence on restricting to the case $\divD$ principal is in fact
a red herring, for deeper reasons than the above proposition.  Take a
more general $\divD$, which we assume for convenience is supported on
$\El$  (although we can often manage with the weaker assumption that
$\bigoplus \divD \in \El$, in the notation of~\eqref{equation2.5}).
We can canonically replace $\divD$ with $\divD - (\deg\divD) (\Pzero)$,
which does not change $\bigoplus \divD$ but now gives us a divisor of
degree zero.

We thus assume in this discussion that $\deg \divD = 0$, but that
$\bigoplus \divD \in \El$ need not be trivial.  Now the divisor
$\myl \divD$ is principal, and we can formally define
$\fD = (\thef_{\myl \divD})^{1/\myl}$ for compatibility
with~\eqref{equation2.16.5}.  Note that if $\bigoplus \divD \neq
\Pzero$, then $\fD$ cannot be an elliptic function with respect to
$L$; its formal logarithmic derivative is nonetheless always periodic
with respect to $L$, and we can simply take $d\fD/\fD =
(1/\myl)d\thef_{\myl \divD}/\thef_{\myl \divD}$ as a definition.  With
this convention, \eqref{equation2.18} continues to hold, and we can
obtain an analog of~\eqref{equation2.17} as a series with good
convergence properties, similarly to our derivation
of~\eqref{equation2.14}.

We can however be more ambitious.  Since $\thef_{\myl \divD}$ has
zeros and poles with multiplicity everywhere divisible by $\myl$, we
see that $\fD$ makes sense as a meromorphic function on $\C$.  We use
this to normalize the choice of $\myl$th root $\fD$ so that its
Laurent series begins with $z^{m_0}$, as in~\eqref{equation2.16}.
With our above conventions (especially in light
of~\eqref{equation2.16.5}), the $\fD$ that we consider are
products of (positive and negative) powers of the $\fP =
(\thef_{\myl(\pointP) - \myl(\Pzero)})^{1/\myl}$, for $\pointP \in \El
- \{\Pzero\}$.  For such a ``basic'' $\fP$, Theorem~\ref{theorem2.8}
then states that
\begin{equation}
\label{equation2.19.5}
\begin{split}
\frac{d\fP}{\fP}
 &= z^{-1}\Bigl(
  -1 - \sum_{\wtj \geq 1}
         \bigl(\Ej(\tau,\pointP) - \Ej(\tau,\Pzero)\bigr) z^{\wtj}
   \Bigr) dz\\
 &= z^{-1}\Bigl(-1 - \Eone(\tau,\pointP)z
              + \bigl(-\Etwo(\tau,\pointP) + \Etwo(\tau,\Pzero)\bigr)z^2
              + \cdots
          \Bigr)dz.
\end{split}
\end{equation}
Note that if $\pointP = \Pzero$, then $\thef_{\Pzero} = 1$,
so~\eqref{equation2.19.5} no longer holds (indeed,
$d\thef_{\Pzero}/\thef_{\Pzero} = 0$, so the first coefficient in the
Laurent expansion is now $0$ instead of $-1$).

Even when $\divD$ is nonprincipal as above, one can show that $\fD$ is
still an elliptic function, however with respect to
the sublattice $\myl L$ of $L$.  When $\divD = \pointP$,
the behavior of $\fP$ under translations by $L$ is described by
a Weil pairing; see Definition~\ref{definition4.1} in
Section~\ref{section4} below, where we work instead with the function
$\gP(z) = \fP(\myl z)$, which is elliptic with respect to the original
lattice $L$.  One can similarly analyze the behavior of an arbitrary
$\fD$ under translations by $L$ in terms of suitable Weil pairings.
The approach of working with $\fP$ that are periodic with respect to
$\myl L$ is used in the work of Borisov and Gunnells on toric modular
forms~\cite{BorisovGunnells}.  They use the function $\vartheta =
\vartheta_{11}$ to write down what amounts to the same
function as $\fP$ when $\pointP = a/\myl + L$ is in the subgroup of
$\El$ generated by $\pointP_{1/\myl}$.  They then use the expansion of
$d\fP/\fP$ at $z = 0$ to define their toric modular forms
$s_{a/\myl}^{(k)}$ (see Section~4.4 of~\cite{BorisovGunnells}).  Thus
their $s_{a/\myl}^{(k)}$ are a special case of our $\Ej(\tau,\divD)$,
where the divisor $\divD$ is of the form $[a]\pointP_{1/\myl} -
\Pzero$.  This means that the $s_{a/\myl}^{(k)}$ are Eisenstein series
with respect to $\Gamma_1(\myl)$ instead of $\Gammal$; Borisov and
Gunnells recognize this from the $q$-expansions, while our approach is
more direct.  Another advantage of our generalization to $\Gammal$ is
that for $\myl \geq 2$, we obtain the full space of holomorphic
Eisenstein series of level $\Gammal$, in all weights, by
Proposition~\ref{proposition2.10}; see also Theorems \ref{theorem3.9}
and~\ref{theorem3.12} below.  In contrast, the ring of toric
modular forms on $\Gamma_1(\myl)$ does not always contain all
Eisenstein series on that group: see Remark~4.13
of~\cite{BorisovGunnellsNonvanishing}.
\end{remark}

\begin{remark}
\label{remark2.12}
One can find the Laurent expansion of $\fD$ by
formally exponentiating the integral of $d\fD/\fD$.  Keeping track of
the algebra, one obtains that $\fD$ has an expansion of the following
form near $z=0$:
\begin{equation}
\label{equation2.20}
\fD = z^{m_0}
      ( 1 + F_1(\tau) z + F_2(\tau) z^2 + \cdots ),
\end{equation}
where $F_\wtj$ is a modular form on $\Gammal$ of weight~$\wtj$, expressible
as a polynomial in the $\Ej(\tau,\divD)$.  This approach is used
extensively in~\cite{BorisovGunnells}.  In the next section, we study the
Laurent series of $\fD$ directly in a purely algebraic setting over a more
general field $\fieldk$, and reformulate and extend the results of this
section algebraically.  For now, we simply note the result for $\fP$,
obtained from~\eqref{equation2.19.5}:
\begin{equation}
\label{equation2.21}
\fP = z^{-1}\left[ 1
               - \Eone z
               + \frac{(\Eone^2 - \tilde{\Etwo})}{2} z^2
               - \left( \frac{\Ethree}{3}
                      - \frac{\Eone \tilde{\Etwo}}{2}
                      + \frac{\Eone^3}{6}
                  \right) z^3
           + \cdots \right]
\end{equation}
where we wrote $\Eone = \Eone(\tau,\pointP)$, $\tilde{\Etwo} =
\Etwo(\tau,\pointP) - \Etwo(\tau,\Pzero)$, and
$\Ethree = \Ethree(\tau,\pointP)$
to save space.  For the ``genuine'' elliptic function
$\thef_{\myl(\pointP) - \myl(\Pzero)} = \fP^\myl$, we have the expansion
\begin{equation}
\label{equation2.22}
\thef_{\myl(\pointP) - \myl(\Pzero)}
   = z^{-\myl}
     \bigl( 1 - \myl \Eone(\tau,\pointP) z + \cdots \bigr).
\end{equation}
Analogous results to~\eqref{equation2.21} and~\eqref{equation2.22} hold for
arbitrary $\divD$.
\end{remark}

\section{Algebraic reformulation and the ring $\ringRl$ of modular
forms}
\label{section3}

Our first step in ``algebrizing'' the results of the previous section is
to normalize the equation of our elliptic curve $\E$.  We embed $\E$ into
the projective plane $\Projective^2$ as follows (note the factor $1/2$):
\begin{equation}
\label{equation3.1}
z \mapsto \pointP_z = [\wp(z;L): (1/2) \wp'(z;L) : 1] = [x(z): y(z): 1].
\end{equation}
As usual, $\Pzero = [0:1:0]$ is the identity element.  The affine algebraic
equation of $\E$ and the invariant differential $\omega$ on $\E$ are
\begin{equation}
\label{equation3.2}
\E: y^2 = x^3 + \acoeff x + \bcoeff,  \qquad \omega = dx/(2y) = dz.
\end{equation}
Here $\acoeff = \acoeff(\tau)$ and $\bcoeff = \bcoeff(\tau)$ are, up to
constant factors, the Eisenstein series of level~$1$ and weights $4$
and~$6$, respectively:
\begin{equation}
\label{equation3.3}
\acoeff(\tau) = -15 \Efour(\tau,0)
              = -15 \sum_{0 \neq \omega \in L_\tau} \omega^{-4},
\qquad
\bcoeff(\tau) = -35 \Esix(\tau,0).
\end{equation}
The symbol $\omega$ in~\eqref{equation3.3} denotes an element of $L$, but
for the rest of this article it will refer almost exclusively to the
invariant differential, as in~\eqref{equation3.2}.

We now regard the
family $\{\E_\tau \mid \tau \in \Half\}$ as a single elliptic curve $\E$
over the rational function field $\C(\acoeff, \bcoeff)$ in two independent
transcendental variables.  We can work with more general fields $\fieldk$
instead of $\C$; in that case, $\E$ is a curve over the field
$\fieldbigK = \fieldk(\acoeff,\bcoeff)$.
It is convenient to
define the following graded rings, where $\acoeff$ and $\bcoeff$ have
weights $4$ and~$6$, respectively:
\begin{equation}
\label{equation3.3.5}
\ringR_1 = \fieldk[\acoeff,\bcoeff], \qquad
\ringR_{1,\Z} = \text{ the image of }\Z[\acoeff,\bcoeff]
                \text{ inside } \ringR_1.
\end{equation}
Here $\ringR_1$ is of course an algebraic analog of the graded ring
$\C[\acoeff(\tau), \bcoeff(\tau)]$ of modular forms on the full modular
group $\Gamma(1)$.  Since we wish to use Weierstrass
normal form for $\E$, and also need to consider the $\myl$-torsion
throughout, we require $6\myl$ to be invertible in $\fieldk$, and for
$\fieldk$ to contain the group $\boldmu_\myl$ of $\myl$th roots of unity
(so as to accommodate the Weil pairing later).  We extend scalars so that
$\E$ is now defined over the $\myl$-torsion extension field
$\fieldbigKl$, a subfield of the algebraic closure $\fieldbigKbar$
of $\fieldbigK$:
\begin{equation}
\label{equation3.4}
\fieldbigKl = \fieldbigK(\El)
  = \fieldbigK\bigl(\bigl\{\xP, \yP \mid
            \pointP = (\xP,\yP)
                 \in \El(\fieldbigKbar) - \{\Pzero\}
              \bigr\}\bigr).
\end{equation}
Over $\C$, it is classical  (Section~2 of \cite{HeckeEisenstein},
especially equations (12--14)) that $\xP$ and $\yP$ are Eisenstein series
of weights $2$ and~$3$, respectively, when viewed as functions of $\tau$.
Specifically, let $\pointP = \pointP_\alpha$ for $\alpha = \alpha_\tau \in
\frac{1}{\myl}L_\tau - L_\tau$.  Then the usual series for $\wp$ and
$\wp'$, along with~\eqref{equation2.9}, immediately give us
\begin{equation}
\label{equation3.5}
\xP = \wp(\alpha; L_\tau) = \Etwo(\tau, \alpha) - \Etwo(\tau, 0),
\qquad
\yP = (1/2)\wp'(\alpha; L_\tau) = -\Ethree(\tau,\alpha).
\end{equation}

We now turn to the algebraic Laurent expansions of meromorphic functions on
$\E$ (i.e., of elements of the function field $\fieldbigKl(\E)$, but we
also view these as elliptic functions with respect to $L_\tau$ when
$\fieldk = \C$).  We fix an algebraic uniformizer $t$ at $\Pzero$:
\begin{equation}
\label{equation3.6}
t = -x/y \quad (= z - 2\acoeff z^5/5 + O(z^7) \text{ when } \fieldk = \C).
\end{equation}
We also write $\Ohat$ for the completion of the local ring of $\E$ at
$\Pzero$; hence $\Ohat$ is canonically isomorphic to the power series ring
$\fieldbigKl[[t]]$ --- we occasionally tacitly extend scalars to work
in $\fieldbigKbar[[t]]$ --- and we can view $\ringR_{1,\Z}[[t]]$ as a
subring of $\Ohat$.
(When $\fieldk$ has characteristic zero, we can still make sense of
the analytic uniformizer $z$ as an element of $\Ohat$, since the relation
$\omega = dz$ means that $z = \int \omega = t + 2a t^5/5 + \cdots$,
from~\eqref{equation3.7} below.)
The meromorphic functions $x,y \in \fieldbigKl(\El)$ then have the
following algebraic Laurent expansions:
\begin{equation}
\label{equation3.7}
\begin{split}
x &= t^{-2} - \acoeff t^2 + \cdots
   = t^{-2}\bigl(1 - \acoeff t^4 + \cdots \bigr)
               \in t^{-2} \ringR_{1,\Z}[[t]],\\
-tx = y &= -t^{-3} + \acoeff t + \cdots
   = t^{-3}\bigl(-1 + \acoeff t^4 + \cdots \bigr),
                \in t^{-3} \ringR_{1,\Z}[[t]],\\
\omega &= (1 + 2\acoeff t^4 + \cdots) dt
                \in \ringR_{1,\Z}[[t]]dt.\\
\end{split}
\end{equation}
Moreover, the coefficient of $t^\wtj$ in the power series inside
each pair of parentheses above is always a weight~$\wtj$ homogeneous
element of the graded ring $\ringR_{1,\Z}$.
For all this, see for example Section~IV.1
in~\cite{SilvermanI}, as well as Lemma~\ref{lemma3.8} below; alternatively,
one can proceed starting from the
usual analytic expansion of $\wp$ in case $\fieldk = \C$ to obtain
expansions of $x$, $y$, and $t$ in terms of $z$, and then revert the series
$t(z)$ to obtain series for $z$, $x$, and $y$ in terms of $t$.

Our goal is now to study the algebraic Laurent expansions of the
meromorphic functions $\fD \in \fieldbigK(\E)$ of
Definition~\ref{definition2.7}.  The second requirement
in~\eqref{equation2.16}, normalizing the constant factor in $\fD$, now
becomes $\fD = t^{m_0}(1 + O(t)) \in t^{m_0} (1 + t\Ohat)$.  This is
compatible with our previous normalization when $\fieldk = \C$, since
$t = z + O(z^2)$ by~\eqref{equation3.6}.
\begin{definition}
\label{definition3.1}
Let $\divD$ be a principal divisor supported on $\El$, with $m_0$ the
multiplicity of $\Pzero$ in $\divD$ as before.  For $\wtj \geq 1$, we define
$\lambdaDj$ to be the following coefficient in the Laurent
expansion of $\fD$ at $\Pzero$:
\begin{equation}
\label{equation3.8}
\fD = t^{m_0}
      ( 1 + \lambdaD^{(1)} t + \lambdaD^{(2)} t^2 + \cdots)
    = t^{m_0}
      ( 1 + \lambdaD t + \muD t^2 + \nuD t^3 + \cdots).
\end{equation}
In the above equation, we have also introduced the useful abbreviations
\begin{equation}
\label{equation3.9}
\lambdaD = \lambdaD^{(1)},
\qquad
\muD = \lambdaD^{(2)},
\qquad
\nuD = \lambdaD^{(3)}.
\end{equation}
We extend the above definitions to arbitrary $\divD$ supported on $\El$ by
the method of Remark~\ref{remark2.11}: we form the degree zero divisor
$\divD - (\deg \divD)(\Pzero)$, and multiply it by $\myl$
to obtain a principal divisor
$\divD' = \myl\cdot[\divD - (\deg \divD) (\Pzero)]$.
We then define
\begin{equation}
\label{equation3.10}
\fD = (\thef_{\divD'})^{1/\myl}
    = t^{m_0 - \deg \divD}(1 + \lambdaD t + \cdots)
                          \in t^{m_0 - \deg\divD}\Ohat,
\end{equation}
using the formal $\myl$th root of the power series, and use this expansion
to define the $\lambdaDj$ in general.  For $\divD$ principal, this
yields the same definition as before, because~\eqref{equation2.16.5} still
holds.  We can further use~\eqref{equation2.16.5}
to deduce various relations among the $\{\lambdaDj\}$, most notably
\begin{equation}
\label{equation3.10.5}
   \thelambda_{\divD+\divE} = \lambdaD + \lambdaE.
\end{equation}
\end{definition}

By the discussion in Remark~\ref{remark2.12}, each $\lambdaDj$ is a
modular form of weight~$\wtj$ on $\Gammal$ when $\fieldk = \C$; the fact
that the expansions in \eqref{equation2.20}--\eqref{equation2.22} are with
respect to $z$ instead of $t$ does not affect this statement.  We
nonetheless prefer to give an independent self-contained algebraic
formulation and proof of this result.  It is sufficient for this article to
work with the following naive algebraic definition of modular forms; in
contrast to the standard definition in, e.g., Section~2 of~\cite{Katz}, we
evaluate our modular forms only on the pair $(\E,\omega)$ with
$\myl$-torsion over the base field $\fieldbigKl$.
\begin{definition}
\label{definition3.2}
An algebraic modular form of level $\Gammal$ and weight $\wtj \geq 0$ is an
element $f \in \fieldbigKl$ satisfying the two properties:
\begin{enumerate}
\item We can write $f = g(\{\xP, \yP\})/h(\{\xP,\yP\})$ as
a quotient of isobaric polynomials (with coefficients in the graded
ring $\ringR_1$) in the variables
$\{\xP, \yP \mid \pointP \in \El - \{\Pzero\}\}$, where $\xP$ has
weight~$2$ and $\yP$ has weight~$3$, so that the resulting weight of
$f$ is $\wtj$;
\item
$f$ satisfies an equation of graded integral dependence over the graded
ring $\ringR_1$.  (Over $\C$, this requirement would ensure that we only
select weight~$\wtj$ elements of $\fieldbigKl$ that are holomorphic at all
$\tau \in \Half$ and at all the cusps of the modular curve $X(\myl)$.)
\end{enumerate}
\end{definition}
Now in light of~\eqref{equation3.5}, we expect that the $\{\xP\}$
and $\{\yP\}$ will turn out to be modular forms of weights $2$ and~$3$ by
the above definition.  We see that this is indeed the case for the
$\{\xP\}$, since the equation of graded integral dependence
that they satisfy is the square\footnote{
We have used the square $\psi^2_\myl$ here so as to avoid
encountering a factor of $y$ when $\myl$ is even; if we did not take
the square, we would need to write $\psi_\myl(x,y)$ to allow for the
presence of $y$.
}
of the $\myl$-division polynomial
$\psi^2_\myl(x) = \myl^2 \prod_{\pointP \in \El-\{\Pzero\}} (x - \xP)
=\myl^2 x^{\myl^2 - 1} + \cdots \in \ringR_{1,\Z}[x]$
(see, for example, Exercise~III.3.7 of~\cite{SilvermanI}).
Similarly, the $\{\yP\}$ are integrally dependent over $\ringR_1$ by
transitivity, using $\yP^2 = \xP^3 + \acoeff \xP + \bcoeff$.

