aainstitutetext: Institute of Mathematical Sciences, Homi Bhabha National
Institute (HBNI)
IV Cross Road, C. I. T. Campus, Taramani, Chennai 600 113, India
bbinstitutetext: Harish-Chandra Research Institute, Homi Bhabha National
Institute (HBNI)
Chhatnag Road, Jhunsi, Allahabad 211 019, India
ccinstitutetext: Department of Theoretical Physics, Tata Institute of
Fundamental Research
Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400 005, India

# Aspects of Hecke Symmetry II:
Anomalies, Curves, and Chazy Equations

Sujay K. Ashok<EMAIL_ADDRESS>b Dileep P. Jatkar<EMAIL_ADDRESS>c and
Madhusudhan Raman<EMAIL_ADDRESS>
###### Abstract

Motivated by their appearance in supersymmetric gauge and string theories, we
study the relations governing quasi-automorphic forms associated to certain
discrete subgroups of SL$(2,\mathbb{R})$ called Hecke groups. The Eisenstein
series associated to a Hecke group H$(m)$ satisfy a set of $m$ coupled linear
differential equations, which are natural analogues of the well-known
Ramanujan identities for modular forms of SL$(2,\mathbb{Z})$. We prove these
identities by appealing to a correspondence with the generalized Halphen
system. Each Hecke group is then associated to a (hyper-)elliptic curve, whose
coefficients are found to be determined by an anomaly equation. The Ramanujan
identities admit a natural geometrical interpretation as a vector field on the
moduli space of this curve. They also allow us to associate a non-linear
differential equation of order $m$ to each Hecke group. These equations are
higher-order analogues of the Chazy equation, and we show that they are solved
by the quasi-automorphic Eisenstein series $E_{2}$ associated to H$(m)$ and
its Hecke orbits. We conclude by demonstrating that these non-linear equations
possess the Painlevé property.

††preprint: TIFR/TH/18-38

##  1 Introduction

The modular group has played an important role in theoretical high-energy
physics, especially in the study of non-perturbative dualities in
supersymmetric gauge and string theories Sen:1994yi ; Seiberg:1994rs . In
these contexts, it acts on a complexified gauge or string coupling and relates
weak-coupling description of one theory to strong-coupling description of
another theory. These dualities typically imply strong constraints on the
spectrum of the physical theory Witten:1995ex .

The modular group has also played an important role in determining the low-
energy effective actions of superconformal gauge theories Seiberg:1994aj .
These insights have been combined with the powerful technology of equivariant
localization Nekrasov:2002qd ; Nekrasov:2003rj  to show that the prepotentials
of $\mathcal{N}=2^{\star}$ theories — these are $\mathcal{N}=2$ gauge theories
coupled to a massive adjoint hypermultiplet — satisfy a modular anomaly
equation Minahan:1997if ; Huang:2013eja ; Billo:2015pjb ; Billo:2015jyt . This
modular anomaly equation can be integrated, ultimately yielding a resummed
expression for the prepotential in terms of quasi-modular forms of
SL$(2,\mathbb{Z})$. Attempts to apply these techniques to $\mathcal{N}=2$
gauge theories with fundamental matter are complicated by the “running” of the
bare coupling constant Billo:2012st . These difficulties can be circumvented
partially by restricting to special loci in the Coulomb moduli space, and
explicit localization calculations have found that S-duality acts on the
“effective” couplings as a Hecke group Ashok:2015cba ; Ashok:2016oyh . This
relation to supersymmetric gauge theories provides us with a strong physical
motivation to study the properties of quasi-automorphic forms of Hecke groups.

An element of a Hecke group is any word made up of the letters $S$ and $T$
that is further circumscribed by the relations

$\displaystyle\begin{split}S^{2}&=1\ ,\\\ (ST)^{m}&=1\ .\end{split}$ (1)

It is easy to see that for $m=3$, the generators of the Hecke group satisfy
the same relations that bind the generators of the modular group. More
concretely, for $\tau$ that takes values in the upper-half plane $\mathbb{H}$,
the generators of the Hecke group act as

$\displaystyle\begin{split}T&:\tau\rightarrow\tau+1\ ,\\\
S&:\tau\rightarrow-\frac{1}{\lambda_{m}\tau}\ ,\end{split}$ (2)

where $\lambda_{m}=4\cos^{2}\left(\frac{\pi}{m}\right)$. The Hecke groups will
be denoted H$(m)$ and are indexed by an integer $m\geq 3$ that will be called
its _height_. For $m\in\\{3,4,6,\infty\\}$ the corresponding
$\lambda_{m}\in\mathbb{Z}$. We will refer to these cases as _arithmetic_ Hecke
groups, while the rest will be referred to as _non-arithmetic_ Hecke groups.
In the interest of uniformity, we will restrict our attention to Hecke groups
with finite heights.

In Section 2, we begin with a brief review of Doran:2013npa  on
(quasi-)automorphic forms of Hecke groups H$(m)$. The ring of quasi-
automorphic forms $\mathbb{C}[E_{2},\cdots,E_{2m}]$ is generated by the
Eisenstein series associated to the Hecke group, which in turn are related to
solutions of a generalized Halphen system in a simple way. The Eisenstein
series are found to satisfy a system of $m$ first order coupled linear
differential equations which are natural analogues of the Ramanujan identities
corresponding to the modular group. These are shown to be consistent with the
identities conjectured in Raman:2018owg .

In Section 3, we associate an algebraic curve to the Hecke group H$(m)$, whose
coefficients $A_{k}$ are quasi-automorphic forms.111As we will see, for
$m=3,4$, this curve is elliptic, and for $m>4$ it is hyperelliptic. An anomaly
equation governing these coefficients is derived; it plays a role analogous to
modular anomaly equations that have appeared in the literature on
supersymmetric gauge theories Huang:2013eja ; Billo:2015pjb ; Billo:2015jyt
in that it fixes the dependence of these coefficients on the quasi-automorphic
Eisenstein series $E_{2}$. A complete solution to these anomaly equations
requires the specification of boundary conditions that fix the dependence of
$A_{k}$ on purely automorphic pieces; this is done by insisting on certain
fall-offs at weak-coupling which we term ‘cuspidal’ boundary conditions. An
interesting alterative choice of boundary conditions is explored in Appendix B
that may have interesting relations to the theory of microdifferential
operators and integrable systems.

The discussion of curves is developed with the goal of interpreting the
Ramanujan identities as corresponding to some natural geometrical object on
the moduli space of these curves, a subject we turn to in Section 4. This has
already been done for the modular group in Movasati2012a ; Movasati2012b . We
review and extend these results to the case of H$(4)$ and construct a
Ramanujan vector field that acts naturally on the period integrals of the
associated curve.

In Section 5, we study the factorization of elliptic curves; the zeroes of the
elliptic curves are expressed in terms of Jacobi $\theta$-constants, and
satisfy a ‘strong’ Halphen system. We rewrite the Halphen system as a flow
equation and show that the matrix which governs the Halphen system is a Cartan
matrix of a Borcherds-Kač-Moody algebra Cheng:2008fc ; Govindarajan:2009qt ;
Cheng:2008kt  of rank $m$. In particular, we present a new Halphen system
corresponding to $m=4$.

While an analogous factorization does not appear possible for taller Hecke
groups, we show that the coefficients $A_{k}$ of the hyperelliptic curve
satisfy simple differential equations which we refer to as ‘weak’ Halphen
systems, which are intimately related to the Ramanujan identities and exist
for all Hecke groups. Along the way we write down the cusp forms which are
‘automorphic’ discriminants of the arithmetic Hecke groups and make some
observations in relation to the counting of BPS states in certain string
theory models Dijkgraaf:1996it ; Jatkar:2005bh ; David:2006ji .

Another mathematical motivation for our studies comes from Takhtajan:1992qb ,
where it was shown that for the modular group, the quasi-modular form $E_{2}$
and its SL$(2,\mathbb{Z})$ orbits satisfy the well-known Chazy equation and
further, that the weight-$12$ modular discriminant plays the role of a
$\tau$-function for the Chazy equation. In Section 6, we observe that for
every Hecke group the corresponding Eisenstein series $E_{2}$ satisfies an
ordinary differential equation of order $m$ that can be systematically
constructed using the Ramanujan identities. We go on to show that Hecke orbits
of $E_{2}$ also satisfy the same differential equation, and that a suitably
defined analogue of the modular discriminant plays the role of the
$\tau$-function for the Hecke group, thereby generalizing the results of
Takhtajan:1992qb  to all Hecke groups.

Finally, in Section 7, we turn to the study of our higher-order Chazy
equations. After a short primer on stability analysis, we explicitly verify
that the higher-order Chazy equations possess the Painlevé property.

For the reader’s convenience, we provide explicit Fourier expansions of some
automorphic forms in Appendix A, for quick verification of formulas.

###### Acknowledgements.

We would like to thank Suresh Govindarajan for discussions and for
collaboration during an early stage of the project. We also thank Renjan John
for helpful comments on an earlier version of the manuscript. MR acknowledges
support from the Infosys Endowment for Research into the Quantum Structure of
Spacetime. This research was supported in part by the International Centre for
Theoretical Sciences (ICTS) during a visit for participating in the program -
Quantum Fields, Geometry and Representation Theory (Code:
ICTS/qftgrt/2018/07).

##  2 Ramanujan Identities

We discuss the Ramanujan identities corresponding to the Hecke group H$(m)$ —
originally conjectured in Raman:2018owg  — and review their relation to the
generalized Halphen system, which we will shortly define. This route to the
Ramanujan identities allows us to furnish a simple proof, in addition to
presenting them in a more elegant closed form.

###  2.1 Generalized Halphen Systems and A Proof

The automorphic forms we study in this paper are Eisenstein series
corresponding to the Hecke group H$(m)$. They will be built out of solutions
to the generalized Halphen system, following Doran:2013npa . The generalized
Halphen system is a set of coupled ordinary differential equations for three
variables $\\{t_{k}(\tau)\\}_{k=1}^{3}$ that satisfy:

$\displaystyle\begin{split}t_{1}^{\prime}&=(a-1)\left(t_{1}t_{2}+t_{1}t_{3}-t_{2}t_{3}\right)+(b+c-1)\,t_{1}^{2}\
,\\\
t_{2}^{\prime}&=(b-1)\left(t_{2}t_{3}+t_{2}t_{1}-t_{1}t_{3}\right)+(a+c-1)\,t_{2}^{2}\
,\\\
t_{3}^{\prime}&=(c-1)\left(t_{3}t_{1}+t_{3}t_{2}-t_{1}t_{2}\right)+(a+b-1)\,t_{3}^{2}\
,\end{split}$ (3)

where the parameters $(a,b,c)$ are specified by the height $m$ of the Hecke
group H$(m)$ as

$\displaystyle\begin{split}a&=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{m}\right)\
,\\\ b&=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{m}\right)\ ,\\\
c&=\frac{1}{2}\left(\frac{3}{2}-\frac{1}{m}\right)\ ,\end{split}$ (4)

and the accent ′ denotes the following derivative:

$f^{\prime}\cong\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}f(\tau)\
.$ (5)

It is at times more convenient to work with Fourier expansions of the
automorphic forms we will introduce. The ‘nome’ in our conventions is
$q=\text{e}^{2\pi\text{i}\,\tau}$, and when working with $q$-series, the
superscript ′ is equivalent to

$\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\cong
q\frac{\text{d}\,}{\text{d}q\,}\ .$ (6)

The solution to the generalized Halphen system can be obtained explicitly in
terms of hypergeometric functions whose arguments depend on the hauptmoduln of
the Hecke group (Doran:2013npa, , Theorem 3(i)). We have included a few
details about these solutions and their Fourier expansions in Appendix A; in
particular, it will be important to keep in mind that some of our
normalizations differ from those of Doran:2013npa .

The Eisenstein series $\\{E_{2k}\\}_{k=2}^{m}$ are holomorphic automorphic
forms of weight $2k$ under the Hecke group, and they have simple expressions
in terms of the solutions to the generalized Halphen system (Doran:2013npa, ,
see p. 707 and Theorem 4(iv)). In order to simplify expressions, we define the
linear combinations

$\displaystyle\mathsf{x}$
$\displaystyle=t_{1}-t_{2}\qquad\text{and}\qquad\mathsf{y}=t_{3}-t_{2}\ .$ (7)

The automorphic forms $E_{2k}$ in these variables are

$E_{2k}=\mathsf{x}^{k-1}\,\mathsf{y}\ ,$ (8)

and in particular, $E_{4}=\mathsf{x}\,\mathsf{y}$. The Hecke groups also come
equipped with a quasi-automorphic weight-$2$ Eisenstein series $E_{2}$, which
is a linear combination of solutions to the generalized Halphen system
(Doran:2013npa, , Theorem 4(iii)). In terms of the variables
$(\mathsf{x},\mathsf{y})$ the Eisenstein series $E_{2}$ can be written as
follows:

$\displaystyle
E_{2}=-\frac{1}{m-2}\left[4\,\mathsf{x}+2m\,\mathsf{y}+(3m+2)\,t_{2}\right]\
.$ (9)

and using the conventions made explicit in Appendix A, we can check that

$\lim_{\tau\rightarrow\text{i}\infty}E_{2}(\tau)=1+O(q)\ .$ (10)

We can perform a linear transformation from the generalized Halphen variables
$(t_{1},t_{2},t_{3})$ to the variables $(\mathsf{x},\mathsf{y},E_{2})$ using
(7) and (9). The generalized Halphen system then takes the form

$\displaystyle\begin{split}E_{2}^{\prime}&=\frac{m-2}{4m}\left(E_{2}^{2}-\mathsf{x}\,\mathsf{y}\right)\
,\\\
\mathsf{x}^{\prime}&=\left(\frac{m-2}{2}\right)\mathsf{x}\,E_{2}+\frac{\mathsf{x}^{2}}{m}-\frac{\mathsf{x}\,\mathsf{y}}{2}\
,\\\
\mathsf{y}^{\prime}&=\left(\frac{m-2}{2}\right)\mathsf{y}\,E_{2}-\left(\frac{m-1}{m}\right)\mathsf{x}\,\mathsf{y}+\frac{\mathsf{y}^{2}}{2}\,.\end{split}$
(11)

Equipped with (8) and (9) along with (11) we are in a position to compute
$\tau$-derivatives of the Eisenstein series. This will yield analogues of the
Ramanujan identities for Hecke groups, which for fixed $m$ and $k\leq m$ take
the form

$\displaystyle
E_{2k}^{\prime}=\frac{k}{2}\left(\frac{m-2}{m}\right)E_{2}E_{2k}-\left(\frac{k-2}{2}\right)E_{4}E_{2k-2}-\left(\frac{m-k}{m}\right)E_{2k+2}\,.$
(12)

It is straightforwardly verified that the case $m=3$ reproduces the well known
Ramanujan identities corresponding to the modular group. For taller Hecke
groups, we get analogous relations; systems of this form will be collectively
referred to as Ramanujan identities.

###  2.2 Relation To Earlier Work

In earlier work Raman:2018owg , similar Ramanujan identities were derived for
cusp forms associated to the Hecke group H$(m)$. In this section we translate
between the two bases of automorphic forms — the Eisenstein series $E_{2k}$
and their cuspidal cousins. Recall the definition (Doran:2013npa, , Theorem
2(i)) of the cusp form $f_{2k}$ of a Hecke group H$(m)$:

$f_{2k}=(-1)^{k}\dot{J}^{k}J^{1-k}(J-1)^{\lceil\frac{k}{2}\rceil-k}\ .$ (13)

Here $J$ is the hauptmodul of the Hecke group H$(m)$, that in turn solves a
Schwarzian differential equation. In order to relate these to the Eisenstein
series $E_{2k}$ in (8), we recall from (Doran:2013npa, , Theorem 4(iii)) that
the solutions to the generalized Halphen system can be written down in terms
of the hauptmodul of the corresponding Hecke group:

$\displaystyle\mathsf{x}=\frac{\dot{J}}{J}\quad\text{and}\quad\mathsf{y}=\frac{\dot{J}}{J-1}\
.$ (14)

Substituting these into the definition of $E_{2k}$, we obtain

$E_{2k}=\dot{J}^{k}J^{1-k}(J-1)^{-1}\ .$ (15)

This yields the following simple relation between the Eisenstein series and
the cusp form:

$(-1)^{k}E_{2k}=f_{2k}\,(J-1)^{d_{2k}}\ ,$ (16)

where $d_{2k}$ is defined as (Doran:2013npa, , Theorem 2(i))

$d_{2k}=k-\bigg{\lceil}\frac{k}{2}\bigg{\rceil}-\bigg{\lceil}\frac{k}{m}\bigg{\rceil}\
,$ (17)

and is related to the dimension of the space of weight-$(2k)$ automorphic
forms $\mathfrak{m}_{2k}$ as

$\displaystyle\text{dim}\,\mathfrak{m}_{2k}$ $\displaystyle=d_{2k}+1\ ,$ (18)
$\displaystyle=k-\left\lceil\frac{k}{2}\right\rceil\ ,$ (19)
$\displaystyle=\left\lfloor\frac{k}{2}\right\rfloor\ .$ (20)

In (19), we have used the fact that $k\leq m$. On using (16) the Ramanujan
identities conjectured in Raman:2018owg  may be related to the ones derived in
the previous subsection.

##  3 Curves and Anomalies

###  3.1 Hyperelliptic Curves and Anomaly Equations

Just as the ring of quasi-modular forms of H$(3)\cong\text{SL}(2,\mathbb{Z})$
is $\mathbb{C}[E_{2},E_{4},E_{6}]$, so it is that the ring of quasi-
automorphic forms of H$(m)$ is identified with
$\mathbb{C}\left[E_{2},E_{4},\cdots,E_{2m}\right]$. To each of these groups we
assign a polynomial $p(x)$ whose degree is the height of the Hecke group

$p(x)=x^{m}+\sum_{k=1}^{m}x^{m-k}\,A_{k}\ ,$ (21)

and an algebraic curve defined by the equation

$y^{2}=p(x)\ .$ (22)

The curves for $m\leq 4$ are elliptic, and for $m>4$ they are hyperelliptic.
Next, assign to the coefficient $A_{k}$ a weight $2k$ under the action of the
relevant Hecke group; for consistency, we need to associate to $x$ the weight
$2$, and to $y$ the weight $m$. With this assignment of weights, $A_{1}$ has
weight $2$, and is thus proportional to $E_{2}$:

$A_{1}=c\,E_{2}\,.$ (23)

The $A_{k}$ are so far characterized solely by their weight. In general, this
makes them quasi-automorphic objects. Drawing from motivations relating to the
sort of algebraic curves that appear in supersymmetric gauge theories
Donagi:1995cf ; DHoker:1999yni ; Ashok:2016ewb , we note that it is often
desirable to have algebraic curves whose coefficients are purely modular — in
this case, automorphic — forms. This requirement is imposed by demanding that
$p(x)$ with shifted argument is such that the coefficients are purely
automorphic:

$p\left(x-\frac{c}{m}E_{2}\right)=x^{m}+\sum_{k=2}^{m}x^{m-k}\widetilde{A}_{k}\
,$ (24)

where under the action of $\gamma\in\text{H}(m)$, the $\widetilde{A}_{k}$
transform as

$\widetilde{A}_{k}(\gamma\cdot\tau)=(c\tau+d)^{2k}\widetilde{A}_{k}(\tau)\ ,$
(25)

i.e., they are automorphic forms of weight $2k$. This would in turn imply that
the coefficients $\widetilde{A}_{k}$ for $k>2$ do not depend on the quasi-
automorphic Eisenstein series $E_{2}$. An interesting outcome of this
requirement can be derived by rewriting $p(x)$ as

$p(x)=\left(x+\frac{c}{m}E_{2}\right)^{m}+\sum_{k=2}^{m}\left(x+\frac{c}{m}E_{2}\right)^{m-k}\widetilde{A}_{k}\
.$ (26)

which in turn allows us to translate our requirement into a constraint of the
form

$\left[\frac{\partial}{\partial E_{2}}-\frac{c}{m}\frac{\partial}{\partial
x}\right]p(x)=0\,,$ (27)

where we now write the polynomial $p(x)$ as in (21). This constraint along
with the relation (23) yields

$\displaystyle\sum_{k=1}^{m}\left[\frac{\partial A_{k}}{\partial
E_{2}}-\frac{c}{m}(m-k+1)\,A_{k-1}\right]x^{m-k}=0\ .$ (28)

This provides a set of $(m-1)$ equations, each expressing the heavier $A_{k}$
in terms of its lighter cousins by constraining their dependence on the quasi-
automorphic Eisenstein series $E_{2}$, and each of the form

$\frac{\partial A_{k}}{\partial E_{2}}=\frac{c\,(m-k+1)}{m}\,A_{k-1}\ .$ (29)

This completes our derivation. Note in particular that the above argument is
worked out in complete generality: it is true for all Hecke groups H$(m)$, and
applies to the coefficients of their associated hyperelliptic curves. For
$m=3$, these kinds of equations have appeared in the context of a
$\mathcal{N}=2$ supersymmetric gauge theories Huang:2013eja ; Billo:2015pjb ;
Billo:2015jyt ; Ashok:2015cba ; Ashok:2016oyh  and are referred to as modular
anomaly equations. Since the above equation constrains the dependence of the
$A_{k}$ on the quasi-automorphic Eisenstein series $E_{2}$, we will refer to
them (more generally) as anomaly equations.

The constant $c$ is a matter of convention, but we will treat it as being
universal (i.e., the same for all Hecke groups) for convenience; for the
special case $m=3$, it is possible to fix this constant unambiguously, as in
this case the corresponding elliptic curve can be fixed independently of the
above discussion.

Integrating the anomaly equations fixes the dependence of $A_{k}$ on the
quasi-automorphic form $E_{2}$. In order to determine $A_{k}$ completely, we
must supply a boundary condition that fixes the purely automorphic pieces. In
the following section, we highlight one possible choice of boundary condition
that will be useful in later sections.