We note for later use a consequence of the above
discussion.  Since the division polynomial $\psi^2_\myl$ is isobaric, the
coefficient of $x^{\myl^2 - 2}$ is a weight~$2$ element of $\ringR_1$, and
must therefore vanish.  This implies that
\begin{equation}
\label{equation3.11}
 \sum_{\pointP \in \El - \{\Pzero\}} \xP = 0.
\end{equation}

\begin{remark}
\label{remark3.3}
Definition~\ref{definition3.2} implies that the graded ring of modular
forms is the (graded) integral closure of $\ringR_1$ in $\fieldbigKl$.  It
is a pleasant exercise to verify that, over $\C$, this produces the usual
graded ring of modular forms.  (Part of the proof involves observing that
$\fieldbigKl$ contains $\acoeff$, $\bcoeff$, and all the $\xP$s and
$\yP$s, which, by Proposition~6.1 of \cite{Shimura}, suffice to
generate the function field of $X(\myl)$ via weight~$0$ meromorphic
ratios of elements of $\ringRl$.)
We reassure the reader who wishes to avoid this
verification that in any case we only use a certain subring $\ringRl$
of the ring of modular forms, given in Definition~\ref{definition3.7}, that
is generated by Eisenstein series of weights $\leq 6$ in case $\fieldk = \C$,
and which turns out to be generated by the Eisenstein series of weight~$1$
(Theorems \ref{theorem3.9} and~\ref{theorem3.12}).
\end{remark}
\begin{remark}
\label{remark3.4}
The weight of a homogeneous element of $\fieldbigKl$ can be defined
intrinsically by considering, for each $\scalarlambda \in \fieldk^\times$,
the automorphism of $\fieldbigKl$ and corresponding isomorphism of
elliptic curves given by:
\begin{equation}
\label{equation3.12}
\begin{split}
& \qquad\qquad
\acoeff \mapsto \scalarlambda^4 \acoeff, \qquad\quad
\bcoeff \mapsto \scalarlambda^6 \bcoeff, \qquad\quad
\omega \mapsto \scalarlambda^{-1} \omega,\qquad\quad
t \mapsto \scalarlambda^{-1} t,\\
&(x,y) \in \E: y^2 = x^3 + \acoeff x + \bcoeff
\mapsto (\scalarlambda^2 x, \scalarlambda^3 y)
   \in \E': y^2 = x^3 + \scalarlambda^4\acoeff x + \scalarlambda^6\bcoeff.\\
\end{split}
\end{equation}
This automorphism naturally sends $\xP \mapsto \scalarlambda^2 \xP$
and $\yP \mapsto \scalarlambda^3 \yP$, and is compatible with the grading
on $\ringR_1$; hence a modular form $f$ of weight $\wtj$ is sent by this
automorphism to $\scalarlambda^j f$.
Incidentally, $\scalarlambda=-1$ corresponds to inversion on $\E$, since
$\ominus \pointP = (\xP, -\yP)$.  This easily distinguishes modular forms
of odd and even weight.  In particular, we have
\begin{equation}
\label{equation3.12.3}
  \thelambda_{\ominus \pointP} = -\lambdaP,
\quad
  \themu_{\ominus\pointP} = \muP,
\quad
  x_{\ominus\pointP} = \xP,
\quad
  \thenu_{\ominus\pointP} = -\nuP,
\quad
  y_{\ominus\pointP} = -\yP.
\end{equation}
The above equations are for $\pointP \in \El-\{\Pzero\}$.  When $\pointP =
\Pzero$, we of course have $\thef_{\Pzero} = 1$, and so
$\thelambda_{\Pzero} = \themu_{\Pzero} = \thenu_{\Pzero} = 0$.  For
convenience, we shall also define $x_{\Pzero} = y_{\Pzero} = 0$ in
this case, even  though the point $\Pzero$, being at infinity, does
not have affine coordinates.  With this convention, we have the
further identities
\begin{equation}
\label{equation3.12.6}
\sum_{\pointP\in\El} \lambdaP
 =  \sum_{\pointP\in\El} \muP
 =  \sum_{\pointP\in\El} \xP
 =  \sum_{\pointP\in\El} \nuP
 =  \sum_{\pointP\in\El} \yP
 = 0.
\end{equation}
The above equations are obvious for the odd weights
($\lambdaP, \nuP, \yP$), while $\sum \xP = 0$ is~\eqref{equation3.11}.
We can however give a uniform proof of all
of these results, including the fact that $\sum \muP = 0$.  The
motivation for the uniform proof is that each sum over all $\pointP
\in \El$ in~\eqref{equation3.12.6} gives a modular form on $\Gamma(1)$
of weight $1$, $2$, or~$3$, which can only be zero.  To see this
algebraically, note that such a sum is invariant under the Galois
group of the extension $\fieldbigKl/\fieldbigK$; this group acts on
the points of $\El$ in a way that preserves the Weil pairing
(since $\boldmu_\myl \subset \fieldk$), and is easily seen to be
isomorphic to $SL(2,\Z/\myl\Z)$.
Thus each such sum is a weight~$\wtj$ element of $\fieldbigK =
\fieldk(\acoeff,\bcoeff)$ for some $\wtj \in \{1,2,3\}$.  Now by
Theorem~\ref{theorem3.9} below (the reader can check that no circular
reasoning is involved), the above sums are all modular forms, and
hence are integral over the subring $\ringR_1 =
\fieldk[\acoeff,\bcoeff]$.  But $\ringR_1$ is integrally closed, and
so the above sums actually belong to $\ringR_1$, which means that they
must vanish due to their weight.

We remark incidentally that an alternative proof of $\sum \muP = 0$ is
contained in the proof of Proposition~\ref{proposition4.3}.
\end{remark}

The following proposition is an easy consequence of the expansions
in~\eqref{equation3.7} and standard facts on elliptic curves:
\begin{proposition}
\label{proposition3.5}
\begin{enumerate}
\item
Let $\pointP = (\xP,\yP) \in \El-\{\Pzero\}$.  Then the divisor $(\pointP) +
(\ominus \pointP) - 2(\Pzero)$ is principal, and we have
\begin{equation}
\label{equation3.13}
\begin{split}
\thef_{(\pointP) + (\ominus \pointP)}
&= \thef_{(\pointP) + (\ominus \pointP) - 2(\Pzero)}
= x - \xP \\
& = t^{-2}(1 - \xP t^2 - \acoeff t^4 + \cdots)
      \in t^{-2} \ringR_1[\xP][[t]].
\end{split}
\end{equation}
In particular, $\thelambda_{(\pointP) + (\ominus \pointP)}
= \thenu_{(\pointP) + (\ominus \pointP)} = 0$ and
$\mu_{(\pointP) + (\ominus \pointP)} = -\xP$.
\item
Let $\pointP, \pointQ, \pointR \in \El - \{\Pzero\}$ satisfy
$\pointP \oplus \pointQ \oplus \pointR = \Pzero$; in other words, they are
collinear in the affine Weierstrass model of $\E$.  Then
$\thelambda_{(\pointP) + (\pointQ) + (\pointR)}$ is the slope of the line
joining the three points.  Specifically, the equation of the line is $y =
\lambdaD x + \nuD$, and we have the following for
$\divD = (\pointP) + (\pointQ) + (\pointR)$:
\begin{equation}
\label{equation3.14}
\begin{split}
\fD &= \thef_{\divD - 3 (\Pzero)} = -y + \lambdaD x + \nuD \\
& =  t^{-3}(1 + \lambdaD t + \nuD t^3 - \acoeff t^4 + \cdots)
  \in t^{-3} \ringR_1[\lambdaD, \nuD][[t]].
\end{split}
\end{equation}
In particular, $\muD = 0$, and we have $\lambdaD = (\yP - \yQ)/(\xP - \xQ)$
if $\pointP \neq \pointQ$ (whence $\xP \neq \xQ$, since we cannot have
$\pointP = \ominus \pointQ$ due to $\pointR \neq \Pzero$).  On the other
hand, if $\pointP = \pointQ$, then
$\lambdaD = (3\xP^2 + \acoeff)/2\yP$; here again, $\yP \neq 0$ since we
again cannot have $\pointP = \pointQ \in \E[2]$.
In both cases, the following standard identity (whose first equality
follows from~\eqref{equation3.10.5}) shows that $\lambdaD$ satisfies
the integrality condition in part~(2) of
Definition~\ref{definition3.2}, and is therefore a modular form of
weight~$1$:
\begin{equation}
\label{equation3.14.3}
(\lambdaP + \lambdaQ + \lambdaR)^2  = \lambdaD^2
  = \xP + \xQ + \xR.
\end{equation}
Finally, we also record the trivial identity
\begin{equation}
\label{equation3.14.6}
\nuD = \yP - \lambdaD \xP = \yQ - \lambdaD \xQ = \yR - \lambdaD \xR.
\end{equation}
\end{enumerate}
\end{proposition}

As mentioned before in Remark~\ref{remark3.3}, we also include a direct
proof in case $\fieldk = \C$ that the form $\lambdaD$ in part~(2) of
the above proposition is a modular form:
\begin{corollary}
\label{corollary3.6}
If $\fieldk = \C$, then in part~(2) of the above proposition take a
principal lift $\tilde{\divD} = (\alpha) + (\beta) + (\gamma) - 3(0)$ of
$\divD - 3(\Pzero)$.  We then obtain
\begin{equation}
\label{equation3.15}
\lambdaD(\tau)
  = -\Eone(\tau,\alpha) - \Eone(\tau,\beta) - \Eone(\tau,\gamma)
  = -\zeta(\alpha) - \zeta(\beta) - \zeta(\gamma).
\end{equation}
More generally, we have $\thelambda_{\divD+\divE} = \lambdaD + \lambdaE$
from~\eqref{equation2.16.5}, and so for all $\divD$ supported on $\El$, we
conclude that
\begin{equation}
\label{equation3.15.5}
\lambdaD(\tau) = - \Eone(\tau, \divD).
\end{equation}
\end{corollary}
\begin{proof}
From~\eqref{equation3.14}, we have that
$d\fD/\fD = t^{-1}(-3 + \lambdaD t + \cdots)$.  By~\eqref{equation3.6}, we
know that $t$ and $z$ agree up to $O(z^4)$, so we obtain the desired result
from~\eqref{equation2.18} (recall that $\Eone(\tau,0) = 0$)
and~\eqref{equation2.13}.  The more general result now follows
from~\eqref{equation3.10.5}.
\end{proof}

We shall now define $\ringRl$ for $\myl \geq 2$, generalizing our
previous definition $\ringR_1 = \fieldk[\acoeff,\bcoeff]$.  Namely,
we let $\ringRl$ be the graded $\fieldk$-subalgebra of the ring of all
modular forms on $\Gammal$ that is generated by:
\begin{itemize}
\item The forms $\acoeff$ and $\bcoeff$, in weights $4$ and $6$,
\item All coordinates $\xP$, $\yP$, in weights $2$ and $3$,
\item All slopes $\lambdaD$ for $\divD = (\pointP) + (\pointQ) + (\pointR)$
  as in Proposition~\ref{proposition3.5}, in weight $1$.
\end{itemize}
(We do not need to include the $\nuD$ in weight $3$, since they already
belong to $\ringRl$ by~\eqref{equation3.14.6}.)  In other words,
\begin{equation}
\label{equation3.16}
\ringRl
   = \fieldk\bigl[\acoeff,\bcoeff,
                \{\xP,\yP \mid \pointP \in \El - \{\Pzero\}\},
                \{\lambdaD \mid \divD = (\pointP) + (\pointQ) + (\pointR)
                               \linequiv 3\Pzero\}
            \bigr].
\end{equation}
We easily have $\ringR_{\myl'} \subset \ringRl$ for $\myl'$ a divisor
of $\myl$ (including $\myl' = 1$).

We observe that the coefficients in the formal Laurent expansions
\eqref{equation3.13} and~\eqref{equation3.14} belong to $\ringRl$ for
all $\myl \geq 2$; moreover, the expansions in \eqref{equation3.13}
and~\eqref{equation3.14} respect the weights of the modular forms, in
the sense that each series has the form
$t^m (1 + c_1 t + c_2 t^2 + \cdots)$ where $c_\wtj$ is a modular form
of weight~$\wtj$.  This observation motivates the following
definition:

\begin{definition}
\label{definition3.7}
Let $\ringR$ be any graded subalgebra of the ring of modular forms (say on
$\Gammal$).  An $\ringR$-balanced Laurent series in $t$ is a
series of the form
\begin{equation}
\label{equation3.17}
t^m \left( 1 + \sum_{\wtj = 1}^{\infty} c_\wtj t^\wtj \right),
\qquad\qquad c_\wtj \in \ringR \text{ of weight } \wtj.
\end{equation}
An analogous definition holds for series expressed in terms of the analytic
uniformizer $z$ (when $\fieldk$ has characteristic zero); as the following
elementary lemma observes, the condition of being $\ringRl$-balanced
does not depend on whether one expands with respect to $t$ or $z$.
\end{definition}

\begin{lemma}
\label{lemma3.8}
Let $\ringR$ be a graded algebra as above.  Then
\begin{enumerate}
\item
If $f(t)$ and $g(t)$ are $\ringR$-balanced Laurent series, then so are
$f(t)g(t)$ and $f(t)/g(t)$.
\item
If $f(t) = t^m(1 + c_1 t + \cdots)$ is $\ringR$-balanced, with $\myl | m$,
then the ``principal branch'' of the $\myl$th root $f(t)^{1/\myl} =
t^{m/\myl}(1 + c_1 t/\myl + \cdots)$ is again $\ringR$-balanced.
\item
Assume that $\fieldk$ has characteristic $0$.  Then
$z = z(t) = t + 2\acoeff t^5/5 + \cdots$
and $t=t(z) = z - 2\acoeff z^5/5 + \cdots$
are both $\ringR_1$-balanced series.  It follows that whenever
$\ringR_1 \subset \ringR$, then a series $f(t)$ is $\ringR$-balanced if and
only if $f(t(z))$ is.
\item
If $f(t) = t^m(1 + c_1 t + \cdots)$ is $\ringR$-balanced, then the
logarithmic differential $df/f$ has the expansion
$df/f = t^{-1}(m + \sum_{\wtj \geq 1} d_\wtj t^\wtj)dt$ with $d_\wtj$ a
weight~$\wtj$ element of $\ringR$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first two assertions are elementary; for the second, recall that $\myl$
is invertible in $\fieldk$ by assumption.  The third follows because the
invariant differential $\omega = dx/(2y) = dz$ has, by the first assertion,
an $\ringR_1$-balanced expansion $\omega = (1 + 2\acoeff t^4 + \cdots) dt$;
now integrate to obtain that $z=z(t)$ is balanced.  The rest is immediate.
\end{proof}