###  3.2 Cuspidal Boundary Conditions

Starting with the lightest $A_{k}$, we solve the anomaly equation, thereby
“integrating in” any $E_{2}$ dependence. For example, on using (23), the
anomaly equation for $A_{2}$ reads

$\frac{\partial A_{2}}{\partial E_{2}}=\frac{c^{2}(m-1)}{m}E_{2}\ ,$ (30)

which is solved by

$A_{2}=\frac{c^{2}(m-1)}{2m}E_{2}^{2}+(\text{automorphic form})\ .$ (31)

The constants of integration are automorphic forms under the group H$(m)$. In
order to determine them, we write down every possible automorphic form
consistent with considerations of weight, accompanied by coefficients that are
to be determined. For $A_{2}$, this means

$A_{2}=\frac{c^{2}(m-1)}{2m}E_{2}^{2}+a\,E_{4}\ ,$ (32)

as for all H$(m)$, the dimension of weight-$4$ forms is unity. For general
$A_{k}$, the number of terms we can write down (i.e., the number of
undetermined coefficients) will by definition be as many terms as the
dimension of the space $\mathfrak{m}_{2k}$, which from (20) is $\lfloor
k/2\rfloor$.

We propose to fix these coefficients by demanding that near the cusp at
$\text{i}\infty$, the $A_{k}$ has a Fourier expansion that starts at
$O\left(q^{\text{dim}\,\mathfrak{m}_{2k}}\right)$. That is,

$\lim_{\tau\rightarrow\text{i}\infty}A_{k}=O\left(q^{\lfloor
k/2\rfloor}\right)\ .$ (33)

This boundary condition provides as many equations as the number of
coefficients to be determined, and is consequently an unambiguous
prescription. Further, by construction these $A_{k}$ satisfy the anomaly
equation. Additionally, we will see in Section 6.2 that the choice of cuspidal
boundary conditions allow us to relate the ‘algebraic’ discriminant $\Delta$
constructed out of the (hyper-)elliptic curve and an ‘automorphic’
discriminant $\Delta_{m}$ defined solely with reference to the ring of quasi-
automorphic forms of H$(m)$ in a simple way.

We mention in passing that there exist alternative choices of boundary
conditions that are also consistent with the anomaly equation. An especially
interesting example of this is related to microdifferential operators and the
theory of integrable systems. We refer the reader to Appendix B for more
details.

###  3.3 Examples

The elliptic curve associated to the modular group is

$y^{2}=x^{3}+A_{1}\,x^{2}+A_{2}\,x+A_{3}\ .$ (34)

Notice that the assignment of weights here implies that the coefficients
$A_{2}$ and $A_{3}$ carry weights $4$ and $6$ respectively. The forms $A_{k}$
will correspond to the generators of the ring of quasimodular forms of H$(3)$.
The polynomial defining the elliptic curve may be factorized

$y^{2}=\prod_{k=1}^{3}\left(x+s_{k}\right)\ ,$ (35)

in which case we find that the $A_{k}$ and $s_{k}$ are related as

$\displaystyle\begin{split}A_{1}&=s_{1}+s_{2}+s_{3}\ ,\\\
A_{2}&=s_{1}s_{2}+s_{2}s_{3}+s_{3}s_{1}\ ,\\\ A_{3}&=s_{1}s_{2}s_{3}\
.\end{split}$ (36)

A consistent assignment of weights leads us to conclude that the roots $s_{k}$
carry weight $2$. Let us focus on the first of these relations, which
expresses $A_{1}=c\,E_{2}$ in terms of a sum of three $s_{k}$. Following
Harnad:1999hi , we define the $s_{k}$ to be logarithmic derivatives of Jacobi
$\theta$-constants:

$\displaystyle\begin{split}s_{1}&=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{2}(\tau)\
,\\\
s_{2}&=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{3}(\tau)\
,\\\
s_{3}&=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{4}(\tau)\
.\end{split}$ (37)

As we know, a $\tau$-derivative raises the weight of a modular (function or
form) by two units. Using this explicit solution, we can solve for $A_{1}$ as

$\displaystyle A_{1}$
$\displaystyle=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{2}(\tau)\theta_{3}(\tau)\theta_{4}(\tau)\
,$ (38)
$\displaystyle=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}\log\eta(\tau)^{3}=\frac{E_{2}}{4}\,.$
(39)

This fixes the constant $c=\frac{1}{4}$. The remaining anomaly equations then
take the form

$\displaystyle\frac{\partial A_{2}}{\partial E_{2}}$
$\displaystyle=\frac{1}{6}A_{1}\ ,$ (40) $\displaystyle\frac{\partial
A_{3}}{\partial E_{2}}$ $\displaystyle=\frac{1}{12}A_{2}\ ,$ (41)

and can be solved by using the cuspidal boundary conditions (33). On doing so,
we find the following quasi-automorphic forms as the coefficients of the
elliptic curve:

$\displaystyle A_{1}$ $\displaystyle=\frac{1}{4}E_{2}\ ,$ $\displaystyle
A_{2}$ $\displaystyle=\frac{1}{48}\left(E_{2}^{2}-E_{4}\right)\ ,$ (42)
$\displaystyle A_{3}$
$\displaystyle=\frac{1}{1728}\left(E_{2}^{3}-3E_{2}E_{4}+2E_{6}\right)\ .$

In concluding this section, we briefly recap what we have accomplished so far.
We started by associating to each Hecke group a curve that was elliptic for
$m\leq 4$ and hyperelliptic for $m>4$. We then derived an anomaly equation by
insisting that in a shifted form of the curve, any dependence of the
coefficients $A_{k}$ on the quasi-automorphic form $E_{2}$ disappeared. We
proposed a natural choice of boundary conditions that allowed us to
unambiguously determine the purely automorphic pieces, i.e., the constants of
integration that are obtained after “integrating in” the $E_{2}$ dependence.
Apart from an overall constant that varies with the height $m$ of the Hecke
group, a few light $A_{k}$ are found to depend on the Eisenstein series as

$\displaystyle\begin{split}A_{1}&\propto\left(E_{2}\right)\ ,\\\
A_{2}&\propto\left(E_{2}^{2}-E_{4}\right)\ ,\\\
A_{3}&\propto\left(E_{2}^{3}-3E_{2}E_{4}+2E_{6}\right)\ ,\\\
A_{4}&\propto\left(E_{2}^{4}-6E_{2}^{2}E_{4}+8E_{2}E_{6}+E_{4}^{2}-4E_{8}\right)\
,\\\
A_{5}&\propto\left(E_{2}^{5}-10E_{2}^{3}E_{4}+20E_{2}^{2}E_{6}+5E_{2}(E_{4}^{2}-4E_{8})-4E_{4}E_{6}+8E_{10}\right)\
,\end{split}$ (43)

and so on. The constant of proportionality is

$A_{k}=\frac{{}^{m-1}C_{k-1}}{k\,4^{k}\,m^{k-1}}\left(E_{2}^{k}+\cdots\right)\
,$ (44)

a simple consequence of the anomaly equation. We will soon see that explicit
solutions for the $s_{k}$ are unavailable for Hecke groups with $m>4$, which
in turn makes it difficult to determine the coefficient $c$ in the relation
(23). We make the simplifying assumption that $c=1/4$ _universally_ , i.e.,
for all heights.222In principle, $c$ could depend on the height of the Hecke
group. It is easy to check that $c$ can always be set to $\frac{1}{4}$ by
rescaling the $s_{k}$. It should be clear that the above procedure will work
for _all_ Hecke groups. We now turn to the question of associating to the
Ramanujan identities a geometrical interpretation.

##  4 Ramanujan Vector Fields

Here we discuss a geometric interpretation of the Ramanujan identities
following Movasati2012a ; Movasati2012b . The goal here will be to associate
to the Ramanujan identities — being as they are a set of ordinary differential
equations — a vector field on the moduli space of the elliptic curve we have
just derived. We then do the same for H$(4)$.

###  4.1 Height Three

We begin with the elliptic curve

$y^{2}=x^{3}+\sum_{k=1}^{3}x^{3-k}A_{k}\ ,$ (45)

with the $A_{k}$ given by (42). Notice that the above elliptic curve (with
this choice of parameters) depends on the generators of the ring of quasi-
modular forms $\\{E_{2},E_{4},E_{6}\\}$. Now, we consider a canonical basis of
differentials on the above elliptic curve

$\Pi=\left(\begin{array}[c]{c}\frac{\text{d}x\,}{y}\\\\[8.53581pt]
\frac{x\,\text{d}x\,}{y}\end{array}\right)\ ,$ (46)

and ask the following question: what happens to this basis of differential
$1$-forms as we vary $\tau$, the modular parameter of the underlying torus?
This leads us to define the Ramanujan vector field:

$\text{R}=\frac{1}{2\pi\text{i}}\frac{\text{d}missing\,}{\text{d}\tau\,}\ .$
(47)

In terms of variations of the parameters that appear in the curve (which are
the Eisenstein series), we therefore have

R
$\displaystyle=\sum_{k=1}^{3}\left(\frac{1}{2\pi\text{i}}\frac{\text{d}missing\,}{\text{d}\tau\,}E_{2k}\right)\frac{\partial}{\partial
E_{2k}}\,.$ (48)

The portion in the brackets above may be replaced with the Ramanujan
identities (12), and we get

$\text{R}=\frac{1}{12}\left(E_{2}^{2}-E_{4}\right)\frac{\partial}{\partial
E_{2}}+\frac{1}{3}\left(E_{2}E_{4}-E_{6}\right)\frac{\partial}{\partial
E_{4}}+\frac{1}{2}\left(E_{2}E_{6}-E_{4}^{2}\right)\frac{\partial}{\partial
E_{6}}\ .$ (49)

It is natural to expect that the basis of differential $1$-forms will rotate
into themselves under the action of the Ramanujan vector field. Before turning
to the explicit computations, we pause to make contact with earlier work in
this direction.

##### Gauss-Manin Connections

The notion of a Gauss-Manin connection formalizes the following observation:
the variation of an elliptic integral — defined using the basis of
differentials in (46) — with respect to a parameter $t$ can be written as a
linear combination of period integrals

$\text{d}\left(\oint\Pi_{a}\right)=\sum_{b}\mathcal{A}_{ab}\left(\oint\Pi_{b}\right)\
.$ (50)

The coefficients $\mathcal{A}$ form a $2\times 2$ matrix, and for multiple
such parameters $\\{t_{k}\\}_{k=1}^{m}$ we define the differential form

$\mathcal{A}=\sum_{k=1}^{m}\mathcal{A}^{(k)}\,\text{d}t_{k}\ ,$ (51)

so the variation of the period integrals with respect to these parameters is
captured in the equations

$\displaystyle\nabla\Pi$
$\displaystyle=\sum_{k=1}^{m}\left(\mathcal{A}^{(k)}\,\Pi\right)\text{d}t_{k}\
,$ (52) $\displaystyle\cong\mathcal{A}\,\Pi\ .$ (53)

More properly, $\mathcal{A}$ should be viewed as a differential $1$-form on
the moduli space of the elliptic curve. The matrix $\mathcal{A}$ is referred
to as a Gauss-Manin connection, and in (Movasati2012b, , Proposition 6) this
Gauss-Manin connection was computed explicitly.