We can now state the first main result of this section.
\begin{theorem}
\label{theorem3.9}
\begin{enumerate}
\item
Let $\divD$ be a divisor supported on $\El$, as in
Definition~\ref{definition3.1}.  Then $\fD(t)$ is an $\ringRl$-balanced
Laurent series, and hence for all $\wtj \geq 1$, $\lambdaDj$ is a
modular form of weight~$\wtj$ on $\Gammal$; furthermore,
$\lambdaDj \in \ringRl$.
\item
The same result holds if we expand $\fD$ with respect to the analytic
uniformizer $z$ in characteristic zero, as well as if we expand the
logarithmic derivative $d\fD/\fD$.  Thus if $\fieldk = \C$, this theorem
combined with Theorem~\ref{theorem2.8} and
Proposition~\ref{proposition2.10} imply that all
Eisenstein series on $\Gammal$ belong to $\ringRl$.
\end{enumerate}
\end{theorem}
\begin{proof}
Proposition~\ref{proposition3.5} already shows that $\fD(t)$ is
$\ringRl$-balanced in the two cases
(i) $\divD = (\pointP) + (\ominus \pointP)$ and
(ii) $\divD = (\pointP) + (\pointQ) + (\pointR)$ with
$\oplus \divD = \Pzero$.  A more general $\divD$ that is supported on
$\El - \{\Pzero\}$, but that still satisfies $\oplus \divD = \Pzero$, can
be written as a $\Z$-linear combination of divisors $\divD$ of types (i)
and (ii).  We can thus use the multiplicativity of the $\fD$
from~\eqref{equation2.16.5} and the first part of Lemma~\ref{lemma3.8} to
conclude that $\fD(t)$ is $\ringRl$-balanced in this case.  For general
$\divD$ supported on $\El$, use if needed the second part of
Lemma~\ref{lemma3.8} to also conclude that $\fD(t)$ is
$\ringRl$-balanced.  The result for $\fD(z)$ is also immediate from the
above lemma.
\end{proof}

\begin{remark}
\label{remark3.9.5}
Statement (2) above can also be proved as in Sections 10.2-10.5
of~\cite{ShimuraElementary}, by expressing the higher derivatives
of $\wp$ in terms of $\wp$, $\wp'$, and $\acoeff(\tau)$; this relates
Eisenstein series of weights $4$ and above to the forms $\xP$, $\yP$, and
$\acoeff$.
\end{remark}

For the cases $\divD = (\pointP) + (\ominus \pointP)$ and $\divD =
(\pointP) + (\pointQ) + (\pointR)$ as in Proposition~\ref{proposition3.5},
we note that the corresponding $\fD$ are polynomials in $x$ and $y$:
namely, $\fD = x - \xP$ and $\fD = -y + \lambdaD x + \nuD$.  In this case,
the value of $\fD$ at a point
$\pointT \in \El - \{\Pzero\}$ is either $\xT - \xP$ or
$-\yT + \lambdaD \xT + \nuD$, which is a weight $2$ or~$3$ element of
$\ringRl$.  More generally, we have the following result:
\begin{corollary}
\label{corollary3.10}
Assume in the setting of Theorem~\ref{theorem3.9}
that $\divD$ is an effective divisor supported on $\El - \{\Pzero\}$, and
assume that $\oplus \divD = \Pzero$, so that $\divD - (\deg \divD) (\Pzero)$
is principal.  Then $\fD$ is a polynomial in $x$ and $y$, whose
coefficients all belong to $\ringRl$.  We can in fact expand
\begin{equation}
\label{equation3.18}
\begin{split}
\fD = x^n - \HD^{(1)} x^{n-2} y + \HD^{(2)} x^{n-1} - \HD^{(3)} x^{n-3} y +
\cdots, & \text{ if } \deg \divD = 2n \geq 2; \\
\fD = -x^n y + \HD^{(1)} x^{n+1} - \HD^{(2)} x^{n-1} y + \HD^{(3)} x^n +
\cdots, & \text{ if } \deg \divD = 2n+3 \geq 3.
\end{split}
\end{equation}
(The choice of signs above ensures that the monomials $x^N = t^{-2N} +
\cdots$ and $-x^N y = t^{-2N-3} + \cdots$ for varying $N$ are normalized
$\ringR_1$-balanced Laurent series in $t$.)
Moreover, $\HDj$ is a weight~$\wtj$ element  of $\ringR$, and
we have $\HDj = \lambdaDj$ for $\wtj=1, 2, 3$, but not for $\wtj \geq 4$.
Finally, for every $\pointT \in \El-\{\Pzero\}$, we have $\fD(\pointT)$ is
an element of $\ringRl$ of weight~$\deg \divD$.
\end{corollary}
\begin{proof}
The coefficients $\HDj$ above can be computed from the Laurent
expansion of $\fD$ by successively subtracting multiples of $x^N$ and
$-x^N y$ for $N$ going from $n$ down to $0$.
\end{proof}

\begin{remark}
\label{remark3.11}
We do not use the results of the above corollary in this article, but we
anticipate that they will be useful in other places.  For example, the
translation $\tau_\pointT^* \fD$ of a function $\fD$ by an element
$\pointT \in \El$ has as divisor the translation $\divD' =
\tau_{\ominus\pointT}(\divD)$ of $\divD$ by $\ominus\pointT$; here
$\tau_\pointT^* \fD$ will not be normalized, but if
$\pointT$ does not belong to the support of $\divD$ we can still write
$\tau_\pointT^* \fD = \fD(\pointT) \thef_{\divD'}$ and deduce
useful formulas.  Another interesting example is the case when we take a
principal divisor $\divD = (\pointP_1) + (\pointP_2) + (\pointP_3) +
(\pointP_4) - 4(\pointP_0)$, such that $\divD \neq \ominus \divD$.  Then
$\fD = x^2 - \lambdaD y + \muD x + \HD^{(4)}$, and $\lambdaD$ cannot equal
zero because $\divD$ is not an ``even'' divisor.  This means that, over
$\C$, $\lambdaD$ is then a weight~$1$ modular form that cannot vanish at
any point of $\Half$, but that can only vanish at the cusps; hence it is
a kind of generalized modular unit constructed from weight~$1$ Eisenstein
series.  A simple example of this is the case $\pointP_0 = \pointP_1 =
\pointP$, $\pointP_2 = \ominus \pointP \ominus \pointQ$, and $\pointP_3 =
\ominus \pointP \oplus \pointQ$, for $\pointP \not\in \E[2] \union
\{\pointQ, \ominus \pointQ\}$.  In this case we obtain
$\lambdaD = \thelambda_{(\pointP) + (\pointQ)
                        + (\ominus \pointP \ominus \pointQ)}
          + \thelambda_{(\pointP) + (\ominus \pointQ)
                        + (\ominus \pointP \oplus \pointQ)}
= 2\yP/(\xP - \xQ)$; this expression appears below in the proof of
Theorem~\ref{theorem3.12}.  The numerator and denominator in this
expression are modular forms that are well known to vanish only at the
cusps, as seen in~\cite{KubertLang}.  Our methods have just shown
that the ratio of these two forms is also a modular form (i.e., it does not
have any poles, even at the cusps), and that this ratio is in
fact $\lambdaD$, an Eisenstein series of weight~$1$.
\end{remark}

Our second main result in this section is the fact that $\ringRl$ is
generated by its elements $\{\lambdaP\}$ of weight~$1$, in other words
(over $\C$) by Eisenstein series of weight~$1$.  This result holds only for
$\myl \geq 3$.  Indeed, if $\myl = 2$, then write as usual $\E[2] =
\{\Pzero, \pointP_1, \pointP_2, \pointP_3\}$ with $\pointP_i = (e_i,0)$ for
$1 \leq i \leq 3$.  Hence $x_{\pointP_i} = e_i$ and $y_{\pointP_i} = 0$ for
$1 \leq i \leq 3$, and all the $\lambdaP$ are zero in this case; moreover,
$(x-e_1)(x-e_2)(x-e_3) = x^3 + \acoeff x + \bcoeff$, as usual.  We easily
obtain that $\ringR_2$ is the full ring of modular forms on $\Gamma(2)$,
namely $ \ringR_2 = \fieldk[e_1, e_2, e_3 \mid e_1 + e_2 + e_3 = 0]$, which
is generated by Eisenstein series of weight~$2$ when $\fieldk = \C$.

\begin{theorem}
\label{theorem3.12}
Assume that $\myl \geq 3$.
Define the subring $\ringR'$ of $\ringRl$ to be the subring generated
by all $\lambdaD$, where $\divD = (\pointP) + (\pointQ) + (\pointR)$ is a
divisor supported on $\El-\{\Pzero\}$ with $\oplus \divD = \Pzero$ as in
part~(2) of Proposition~\ref{proposition3.5}.  Then
the forms $\acoeff, \bcoeff, \{\xP\}, \{\yP\}$, for $\pointP \in \El -
\{\Pzero\}$,
all belong to $\ringR'$.  In particular, $\ringRl = \ringR'$ and is
hence generated by the $\lambdaD$ of the above form.
\end{theorem}

\begin{proof}
We begin by showing that all the $\{\xP\}$ belong to $\ringR'$.  This
boils down to a judicious use of~\eqref{equation3.14.3}, and
involves three cases, depending on $\myl$:
\begin{enumerate}
\item
If $\myl \geq 5$, let $\pointP$ be a point of exact order $\myl$, and
consider the following four elements of $\ringR'$ (recall also that
$x_{\ominus P} = \xP$):
\begin{equation}
\label{equation3.19}
\begin{split}
(\thelambda_{(\pointP) + (\pointP) + ([-2]\pointP)})^2
       = \xP + \xP + x_{[-2]\pointP} &= 2\xP + x_{[2]\pointP}\\
(\thelambda_{(\pointP) + ([2]\pointP) + ([-3]\pointP)})^2
       = \xP + x_{[2]\pointP} + x_{[-3]\pointP}
              &= \xP + x_{[2]\pointP} + x_{[3]\pointP}\\
(\thelambda_{(\pointP) + ([3]\pointP) + ([-4]\pointP)})^2
      \qquad\qquad\qquad\qquad
      &= \xP \qquad\quad + x_{[3]\pointP} + x_{[4]\pointP}\\
(\thelambda_{([2]\pointP) + ([2]\pointP) + ([-4]\pointP)})^2
      \qquad\qquad\qquad\qquad
      &= \qquad \quad 2x_{[2]\pointP} \qquad + x_{[4]\pointP}.\\
\end{split}
\end{equation}
Here the determinant
$\det \begin{pmatrix} 2&1&0&0\\ 1&1&1&0\\ 1&0&1&1\\0&2&0&1\\ \end{pmatrix}
= 6$ is invertible in $\fieldk$, and so each of $\xP, x_{[2]\pointP},
x_{[3]\pointP}, x_{[4]\pointP}$ can be expressed in terms of $\thelambda$'s,
and so belongs to $\ringR'$.  Now if $\pointP \in \El$ is a point of order
less than $\ell$, we can find a basis $\{\pointQ, \pointR\}$ for $\El
\isomorphic (\Z/\ell\Z)^2$, such that $\pointP = [d]\pointQ$ for some
$d > 1$.
In that case, the points $\pointP' = (\ominus \pointP) \oplus \pointR =
[-d] \pointQ \oplus \pointR$ and $\pointP'' = \ominus \pointR$ both have
exact order $\myl$, so $x_{\pointP'}$ and $x_{\pointP''}$ both belong to
$\ringR'$.  The points $\pointP, \pointP', \pointP''$ are collinear, and so
$(\thelambda_{(\pointP)+(\pointP')+(\pointP'')})^2
     =\xP + x_{\pointP'} + x_{\pointP''}$ belongs to $\ringR'$, whence $\xP
\in \ringR'$.  (Alternatively, we can deal with a point $\pointP =
[d]\pointQ$ of order less than $\myl$ by using identities
analogous to~\eqref{equation3.19} to see that
$\xQ + x_{[n]\pointQ} + x_{[n+1]\pointQ} \in \ringR'$, and to deduce
inductively that the $x$-coordinates of all multiples $[n]\pointQ$ belong
to $\ringR'$ whenever $\pointQ$ has exact order $\myl$.)
\item
If $\myl = 3$, we simply note that
$(\thelambda_{3(\pointP)})^2  = 3\xP$ for all $\pointP \in
\E[3] - \{\Pzero\}$.
\item
If $\myl = 4$, let $\{\pointQ, \pointR\}$ be a basis for $\E[4] \isomorphic
(\Z/4\Z)^2$.  By the same technique as in the first case above, we see that
the following sums belong to $\ringR'$, being squares of suitable
$\thelambda$'s:
\begin{equation}
\label{equation3.20}
\begin{matrix}
2\xQ & + x_{[2]\pointQ}, &&&&\\
&& 2\xR & + x_{[2]\pointR}, &&\\
\xQ && + \xR && + x_{\pointQ \oplus \pointR}, &\\
\xQ && + \xR &&& + x_{\pointQ \ominus \pointR}, \\
& x_{[2]\pointQ} &&& +  x_{\pointQ \oplus \pointR} &
                                 +  x_{\pointQ \ominus \pointR}, \\
&&& x_{[2]\pointR} & +  x_{\pointQ \oplus \pointR} &
                                 +  x_{\pointQ \ominus \pointR}.\\
\end{matrix}
\end{equation}
(For example, the fourth sum above is
$(\thelambda_{(\pointQ)+(\ominus\pointR)+(\pointR\ominus\pointQ)})^2$.)
The corresponding determinant is $-12$, again invertible, so we deduce in
particular that $\xQ, x_{[2]Q} \in \ringR'$.  Now any $\pointP \in \E[4] -
\{\Pzero\}$ has exact order either $4$ or~$2$.  So we can choose our basis
$\{\pointQ, \pointR\}$ so as to have $\pointP = \pointQ$ in the former
case, and $\pointP = [2]\pointQ$ in the latter case, thereby concluding
that $\xP \in \ringR'$.
\end{enumerate}
Now that we have shown that all the $\xP$ belong to $\ringR'$, let us show
that all the $\yP$ also belong to $\ringR'$.
Fix $\pointP \in \El - \{\Pzero\}$, and
take any $\pointQ \in \El - \{\Pzero, \pointP, \ominus\pointP \}$.  Then
$(\yP - \yQ)/(\xP - \xQ)$ and $(\yP + \yQ)/(\xP - \xQ)$ are among our
$\thelambda$'s (the latter being the slope of the line through $\pointP$
and $\ominus \pointQ$), and so their sum $2\yP/(\xP - \xQ)$ belongs to
$\ringR'$.  Multiplying by $\xP - \xQ \in \ringR'$ shows that $\yP \in
\ringR'$.  Observe that at this point we know by~\eqref{equation3.14.6}
that the forms $\{\nuD\}$ for $\divD = (\pointP) + (\pointQ) + (\pointR)$
also belong to $\ringR'$.