We can contract this differential $1$-form on the moduli space of the elliptic
curve with a vector from the same space. The Ramanujan identities define the
vector field (49) on the moduli space of the elliptic curve, and we define its
contraction with the connection as

$\nabla_{\text{R}}=\iota_{\text{R}}\nabla\ ,$ (54)

using the rule

$\frac{\partial}{\partial t_{k}}\left(\text{d}t_{\ell}\right)=\delta_{k\ell}\
.$ (55)

We now ask how $\nabla_{\text{R}}$ acts on a basis of differential $1$-forms
associated to our elliptic curve, and in (Movasati2012b, , Proposition 7) it
was demonstrated that

$\nabla_{\text{R}}\,\Pi=\mathcal{A}_{\text{R}}\,\Pi\ ,$ (56)

with $\mathcal{A}_{\text{R}}$ determined by explicit calculation.

To summarize briefly, all the Eisenstein series depend on a single complex
structure modulus $\tau$. For H$(3)$, this parameter coincides with the
modular parameter of the underlying torus. Indeed, our earlier discussion of
Gauss-Manin connections treated the $t_{k}$ as independent parameters, and in
this case it is useful to keep in mind that the Eisenstein series are not
independent in the same sense. This was implicit in our definition of the
Ramanujan vector field in (47). The appealing feature of (49) is that in this
presentation, the coefficients of the $E_{2k}$-derivatives are precisely the
terms that appear in the Ramanujan identities. For this reason, and despite
the abuse of terminology, we will occasionally refer to the “vectors”
$\partial_{E_{2k}}$ as being components of the Ramanujan vector field.

##### Shifted Curves

Let us quickly review the manner in which this computation is performed,
albeit in a shifted form of the curve. The coefficient of the quadratic term
in the elliptic curve can be set to zero through a shift of the form

$x\rightarrow x-\frac{1}{3}A_{1}\ .$ (57)

Then, the curve (in terms of the Eisenstein series) takes its Weierstrass
normal form

$y^{2}=\widetilde{p}(x)=x^{3}-\frac{E_{4}}{48}\,x+\frac{E_{6}}{864}\ .$ (58)

The coefficients of the polynomial $\widetilde{p}$ are precisely the
automorphic forms $\widetilde{A}_{k}$ that we encountered in (24). Our goal
will be to determine the action of the Ramanujan vector fields on the period
integrals; this computation is built out of atomic constituents that have the
general form

$\frac{\partial}{\partial
E_{2k}}\oint\frac{x^{\ell}\text{d}x\,}{y}=\oint\frac{\text{d}x\,}{\widetilde{p}y}\left(-\frac{x^{\ell}}{2}\frac{\partial\widetilde{p}}{\partial
E_{2k}}\right)$ (59)

At this stage, a happy consequence of using the shifted polynomials
$\widetilde{p}$ is that we can conclude by construction

$\frac{\partial}{\partial E_{2}}\oint\frac{x^{\ell}\text{d}x\,}{y}=0\ ,$ (60)

i.e., the period integrals do not vary in the direction $\partial_{E_{2}}$ as
the curve carries no dependence on $E_{2}$. For the other components of the
Ramanujan vector field $\partial_{E_{2k}}$ (for $k\in\\{2,3\\}$) and each
independent differential form $\frac{x^{\ell}\text{d}x\,}{y}$ (for
$\ell\in\\{0,1\\}$), we follow a technique of Movasati2012b  and look for
polynomials $\alpha$ and $\beta$ that satisfy the constraint

$-\frac{x^{\ell}}{2}\frac{\partial}{\partial
E_{2k}}\widetilde{p}=\alpha\,\frac{\text{d}\widetilde{p}\,}{\text{d}x\,}+\beta\,\widetilde{p}\
.$ (61)

Once determined, we plug this into the variation in question (59) and after
some elementary manipulations, it is then easy to see that the result of the
variation w.r.t. $E_{2k}$ is simply

$\frac{\partial}{\partial
E_{2k}}\oint\frac{x^{\ell}\text{d}x\,}{y}=\oint\frac{\text{d}x\,}{y}\left(2\frac{\text{d}\alpha\,}{\text{d}x\,}+\beta\right)\
.$ (62)

Using this technique, we find that for the shifted H$(3)$ curve

$\displaystyle\begin{split}\nabla_{\text{R}}\Pi_{1}&=-\frac{E_{2}}{12}\,\Pi_{1}+\Pi_{2}\
,\\\
\nabla_{\text{R}}\Pi_{2}&=-\frac{E_{4}}{144}\,\Pi_{1}+\frac{E_{2}}{12}\,\Pi_{2}\
.\end{split}$ (63)

Alternatively, we may write down the following connection equation

$\nabla_{\text{R}}\,\Pi=\mathcal{A}_{\text{R}}\,\Pi\ ,$ (64)

where

$\mathcal{A}_{\text{R}}=\begin{pmatrix}-\frac{E_{2}}{12}&1\\\
-\frac{E_{4}}{144}&\frac{E_{2}}{12}\end{pmatrix}\ .$ (65)

This computation may be performed in the original (unshifted) basis as well,
and in that case the result is

$A_{\text{R}}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\ ,$ (66)

which agrees with Movasati2012b  up to a sign.

###  4.2 Height Four

The above analysis can be repeated for the Hecke group H$(4)$ by geometrizing
the corresponding Ramanujan identities, which we write below for convenience:

$\displaystyle\begin{split}E_{2}^{\prime}&=\frac{1}{8}\left(E_{2}^{2}-E_{4}\right)\
,\\\ E_{4}^{\prime}&=\frac{1}{2}\left(E_{2}E_{4}-E_{6}\right)\ ,\\\
E_{6}^{\prime}&=\frac{1}{4}\left(3E_{2}E_{6}-2E_{4}^{2}-E_{8}\right)\ ,\\\
E_{8}^{\prime}&=\left(E_{2}E_{8}-E_{4}E_{6}\right)\ .\end{split}$ (67)

As we have discussed at great length, the first step in the geometrization
process is to find the relation between the $A_{k}$ that appear in the curve
and the Eisenstein series associated to H$(4)$. Once that is done, the goal
will be to associate to the Ramanujan identities a vector field on the moduli
space of an elliptic curve, just as we had done in the case of H$(3)$. Since
both these groups are associated to curves that are elliptic, it is natural to
expect that much of our earlier discussion will go through. We begin with the
quartic curve

$y^{2}=p(x)=x^{4}+\sum_{k=1}^{4}x^{4-k}A_{k}\,.$ (68)

Having assumed the constant $c=\frac{1}{4}$, the $A_{k}$ satisfy our anomaly
equation (29) with $m=4$:

$\frac{\partial A_{k}}{\partial E_{2}}=\frac{5-k}{16}A_{k-1}\ .$ (69)

The cuspidal boundary conditions discussed in Section 3.2 give the following
solutions to the anomaly equations

$\displaystyle\begin{split}A_{1}&=\frac{1}{4}E_{2}\ ,\\\
A_{2}&=\frac{3}{128}\left(E_{2}^{2}-E_{4}\right)\ ,\\\
A_{3}&=\frac{1}{1024}\left(E_{2}^{3}-3E_{2}E_{4}+2E_{6}\right)\ ,\\\
A_{4}&=\frac{1}{65536}\left(E_{2}^{4}-6E_{2}^{2}E_{4}+8E_{2}E_{6}+E_{4}^{2}-4E_{8}\right)\
.\end{split}$ (70)

Notice that the above elliptic curve (with this choice of parameters) depends
on the generators of the ring of quasi-modular forms
$\\{E_{2},E_{4},E_{6},E_{8}\\}$. Now that we have derived the curve, we should
consider a canonical basis of differentials associated to the above elliptic
curve. Since both H$(3)$ and H$(4)$ are elliptic curves — each with two
independent cycles — we might think that the earlier choice of basis (46) is
acceptable. This is not true.

While the holomorphic differential is still $\frac{\text{d}x\,}{y}$ — and this
is true for all hyperelliptic curves — the differential
$\frac{x\,\text{d}x\,}{y}$ is no longer a good candidate as it has a simple
pole at infinity, making it a differential of the third kind Broedel:2017kkb .
Instead, a valid differential of the second kind is given by

$\Pi_{2}=\frac{x^{2}\,\text{d}x\,}{y}\ .$ (71)

This choice of differential doesn’t have a pole at infinity, and while it is
not the differential presented in Broedel:2017kkb , the discrepancy is not a
problematic as we are considering a shifted form of the curve, in analogy with
the case of H$(3)$. In conclusion, we will see that a good basis of
differential $1$-forms for an elliptic curve defined by a quartic polynomial
is given by

$\Pi=\left(\begin{array}[c]{c}\frac{\text{d}x\,}{y}\\\\[8.53581pt]
\frac{x^{2}\,\text{d}x\,}{y}\end{array}\right)\ ,$ (72)

Just as before, we can ask what happens to this basis of differential
$1$-forms as we vary $\tau$, the complex structure modulus. The Ramanujan
vector field is

$\displaystyle\text{R}=\frac{1}{2\pi\text{i}}\frac{\text{d}missing\,}{\text{d}\tau\,}$
$\displaystyle=\sum_{k=1}^{4}\left(\frac{1}{2\pi\text{i}}\frac{\text{d}missing\,}{\text{d}\tau\,}E_{2k}\right)\frac{\partial}{\partial
E_{2k}}\ ,$ (73)

and as before, the expression in brackets may be replaced with the Ramanujan
identities, giving

$\displaystyle\begin{split}\text{R}&=\frac{1}{8}\left(E_{2}^{2}-E_{4}\right)\frac{\partial}{\partial
E_{2}}+\frac{1}{2}\left(E_{2}E_{4}-E_{6}\right)\frac{\partial}{\partial
E_{4}}\\\
&\qquad+\frac{1}{4}\left(3E_{2}E_{6}-2E_{4}^{2}-E_{8}\right)\frac{\partial}{\partial
E_{6}}+\left(E_{2}E_{8}-E_{4}E_{6}\right)\frac{\partial}{\partial E_{8}}\
.\end{split}$ (74)

The basis of differential $1$-forms will rotate into themselves under the
action of the Ramanujan vector field. After a shift of the form

$x\rightarrow x-\frac{1}{4}A_{1}\ .$ (75)

the curve (in terms of Eisenstein series) takes the form

$y^{2}=\widetilde{p}=x^{4}-\frac{3E_{4}}{128}\,x^{2}+\frac{E_{6}}{512}\,x+\frac{E_{4}^{2}-4E_{8}}{65536}\
.$ (76)

Employing the technique of solving for polynomials $\alpha$ and $\beta$ allows
us to determine the effect of the Ramanujan vector field on the basis of
differential $1$-forms. We find that

$\displaystyle\begin{split}\nabla_{\text{R}}\Pi_{1}&=-\frac{E_{2}}{4}\,\Pi_{1}\
,\\\
\nabla_{\text{R}}\Pi_{2}&=-\frac{E_{6}}{512}\,\Pi_{1}+\frac{E_{2}}{4}\,\Pi_{2}\
.\end{split}$ (77)

Alternatively, we may write down the following connection equation

$\nabla_{\text{R}}\,\Pi=\mathcal{A}_{\text{R}}\,\Pi\ ,$ (78)

where

$\mathcal{A}_{\text{R}}=\begin{pmatrix}-\frac{E_{2}}{4}&0\\\
-\frac{E_{6}}{512}&\frac{E_{2}}{4}\end{pmatrix}\ .$ (79)

##  5 Halphen Systems Revisited

Our discussion began with an elliptic curve corresponding to the modular
group, whose coefficients we determined via the anomaly equation. In this
section, we use the factorized form of this elliptic curve in (35), with the
$s_{k}$ — which we will refer to as Halphen variables — given by (37), and
determine the relations that bind them.333The reader should be careful to
distinguish between Halphen variables — used in this section, and denoted
$s_{k}$ — and generalized Halphen variables — used in Section 2 and denoted
$t_{k}$. We will refer to these relations as a Halphen system. We start by
reviewing the Halphen system corresponding to H$(3)$ and then do the same for
the Hecke group H$(4)$, introducing a new Halphen system corresponding to this
group.