Finally, take any $\pointP \in \El - \E[2]$.  Then $\acoeff =
2\yP \thelambda_{(\pointP) + (\pointP) + ([-2]\pointP)} - 3\xP^2$
also belongs to
$\ringR'$, as does $\bcoeff = \yP^2 - \xP^3 - \acoeff \xP$.  Alternatively,
we can deduce that $\acoeff, \bcoeff \in \ringR'$ from the polynomial
identity $(x-\xP)(x-\xQ)(x-\xR) = x^3 - (\lambdaD x + \nuD)^2 + \acoeff x +
\bcoeff$ whenever $\divD = (\pointP) + (\pointQ) + (\pointR)$ with
$\pointP, \pointQ, \pointR$ collinear as usual.
\end{proof}

\begin{remark}
\label{remark3.13}
We can also define a subring $\ringR'_A$ of $\ringR'$, corresponding to a
subgroup $A \subset \El$: let $\ringR'_A$ be generated by the forms
$\thelambda_{(\pointP) + (\pointQ) + (\pointR)}$, for $\pointP, \pointQ,
\pointR \in A - \{\Pzero\}$ with $\pointP \oplus \pointQ \oplus \pointR =
\Pzero$.
Assume that $A \isomorphic \Z/m\Z \oplus \Z/\myl \Z$ with $m | \myl$ and
$\myl \geq 5$ (possibly $m = 1$).
Then our methods of proof show that $\acoeff, \bcoeff, \{\xP, \yP \mid
\pointP \in A - \{\Pzero\}\}$ all belong to $\ringR'_A$, as do the
appropriate $\thenu$'s coming from points in $A$.  Compare this to
Proposition~4.9 in~\cite{BorisovGunnells}.
\end{remark}

\begin{remark}
\label{remark3.14}
The above two theorems show that when $\myl \geq 3$, all the modular forms
that we have constructed through Laurent expansions can be expressed as
polynomials in the $\lambdaD$, which are special Eisenstein series of
weight~$1$ when $\fieldk = \C$.  It is equally useful to consider a
different set of generators of $\ringRl$, namely the
$\{\lambdaP \mid \pointP \in \El - \{\Pzero\}\}$.  We have the relation
$\thelambda_{(\pointP)+(\pointQ)+(\pointR)}
= \lambdaP + \lambdaQ + \lambdaR$, which shows that the $\{\lambdaP\}$ for
single points generate the $\{\lambdaD\}$ as above.  Our proof above gives a
rather indirect proof of the converse statement, that the $\{\lambdaD\}$
generate the $\{\lambdaP\}$.  One can also see this converse directly by
observing that $\myl$ is invertible in $\fieldk$ and that
$\myl \lambdaP
   = \sum_{n = 1}^{\myl-2}
       \thelambda_{(\pointP) + ([n]\pointP) + ([-n-1]\pointP)}$.
Alternatively, one can express $\lambdaP$ as a linear combination of
$O(\log \myl)$
different $\lambdaD$s using values of $n$ starting from $1$ and increasing
by a ``double-and-add'' approach until we reach $n = \myl - 1$.  This
is left to the reader.
\end{remark}

We conclude this section by noting a couple of useful algebraic
relations between the modular forms in $\ringRl$.
We note that~\eqref{equation3.22} below has already appeared for
$\Gamma_1(\myl)$
in~\cite{BorisovGunnellsNonvanishing,BorisovGunnellsPopescuEquations}.
The approach of obtaining relations by taking a sum of residues over
all points of $\E$ is taken from~\cite{BorisovGunnells}.
\begin{lemma}
\label{lemma3.15}
\begin{enumerate}
\item
Let $\pointP \in \El - \{\Pzero\}$.  Then the Laurent expansion of the
logarithmic differential $d\fP/\fP$ begins with
\begin{equation}
\label{equation3.20.5}
d\fP/\fP = t^{-1}[-1 + \lambdaP t - \xP t^2 + \yP t^3 + \cdots]dt.
\end{equation}
(This is the algebraic analog of~\eqref{equation2.19.5}, taking into
account \eqref{equation3.5}, \eqref{equation3.6},
and~\eqref{equation3.15.5}.)
We deduce the following equations, which over $\C$ can also be seen
from~\eqref{equation2.21}:
\begin{equation}
\label{equation3.21}
\xP = \lambdaP^2 - 2\muP, \qquad
\yP = 3\nuP - 3\muP\lambdaP + \lambdaP^3.
\end{equation}
\item
Let $\divD = (\pointP) + (\pointQ) + (\pointR)$ be as usual a divisor
supported on $\El - \{\Pzero\}$ with $\oplus \divD = \Pzero$.  Then
\begin{equation}
\label{equation3.22}
\lambdaP\lambdaQ + \lambdaQ\lambdaR + \lambdaP\lambdaR
 + \muP + \muQ + \muR = 0.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
For \eqref{equation3.20.5} and~\eqref{equation3.21}, consider the
meromorphic differential form $d\fP/\fP$ on $\E$.  Recall that
$\fP = (\thef_{\myl(\pointP) - \myl(\Pzero)})^{1/\myl}$ exists in
$\Ohat$ but is not a meromorphic function on $\E$; however, its
logarithmic differential is globally defined since
$\thef_{\myl(\pointP) - \myl(\Pzero)}$ is a global meromorphic
function on $\E$, and $d\fP/\fP$ has simple poles at each of $\Pzero$
and $\pointP$, with residues $-1$ and $1$, respectively.  Now use the
fact that the sum of the residues of the global meromorphic
differential $x\,d\fP/\fP$ (respectively, $y\,d\fP/\fP$) at all points
of $\E(\fieldbigKbar)$ is zero.  Taking into account the fact that $x
= t^{-2}(1+O(t^4))$ and $y = -t^{-3}(1+O(t^4))$, this
yields the coefficients $\xP$ and $\yP$ in~\eqref{equation3.20.5}.
On the other hand, we can directly compute the logarithmic
differential of $\fP = t^{-1}(1+\lambdaP t + \muP t^2 + \nuP t^3 +
\cdots)$, and this yields the coefficient $\lambdaP$
in~\eqref{equation3.20.5}, as well as~\eqref{equation3.21}.
Finally, to see~\eqref{equation3.22}, combine the
equations $\xP = \lambdaP^2 - 2\muP$ for $\pointP$, $\pointQ$, and
$\pointR$ with~\eqref{equation3.14.3}.
\end{proof}

\section{Relations involving the Weil pairing and Hecke operators}
\label{section4}

In this section, we prove deeper algebraic relations between the
modular forms $\lambdaDj$ than those in Lemma~\ref{lemma3.15}.  The
first few relations owe their existence to the Weil pairing on the
$\myl$-torsion group $\El$ of our elliptic curve.  Others are related
to the action of the full Hecke algebra of $\Gammal$ on modular forms
in $\ringRl$.  We eventually obtain enough relations to be able to
show in essence that, over $\C$, the weight $2$ and~$3$ parts of
$\ringRl$ are stable under the action of the Hecke algebra.
(Actually, in the case of weight~$3$ we obtain only a partial result
at this stage of the proof.)
We use this in Section~\ref{section5} to conclude over $\C$ that
the ring $\ringRl$ contains all modular forms of weights $2$ and
above.  This of course implies Hecke stability in all weights, and
supersedes the previous result.  Thus the only modular forms
that do not appear in $\ringRl$ are the cusp forms of weight~$1$; all
other modular forms of all weights are expressible as polynomials in
the $\lambdaP$, or equivalently as polynomials in the
$\thelambda_{(\pointP)+(\pointQ)+(\pointR)}$ which are slopes of lines
through torsion points of the Weierstrass model of $\E$.

The overall shape of the formulas giving the action of the Hecke
operators is similar to the results in the articles of Borisov and
Gunnells~\cite{BorisovGunnells,BorisovGunnellsNonvanishing,BorisovGunnellsHigherWeight}.
The treatment in those articles concerns only the group
$\Gamma_1(\myl)$, and proceeds via $q$-expansions and periods of
modular forms (the reader is referred also to~\cite{Pasol}).  Our
formulation in terms of $\Gammal$ involves neither of the above
techniques, but focuses instead on the modular parametrization given
by the modular curve.  We hope to treat some of the connections
between this article and those previous articles in later work; it
would also be desirable to understand the Hecke action better by
directly relating our relations coming from Laurent expansions of
elliptic functions to the geometry of toric varieties used in
\cite{BorisovGunnells}.

In order to introduce the Weil pairing on $\El$, we also need to
discuss pullbacks (i.e., composition) of elements $\Ohat$ by the
multiplication map $[n]:\E\to\E$; our main concern is to define the
element $\fQ\compose[n]\in \Ohat$, in the sense of controlling its
algebraic Laurent expansion in terms of $t$.  This can be done
entirely inside the formal group, since we have an expansion of the form
$t \compose [n] = nt + 2at^5(n-n^5)/5 + O(t^7) \in
\ringR_{1,\Z}[[t]]$, so we can formally obtain
$\fQ\compose[n]
  = n^{-1}t^{-1}(1 + \lambdaQ nt + \cdots)$.
At the same time, we can identify $\fQ \compose [n]$ by its
formal zeros and poles as below, and normalize it by a constant factor
so that its expansion begins with $n^{-1}t^{-1}$.  We thus obtain the
first part of the following definition.

\begin{definition}
\label{definition4.1}
\begin{enumerate}
\item
Let $\pointQ \in \El - \{\Pzero\}$ and let $1 \leq n \in \Z$, with $n$
invertible in $\fieldk$.  Choose
a point $\pointQ' \in \E[n\myl]$ such that $[n]\pointQ' = \pointQ$.
Then define the element $\fQ \compose [n] := n^{-1}\fD \in \Ohat$, where
$\divD
   = \sum_{\pointT\in\E[n]} (\pointQ' \oplus \pointT)
       - \sum_{\pointT \in \E[n]}(\pointT)$.
Here $\fQ \compose[n]$ is usually not an element of the function field
of $\E$, but we have the Laurent expansion
\begin{equation}
\label{equation4.1}
\fQ \compose [n]
   = n^{-1} t^{-1} (1 + \lambdaQ n t + \muQ n^2 t^2
       + \nuQ n^3 t^3 + O(t^4)),
\end{equation}
where the remaining terms after $t^3$ do \emph{not} follow the simple
initial pattern.
(In case $\pointQ = \Pzero$, we have $\thef_{\Pzero} =
\thef_{\Pzero}\compose[n] = 1$.)
\item
In the special case $n = \myl$, we introduce the notation
$\gQ = \fQ \compose[\myl]$.  In this setting, $\gQ$ is a genuine
element of $\fieldbigKl(\E)$, since the divisor $\divD$ of part~(1) is
now principal.
\item
The Weil pairing $e_\myl:\El\times\El \to \boldmu_\myl$ is given (as
usual) by the behavior of the functions $\gQ$ under translation by
elements of $\El$: namely,
\begin{equation}
\label{equation4.2}
\gQ(\pointP \oplus \pointR) = e_\myl(\pointQ,\pointR)
\gQ(\pointP), \text{ where } \pointQ, \pointR \in \El \text{ and }
\pointP \in  \E(\fieldbigKbar).
\end{equation}
\end{enumerate}
\end{definition}
\begin{remark}
\label{remark4.2}
If $\fieldk=\C$, consider the case when
$\pointQ = \pointP_{1/\myl}$ and $\pointR = \pointP_{\tau/\myl}$.
One can then show that our normalization gives
$e_\myl(\pointP_{1/\myl},\pointP_{\tau/\myl}) = e^{2\pi i/\myl}$.
(The easiest way to do this calculation is to avoid the Weierstrass
$\sigma$-function; instead, begin by showing that
$\theg_{\pointP_{1/\myl}}(z) = C \cdot \vartheta(\myl
z-1/\myl)/\vartheta(\myl z)$ for some nonzero constant $C$, where
$\vartheta = \vartheta_{11}$.)
\end{remark}

We are now ready for the relations arising from the Weil pairing.
In weight~$1$, they imply a subtle symmetry between the
$\{\lambdaP\}$, essentially a duality under the Fourier transform on
$\El$ induced by the pairing $e_\myl$.  When $\fieldk = \C$, this
subtle symmetry motivates Hecke's result that the dimension of the
space of Eisenstein series of weight~$1$ on $\Gammal$ is half the
number of cusps of $X(\myl)$ (see the end of Section~2
of~\cite{HeckeEisenstein}).  This symmetry is usually
expressed in terms of $q$-expansions of weight~$1$ Eisenstein series;
see the second identity at the beginning of Section~7
of~\cite{HeckeZurTheorie}, which is also derived in Sections 3.4
and~3.5 of~\cite{Katz}.

\begin{proposition}
\label{proposition4.3}
The following identities hold for all $\pointR \in \El$, where we use
the conventions of Remark~\ref{remark3.4} (thus the sums over
$\pointQ$ below are unchanged if we sum instead over
$\pointQ \in \El - \{\Pzero\}$):
\begin{equation}
\label{equation4.3}
\begin{split}
\lambdaR &=
 \frac{-1}{\myl} \sum_{\pointQ \in \El}
                           \lambdaQ e_\myl(\pointQ,\pointR),\\
\xR &= - \sum_{\pointQ \in \El}
                           \muQ e_\myl(\pointQ,\pointR),\\
\yR &= -\myl \sum_{\pointQ \in \El}
                           \nuQ e_\myl(\pointQ,\pointR).\\
\end{split}
\end{equation}
By Fourier inversion on the finite group $\El$, we obtain
from~\eqref{equation4.3} the identities
\begin{equation}
\label{equation4.4}
 \muR = \frac{-1}{\myl^2} \sum_{\pointQ \in \El}
                             \xQ e_\myl(\pointQ,\pointR),
\qquad\qquad
 \nuR = \frac{-1}{\myl^3} \sum_{\pointQ \in \El}
                             \yQ e_\myl(\pointQ,\pointR).
\end{equation}
\end{proposition}
\begin{proof}
Let $\pointQ \in \El - \{\Pzero\}$, and consider the meromorphic
function $\gQ = \fQ \compose [\myl] \in \fieldbigKl(\E)$, whose
Laurent expansion is of the form~\eqref{equation4.1}.  Define the global
meromorphic differential form $\eta_\pointQ = g_\pointQ \omega$ on
$\E$, where $\omega = (1+O(t^4))dt$ is the invariant differential; the
only singularities of $\eta_\pointQ$ are simple poles at the points of
$\El$.  Now the residue of $\eta_\pointQ$ at $\Pzero$ is
$\myl^{-1}$, and~\eqref{equation4.2} says that
$\trstar \eta_\pointQ = e_\myl(\pointQ, \pointR) \eta_\pointQ$, where
$\tau_\pointR: \E \to \E$ is translation by $\pointR$.  Thus the
residue of $\eta_\pointQ$ at any $\pointR \in \El$ is
$\myl^{-1} e_\myl(\pointQ, \pointR)$.  Now define the differential
form $\eta = -\myl \sum_{\pointQ \in \El - \{\Pzero\}} \eta_\pointQ$.
We see that $\eta$ has simple poles at all the points of $\El$, and
that the residue of $\eta$ at $\Pzero$ is $-\myl^2 + 1$, while the
residue at $\pointR \in \El - \{\Pzero\}$ is $1$,
by the nondegeneracy of the Weil pairing.
More precisely, the series expansions of $\eta$ and $\trstar \eta$ for
$\pointR \neq \Pzero$ have the following form (the sums below are over
$\pointQ \in \El - \{\Pzero\}$):
\begin{equation}
\label{equation4.5}
\begin{split}
\eta &= t^{-1}\Bigl[(-\myl^2 + 1)
                      + \sum_\pointQ \lambdaQ \myl t
                      + \sum_\pointQ \muQ \myl^2 t^2
                      + \sum_\pointQ \nuQ \myl^3 t^3
                      + \cdots\Bigr] dt\\
       &\qquad = t^{-1} \Bigl[(-\myl^2 + 1) + O(t^4)\Bigr] dt,\\
\trstar \eta
  & = t^{-1}\Bigl[1
              - \sum_\pointQ \lambdaQ e_\myl(\pointQ, \pointR) \myl t
              - \sum_\pointQ \muQ e_\myl(\pointQ, \pointR) \myl^2 t^2
              - \sum_\pointQ \nuQ e_\myl(\pointQ, \pointR) \myl^3 t^3
              + \cdots \Bigr] dt.\\
\end{split}
\end{equation}
The second equality above in the expansion of $\eta$ follows
from~\eqref{equation3.12.6} (which incidentally
yields~\eqref{equation4.3} in the special case $\pointR = \Pzero$; see
however the upcoming footnote in this proof).  The expansion of
$\trstar \eta$  holds because
$\trstar \eta =
  - \myl \sum_\pointQ e_\myl(\pointQ, \pointR) \eta_\pointQ$.