###  5.1 Halphen Systems for Elliptic Curves

####  5.1.1 Height Three

As we have seen, the elliptic curves corresponding to the modular group admit
at least two presentations, both of which we have encountered:

$\displaystyle\text{Ramanujan}:$ $\displaystyle\qquad
y^{2}=\left(x+\frac{E_{2}}{12}\right)^{3}-\frac{E_{4}}{48}\left(x+\frac{E_{2}}{12}\right)+\frac{E_{6}}{864}\
,$ (80) $\displaystyle\text{Halphen}:$ $\displaystyle\qquad
y^{2}=(x+s_{1})(x+s_{2})(x+s_{3})\ ,$ (81)

where the $s_{k}$ are defined in (37). The first curve allowed us to
geometrize the Ramanujan identities in Section 4. The coefficients of the
Ramanujan curve are combinations of the Eisenstein series $E_{2k}$. Given that
the Ramanujan identities express $E_{2k}^{\prime}$ in terms of combinations of
Eisenstein series, it is natural to ask if similar relations can be found that
express $s_{k}^{\prime}$ in terms of (quadratic) combinations of Halphen
variables. Such relations are well-known Darboux1878 ; Halphen1881  and take
the form:

$\displaystyle\begin{split}s_{1}^{\prime}&=s_{1}s_{2}+s_{1}s_{3}-s_{2}s_{3}\
,\\\ s_{2}^{\prime}&=s_{1}s_{2}-s_{1}s_{3}+s_{2}s_{3}\ ,\\\
s_{3}^{\prime}&=-s_{1}s_{2}+s_{1}s_{3}+s_{2}s_{3}\ .\end{split}$ (82)

The solution to the above equations were already given in (37). Given the
above Halphen system and the equivalence of the curves, it is a
straightforward exercise to derive the Ramanujan identities. This is a strong
consistency check.

There exists elegant way of characterizing this Halphen system due to
Takhtajan:1992qb , which rewrites the above relations as a flow equation

$s_{k}^{\prime}=C_{k\ell}\frac{\partial F}{\partial s_{\ell}}\ ,$ (83)

where $C$ is a $3\times 3$ matrix, and $F$ is a cubic function of the $s_{k}$
as required by considerations of weight. For the modular group, the Halphen
system admits a characterization in terms of the matrix

$C=\begin{pmatrix}+2&-2&-2\\\ -2&+2&-2\\\ -2&-2&+2\end{pmatrix}\ ,$ (84)

and the function

$F=-\frac{1}{2}s_{1}s_{2}s_{3}\ .$ (85)

##### Borcherds-Kač-Moody Algebras and BPS State Counting:

It is interesting to note that the matrix $C$ is the Cartan matrix of a
Borcherds-Kač-Moody (BKM) algebra Gritsenko:1995di , and furthermore, the BKM
algebra with this Cartan matrix also appears in the dyon degeneracy formula of
four dimensional $N=4$ heterotic string theory Cheng:2008fc ;
Govindarajan:2009qt ; Cheng:2008kt . The denominator formula for counting dyon
degeneracy in the toroidally compactified heterotic string theory is given by
the Igusa cusp form $\Phi_{10}$, which is a genus-$2$, weight-$10$ Siegel form
Dijkgraaf:1996it . The walls of marginal stability of this denominator formula
are the walls of the Weyl chambers of the BKM algebra with Cartan matrix given
by (84). On the $z\to 0$ wall the Siegel form degenerates into a product of
two genus-$1$ cusp forms

$\Phi_{10}(\rho,\sigma,z)\sim z^{2}\eta^{24}(\rho)\,\eta^{24}(\sigma)\ .$ (86)

We see here the appearance of the modular discriminant
$\Delta_{3}=\eta^{24}(\tau)$, which is the unique weight-$12$ cusp form of
H$(3)\cong\text{SL}(2,\mathbb{Z})$. We pause to recall that the weight-$2$
Eisenstein series may be written in terms of the discriminant in the following
suggestive form (Doran:2013npa, , Theorem 2(ii)) :

$E_{2}=4\left(s_{1}+s_{2}+s_{3}\right)=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\Delta_{3}\
.$ (87)

Such a relation, as we will see shortly, exists for all Hecke groups. Further,
$E_{2}$ satisfies a non-linear differential equation of order three: the Chazy
equation Takhtajan:1992qb . This suggests a possible connection between the
BKM algebras and the Halphen (equivalently, Chazy) systems associated to the
Hecke groups. In the next subsection we will see another example of this
correspondence.

####  5.1.2 Height Four

It is natural to ask if a similar reformulation of the Ramanujan identities is
possible in terms of a Halphen system for the Hecke group H$(4)$. This would
correspond to a factorization of the quartic curve:

$x^{4}+\sum_{k=1}^{4}A_{k}\,x^{4-k}=\prod_{k=1}^{4}(x+s_{k})\,,$ (88)

where the $A_{k}$ are given in terms of the Eisenstein series in (70) Now, the
$A_{k}$ are the elementary symmetric polynomials

$\displaystyle\begin{split}A_{1}&=s_{1}+s_{2}+s_{3}+s_{4}\ ,\\\
A_{2}&=s_{1}s_{2}+s_{3}s_{2}+s_{4}s_{2}+s_{1}s_{3}+s_{1}s_{4}+s_{3}s_{4}\ ,\\\
A_{3}&=s_{1}s_{2}s_{3}+s_{1}s_{4}s_{3}+s_{2}s_{4}s_{3}+s_{1}s_{2}s_{4}\ ,\\\
A_{4}&=s_{1}s_{2}s_{3}s_{4}\ .\end{split}$ (89)

We claim that a choice of the $s_{k}$ that satisfy these equations is given as
follows Govindarajan2012 :

$\displaystyle\begin{split}s_{1}&=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{2}(2\tau)\
,\\\
s_{2}&=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{3}(2\tau)\
,\\\
s_{3}&=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{3}(\tau)\
,\\\
s_{4}&=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\theta_{4}(\tau)\
.\end{split}$ (90)

We can now start with the Ramanujan identities and trade variables to obtain a
system of differential equations relating $s_{k}^{\prime}$ to quadratic
combinations of the $s_{k}$. That is, we have here a new, four-member Halphen
system; instead of writing out this new Halphen system in the usual way, we
find it profitable to represent it, once again, as a flow equation:

$s_{k}^{\prime}=C_{k\ell}\,\frac{\partial F}{\partial s_{\ell}}\ ,$ (91)

where $C_{k\ell}$ is the matrix

$C=\begin{pmatrix}+2&-1&-1&-1\\\ -1&+2&-1&-1\\\ -1&-1&+2&-1\\\
-1&-1&-1&+2\end{pmatrix}\ ,$ (92)

and the function $F$ is given by

$F=-\frac{2}{3}(s_{1}s_{2}s_{3}+s_{1}s_{2}s_{4}+s_{1}s_{3}s_{4}+s_{2}s_{3}s_{4})\
.$ (93)

The novelty of this system is that it consists of four variables $s_{k}$
instead of the usual three. This is fruitfully contrasted with another
seemingly four-member Halphen-like system Ohyama1996  made up of four
functions $\widetilde{s}_{i}$ that satisfy

$\displaystyle\begin{split}\frac{\text{d}\,}{\text{d}\tau\,}(\widetilde{s}_{1}+\widetilde{s}_{2}+\widetilde{s}_{3})&=\widetilde{s}_{1}\widetilde{s}_{2}+\widetilde{s}_{2}\widetilde{s}_{3}+\widetilde{s}_{1}\widetilde{s}_{3}\
,\\\
\frac{\text{d}\,}{\text{d}\tau\,}(\widetilde{s}_{1}+\widetilde{s}_{3}+\widetilde{s}_{4})&=\widetilde{s}_{1}\widetilde{s}_{3}+\widetilde{s}_{3}\widetilde{s}_{4}+\widetilde{s}_{1}\widetilde{s}_{4}\
,\\\
\frac{\text{d}\,}{\text{d}\tau\,}(\widetilde{s}_{1}+\widetilde{s}_{2}+\widetilde{s}_{4})&=\widetilde{s}_{1}\widetilde{s}_{2}+\widetilde{s}_{2}\widetilde{s}_{4}+\widetilde{s}_{1}\widetilde{s}_{4}\
,\\\
\frac{\text{d}\,}{\text{d}\tau\,}(\widetilde{s}_{2}+\widetilde{s}_{3}+\widetilde{s}_{4})&=\widetilde{s}_{2}\widetilde{s}_{3}+\widetilde{s}_{3}\widetilde{s}_{4}+\widetilde{s}_{2}\widetilde{s}_{4}\
,\end{split}$ (94)

supplemented by the condition

$e^{\frac{4\pi\text{i}}{3}}(\widetilde{s}_{2}\widetilde{s}_{4}+\widetilde{s}_{1}\widetilde{s}_{3})+e^{\frac{2\pi\text{i}}{3}}(\widetilde{s}_{1}\widetilde{s}_{2}+\widetilde{s}_{3}\widetilde{s}_{4})+(\widetilde{s}_{2}\widetilde{s}_{3}+\widetilde{s}_{1}\widetilde{s}_{4})=0\
.$ (95)

The final algebraic equation imposes a quadratic constraint, leading us to
conclude that the $\widetilde{s}_{i}$ are not independent. Further, while this
example is interesting, it lacks a straightforward geometrical interpretation.

##### Another Borcherds-Kač-Moody Algebra:

The matrix (92) is once again the Cartan matrix of a BKM algebra. As was the
case with the modular group, it turns out that the weight-$2$ Eisenstein
series of H$(4)$ may be written as the logarithmic derivative of a cusp form:

$E_{2}=4\left(s_{1}+s_{2}+s_{3}+s_{4}\right)=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\Delta_{4}\,,$
(96)

where $\Delta_{4}=\eta^{8}(\tau)\,\eta^{8}(2\tau)$. This cusp form appears in
the counting of $\frac{1}{2}$-BPS states in the $\mathbb{Z}_{2}$ CHL models
toroidally compactified down to four dimensions. The corresponding Siegel form
which appears as the denominator formula for $\mathbb{Z}_{2}$ CHL dyon
counting is a genus-$2$, weight-$6$ cusp form $\Phi_{6}(\rho,\sigma,z)$
Jatkar:2005bh ; David:2006ji . On the wall of marginal stability this form
decomposes into a product of $\eta^{8}(\rho)\,\eta^{8}(2\rho)$ and
$\eta^{8}(\sigma)\,\eta^{8}(\sigma/2)$. However, the corresponding BKM algebra
has a different Cartan matrix Cheng:2008kt . It would be interesting to find
the denominator formula for the BKM algebra with the Cartan matrix (92) and
check whether its degeneration contains the genus-$1$ cusp form
$\eta^{8}(\tau)\,\eta^{8}(2\tau)$.