We now relate the differential form $\eta$ to the function $\fD$,
corresponding to the divisor $\divD = \sum_{\pointQ \in \El} (\pointQ)$.
The divisor of $\fD$ is $(\fD)
   = \Bigl(\sum_{\pointQ \in \El} (\pointQ) \Bigr) - \myl^2 (\Pzero)
   = \Bigl( \sum_{\pointQ \in \El - \{\Pzero\}} (\pointQ) \Bigr)
        + (-\myl^2 + 1) (\Pzero)$.
Thus $\eta$ and $d\fD/\fD$ have poles at
the same locations, with the same residues.  We claim that in fact
$\eta = d\fD/\fD$, since the difference is not only everywhere
holomorphic, but also vanishes at $\Pzero$, by looking beyond the
first term in the Laurent expansions at $\Pzero$.  Indeed,
$\fD = \pm \myl^{-1}\psi_\myl(x,y)$ where $\psi_\myl$ is the $\myl$th
division polynomial, and hence $\fD$ has an $\ringR_1$-balanced
Laurent expansion of the form $\fD = t^{-\myl^2 + 1}(1 + O(t^4))$,
which implies that $d\fD/\fD = t^{-1}[(-\myl^2 + 1) + O(t^4)]$; on the
other hand, $\eta$ has a similar expansion by~\eqref{equation4.5}, and
our claim follows.\footnote{
The alert reader will note that we needed only the simple
result $\sum_\pointQ \lambdaQ = 0$ of~\eqref{equation3.12.6} to deduce
that $\eta = d\fD/\fD$.  The form of the expansion of $\fD$ then
allows us to conclude the identity $\sum_\pointQ \muQ = 0$ --- as well
as the simple identity $\sum_\pointQ \nuQ = 0$ --- thereby giving a
second way to complete the proof of~\eqref{equation3.12.6}, and hence
of~\eqref{equation4.3} when $\pointR = \Pzero$.
}

We now consider the translation of the identity $\eta = d\fD/\fD$ by
the point $\pointR$, when $\pointR \neq \Pzero$.  This gives us
$\trstar \eta = \trstar\bigl(d\fD/\fD \bigr)
              = d(\trstar \fD)/\trstar \fD$.
We shall compare the expansion of $\trstar \eta$
from~\eqref{equation4.5} to the expansion of the logarithmic
differential of $\trstar \fD$.  Comparing the locations of zeros and
poles, we see that
$\trstar \fD = C \cdot \fD \cdot (\thef_{(\ominus \pointR)})^{-\myl^2}$
for some nonzero constant $C$.  (Here $\thef_{(\ominus \pointR)}$ is not
a genuine meromorphic function on $\E$, but its $\myl$th power is, so
$(\thef_{\ominus \pointR})^{-\myl^2}$ is also a genuine meromorphic
function.)  We obtain that $d(\trstar \fD)/\trstar \fD =
d\fD/\fD - \myl^2 d\thef_{\ominus\pointR}/\thef_{\ominus\pointR}$.
However, from \eqref{equation3.20.5} and~\eqref{equation3.12.3}, we have
\begin{equation}
\label{equation4.6}
d\thef_{{\ominus\pointR}}/\thef_{{\ominus\pointR}}
 = t^{-1}[-1 - \lambdaR t - \xR t^2 - \yR t^3 + \cdots]dt.
\end{equation}
Combining all this and comparing the Laurent expansions, we
obtain~\eqref{equation4.3} as desired.  Equation~\eqref{equation4.4}
then follows immediately.
\end{proof}

The relations~\eqref{equation4.4}, when combined with
\eqref{equation3.5}, imply that the forms $\{\muP,\nuP\}$
are Eisenstein series of weights $2$ and~$3$, when $\fieldk = \C$.  It
will be useful for us to formalize this algebraically, while also
taking into account~\eqref{equation3.15.5}.

\begin{definition}
\label{definition4.4}
For $\wtj \in \{1,2,3\}$, we define the algebraic space $\Espace_\wtj$
of Eisenstein series of weight~$\wtj$ by
\begin{equation}
\label{equation4.7}
\Espace_1 = \linalgspan\{\lambdaP \mid \pointP \in \El\},
\qquad
\Espace_2 = \linalgspan\{\xP\},
\qquad
\Espace_3 = \linalgspan\{\yP\}.
\end{equation}
(If we wish to draw attention to the level $\myl$, we will write
$\Espace_\wtj^{\myl}$.)

We deduce from \eqref{equation4.4} and~\eqref{equation3.21} that for
all $\pointP \in \El$,
\begin{equation}
\label{equation4.8}
\muP, \lambdaP^2 \in \Espace_2,
\qquad
\nuP \in \Espace_3.
\end{equation}
From~\eqref{equation3.22}, we also obtain that for $\pointP, \pointQ,
\pointR \in \El$ with $\pointP \oplus \pointQ \oplus \pointR =
\Pzero$,
\begin{equation}
\label{equation4.9}
 \lambdaP \lambdaQ + \lambdaQ \lambdaR + \lambdaP \lambdaR \in \Espace_2.
\end{equation}
Note that in the above equation, the points $\pointP, \pointQ, \pointR$
are allowed to take the value $\Pzero$; for example, if $\pointQ =
\Pzero$, then $\lambdaR = - \lambdaP$, in which
case~\eqref{equation4.9} becomes the statement
$-\lambdaP^2 \in \Espace_2$ that we know from~\eqref{equation4.8}.
(The result that $\muP$ and $\lambdaP^2$ are Eisenstein series, as
well as the result~\eqref{equation4.9}, were already observed for
$\Gamma_1(\myl)$ in~\cite{BorisovGunnellsNonvanishing}).
\end{definition}

In our treatment of Hecke operators, we shall need the following
identities, which are related to the fact that the trace from
$\Gamma(n\myl)$ to $\Gamma(\myl)$ of an Eisenstein series on
$\Gamma(n\myl)$ is again an Eisenstein series.
\begin{lemma}
\label{lemma4.5}
Let $n \geq 1$ be invertible in $\fieldk$.  Let $\pointP \in
\Enl$ (typically, $\pointP \in \El$), and let $\pointT \in \En$.
Consider the modular forms
$\thelambda_{\pointP \oplus \pointT}$, $x_{\pointP \oplus \pointT}$, and
$y_{\pointP \oplus \pointT}$ on $\Gamma(n\myl)$.  We then have
\begin{equation}
\label{equation4.10}
\sum_{\pointT \in \En} \thelambda_{\pointP \oplus \pointT}
           = n \thelambda_{[n]\pointP},
\quad
\sum_{\pointT \in \En} x_{\pointP \oplus \pointT}
           = n^2 x_{[n]\pointP},
\quad
\sum_{\pointT \in \En} y_{\pointP \oplus \pointT}
           = n^3 y_{[n]\pointP}.
\end{equation}
We also have
\begin{equation}
\label{equation4.11}
\sum_{\pointT \in \En} \themu_{\pointP \oplus \pointT}
           =  \themu_{[n]\pointP},
\qquad
\sum_{\pointT \in \En} \thenu_{\pointP \oplus \pointT}
           = \frac{1}{n} \, \thenu_{[n]\pointP}.
\end{equation}
\end{lemma}
\begin{proof}
Over $\C$, equation~\eqref{equation4.10} is immediate from the
definition of $\Ej$ in \eqref{equation2.1} and~\eqref{equation2.2} ---
in verifying this, the reader should bear in mind that $\xP$ is a
difference between two $\Etwo$s.  Let us however give a proof in our
algebraic setting.  Now~\eqref{equation4.10} is trivial for
$\pointP = \Pzero$.  If $\pointP \neq \Pzero$, we begin by noting the
following identity, which is obtained by comparing zeros and poles, as
well as the leading coefficient of the Laurent expansion:
\begin{equation}
\label{equation4.12}
\thef_{[n]\pointP} \compose [n]
   = n^{-1}
     \Bigl(\prod_{\pointT \in \En}
               \thef_{\pointP \oplus \pointT}
     \Bigr) / \fD.
\end{equation}
Here $\divD = \sum_{\pointT \in \En} (\pointT)$ is the divisor
supported on the $n$-torsion points.  Hence the principal divisor of
$\fD$ is $(\fD) = \divD - n^2 (\Pzero)$, similarly to the proof of
Proposition~\ref{proposition4.3}; this also implies the expansion
$\fD = t^{-n^2+1}(1+O(t^4))$.  Now taking the logarithmic differential of
both sides of~\eqref{equation4.12} and comparing the first few
coefficients yields~\eqref{equation4.10}, as desired.

As for~\eqref{equation4.11}, we prove it using the Fourier duality of
Proposition~\ref{proposition4.3}.  (This approach also yields a
different proof of~\eqref{equation4.10}.)  For instance,
use~\eqref{equation4.4} to express each $\themu$ in the first sum
in~\eqref{equation4.11} in terms of an $x$.  This yields
\begin{equation}
\label{equation4.13}
\sum_{\pointT \in \En} \themu_{\pointP \oplus \pointT}
  = \sum_{\pointT \in \En} \frac{-1}{n^2\myl^2}
                   \sum_{A \in \Enl}
                        x_A e_{n\myl}(A,\pointP \oplus \pointT).
\end{equation}
Rearrange the sum as $\sum_A \sum_\pointT$, and use the property of
the Weil pairing
\begin{equation}
\label{equation4.14}
  A \in \Enl, \quad \pointT \in \En
  \implies
 e_{n\myl}(A,\pointT) = e_n([\myl]A,\pointT)
\end{equation}
to conclude that the only surviving terms are those when $[\myl]A =
\Pzero$, in other words, for $A \in \El$.  Thus we obtain
\begin{equation}
\label{equation4.15}
\sum_{\pointT \in \En} \themu_{\pointP \oplus \pointT}
  = \frac{-n^2}{n^2\myl^2} \sum_{A \in \El} x_A e_{n\myl}(A,\pointP)
  = \frac{-1}{\myl^2} \sum_{A \in \El} x_A e_{\myl}(A,[n]\pointP),
\end{equation}
where the last equality is analogous to~\eqref{equation4.14}.  This
implies the first part of~\eqref{equation4.11}.  The second part,
involving $\thenu$, is proved similarly.
\end{proof}

We are now ready for the main ingredient in the proof that the
degree~$2$ part of $\ringRl$ is stable under the Hecke algebra.  This proof
involves an interesting induction on the level.  One starts with forms on
$\Gamma(n\myl)$, ``raises the level'' to rewrite them in terms of forms of
higher level $\Gamma(sn\myl)$ with $s < n$, then ``lowers the level''
back to level $\Gamma(s\myl)$.  Repeating this process reduces the value of
$s$, and one eventually reaches $s = 0$, which can be dealt with using
Lemma~\ref{lemma4.5}.

\begin{proposition}
\label{proposition4.6}
Let $n \geq 1$ and assume that $n!$ is invertible in $\fieldk$.  Let
$A,B \in \Enl$ (as before, typically $A,B \in \El$), and let $s \in
\Z$.  Then
\begin{equation}
\label{equation4.16}
\begin{split}
&\sum_{T \in \En}
   \thelambda_{A \oplus T}
   \thelambda_{B \ominus [s]T} \\
&\quad=
\bigl( \text{a linear combination of terms of the form }
         \thelambda_{[a]A \oplus [b]B} \thelambda_{[c]A \oplus [d]B}
\bigr)\\
&\qquad+
\bigl(
\text{ an element of } \Espace^{n!\myl}_2
\bigr),\\
\end{split}
\end{equation}
where the linear combination above is over finitely many
$(a,b,c,d) \in \Z^4$ satisfying
\begin{equation}
\label{equation4.17}
\det \twomatr{a}{b}{c}{d} = \pm n,
\qquad\qquad
a - sb \equiv c - sd  \equiv 0 \pmod{n}.
\end{equation}
\end{proposition}
\begin{proof}
The proof is by induction on $n$, the case $n=1$ (so $T =
\Pzero$) being trivial.  Note that the value of $s$ only matters
modulo $n$, so we henceforth assume that $0 \leq s < n$.  If $s = 0$,
then the sum over $T$ is $n \thelambda_{[n]A} \thelambda_B$
by~\eqref{equation4.10}, so we are done.  In general, we shall invoke
an inductive step analogous to the Euclidean algorithm,
reducing~\eqref{equation4.16} for the pair $(n,s)$ to the analogous
statement for $(s,n)$, which amounts to the same as $(s, n \bmod s)$.
To this end, choose a point $B' \in \E[sn\myl]$ for which
$[s]B' = B$.  We then see from~\eqref{equation4.10} that
\begin{equation}
\label{equation4.18}
\thelambda_{B \ominus [s]T}
= s^{-1} \sum_{U \in \E[s]} \thelambda_{B' \ominus T \oplus U}.
\end{equation}
Hence, up to the factor $s^{-1}$, our sum in~\eqref{equation4.16}
becomes
\begin{equation}
\label{equation4.19}
\begin{split}
\sum_{T \in \En, \, U \in \E[s]} &\thelambda_{A \oplus T}
                              \thelambda_{B' \ominus T \oplus U}\\
\equiv
\sum_{T,U} &\thelambda_{A \oplus T} \thelambda_{A \oplus B' \oplus U}
  - \sum_{T,U} \thelambda_{\ominus B' \oplus T \ominus U}
             \thelambda_{A \oplus B' \oplus U}
                      \quad \pmod{\Espace_2^{sn\myl}},\\
\end{split}
\end{equation}
where the congruence is obtained from~\eqref{equation4.9} with
$\pointP = A \oplus T$, $\pointQ = B' \ominus T \oplus U$,
and $\pointR = \ominus A \ominus B' \ominus U$; we have also
used~\eqref{equation3.12.3}.  Now the first sum on the right hand side
of equation~\eqref{equation4.19} is a constant (namely, $ns$) times
$\thelambda_{[n]A} \thelambda_{[s](A \oplus B')}
= \thelambda_{[n]A} \thelambda_{[s]A \oplus B}$, which has the desired
form.  On the other hand, the second sum on the right hand side can be
summed first over all $T \in \En$, which by~\eqref{equation4.10}
yields a constant times
\begin{equation}
\label{equation4.20}
\sum_{U \in \E[s]}
 \thelambda_{[-n]B' \ominus [n]U} \thelambda_{A \oplus B' \oplus U}.
\end{equation}
By the inductive hypothesis, the above sum is congruent modulo
$\Espace_2^{s! n\myl}$ to a linear combination of terms of the form
\begin{equation}
\label{equation4.21}
 \thelambda_{[a'](A \oplus B') \oplus [-nb'] B'}
 \thelambda_{[c'](A \oplus B') \oplus [-nd'] B'}
=
 \thelambda_{[a']A \oplus [\frac{a'-nb'}{s}]B}
 \thelambda_{[c']A \oplus [\frac{c'-nd'}{s}]B}
\end{equation}
where $(a',b',c',d')$ satisfy~\eqref{equation4.17} with the roles of
$s$ and $n$ interchanged; in particular, $\frac{a'-nb'}{s},
\frac{c'-nd'}{s} \in \Z$, and we get that each term is of the form
$\thelambda_{[a]A \oplus [b]B} \thelambda_{[c]A \oplus [d]B}$,
satisfying the original requirements of~\eqref{equation4.17}.
Finally, we remark that $\Espace_2^{sn\myl}$ and
$\Espace_2^{s!n\myl}$ are both subspaces of $\Espace_2^{n!\myl}$.
\end{proof}
\begin{remark}
\label{remark4.7}
The element of $\Espace_2^{n!\myl}$ above actually belongs to
$\Espace_2^{n\myl}$, but we shall not prove this in our algebraic
context; it is obvious over $\C$, since it is an Eisenstein series
with level $n!\myl$ that happens to transform under $\Gamma(n\myl)$.
(Similarly, if $A,B \in \El$, then the element of $\Espace_2$ above actually
belongs to $\Espace_2^\myl$.) It is possible to specify this element
more precisely by applying~\eqref{equation3.22} instead
of~\eqref{equation4.9} in the above proof.  Typically, this yields an
element of $\Espace_2$ that is a linear combination of terms
$\themu_{[a]A+[b]B}$ where $a-sb \equiv 0 \pmod{n}$, after one also
invokes~\eqref{equation4.11}.  However, one must be careful not to
apply~\eqref{equation3.22} when one of the torsion points $\pointP,
\pointQ, \pointR$ is $\Pzero$.

On another topic, we observe that the linear combination
in~\eqref{equation4.16} is $\Z$-linear, and the coefficients are all
divisible by $n$.  This we leave to the reader.
\end{remark}

We need a few more standard observations before we prove the Hecke
stability of the weight~$2$ part of $\ringRl$.  Starting from this
point, we shall for convenience work exclusively over $\C$; also,
since $\ringR_1$ and $\ringR_2$ are the full rings of modular forms on
$\Gamma(1)$ and $\Gamma(2)$, we can restrict to $\myl \geq 3$.  We use
the standard notation for the spaces of cusp forms and modular forms
of weight~$\wtj$ on a congruence subgroup $\Gamma$:
\begin{equation}
\label{equation4.22}
  \cuspforms_\wtj(\Gamma) = \{\text{cusp forms}\}
\subset
  \modforms_\wtj(\Gamma) = \{\text{holomorphic modular forms over }\C \}.
\end{equation}
We also make use of the usual group action
$f \mapsto f |_{\wtj} \gamma$ of $\gamma \in \Gamma(1)$ on the space
$\modforms_\wtj(\Gammal)$.  This action of course preserves the spaces
of Eisenstein series and cusp forms in all weights.  We
interchangeably view $\gamma$ as an element of $\Gamma(1)$ or of
$\Gamma(1)/\Gammal \isomorphic SL(2,\Z/\myl\Z)$.
This group also acts on the torsion group $\El$ while preserving the
Weil pairing, and on our ring $\ringRl$ via ring isomorphisms.  Indeed,
for $\gamma \in SL(2,\Z/\myl \Z)$, we have
$\pointP \mapsto \pointP \cdot \gamma$, where
\begin{equation}
\label{equation4.23}
\begin{split}
\zP = \frac{\torsi \tau + \torsj}{\myl}
  &\implies
z_{\pointP \cdot \gamma} = \frac{\torsi' \tau + \torsj'}{\myl}
\text{ with }
(\torsi' \quad \torsj') = (\torsi \quad \torsj) \gamma,
\\
\pointP \in \El
  &\implies
\lambdaP |_1 \gamma = \thelambda_{\pointP \cdot \gamma}.
\\
\end{split}
\end{equation}

We briefly review the well-known interpretation of Hecke operators in
terms of a trace between congruence subgroups.  Given a Hecke operator
described as a double coset $\Gammal \alpha \Gammal$ with
$\alpha \in GL^+(2,\Q)$, we can harmlessly multiply $\alpha$ by a
scalar to obtain a primitive integral matrix; then composing this
double coset on the left and right by the action of elements
$\gamma_1, \gamma_2 \in \Gamma(1)$ allows us to assume without loss of
generality that $\alpha = \stwomatr{n}{}{}{1}$ for some $n \geq 1$.
We then have, for $f(\tau) \in \modforms_\wtj(\Gammal)$:
\begin{equation}
\label{equation4.24}
f |_\wtj \Gammal \twomatr{n}{}{}{1} \Gammal
 = C \sum_{\gamma \in \Gamma(n\myl)\backslash\Gammal}
                \bigl(f(n\tau) \bigr) |_\wtj \gamma,
\end{equation}
where $C = C_{n,\myl,\wtj}$ is a suitable normalizing constant.  Note
that if $f(\tau) \in \ringRl$, then $f(n\tau) \in \ringR_{n\myl}$;
indeed, the map $f \mapsto f(n\tau)$ respects multiplication of forms,
so it is enough to check the above statement for the weight~$1$
Eisenstein series $\lambdaP = -\Eone(\tau,\pointP)$ that generate
$\ringRl$.  This is just the identity
\begin{equation}
\label{equation4.25}
\Eone(n\tau, \frac{\torsi \tau + \torsj}{\myl})
  = n^{-1}
    \sum_{k \bmod n}
        \Eone(\tau, \frac{\torsi n\tau + \torsj + k\myl}{n\myl}).
\end{equation}
The sum over representatives $\gamma \in \Gamma(n\myl)\backslash\Gammal$
in~\eqref{equation4.24} is a trace from
$\modforms_\wtj(\Gamma(n\myl))$ to $\modforms_\wtj(\Gamma(\myl))$, and
we shall henceforth work with it instead of with double cosets.

With these preliminaries out of the way, we can state and prove our
result for weight~$2$, which will be superseded later when we show
that $\ringRl$ contains all of $\modforms_2(\Gammal)$.

\begin{proposition}
\label{proposition4.8}
Let $\fieldk = \C$.  Then the trace of a weight~$2$ element of
$\ringR_{n\myl}$ from $\modforms_2(\Gamma(n\myl))$ to
$\modforms_2(\Gamma(\myl))$ actually belongs to $\ringRl$.
(\emph{A priori,} this trace merely belongs to
$\ringR_{n\myl} \intersect \modforms_2(\Gammal)$.)
\end{proposition}
\begin{corollary}
\label{corollary4.9}
Over $\C$, the weight~$2$ part of $\ringRl$ is stable under the action
of the Hecke algebra for $\Gammal$.
\end{corollary}
\begin{proof}[Proof of Proposition~\ref{proposition4.8}]
By the observation immediately following Remark~\ref{remark4.7}, we
can assume that $\myl \geq 3$; it is enough to show in that case
that the trace of any product $\lambdaP \lambdaQ = \Eone(\tau,\pointP)
\Eone(\tau,\pointQ)$ with $\pointP, \pointQ \in \Enl - \{\Pzero\}$
belongs to $\ringRl$.  Now $\ringRl$ already contains all the
Eisenstein series on $\Gammal$ in weight~$2$ (indeed, in
all weights~$\wtj$, by Theorem~\ref{theorem3.9}), so we can work
modulo Eisenstein series in our proof.  As we mentioned in
Remark~\ref{remark4.7}, this can be done even if we encounter
Eisenstein series of higher level in some intermediate steps.
Furthermore, the trace down from level $n\myl$ to level $\myl$ can be
done one prime factor at a time, so we may harmlessly assume that $n$
is a prime number.  There are two cases to consider: (i) $n$ is prime
and $n \notdividing \myl$, and (ii) $n$ is prime and $n | \myl$.

In case (i), we take a direct sum decomposition $\Enl = \El \bigoplus
\En$, and note that $\Gamma(n\myl)\backslash\Gamma(\myl)$ is isomorphic
to $SL(2,\Z/n\Z)$ and that it affects only the $\En$ part.  Decompose
$\pointP = A \oplus T_0$ and $\pointQ = B \oplus U_0$, with $A,B \in
\El$ and $T_0, U_0 \in \En$.  We may suppose that one of $\{T_0, U_0\}$
--- let us say, $T_0$ --- is not equal to $\Pzero$, since otherwise
$\lambdaP \lambdaQ \in \ringRl$ already.  Then there are two
subcases: (i.a) there exists $s \in \Z/n\Z$ ($s=0$ is allowed) such that
$U_0 = [s]T_0$, and (i.b) $\{T_0, U_0\}$ are a basis for $\E[n]$.
In subcase (i.a), the trace of
$\thelambda_{A \oplus T_0} \thelambda_{B \oplus [s] T_0}$ is equal to a
multiple of
$\sum_{T \in \En - \{\Pzero\}}
    \thelambda_{A \oplus T} \thelambda_{B \oplus [s]T}$.
By Proposition~\ref{proposition4.6}, this is congruent modulo
$\Espace_2$ to an element of $\ringRl$ (the ``missing term'' in the sum,
corresponding to $T = \Pzero$, is $\thelambda_A \thelambda_B$, which
already belongs to $\ringRl$).  Hence the trace itself
belongs to $\ringRl$, as we have observed before.

In subcase (i.b), let
$\zeta = e_n(T_0, U_0)$, which is a primitive $n$th root of unity.  Then
the trace that we wish to compute is
\begin{equation}
\label{equation4.26}
 \sum_{\substack{T,U \in \En \\ e_n(T,U) = \zeta}}
    \thelambda_{A \oplus T} \thelambda_{B \oplus U}
=  \frac{-1}{n\myl}
   \sum_{\substack{T,U \in \En \\ e_n(T,U) = \zeta\\ V \in \En, C \in \El}}
    \thelambda_{A \oplus T} \thelambda_{C \oplus V}
      e_{n\myl}(C \oplus V, B \oplus U).
\end{equation}
Here we have invoked~\eqref{equation4.3}, where we let $C \oplus V$ range
over the elements of $\Enl$.  Now
$e_{n\myl}(C \oplus V, B \oplus U) = e_{\myl}([n]C,B) e_{n}([\myl]V,U)$, so
the quantity in~\eqref{equation4.26} is a linear combination of terms
(indexed by $C$) of the form
\begin{equation}
\label{equation4.27}
  \sum_{\substack{T,U,V \in \En \\ e_n(T,U) = \zeta}}
     \thelambda_{A \oplus T} \thelambda_{C \oplus V}
     e_{n}([\myl]V,U).
\end{equation}
For fixed $T$ and $V$, we must hence study the sum over those $U$ for
which $e_n(T,U) = \zeta$.  Such a $U$ exists if and only if $T \neq \Pzero$
(here we use the facts that $n$ is prime and $\zeta \neq 1$), in which case
$U$ ranges over the set of torsion points
$\{U_T \oplus [t]T \mid t \in \Z/n\Z\}$ for some particular choice of $U_T$
(depending on $T$) with $e_n(T,U_T) = \zeta$.  The sum over $U$ thus
contains a factor $\sum_{t \in \Z/n\Z} e_n([\myl]V,U_T \oplus [t]T)$, which
vanishes unless $V$ belongs to the cyclic subgroup generated by $T$ (recall
that $\myl$ is relatively prime to $n$).  We obtain
that~\eqref{equation4.27} is equal to
\begin{equation}
\label{equation4.28}
\begin{split}
  &\sum_{T \in \En - \{\Pzero\}}\quad
  \sum_{V \text{ of the form } V = [s]T}
          \thelambda_{A \oplus T} \thelambda_{C \oplus [s]T}
           \cdot n e_n([\myl][s]T,U_T)\\
 &\quad= \sum_{s \in \Z/n\Z}
           n \zeta^{\myl s}
         \sum_{T \neq \Pzero}
            \thelambda_{A \oplus T} \thelambda_{C \oplus [s]T},\\
\end{split}
\end{equation}
which brings us back to subcase (i.a).

We now turn to case (ii).  We write $\myl = Ln^k$ with
$n \notdividing L$ and $k \geq 1$, and decompose
$P = A \oplus T_0$ and $Q = B \oplus U_0$ with $A,B \in \E[L]$
and $T_0,U_0 \in \E[n^k]$.  We wish to compute a trace using
representatives for $\Gamma(Ln^{k+1})\backslash\Gamma(Ln^k)$.  Such
representatives again do not affect $A$ or $B$, and their action on $T_0$
and $U_0$ can be described by matrices in $SL(2,\Z/n^{k+1}\Z)$ that are
congruent to the identity modulo $n^k$; thus such matrices have the form
\begin{equation}
\label{equation4.29}
\twomatr{1+n^k \alpha}{n^k \beta}{n^k\gamma}{1-n^k\alpha}
= I + n^k M,
\qquad
M = \twomatr{\alpha}{\beta}{\gamma}{-\alpha}
               \in M_2^{\text{trace } 0}(\Z/n\Z).
\end{equation}
The reader should note that we view the entries of $M$ as being in
$\Z/n\Z$, but that multiplying them by $n^k$ yields elements of
$n^k\Z/n^{k+1}\Z$, not zero.  We also point out that we shall feel free to
use other bases for $\E[n^{k+1}] \isomorphic (\Z/n^{k+1}\Z)^2$ than the
standard basis $\{ \pointP_{\tau/n^{k+1}}, \pointP_{1/n^{k+1}} \}$; even if
the change of basis does not have determinant~$1$ (and hence changes the
Weil pairing), our description of $M$ in~\eqref{equation4.29} remains
valid.  Let us write $\hat{T_0} = [n^k]T_0$ and $\hat{U_0} = [n^k]U_0$.  We
have $\hat{T_0},\hat{U_0} \in \E[n]$, and the trace that we wish to compute
is then
\begin{equation}
\label{equation4.30}
\sum_{M \in M_2^{\text{trace } 0}(\Z/n\Z)}
  \thelambda_{A \oplus T_0 \oplus (\hat{T_0} \cdot M)}
  \thelambda_{B \oplus U_0 \oplus (\hat{U_0} \cdot M)},
\end{equation}
where the action of $M$ is analogous to that in~\eqref{equation4.23}.  Once
again we may suppose that $\hat{T_0}$ and $\hat{U_0}$ are not both $\Pzero$
(otherwise, $T_0, U_0 \in \E[n^k]$ and we are already in $\ringR_{n^k L} =
\ringRl$), and that without loss of generality $\hat{T_0} \neq \Pzero$.  We
face analogous subcases: (ii.a) there exists $s \in \Z/n\Z$ such that
$\hat{U_0} = [s] \hat{T_0}$ and (ii.b) $\{\hat{T_0}, \hat{U_0}\}$ are a
basis for $\E[n]$.

In subcase (ii.a), the points $\hat{T_0}\cdot M$ cover
all of $\E[n]$ (including $\Pzero$), each point $\hat{T} \in \E[n]$
occurring $n$ times.  (The easiest way to see this is to write $M$ with
respect to a basis for $\E[n]$ that includes $\hat{T_0}$.)  Hence we obtain
that~\eqref{equation4.30} is a multiple of
\begin{equation}
\label{equation4.31}
 \sum_{\hat{T} \in \E[n]}
   \thelambda_{A \oplus T_0 \oplus \hat{T}}
   \thelambda_{B \oplus U_0 \oplus [s]\hat{T}}.
\end{equation}
Modulo Eisenstein series, this last expression is a linear combination of
terms of the form
\begin{equation}
\label{equation4.32}
 \thelambda_{[a](A \oplus T_0) \oplus [b](B \oplus U_0)}
 \thelambda_{[c](A \oplus T_0) \oplus [d](B \oplus U_0)},
\text{ for }
a + sb \equiv c + sd  \equiv 0 \pmod{n}.
\end{equation}
We observe that $[n^k]([a]T_0 \oplus [b] U_0)
 = [a]\hat{T_0} \oplus [b]\hat{U_0} = [a + sb] \hat{T_0} = \Pzero$, whereas
$[a]A \oplus [b]B \in \E[L]$, so the first factor in~\eqref{equation4.32}
involves torsion points in $\E[n^k L] = \El$; an analogous statement holds
 for the second factor, and we obtain an element of $\ringRl$, as desired.

In subcase (ii.b), we write $M$ in terms of the basis
$\{\hat{T_0}, \hat{U_0}\}$, and obtain that we wish to study
\begin{equation}
\label{equation4.33}
 \sum_{\alpha,\beta,\gamma\in\Z/n\Z}
\thelambda_{A \oplus T_0
                 \oplus [\alpha] \hat{T_0} \oplus [\beta] \hat{U_0}}
\thelambda_{B \oplus U_0
                 \oplus [\gamma] \hat{T_0} \oplus [-\alpha] \hat{U_0}}.
\end{equation}
Similarly to subcase (i.b), we rewrite the factor
$\thelambda_{B \oplus U_0
                 \oplus [\gamma] \hat{T_0} \oplus [-\alpha] \hat{U_0}}$
in terms of the Weil pairing and $\thelambda_{C \oplus V}$,
for $C \in \E[L]$ and $V \in \E[n^{k+1}]$.  We obtain a linear
combination of terms of the following form (here the triples
$(\alpha,\beta,\gamma) \in (\Z/n\Z)^3$ are analogous to the pairs
$\{(T,U) \in \E[n]\times\E[n] \mid e_n(T,U) = \zeta\}$
of~\eqref{equation4.27}):
\begin{equation}
\label{equation4.34}
\sum_{\substack{\alpha,\beta,\gamma\in\Z/n\Z \\ V \in \E[n^{k+1}]}}
  \thelambda_{A \oplus T_0
                 \oplus [\alpha] \hat{T_0} \oplus [\beta] \hat{U_0}}
  \thelambda_{C \oplus V}
  e_{n^{k+1}}(
       [L]V,
       U_0 \oplus [\gamma] \hat{T_0} \oplus [-\alpha] \hat{U_0}
             ).
\end{equation}
Now perform the sum over $\gamma$ first: the inner factor
$\sum_\gamma e_{n^{k+1}}([L]V, [\gamma] \hat{T_0})$
can be rewritten as
$\sum_\gamma e_{n}\Bigl([L]([n^k]V), [\gamma] \hat{T_0}\Bigr)$
with $[n^k]V \in \E[n]$.  Thus, as in case (i.b), the only terms that
survive are those where $[n^k]V = [s] \hat{T_0} = [sn^k] T_0$ for some
$s \in \Z/n\Z$.  Equivalently, we can write $V = [s] T_0 \oplus W$ for
some $s$ and for some $W \in \E[n^k]$.  In such a situation, we have
$e_{n^{k+1}}([L]V, [-\alpha] \hat{U_0})
 = e_n([n^k L]V, [-\alpha]\hat{U_0})
 = e_n([Ls] \hat{T_0},[-\alpha]\hat{U_0})
 = e_n\Bigl([-Ls]([\alpha]\hat{T_0} \oplus [\beta]\hat{U_0}),
            \hat{U_0}\Bigr)$.
At this point, we note that as $\alpha$ and $\beta$ vary, the point
$\hat{T} := [\alpha]\hat{T_0} \oplus [\beta]\hat{U_0}$ runs over all
points of $\E[n]$.
Putting all this together, we obtain that our expression is
a linear combination of terms (indexed by $C$ and $s$) of the form
\begin{equation}
\label{equation4.35}
\sum_{\substack{\hat{T} \in \E[n] \\ W \in \E[n^k]}}
  \thelambda_{A \oplus T_0
                 \oplus \hat{T}}
  \thelambda_{C \oplus [s] T_0 \oplus W}
  e_{n^{k+1}}([Ls]T_0 \oplus [L]W, U_0)
  e_n([-Ls]\hat{T}, \hat{U_0}).
\end{equation}
The above expression contains a common factor
$e_{n^{k+1}}([Ls]T_0,U_0)$.  We can also rewrite
\begin{equation}
\label{equation4.36}
e_{n^{k+1}}([L]W, U_0) e_n([-Ls]\hat{T}, \hat{U_0})
= e_{n^{k+1}}\Bigl([L](W \ominus [s]\hat{T}), U_0\Bigr).
\end{equation}
We define $X := W \ominus [s]\hat{T} \in \E[n^k]$; this yields
a bijection from the set $\E[n]\times \E[n^k]$ to itself,
sending the pair $(\hat{T},W)$ to the pair $(\hat{T},X)$.
Our expression~\eqref{equation4.35} then becomes
\begin{equation}
\label{equation4.37}
e_{n^{k+1}}([Ls]T_0,U_0)
\sum_{\substack{\hat{T} \in \E[n] \\ X \in \E[n^k]}}
  \thelambda_{A \oplus T_0
                 \oplus \hat{T}}
  \thelambda_{C \oplus [s] T_0 \oplus X \oplus [s]\hat{T}}
 e_{n^{k+1}}([L]X, U_0).
\end{equation}
Writing the sum in the order $\sum_X \sum_{\hat{T}}$, we see that the
inner sum over $\hat{T}$ is now exactly analogous
to~\eqref{equation4.31}.  In our setting, $C$ and $[s]T_0 \oplus X$
play the roles of $B$ and $U_0$ from~\eqref{equation4.31}\footnote{
Recall that $U_0$ in~\eqref{equation4.31} had the property that
$[n^k]U_0 = \hat{U_0} = [s]\hat{T_0}$.  The analogous observation in
our setting is that $[n^k]([s]T_0 \oplus X) = [s]\hat{T_0}$.
}, and we
obtain by an identical argument to the subcase (ii.a) that our final
expression is congruent modulo Eisenstein series to an element of
$\ringRl$.  This completes the proof.
\end{proof}