###  5.2 Taller Hecke Groups and Weak Halphen Systems

Bolstered by the success of the previous section, we might be tempted to
conjecture that such an $m$-member Halphen system exists for all Hecke groups
H$(m)$. Indeed, it might be natural to propose that the flow function $F$ is
always

$F=\sum_{i\neq j\neq k}^{m}\bar{C}_{ijk}\,s_{i}s_{j}s_{k}\ ,$ (97)

this assignment being the only possibility consistent with considerations of
weight and symmetry. We have been unsuccessful in determining systematically
what the matrix $\bar{C}$ should be for arbitrary height $m$, which in turn
limits our ability to state definitively whether such Halphen systems exist
for the taller Hecke groups.

Let us consider the only other arithmetic Hecke group with finite height,
namely H$(6)$, in order to highlight an interesting point. The hyperelliptic
curve for this case takes the form

$y^{2}=x^{6}+\sum_{k=1}^{6}A_{k}\,x^{6-k}\,.$ (98)

As discussed in detail, the $A_{k}$ can be written in terms of the Eisenstein
series by solving the anomaly equation and using the cuspidal boundary
conditions for $m=6$; in particular, we find as before that
$A_{1}=\frac{1}{4}E_{2}$. By simply comparing the $q$-expansions, we find
that:

$\displaystyle A_{1}$
$\displaystyle=\frac{1}{4\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\left[\theta_{2}(\tau)\theta_{3}(\tau)\theta_{4}(\tau)\theta_{2}(3\tau)\theta_{3}(3\tau)\theta_{4}(3\tau)\right]\
,$ (99)
$\displaystyle=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\left[\eta^{6}(\tau)\,\eta^{6}(3\tau)\right]\
.$ (100)

While this is certainly suggestive, the heavier $A_{k}$ could not be written
in terms of the logarithmic derivatives of the Jacobi $\theta$-constants. For
the non-arithmetic Hecke groups as well, we are faced with the difficulty of
being unable to explicitly define appropriate $s_{k}$, whose elementary
symmetric combinations form the $A_{k}$ that appear naturally in the
corresponding hyperelliptic curves.

In light of these difficulties we turn our attention to a weaker form of the
Halphen system that can be written down for all Hecke groups H$(m)$. The idea
is to invert the relation between the $A_{k}$ and the $E_{2k}$, then consider
$A_{k}^{\prime}$, which can via the Ramanujan identities (12) be re-expressed
solely in terms of the $A_{k}$.

For each Hecke group H$(m)$, we define a _weak_ Halphen systems to be a system
of $m$ ODEs that schematically take the form

$A_{k}^{\prime}=g_{k}(A_{\ell})\ ,$ (101)

where $g_{k}$ is a polynomial in the $A_{\ell}$ and a linear combination of
weight $(2k+2)$ quasi-automorphic forms. Since the generalized Ramanujan
identities are well-defined for all Hecke groups, a weak Halphen system will,
by the same token, be well-defined. In fact, since $A_{k}\propto
E_{2k}+\ldots$ for all $k\leq m$, the weak Halphen system is in fact merely a
restatement of the Ramanujan identities.

We refer to these systems as _weak_ as they only constrain elementary
symmetric combinations of the Halphen variables and not the Halphen variables
themselves. There is no unique way to delineate each individual Halphen
variable $s_{k}$ when presented with their symmetric combinations $A_{k}$.
When it becomes possible to constrain the Halphen variables as well — as in
the specific cases of the $3$\- and $4$-member Halphen systems we encountered
associated to H$(3)$ and H$(4)$ — we will refer to this as a _strong_ Halphen
system.

As a test of consistency, it is easily verified that if one starts with the
weak Halphen system corresponding to H$(m)$ and plugging in the ansatz that
$A_{k}$ is an elementary symmetric combination of $m$ Halphen variables with
weight $2k$, then one (in principle) arrives at a system of equations that
schematically takes the form

$s_{k}^{\prime}=h_{k}(s_{\ell})\ .$ (102)

In the cases $m=3$ and $m=4$ the $h_{k}$ are simply quadratic in the
$s_{\ell}$ — thereby recovering the Halphen systems we encountered in the
previous sections — while in general $h_{k}$ is a _rational_ function of the
$s_{k}$ with weight $4$.

Define the algebraic discriminant $\Delta$ of a hyperelliptic curve in the
usual way:

$\Delta=\prod_{i<j}(s_{i}-s_{j})^{2}\ .$ (103)

Since each of the $s_{k}$ have weight $2$, the discriminant has weight
$2m(m-1)$. Our investigations suggest that in general, $h_{k}$ is the ratio of
an automorphic form of weight $2(m^{2}-m+2)$ and the discriminant of the
hyperelliptic curve. In the cases $m=3$ and $m=4$, this discriminant cancels,
although the same is not true for taller Hecke groups.

The weak Halphen system associated to the modular group has appeared in the
literature Harnad:1999hi ; we have rederived it using our methods, and we find
perfect agreement. As a proof of principle that highlights the fact that all
Hecke groups may be associated to a weak Halphen system, we give below the
weak Halphen system associated to the group H$(5)$:

$\displaystyle\begin{split}A_{1}^{\prime}&=\frac{3}{2}A_{2}\ ,\\\
A_{2}^{\prime}&=6A_{3}\ ,\\\
A_{3}^{\prime}&=-\frac{3}{2}A_{2}^{2}+4A_{1}A_{3}+8A_{4}\ ,\\\
A_{4}^{\prime}&=-A_{2}A_{3}+6A_{1}A_{4}+10A_{5}\ ,\\\
A_{5}^{\prime}&=\frac{3A_{2}^{3}}{250}-\frac{6}{125}A_{1}A_{3}A_{2}+\frac{3A_{3}^{2}}{25}+\left(\frac{12A_{1}^{2}}{125}-\frac{37A_{2}}{50}\right)A_{4}+8A_{1}A_{5}\
.\end{split}$ (104)

One can proceed in similar fashion for all Hecke groups and derive such weak
Halphen systems in a systematic manner. As we have emphasized, whether one is
able to write simpler (quadratic, as opposed to rational) systems for the
(weight two) zeros of the hyperelliptic curve remains an outstanding problem.

##  6 Chazy Equations

In this section we derive the Chazy equation and its higher-order analogues,
each of them canonically associated to a set of Ramanujan identities that are
in a one-to-one correspondence with the Hecke groups.

We will also show that like the Chazy equation, its higher-order
generalizations also possess the Painlevé property. In particular, the Chazy
equation and its generalizations possess negative resonances, which in turn
naively imply the instability of linear perturbations about its solutions. We
will show that the negative resonances vanish “on-shell,” i.e., when the Chazy
equation is satisfied. This demonstrates the stability of these solutions
against linear perturbations.

Let us quickly review the relation between the Chazy equation and the
Ramanujan identities corresponding to the modular group. The Ramanujan
identities for H$(3)$ take the following form:

$\displaystyle E_{2}^{\prime}$
$\displaystyle=\frac{1}{12}\left(E_{2}^{2}-E_{4}\right)\ ,$ (105)
$\displaystyle E_{4}^{\prime}$
$\displaystyle=\frac{1}{3}\left(E_{2}E_{4}-E_{6}\right)\ ,$ (106)
$\displaystyle E_{6}^{\prime}$
$\displaystyle=\frac{1}{2}\left(E_{2}E_{6}-E_{4}^{2}\right)\ .$ (107)

A well-known strategy (outlined for example in Chanda:2015baj ) consists of
using the above equations to find a differential equation satisfied by the
weight-$2$ Eisenstein series $E_{2}$, which we will denote by $y$. This is
done straightforwardly — differentiate (105) and plug in (106), and so on, all
the while systematically eliminating heavier Eisenstein series — and we find
the following equation:

$2y^{(3)}-2yy^{\prime\prime}+3y^{\prime 2}=0\ .$ (108)

which is almost, but not quite, the Chazy equation. The Chazy equation is a
nonlinear third-order differential equation which takes the form

$C_{3}\quad:\quad y^{(3)}-2yy^{\prime\prime}+3(y^{\prime})^{2}=0\ .$ (109)

The difference between these two equations just corresponds to different
conventions for the elliptic nome; the derivatives in (108) are with respect
to $\log q=2\pi\text{i}\,\tau$, while those in (109) are with respect to, say,
some variable $t$. It may be readily verified that the variable change

$q=\text{e}^{2t}$ (110)

allows us to interpolate between the two equations. Thus (108) is fully
equivalent to the Chazy equation (109), and we will refer to the latter as
$C_{3}$. In order to make contact with the Chazy equation for H$(3)$, we will
henceforth use the accent ′ to denote

$y^{\prime}=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}y=2q\frac{\text{d}\,}{\text{d}q\,}y\
.$ (111)

###  6.1 Higher-Order Chazy Equations

Now that we have a procedure for deriving the Chazy equation corresponding to
Ramanujan’s identities, it is natural to ask: if the Ramanujan identities
admit a generalization to the case of Hecke groups, what do the corresponding
Chazy equations $C_{m}$ look like?

##### An Example: Height Four

Let us consider the Ramanujan identities (67) for the group H$(4)$. Using the
symbol $y$ to denote $E_{2}$ in analogy with the case of the modular group,
and following the procedure outlined in the previous section yields a new,
fourth-order cousin of the Chazy equation:444This equation was independently
derived in Govindarajan2012 .

$C_{4}\quad:\quad
y^{(4)}-5y^{(3)}y+6y^{2}y^{\prime\prime}-9y\left(y^{\prime}\right)^{2}+6y^{\prime}y^{\prime\prime}=0\
.$ (112)

The structure of the Ramanujan identities is uniform across all heights, so it
is natural to expect that the order of the differential equation matches the
number of generalized Ramanujan identities there are, which is the same as the
height $m$ of the Hecke group H$(m)$ in question.

##### Chazy Equations for H$(m)$

By following the same logic one can construct higher-order analogues of the
Chazy equation, corresponding to taller Hecke groups. Below, we list a couple
of members of this hierarchy, for future reference.

$\displaystyle C_{5}\quad:\quad$ $\displaystyle
75y^{(5)}-675y^{(4)}y+1932y^{(3)}y^{2}-1764y^{3}y^{\prime\prime}+460\left(y^{\prime\prime}\right)^{2}$
$\displaystyle\quad+2646y^{2}\left(y^{\prime}\right)^{2}-168\left(y^{\prime}\right)^{3}+y^{\prime}\left(310y^{(3)}-2772yy^{\prime\prime}\right)=0\
,$ (113) $\displaystyle C_{6}\quad:\quad$ $\displaystyle
9y^{(6)}-126y^{(5)}y+632y^{(4)}y^{2}-1344y^{(3)}y^{3}-624y\left(y^{\prime\prime}\right)^{2}$
$\displaystyle\quad+384y\left(y^{\prime}\right)^{3}+\left(156y^{(3)}+1024y^{4}\right)y^{\prime\prime}+\left(y^{\prime}\right)^{2}\left(-192y^{\prime\prime}-1536y^{3}\right)$
$\displaystyle\quad\quad+y^{\prime}\left(-6y^{(4)}-272y^{(3)}y+1920y^{2}y^{\prime\prime}\right)=0\
,$ (114)

Note that each term in the Chazy equation corresponding to Hecke group H$(m)$
has weight $2m+2$. This will be important in what follows.