Having disposed of weight~$2$, we now turn our attention to
weight~$3$.  We shall prove weight~$3$ analogs of Propositions
\ref{proposition4.6} and~\ref{proposition4.8}, but only for modular
forms of the form $\xP\lambdaQ$, i.e., for products of an Eisenstein
series of weight~$2$ with an Eisenstein series of weight~$1$.  We
shall continue to work modulo Eisenstein series, i.e., modulo the
space $\Espace_3$.  In this context, the analog of~\eqref{equation4.9}
is the following statement, which holds whenever
$\pointP \oplus \pointQ \oplus \pointR = \Pzero$:
\begin{equation}
\label{equation4.38}
 (\xP - \xR)(\lambdaP + \lambdaQ + \lambdaR) \in \Espace_3.
\end{equation}
To see this, first observe by Proposition~\ref{proposition3.5} that if
none of $\pointP$, $\pointQ$, and $\pointR$ is equal to $\Pzero$, then
the above expression is equal to $\yP - \yR$, which is in $\Espace_3$.
On the other hand, if one of the points is $\Pzero$, then $\lambdaP +
\lambdaQ + \lambdaR = 0$ by our conventions, and the above expression
is equal to~$0$.

The next lemma is the weight~$3$ analog of the key computational step that
we did in~\eqref{equation4.19}.  We note incidentally that we could have
applied the techniques of this lemma to the weight~$2$ identity
$(\lambdaP + \lambdaQ + \lambdaR)^2 = \xP + \xQ + \xR$.  This would have
yielded a slightly weaker result than Proposition~\ref{proposition4.6}
(analogous to the proof below in weight~$3$) that would also have been
sufficient for our purposes.

\begin{lemma}
\label{lemma4.10}
Let $\fieldk = \C$, let $n \geq 1$, and let $A,B \in \E[n\myl]$ (typically
with $A,B \in \El$).  Then we have the following congruences modulo
$\Espace_3$:
\begin{equation}
\label{equation4.39}
 \sum_{T \in \E[n]} x_{A\oplus T} \thelambda_{A \oplus T}
        \equiv
    n x_{[n]A} \thelambda_{[n]A},
\end{equation}
\begin{equation}
\label{equation4.40}
\begin{split}
 \sum_{T \in \E[n]} x_{A \oplus T} \thelambda_{B \ominus T}
        &\equiv
    - nx_{[n]A}\thelambda_{[n]A}
    + n^2 x_{[n]A}\thelambda_{A \oplus B}\\
        & \qquad
    + n x_{A \oplus B} \thelambda_{[n]A}
    + n x_{A \oplus B} \thelambda_{[n]B}
    - n^2 x_{A \oplus B} \thelambda_{A \oplus B}.\\
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
To show~\eqref{equation4.39}, we take
$\pointP = A \oplus T$, $\pointQ = \ominus A \oplus U$, and
$\pointR = \ominus T \ominus U$ in~\eqref{equation4.38}, and we sum the
result over all $T,U \in \E[n]$, knowing that the final result will be
$\equiv 0$ modulo $\Espace_3$.  We now observe that
\begin{equation}
\label{equation4.41}
\begin{split}
\sum_{T,U \in \E[n]} x_{A \oplus T} \thelambda_{A \oplus T}
        &= n^2 \sum_T x_{A \oplus T} \thelambda_{A \oplus T},\\
\sum_{T,U} x_{A \oplus T}\thelambda_{\ominus A \oplus U}
        &= n^2 x_{[n]A} \cdot n \thelambda_{[-n]A}
        = -n^3 x_{[n]A} \thelambda_{[n]A},\\
\sum_{T,U} x_{A \oplus T}\thelambda_{\ominus T \ominus U}
        &= \sum_{T,V \in \E[n]}  x_{A \oplus T}\thelambda_{V} = 0,\\
\sum_{T,U} x_{\ominus T \ominus U} \thelambda_{A \oplus T}
        &= \sum_{T,V} x_V \thelambda_{A \oplus T} = 0;
\text{ similarly, }\sum_{T,U} x_{\ominus T \ominus U}
        \thelambda_{\ominus A \ominus U} = 0,\\
\sum_{T,U} x_{\ominus T \ominus U}  \thelambda_{\ominus T \ominus U}
        &= n^2 \sum_V x_V \thelambda_V
        = n^2 \sum_V x_{\ominus V} \thelambda_{\ominus V}
        = -(\text{itself}) = 0,
\end{split}
\end{equation}
where we have used~\eqref{equation4.10} and~\eqref{equation3.12.6} as
needed.

For the proof of~\eqref{equation4.40}, we take the sum over
all $T$ in $\E[n]$ of~\eqref{equation4.38} with
$\pointP = A \oplus T$, $\pointQ = B \ominus T$, and
$\pointR = \ominus A \ominus B$
(so $\lambdaR = -\thelambda_{A \oplus B}$
and $\xR = x_{A \oplus B}$).  We then proceed as in the
proof of~\eqref{equation4.39}, while using~\eqref{equation4.39} at one
point, to obtain the desired result.
\end{proof}

At this point, the generalization of Propositions \ref{proposition4.6}
and~\ref{proposition4.8} to weight~$3$ is straightforward.

\begin{proposition}
\label{proposition4.11}
Make the same hypotheses as Lemma~\ref{lemma4.10}, and let $s \in
\Z$.  Then we have the following congruence modulo $\Espace_3$:
\begin{equation}
\label{equation4.42}
\begin{split}
&\sum_{T \in \E[n]} x_{A \oplus T} \thelambda_{B \ominus [s]T}\\
&\quad \equiv
\bigl( \text{a linear combination of terms of the form }
         x_{[a]A \oplus [b]B} \thelambda_{[c]A \oplus [d]B}
\bigr),\\
&\qquad\qquad\text{with }
  a - sb \equiv c - sd \equiv 0 \pmod{n}.\\
\end{split}
\end{equation}
(An analogous statement holds for sums
$\sum_T x_{A \ominus [s]T} \thelambda_{B \oplus T}$, in which case the
congruence condition modulo $n$ becomes $-sa + b \equiv -sc + d \equiv
0$.)
We remark incidentally that the determinant of $\stwomatr{a}{b}{c}{d}$
above need not be equal to $n$.

Furthermore, if $\pointP, \pointQ \in \E[n\myl]$, then the trace of the
weight~$3$ element $\xP \lambdaQ \in \ringR_{n\myl}$ down to level
$\Gammal$ is congruent modulo $\Espace_3$ to a linear combination of
terms $\xR \lambdaS \in \ringRl$, with $\pointR, \pointS \in \El$.
\end{proposition}
\begin{proof}
The proof of~\eqref{equation4.42} follows the same lines as the proof
of Proposition~\ref{proposition4.6}, with the same type of induction
on $s$.  For $s=0$, it follows as usual from~\eqref{equation4.10}, and
we have already proved the case $s=1$ in~\eqref{equation4.40}.  The
key step in the induction (analogous to~\eqref{equation4.19}) amounts
to applying~\eqref{equation4.40} to the $T$-part of the sum
$\sum_{T,U} x_{A\oplus T} \thelambda_{B' \ominus T \oplus U}$.  The
ideas are essentially the same as before, with the use
of~\eqref{equation4.39} thrown in for good measure.  (It is worth
pointing out that while carrying out the same proof in the case of
$\sum_T x_{A \ominus [s]T} \thelambda_{B \oplus T}$, we encounter at
one stage the sum
$\sum_{U \in \E[s]}
    x_{[n]A' \oplus [n]U} \thelambda_{[n]A' \oplus [n]U}$,
where $[s]A' = A$.  Write $d=gcd(n,s)$ and $\hat{s} = s/d$;
then our sum becomes
$d^2 \cdot \sum_{\hat{U} \in \E[\hat{s}]}
   x_{[n]A' \oplus \hat{U}} \thelambda_{[n]A' \oplus \hat{U}}$,
which we simplify using~\eqref{equation4.39}.)

As for the proof of the statement about the trace of $\xP\lambdaQ$, it
follows the argument of Proposition~\ref{proposition4.8} with only
trivial changes.  The only point worth mentioning is that the roles of
$T_0$ and $U_0$ are no longer symmetric, so we cannot simply assume
that $T_0$ in case (i) (respectively, $\hat{T_0}$ in case (ii)) is not
equal to $\Pzero$.  However, if $T_0$ (respectively, $\hat{T_0}$) is equal
to $\Pzero$, then $\pointP \in \El$ already, and the trace is then equal
to $\xP \tr(\lambdaQ)$, which is easy to analyze
using~\eqref{equation4.10}, or for that matter by noting that
the trace of the Eisenstein series $\lambdaQ$ is again an Eisenstein
series.
\end{proof}

\section{Generating all modular forms in weights $\geq 2$, and a model
  for $X(\myl)$}
\label{section5}

We are now ready to use Propositions \ref{proposition4.8}
and~\ref{proposition4.11} to show that the ring of modular forms over~$\C$
generated by the Eisenstein series of weight~$1$ contains all modular forms
in weights $2$ and above.  This is Theorem~\ref{theorem5.1} below.  We
then apply the result to obtain a convenient method to find explicit
models for the modular curve $X(\myl)$, in Theorem~\ref{theorem5.5}
below.

We prove Theorem~\ref{theorem5.1} via relating the result to the
nonvanishing of a special value of an $L$-function, which is also the
strategy
of~\cite{BorisovGunnellsNonvanishing,BorisovGunnellsHigherWeight}.
Our proof brings in the $L$-function via a Rankin-Selberg integral, in
contrast to the calculations in the articles by Borisov and Gunnells, which
use $q$-expansions whose coefficients are modular symbols.  It is worth
noting that one can give a much simpler proof of the (rather weaker)
fact that $\ringRl$ contains all modular forms in sufficiently high
weights.  To see this, note that the ring of all modular forms is the
graded integral closure of $\ringRl$ in its own field of fractions, by
Definition~\ref{definition3.2} and Remark~\ref{remark3.3}.  Hence
$X(\myl) = \mathbf{Proj}~\ringRl$; since $X(\myl)$ is nonsingular, it is
then a standard fact that the graded components of the two rings ($\ringRl$
and the ring of modular forms) agree in sufficiently high weights --- see
for example~\cite{Hartshorne}, Section II.5.19 and Exercises II.5.9,
II.5.14.  Precise but large bounds for the meaning of ``sufficiently high''
for arbitrary  curves are given in~\cite{GrusonLazarsfeldPeskine}, but
they of course grow with the genus of the curve, which for $X(\myl)$
is $O(\myl^3)$.  The interest of our results, as well as those of
Borisov-Gunnells, is that they give a fixed value for ``sufficiently
high'': $2$ in our result for $\Gammal$, and $3$ for their result for
$\Gamma_1(\myl)$ (where they obtain all cusp forms modulo Eisenstein
series, but potentially miss some Eisenstein series).

\begin{theorem}
\label{theorem5.1}
Let $\fieldk = \C$.  Then $\ringRl$ contains all modular forms on $\Gammal$
of weight $2$ and above.  In other words, $\ringRl$ ``misses'' precisely
the cusp forms in weight~$1$.
\end{theorem}
\begin{proof}
Since $\ringRl$ contains all modular forms for $\myl \leq 2$, we as usual
restrict to the case $\myl \geq 3$.
Our first claim is that it is enough to show that $\ringRl$ contains all
of $\modforms_2(\Gammal)$ and $\modforms_3(\Gammal)$.  To see this claim,
observe that $\Gammal$ has no elliptic elements or irregular cusps;
hence there exists a line bundle $\LL$ on $X(\myl)$ such that for all
weights $\wtj$, we have
$\modforms_\wtj(\Gammal) = H^0(X(\myl),\LL^{\tensor \wtj})$.  Moreover,
elements of $\modforms_2$ can be viewed as $1$-forms on $X(\myl)$ with at
worst a simple pole at each cusp.  Hence the degree of $\LL^{\tensor 2}$ is
equal to $2g-2 + \kappa$, where $g$ is the genus of
$X(\myl)$, and $\kappa$ is the number of cusps.  Since $\kappa \geq 4$ for
$\myl \geq 3$, by standard formulas for modular curves (e.g., Section~1.6
of~\cite{Shimura}), we obtain that $2 \deg \LL \geq 2g + 2$.  This is
enough to show that the multiplication map $\modforms_{\wtj}(\Gammal)
\tensor \modforms_{\wtj'}(\Gammal) \to \modforms_{\wtj+\wtj'}(\Gammal)$ is
surjective for $\wtj, \wtj' \geq 2$, since the degrees of
$\LL^{\tensor \wtj}$ and $\LL^{\tensor \wtj'}$ are both $\geq 2g+1$ (for a
sketch of this standard result, see Lemma~2.2 of~\cite{KKM}; the survey
in Section~1 of~\cite{Lazarsfeld} is also a particularly useful
reference).  This implies that any ring of modular forms that contains
$\modforms_2(\Gammal)$ and $\modforms_3(\Gammal)$ must contain all forms in
higher weights, thereby establishing our claim.

We therefore turn to the situation in weights $2$ and~$3$, which we study
using a result of Shimura~\cite{ShimuraSpecialValues}.  This result
states that a suitable Rankin-Selberg convolution of a newform $F$ with an
Eisenstein series gives a product of two special values of Hecke
$L$-functions associated to $F$ and to Dirichlet characters $\xi, \psi$.
More precisely, Theorem~2 (with $r=0$) of~\cite{ShimuraSpecialValues}, and
equation~(4.3) of the same article (with $k=\wtj \geq 2$, $l=1$, and
$m=\wtj-1$) imply the following statement, which holds for any
$\wtj \geq 2$:
let $F \in \cuspforms_\wtj$ be a newform with character $\chi$, and let
$\xi, \psi$ be Dirichlet characters with $(\xi\psi)(-1) = -1$; then there
exists a product $\Eis \Eis'$ of two Eisenstein series, with $\Eis \in
\Espace_1$ and $\Eis' \in \Espace_{\wtj-1}$, such that the Petersson inner
product of $F$ with $\Eis \Eis'$ gives
\begin{equation}
\label{equation5.1}
\langle F, \Eis \Eis'\rangle = C \cdot L(\wtj-1, F, \xi) L(\wtj-1, F, \psi)
\end{equation}
with an explicit nonzero constant $C$.  (Here, if $\wtj=3$, we must have
$\chi \xi \psi \neq 1$ in order for $\Eis' \in \Espace_2$ to be
holomorphic.)  Note that we have normalized the Petersson inner product so
that it is insensitive to the choice of common congruence subgroup $\Gamma$
with respect to which $F$, $\Eis$, and $\Eis'$ are all invariant.

We deduce from~\eqref{equation5.1} that for a given $F$, we can choose
$\xi$ and $\psi$ (and, with them, $\Eis$ and $\Eis'$) so as to make the
above inner product nonzero.  Indeed, when $\wtj \geq 3$, then, regardless
of $\xi$ and $\psi$, the $L$-functions on the right side are nonzero, since
they are evaluated outside the critical strip if $\wtj \geq 4$, and at
the edge of the critical strip if $\wtj = 3$ (see, e.g., Proposition~2
of~\cite{ShimuraSpecialValues}, or~\cite{JacquetShalikaNonvanishing} for a
more general result).  Thus we can also ensure that $\chi\xi\psi \neq 1$ as
needed in the special case $\wtj=3$.  On the other hand, if $\wtj=2$, then,
by Theorem~2 of~\cite{ShimuraPeriods}, there exist $\xi$ and $\psi$ for
which the right side of~\eqref{equation5.1} is nonzero.

We can now show that $\ringRl$ contains all of $\modforms_2(\Gammal)$ and
$\modforms_3(\Gammal)$.  Since $\ringRl$ contains all Eisenstein series on
$\Gammal$, we are reduced to checking whether $\ringRl$ contains all of
$\cuspforms_\wtj(\Gammal)$ for $\wtj \in \{2,3\}$, or alternatively to
checking that the orthogonal complement $[\ringRl \intersect
\cuspforms_\wtj(\Gammal)]^\perp$ in $\cuspforms_\wtj(\Gammal)$ is zero.
Let $0 \neq f \in \cuspforms_\wtj(\Gammal)$ be any nonzero cuspform in this
orthogonal complement.  Then there exist constants $c_1,\dots, c_N \in \C$
and matrices $\alpha_1, \dots, \alpha_N \in GL^+(2,\Q)$ such that the
linear combination $F = \sum_i c_i f| \alpha_i$ is actually a newform (for
instance, use an element of the Hecke algebra to project to a single
automorphic representation, and then move around within it to reach the
newform).  We can find $\Eis, \Eis'$ as above such that
$\langle F, \Eis\Eis' \rangle \neq 0$.  But this means that
\begin{equation}
\label{equation5.2}
 0 \neq \langle \sum_i c_i f| \alpha_i, \Eis \Eis' \rangle
 = \sum_i c_i \langle f, (\Eis \Eis')|\alpha_i^{-1} \rangle.
\end{equation}
In the above expression, each form $(\Eis \Eis')|\alpha_i^{-1}
= (\Eis|\alpha_i^{-1})(\Eis'|\alpha_i^{-1})$ is still
the product of an Eisenstein series of weight $1$ with an Eisenstein series
of weight $j-1 \in \{1,2\}$; hence it can be written as a linear
combination of modular forms of the form $\lambdaP \lambdaQ$ or $\lambdaP
\xQ$, with $\pointP, \pointQ \in \E[n\myl]$ for some (possibly rather
large) $n$.  We obtain a linear combination of inner products of the
form $\langle f, \lambdaP \lambdaQ \rangle$ or
 $\langle f, \lambdaP \xQ \rangle$, which can in turn be
reexpressed (up to a constant factor) as an inner product of the form
$\langle f, \tr^{\Gamma(n\myl)}_\Gammal \lambdaP \lambdaQ\rangle$
or $\langle f, \tr^{\Gamma(n\myl)}_\Gammal \lambdaP \xQ\rangle$,
and the traces belong to $\ringRl$ by Propositions \ref{proposition4.8}
and~\ref{proposition4.11}.  Furthermore, the Eisenstein part of each
such trace, and therefore also the cuspidal part, must then belong to
$\ringRl$.  Thus the inner products must all be zero if $f$ belongs to the
orthogonal complement $[\ringRl \intersect \cuspforms_\wtj(\Gammal)]^\perp$
in question.  This contradicts the fact that $\langle F, \Eis \Eis' \rangle
\neq 0$, and we deduce that the orthogonal complement is zero after all.
This concludes our proof.
\end{proof}

Theorem~\ref{theorem5.1} makes it possible to compute nice
models for the modular curve $X(\myl)$.  These models are defined over
$\Q(\boldmu_\myl)$ (we suspect that a more careful investigation would
yield models over $\Q$), and are in the form called
``Representation~B'' in~\cite{KKMasymptotic}.  The basic idea is to work
implicitly with the projective embedding of $X$ given by a line bundle
$\LLhat$, with $\deg \LLhat \geq 2g+2$, for which it is a standard fact
that the ideal of equations defining the image of $X$ is generated by
quadrics (see, for example, Section~1 of~\cite{Lazarsfeld}).  (In our
setting, we will have $X = X(\myl)$ and $\LLhat = \LL^{\tensor 2}$, in the
notation of the proof of Theorem~\ref{theorem5.1}.)  We then define $V =
H^0(X,\LLhat)$ and $V' = H^0(X,\LLhat^{\tensor 2})$; in our setting, this
means that $V = \modforms_2(\Gammal)$ and $V' = \modforms_4(\Gammal)$.
Let $\mu$ be the multiplication map $\mu: V \tensor V \to V'$, and note
that $\mu$ factors through a map $\mubar: \Sym^2 V \to V'$.  Then the
kernel of $\mubar$ describes exactly the quadric equations that define $X$
in its projective embedding, and hence $X$ can be recovered from a
knowledge of the spaces $V$ and $V'$ and of the multiplication map $\mu$.
Now it is possible to represent $\mu$ by a multiplication table in terms of
bases for $V$ and $V'$ (this was called ``Representation~A''
in~\cite{KKMasymptotic}), but a superior method is to take a collection of
points $\p_1, \dots, \p_N$ of points on $X$, with $N > 2\deg \LLhat$, and to
represent elements of $V$ and $V'$ by their ``values'' at the points $\p_i$;
this presupposes some fixed choice of trivialization of $\LLhat$ in a
neighborhood of each $\p_i$.  It turns out that the points $\p_i$ need not be
distinct, provided we replace the value of an element $s \in V$ (or $V'$)
by its $n$th order Taylor expansion at a point that appears with
multiplicity $n$.  A better point of view is to replace the points $\p_i$ by
the effective divisor $D = (\p_1) + \cdots + (\p_N)$ on $X$, and
reformulating the value of $s \in V = H^0(X,\LLhat)$ at the points of $D$
in terms of the image of $s$ in $H^0(X,\LLhat)/H^0(X,\LLhat(-D))
\isomorphic H^0(X, \LLhat/\LLhat(-D))$.  The local trivialization of
$\LLhat$ then amounts to fixing once and for all an isomorphism between the
sheaves of $\mathcal{O}_X$-modules $\LLhat/\LLhat(-D)$ and
$\mathcal{O}_X/\mathcal{O}_X(-D) = \mathcal{O}_D$, which are
supported on the possibly nonreduced zero-dimensional subscheme $D$ of $X$.
Thus our ``values'' in $H^0(X,\LLhat/\LLhat(-D))$ are interpreted as
elements of the finite-dimensional algebra $\mathcal{A} =
H^0(X,\mathcal{O}_D)$.  A similar assertion works for the
values of an element in $V'$ (using the induced isomorphism between
$\LLhat^{\tensor 2} / \LLhat^{\tensor 2}(-D)$ and
$\mathcal{O}_D$), and in this case the multiplication map
$\mu$ amounts to the multiplication in $\mathcal{A}$.  All our ingredients
are now in place to give the definition of Representation~B.

\begin{definition}
\label{definition5.2}
Let $X$ be a smooth projective curve over a base field $F$, and
choose a line bundle $\LLhat$ and an effective divisor $D$ on $X$
that are both $F$-rational.  Assume moreover that
$\deg \LLhat \geq 2g+2$ and that $\deg D > 2\deg \LLhat$, as discussed
above.   Then Representation~B of the curve $X$ is given by the
finite-dimensional $F$-algebra
$\mathcal{A} = H^0(X,\mathcal{O}_D)$, along with $F$-subspaces $V, V'
\subset \mathcal{A}$.  Here we have replaced $V$ by its image under
the $F$-linear map
$H^0(X,\LLhat) \to H^0(X,\LLhat/\LLhat(-D)) \isomorphic \mathcal{A}$,
and similarly for $V'$ with respect to $\LLhat^{\tensor2}$.  (The
condition on $\deg D$ ensures that these two
$F$-linear maps are injections.)  The multiplication map $\mu$ is
simply the restriction to $V$ of the  multiplication in $\mathcal{A}$,
and this yields sufficient information to deduce the set of quadric
equations that define the image of $X$ in projective space defined by
the embedding associated to $\LLhat$.
\end{definition}

We point out that the precise definition of Representation~B
in~\cite{KKMasymptotic} also specifies that $\mathcal{A}$ is represented as
a product of rings that are explicitly given in the form
$F[x]/(f(x))$; this shall not concern us here.

\begin{remark}
\label{remark5.3}
In our setting, where $V$ and $V'$ are spaces of modular forms, we can take
$D$ to be a multiple of the cusp at infinity; then the values at $D$ are
$q$-expansions of modular forms up to $O(q^N)$, and these $q$-expansions
give rise to equations for modular curves via the approach sketched above.
This approach has already appeared in the literature;
see~\cite{Galbraiththesis} and Section~2 of~\cite{BakerGonzalez2Poonen},
where $\LLhat$ is replaced by the canonical bundle, and the projective
embedding is then replaced with the canonical embedding (if the curve is
not hyperelliptic), with some modifications since the ideal of a
canonical curve occasionally requires generators going up to degree~$4$.

One novel aspect of our approach is that we evaluate the modular forms at
noncuspidal points; we hope that this approach, suitably developed, can
eventually also yield equations of Shimura curves.
\end{remark}

\begin{remark}
\label{remark5.4}
Here is a more concrete way to describe what is going on in
Representation~B.  Let $\{s_0, \dots, s_L\}$ be a basis for $V$; then each
vector of values $\p'_i = [s_0(\p_i):\cdots:s_L(\p_i)] \in \Projective^L$
gives the image of the point $\p_i \in X$ under the projective embedding.
(For convenience, suppose in this remark that the points $\p_i$ are all
distinct, and that the field $F$ is perfect; it is easy to modify the
argument for the general case).  We obtain sufficiently many points
to be able to identify $X$ as the unique projective curve that interpolates
the $\{\p'_i\}$, in the sense that $X$ is defined by all the quadric
equations vanishing at the $\{\p'_i\}$.  The quadrics that generate the
ideal of $X$ are of the form $\sum_{j,k} c_{jk} X_j X_k$, and can be found
by solving for the $c_{ij}$ in the linear system
$\{\sum_{j,k} c_{jk} s_j(\p_i) s_k(\p_i) = 0
\mid 1\leq i \leq N\}$.  Now the individual $\p_i$ need not be defined over
$F$, even though the divisor $D$ is $F$-rational; still, the set of points
$\{\p'_i\}$ is stable under $\Gal(\overline{F}/F)$, and so the linear
system of equations for the $c_{jk}$ is unaffected by the Galois group.
This implies that $X$ can be defined by quadrics with $F$-rational
coefficients; for example, take an echelon basis for the solution space of
the linear system.
\end{remark}

We are now ready for the last result of this article.

\begin{theorem}
\label{theorem5.5}
Let $\myl \geq 3$.  Fix a number field $F \subset \C$ and an elliptic
curve $\E_0$ over $F$ given by a Weierstrass equation
$y^2 = x^3 + a_0 x + b_0$, with
$a_0, b_0 \in F - \{0\}$.  Then consider all torsion points
$\{(x_{0,\pointP}, y_{0,\pointP})
     \mid \pointP \in \E_0[\myl](\overline{F}) - \{\Pzero\}\}$,
and the slopes
$\thelambda_{0, (\pointP)+(\pointQ)+(\ominus \pointP \ominus \pointQ)} =
(y_{0,\pointP} - y_{0,\pointQ})/(x_{0,\pointP} - x_{0,\pointQ})
      \in F(\E_0[\myl])$
of lines through pairs of torsion points (with the appropriate modification
when $\pointP = \pointQ$).  These slopes for the one elliptic curve $\E_0$
contain enough information to reconstruct the projective embedding of
$X(\myl)$ coming from the linear system $\modforms_2(\Gammal)$.  This
embedding is defined over $F(\boldmu_\myl)$.
\end{theorem}
\begin{proof}
We first observe that the condition $a_0, b_0 \neq 0$ implies that $\E_0$
does not have nontrivial automorphisms, and hence does not correspond to an
elliptic point for $\Gamma(1)$ in the upper half plane $\Half$.  Thus the
projection map $\pi: X(\myl) \to  X(1)$ is unramified over the point
$q_0 \in X(1)(F)$ corresponding to
$\E_0$, and hence the preimages $\{\p_1, \dots, \p_N\} = \pi^{-1}(\{q_0\})$
are distinct points of $X(\myl)$, which are rational over $F(\E_0[\myl])$.
We claim that $N$ (which is incidentally $\abs{PSL(2,\Z/\myl\Z)}$) is
large enough that we can identify modular forms of weight~$< 12$ via their
``values'' at the $\{\p_i\}$.  To see this claim, either use standard
formulas for the degree of the line bundle $\LL^{\tensor \wtj}$, whose
global sections are $\modforms_\wtj(\Gammal)$, or note
that one section of the line bundle $\LL^{\tensor 12}$ is the
$\Gamma(1)$-invariant modular form $b_0^2 a(\tau)^3 - a_0^3 b(\tau)^2$,
which vanishes precisely to order~$1$ at each point $\p_i$; this
last statement holds because modular forms in $\modforms_{12}(\Gamma(1))$
have precisely one zero (counted appropriately) in the fundamental domain
for the $\Gamma(1)$-action on~$\Half$.  Thus $N = 12 \deg \LL$, and
our claim is proved.

Hence, as mentioned in our discussion preceding the theorem, we can
represent $X(\myl)$ in Representation~B using the line bundle
$\LLhat = \LL^{\tensor 2}$ and the divisor $D = \sum_i (\p_i)$; this
amounts to representing the spaces $V$ and $V'$ by the ``values'' of modular
forms of weights $2$ and~$4$ at the points $\{\p_i\}$.  Concretely, such a
point $\p_i$ corresponds to a choice of symplectic basis
$\{T_0,U_0\}$ for the $\myl$-torsion $\E_0[\myl]$, with
$e_\myl(T_0, U_0) = e^{2\pi i/\myl} \in F(\E_0[\myl])$.  We know
how to ``evaluate'' an Eisenstein series of weight~$1$ at $\p_i$; this
amounts to computing slopes between the torsion points to get the
$\lambda_0$s appearing in the statement of the theorem.  Here, the local
trivialization of each line bundle $\LL^{\tensor \wtj}$ near $\p_i$
corresponds to the particular choice of Weierstrass model of $\E_0$ and of
its global differential $\omega_0$.  To define this trivialization more
precisely, let $\tau_1 \in \Half$ be such that the elliptic curve
$\E_1 = \E_{\tau_1} = \C/L_{\tau_1}$ and its symplectic $\myl$-torsion
basis $\{\pointP_{1/\myl}, \pointP_{\tau_1/\myl}\}$ are isomorphic to our
given triple $(\E_0, T_0, U_0)$.  Then there exists a unique
$\scalarlambda \in \C^\times$ such
that $a_0 = \scalarlambda^4 \acoeff(\tau_1)$
and  $b_0 = \scalarlambda^6 \bcoeff(\tau_1)$,
and which is also compatible with the level structures.
Hence each $\lambda_0$ is equal to $\scalarlambda \lambda_1(\tau_1)$
for a corresponding classical modular form
$\lambda_1(\tau) \in \Espace_1^{\myl}$, and similarly for other
weights~$\wtj$.  It follows that our trivialization of $\LL^{\tensor \wtj}$
near $\p_i$ is $u^\wtj$ times the trivialization induced by evaluating
modular forms in a neighborhood of $\tau_1$.

At this point, we see that if we work over the field $F_\myl =
F(\E_0[\myl])$, which contains the values of all the $\lambda_0$s, then our
algebra $\mathcal{A}$ is isomorphic to the direct product $F_\myl^N$, and
our space $V$ (respectively, $V'$) can be obtained as the span of all
products of the values of two (respectively, four) of the $\lambda_0$s at
each $p_i$.  This follows from Theorems \ref{theorem5.1}
and~\ref{theorem3.12}.  We thus obtain equations for $X(\myl)$ from the
kernel of $\mubar: \Sym^2 V \to V'$.  These equations are actually defined
over the smaller cyclotomic extension $F(\boldmu_\myl)$, because our whole
setup is invariant under the full $SL(2,\Z/\myl\Z)$-action on modular
forms, which permutes the possible symplectic bases for $\E_0[\myl]$.  Now
the action of $\Gal (F_\myl/F(\boldmu_\myl))$ arises from a subgroup $H$ of
$SL(2,\Z/\myl\Z)$ (in fact, when $\E_0$ does not have complex
multiplication, then for almost all $\myl$, $H = SL(2,\Z/\myl\Z)$), so the
equations that we obtain can be set up over the smaller field
$F(\boldmu_\myl)$, as mentioned in Remark~\ref{remark5.4}.
\end{proof}

We note in closing that an analog of Theorem~\ref{theorem5.5} holds
for the projective embedding of $X(\myl)$ coming from the (usually
incomplete) linear system
$\Espace_1^\myl \subset \modforms_1(\Gammal)$.  By
Theorem~\ref{theorem5.1} and a computation of Castelnuovo-Mumford
regularity, that projective model is defined by equations in degrees
$2$ and~$3$.

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