###  6.2 Hecke Orbits

In the previous section, we have demonstrated constructively that

$\displaystyle y$ $\displaystyle=\,E_{2}\ ,$ (115)
$\displaystyle=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\Delta_{m}\
,$ (116)

satisfies a higher-order Chazy equation, which for Hecke group H$(m)$ is a
non-linear ordinary differential equation of order $m$. In (116), we have used
the representation of $E_{2}$ in terms of the ‘automorphic’ discriminant
$\Delta_{m}$ following (Doran:2013npa, , Theorem 2(ii)), and in doing so
arrive at a generalization of (Takhtajan:1992qb, , eq. 8). We conclude in a
similar fashion that the discriminant $\Delta_{m}$ plays the role of the
$\tau$-function for the Chazy equation $C_{m}$.

It is natural to wonder what the relationship between $\Delta$, the
‘algebraic’ discriminant of the (hyper-)elliptic curve which we have
determined earlier to have weight $w=2m(m-1)$, and the automorphic
discriminant $\Delta_{m}$ is. From (Doran:2013npa, , Theorem 2(ii)) we see
that the weight of $\Delta_{m}$ is

$w_{m}=\begin{cases}4m&\text{for}\ m\ \text{odd}\ ,\\\ 2m&\text{for}\ m\
\text{even}\ .\end{cases}$ (117)

It is easy to check that the algebraic and automorphic discriminants are
related as

$\left(\Delta\right)^{w_{m}}\propto\left(\Delta_{m}\right)^{w}\ ,$ (118)

which serves as a strong, non-trivial consistency check on the web of
relationships we have uncovered. In particular, it means we are using the
correct (hyper-)elliptic curve, and justifies the use of cuspidal boundary
conditions.

In this section, we show that orbits of $y$ in H$(m)$ also solve the same
differential equation. That is, when $y$ solves the higher-order Chazy
equation, then so does

$\widetilde{y}=E_{2}(\gamma\cdot\tau)\ ,$ (119)

where $\gamma\in\text{H}(m)$. When coupled with the solution (116), this
provides for a general solution to the Chazy equation $C_{m}$.

The strategy of our proof is inspired by (Takhtajan:1992qb, , Lemma 3) and
proceeds by identifying $C_{m}$ as an automorphic form. Let us see how this
works in the case of the usual Chazy equation corresponding to H$(3)$, thereby
reviewing (Takhtajan:1992qb, , Theorem 2). The expression

$Z=E_{2}^{\prime}-\frac{1}{6}E_{2}^{2}\ ,$ (120)

is a weight-$4$ modular form. We want to generate heavier forms, and a well-
known procedure to do this is to use the Ramanujan-Serre derivatives
Zagier2008  that send

$\text{D}:\mathfrak{m}_{k}\rightarrow\mathfrak{m}_{k+2}\ .$ (121)

Explicitly, this derivative takes the form (with our normalizations):

$\text{D}=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}-\frac{k}{3}E_{2}\
.$ (122)

We now act with D on $Z$ until we get a weight-$8$ form, since each term in
the Chazy equation we are interested in has weight-$8$. There are essentially
two terms we can write down, and we consider a linear combination of them:

$a\,\text{D}^{2}Z+b\,Z^{2}\ ,$ (123)

which is guaranteed to be an automorphic form of weight-$8$. Finally, we can
check that

$C_{3}\propto\text{D}^{2}Z+16\,Z^{2}\ .$ (124)

We have thus demonstrated that $C_{3}$ is a weight-$8$ automorphic form; thus,
under the action of a $\gamma\in\text{H}(3)$, $C_{3}$ will transform as

$C_{3}\mapsto(c\tau+d)^{8}\,C_{3}\ .$ (125)

With this, we can conclude that all H$(3)\cong\text{SL}(2,\mathbb{Z})$ orbits
of $y$ solve the Chazy equation. More generally, for H$(m)$ we will define

$Z=E_{2}^{\prime}-\left(\frac{m-2}{2m}\right)E_{2}^{2}\ ,$ (126)

and invoke a result of Raman:2018owg  that defines an appropriate analogue of
the Ramanujan-Serre derivatives for the Hecke group H$(m)$. In our present
normalization this takes the form

$\text{D}=\frac{1}{\text{i}\pi}\frac{\text{d}\,}{\text{d}\tau\,}-k\left(\frac{m-2}{m}\right)E_{2}\
.$ (127)

It will turn out that under the action of a $\gamma\in\text{H}(m)$, $C_{m}$
will transform as

$C_{m}\mapsto(c\tau+d)^{2m+2}\,C_{m}\ .$ (128)

This can be explicitly verified for relatively short Hecke groups. For
example, we find

$\displaystyle C_{4}$ $\displaystyle\propto\text{D}^{3}Z+48\,Z\,\text{D}Z\ ,$
(129) $\displaystyle C_{5}$ $\displaystyle\propto
3\,\text{D}^{4}Z+208\,Z\,\text{D}^{2}Z+160\left(\text{D}Z\right)^{2}+768\,Z^{3}\
,$ (130) $\displaystyle C_{6}$
$\displaystyle\propto\text{D}^{5}Z+96\,Z\,\text{D}^{3}Z+192\,\text{D}^{2}Z\,\text{D}Z+2048\,Z^{2}\,\text{D}Z\
,$ (131)

and so on. In each of these cases, the proof goes through as before.

It remains for us to understand why it is reasonable to expect that $C_{m}$ is
zero. Once again, let us look to H$(3)$ as a guiding example. The space of
weight-$8$ forms is $1$-dimensional. The above procedure generates two
weight-$8$ forms. It follows that there must be a relation between them, and
so some linear combination of the two must vanish. This combination is
precisely the Chazy equation.

More generally, we know that for the Hecke group H$(m)$ there are $m$ of
generators of the ring of quasi-automorphic forms
$\mathbb{C}[E_{2},\cdots,E_{2m}]$. Thus, at weight-$(2m+2)$ we will have no
new forms, and all forms that span this vector space will be products of
lighter forms and their modular covariant derivatives. Any weight-$(2m+2)$
form must be expressible as a linear combination of these
products/derivatives. We may thus conclude on general grounds that these Chazy
equations are statements of linear dependence.

##  7 Painlevé Analysis

We now test the Chazy equations for the Painlevé property, which roughly
corresponds to a statement that the only movable singularities of the
differential equations are poles. We begin with a brief primer on stability
analysis and go on to apply the methods of Conte:1989pi ; Conte:1993pp  to the
higher-order Chazy equations.

###  7.1 A Primer on Stability

Consider a non-linear ordinary differential equation of order $n$. The
analysis due to Painlevé Painleve1902  involves expanding the independent
variable near the singular point — this “nearness” is parametrized by a small
parameter $\alpha$ — and expanding the dependent variable as a formal power
series in $\alpha$. The method due to Kowalevskaya Sophie1890  on the other
hand involves a Laurent expansion of the dependent variable around the
singular point, leading to algebraic equations for the coefficients.

These tests were subsequently refined in Ablowitz:1980ca , where the methods
of Kowalevskaya and Painlevé were combined. It was demonstrated that the
Painlevé property is a necessary condition for the differential equation to be
integrable. This method proposes that the local solution around the
singularity is of the the Frobenius form:

$y(\tau)=\sum_{i=0}^{\infty}y_{i}\ (\tau-\tau_{0})^{i-a}\ .$ (132)

In Ablowitz:1980ca  a set of criteria were identified, and differential
equations that satisfied all of them were said to possess the Painlevé
property. The first of these criteria is that $a$ is a positive integer, which
in effect reduces the solution to the Laurent series form. This part of the
test amounts to studying the indicial equation, and to go further, one
linearizes the equation around the movable singularity. Let the original
equation be

$K\big{[}y\big{]}=0\,,$ (133)

where $K[y]$ is a polynomial function of $y$, $y^{\prime}$, etc. up to the
$n^{\text{th}}$ derivative, then the linearized equation is obtained as

$\frac{\text{d}\,}{\text{d}\epsilon\,}K[y+\epsilon\,w]\bigg{|}_{\epsilon=0}=0\
.$ (134)

Substituting the Frobenius form (132) into the above equation, one can
equivalently write the linearized equation as

$K_{\text{L}}\left[\frac{\text{d}\,}{\text{d}\tau\,}\right]w(\tau)=0\,,$ (135)

for some polynomial $K_{\text{L}}$ whose coefficients are given by the
$y_{i}$. We assume the following ansatz for the linearized solution:

$w(\tau)=w_{0}\ (\tau-\tau_{0})^{-a}+\sum_{i=1}^{\infty}w_{i}\
(\tau-\tau_{0})^{i-a}\ .$ (136)

This ansatz supposes that the linearized equation also has a Frobenius series
solution. For a linear differential equation of order $n$, the Frobenius
analysis tells us that the series is a solution if the coefficient of
$(\tau-\tau_{0})^{i-a-n}$ vanishes for each $i$. This condition gives a
recursion relation, which determines $w_{i}$ in terms of $w_{j}$ with $j<i$,
and the parameters appearing in the linear differential equation. We write
this equation as

$P(i)w_{i}-f\left(\tau_{0},a,y_{k};w_{1},\cdots,w_{i-a}\right)=0\ ,$ (137)

Let $r_{1}\leq r_{2}\leq\cdots\leq r_{n}$ be roots of the polynomial $P(i)$;
then we can determine $w_{i}$ in terms of $w_{j}$ (with $j<i$) as long as
$i\neq r_{1}$. When $i=r_{1}$ then we have $P(i)=0$ and we end up with a
condition $f(\tau_{0},a,y_{k},w_{1},\cdots,w_{i-a})=0$. If this condition is
satisfied then $w_{r_{1}}$ is indeterminate and the linearized equation is
said to pass the Painlevé test. If, however,
$f(\tau_{0},a,y_{k},w_{1},\cdots,w_{i-a})\not=0$ then we have a logarithmic
branch; in this case we say that the linearized equation does not possess the
Painlevé property.

If the linearized equation passes the Painlevé test at $i=r_{1}$, we continue
the test for $i=r_{k}$ for $k=\\{2,\cdots,n\\}$. The linearized equation is
said to possesses the Painlevé property if and only if it passes the Painlevé
test at each root of $P(i)$. These criteria are collectively known as the
Ablowitz-Ramani-Segur (ARS) stability conditions Ablowitz:1980ca . If the
linearized equation satisfies the ARS stability conditions then we say that
the original non-linear equation possesses the Painlevé property.

The procedure we have outlined above assumes that the linearized equation has
positive resonances, i.e., the solutions to the linearized equation have
singularities that are less severe than those of the nonlinear equation. This
is not true in general; in fact, the Chazy equation is a well-known counter-
example. While the Chazy equation passes the first condition of having only
movable poles, the linearized equation turns out to have negative resonances,
and in these situations the techniques of Ablowitz:1980ca  are insufficient.
Luckily, the analysis of negative resonances has been carried out in
Conte:1989pi ; Conte:1993pp  and has the added advantage of being applicable
to non-linear partial differential equations, thereby subsuming the analysis
of Weiss:1983pt .

We now turn to the notion of stability for systems with negative resonances.
In a nutshell, the argument of Conte:1989pi ; Conte:1993pp  is that if the
coefficients of the negative resonances vanish identically when the zeroth
order non-linear equation is satisfied, we are permitted to conclude that the
equation possesses the Painlevé property.

###  7.2 The Painlevé Property

We will now consider the Chazy equations $C_{m}$, examples of which are
presented in (109) and eqs. (112) to (114). These equations are nonlinear
ordinary differential equations of order $m$. All these equations satisfy the
first criterion of ARS with $a=1$, implying that every nonlinear equation has
a solution with simple movable poles. Before discussing the general case we
begin by reviewing the analysis of the original Chazy equation $C_{3}$ in
(109), following Fordy:1991nd .

In order to illustrate the procedure, we first seek a solution to the equation
in the Frobenius form, i.e., (132). The indicial equation gives $a=1$ for the
“maximal” case: the case in which all the terms in the Chazy equation scale in
the same fashion as we scale $\tau\to\lambda\tau$ and $y\to\lambda^{a}y$. The
integrality of $a$ ensures that the Chazy equation passes the first criterion
of ARS. We then proceed to determine the coefficients $y_{i}$ by recursively
solving the equation (108). Some low order coefficients are

$y_{0}=-6\ ,\quad y_{1}=y_{2}=y_{3}=0\ .$ (138)

To see if this solution is stable against perturbation we substitute
$y(\tau)=y_{s}(\tau)+\epsilon\,w(\tau)$ in the Chazy equation, where
$y_{s}(\tau)$ is the solution to the Chazy equation. The linearized equation
is obtained by picking up terms linear in $\epsilon$, as in (134), and we find

$6y^{\prime}(\tau)w^{\prime}(\tau)-2y^{\prime\prime}(\tau)w(\tau)-2y(\tau)w^{\prime\prime}(\tau)+w^{\prime\prime\prime}(\tau)=0\
.$ (139)

Substituting the ansatz (136) into this equation, one finds that the
linearized solutions of equation (139) has poles of order higher than those of
the solution to the Chazy equation, in particular poles of order two and three
in addition to the usual simple pole. As discussed earlier, this violates the
criterion of Ablowitz:1980ca , which implicitly assumes that the resonances
are less singular than the original solution. In order to perform the
stability analysis we now outline a strategy to circumvent this difficulty
Fordy:1991nd . This involves a Frobenius expansion of the solution written in
terms of a function which reflects the fact that the solution has movable
poles. This is done in two steps.

First we define a function $\phi(\tau)$, which parametrizes the singular
manifold when $\phi(\tau)=0$. The solution, however, is written in terms of a
Frobenius series in another function $\chi(\tau)$, related to $\phi(\tau)$ as
follows:

$\chi(\tau)=\frac{2\phi\,\phi^{\prime}}{2(\phi^{\prime})^{2}-\phi\phi^{\prime\prime}}\,.$
(140)

This provides us with a germ which is formally independent of the function
defining the singular manifold. The function $\chi(\tau)$ satisfies a Ricatti
type equation:

$\chi^{\prime}(\tau)=1-\frac{1}{2}S\chi^{2}(\tau)\ ,$ (141)

where the Schwarzian $S$ is defined to be

$S=\frac{\phi^{\prime\prime\prime}(\tau)}{\phi(\tau)}-\frac{3}{2}\left(\frac{\phi^{\prime\prime}(\tau)}{\phi^{\prime}(\tau)}\right)^{2}\,.$
(142)

We now make the following ansatz for the leading order solution:

$y(\tau)=\sum_{i=0}^{\infty}y_{i}\ \left(\chi(\tau)\right)^{i-a}\,,$ (143)

and find that $a=1$ as before but now find the following solutions for the
coefficients:

$y_{0}=-6\,,\quad y_{1}=0\,,\quad y_{2}=S\,,\quad y_{3}=0\,,$ (144)

and so on. Note that we now get a non-zero value for $y_{2}$, unlike the
earlier (naive) Laurent expansion. For the next-to-leading order we now use
the following ansatz:

$w(\tau)=\sum_{i=0}^{\infty}w_{i}\ (\chi(\tau))^{i-b}\ .$ (145)

Substituting this in the linearized Chazy equation (139) gives $b=\\{3,2,1\\}$
— which shows the existence of higher-order poles for the solution to the
linearized equation compared to the solution of the Chazy equation itself.
However, when we substitute the coefficients $y_{i}$ in (144) — in other
words, go “on shell” — we find that the coefficients of the resonances at
$b=3$ and $b=2$ vanish identically. This implies in turn that these resonances
do not destabilise the solution of the Chazy equation.

We will now carry out a similar analysis for the Chazy equation $C_{4}$ (see
(112)) associated to the Hecke group H$(4)$. The solution to the indicial
equation again gives $a=1$. Using the leading-order ansatz (143), we find the
following results for the coefficients $y_{i}$:

$y_{0}=-4\ ,\quad y_{1}=0\ ,\quad y_{2}=\frac{2}{3}S\ ,\quad y_{3}=0\ ,\ $
(146)

and so on. The linearized system for the stability analysis takes the form

$6w^{\prime}(\tau)\left(-3y(\tau)y^{\prime}(\tau)+y^{\prime\prime}(\tau)\right)+6w^{\prime\prime}(\tau)\left(y^{2}(\tau)+y^{\prime}(\tau)\right)-5y(\tau)w^{\prime\prime\prime}(\tau)\\\
+w(\tau)\left(-9(y^{\prime}(\tau))^{2}+12y(\tau)y^{\prime\prime}(\tau)-5y^{\prime\prime\prime}(\tau)\right)+w^{(4)}(\tau)=0\
.$ (147)

Once again, we use the ansatz (145) and find that this equation has four
resonances $b=\\{4,3,2,1\\}$. The coefficients of these resonances depend on
$y_{i}$ and we find that on the solution of the H$(4)$ Chazy equation in
(146), the coefficients of the resonances at $b=4$, $b=3$, and $b=2$ vanish
identically. We may then conclude that the Chazy equation corresponding to the
Hecke group H$(4)$ possesses the Painlevé property.

We have carried out this analysis for the Chazy equations corresponding the
Hecke groups H$(m)$ for $3\leq m\leq 10$ and we find that all of them possess
the Painlevé property. The analysis of Conte:1989pi ; Conte:1993pp  leads us
to conclude that the leading order ansatz (143) satisfies the Chazy equation
$C_{m}$ for the following values of the coefficients:

$\displaystyle
y(\tau)=\frac{m}{m-2}\left(-\frac{2}{\chi(\tau)}+\frac{S}{3}\chi(\tau)+\frac{2S^{2}}{45}\chi(\tau)^{3}+\frac{11S^{3}}{945}\chi(\tau)^{5}+\cdots\right)\,.$
(148)

The next-to-leading order ansatz (145) shows that the Chazy equation has
negative resonances, with $b\in\\{1,2,\cdots,m\\}$ for $C_{m}$. The number of
resonances is equal to the order of the linearized equation and therefore the
set of resonances is maximal. Exactly as we saw in the case of $C_{3}$ and
$C_{4}$, we find that when evaluated on the leading order solution, the
coefficients of these resonances vanish identically, guaranteeing that these
equations have the Painlevé property. More generally, we expect that _all_
higher-order Chazy equations possess the Painlevé property, which in turn
raises the interesting possibility that they may be integrable.

It is straightforward to see that by a simple $m$-dependent rescaling, the
solution $y(\tau)$ of the Chazy equation for any of the Hecke groups H$(m)$
shows a universal singular behaviour, i.e., the coefficients of the Laurent
expansion in $\chi(\tau)$ are independent of $m$. This universality hints at
the possibility that $C_{m}$ form an integrable hierarchy. At this point,
however, more work is required to ascertain the integrability of these
equations.

##  8 Summary and Discussion

The ring of quasi-modular forms of SL$(2,\mathbb{Z})$ is generated by the
Eisenstein series $\\{E_{2},E_{4},E_{6}\\}$. In Ramanujan1916 , a set of
identities relating $\tau$-derivatives of these generators to their various
combinations were determined and in Raman:2018owg , analogues of these
identities for cusp forms were conjectured for all Hecke groups with finite
heights. Our work here is an attempt to further explore this automorphic
structure. Our investigations in this paper have proceeded along a number of
complementary directions:

  1. (i)

We provided a proof of the Ramanujan identities for Eisenstein series
corresponding to taller Hecke groups, which for any finite $m$ and $2\leq
k\leq m$ take the form

$E_{2k}^{\prime}=\frac{k}{2}\left(\frac{m-2}{m}\right)E_{2}E_{2k}-\left(\frac{k-2}{2}\right)E_{4}E_{2k-2}-\left(\frac{m-k}{m}\right)E_{2k+2}\,.$

  2. (ii)

We identified an algebraic curve naturally associated to each Hecke group
H$(m)$

$y^{2}=x^{m}+\sum_{k=1}^{m}x^{m-k}\,A_{k}\ ,$

whose coefficients $A_{k}$ are determined by an anomaly equation

$\frac{\partial A_{k}}{\partial E_{2}}=\frac{(m-k+1)}{4m}\,A_{k-1}\ ,$

and supplemented by a natural choice of boundary conditions

$\lim_{\tau\rightarrow\text{i}\infty}A_{k}=O\left(q^{\lfloor
k/2\rfloor}\right)\ .$

These boundary conditions naturally relate the algebraic discriminant $\Delta$
of the algebraic curve with the automorphic discriminant $\Delta_{m}$ defined
independently in Doran:2013npa  via the relation

$\left(\Delta\right)^{w_{m}}\propto\left(\Delta_{m}\right)^{w}\ .$

  3. (iii)

We used these algebraic curves to explore the geometrical interpretation of
the Ramanujan identities as a vector field on the moduli space of the
algebraic curve.

  4. (iv)

We outlined a systematic derivation of Halphen systems, used it to uncover a
new $4$-member Halphen system, and introduced the notion of a ‘weak’ Halphen
system.

  5. (v)

We showed that quasi-modular forms $E_{2}$ corresponding to the Hecke group
H$(m)$ and its corresponding orbits satisfy a higher-order analogue of the
Chazy equation, whose construction we outline explicitly. Here $E_{2}$ admits
a representation in terms of an automorphic discriminant

$E_{2}=\frac{1}{2\pi\text{i}}\frac{\text{d}\,}{\text{d}\tau\,}\log\Delta_{m}\
,$

thus playing the role of a $\tau$-function for the higher-order Chazy
equation.

  6. (vi)

We showed that the higher-order Chazy equations we constructed possessed the
Painlevé property.

We now list some future directions for research motivated by these
developments.

##### Integrability

Our work may be viewed as being related to the theory of integrable systems in
more than one way. For the anomaly equation to specify the coefficients of the
(in general hyperelliptic) curve alluded to above, we must in addition to
“integrating in” the dependence on $E_{2}$ supply a principle that fixes the
purely automorphic pieces. While we find that the cuspidal boundary conditions
introduced in Section 3.2 are appropriate for our purposes, we find in
Appendix B that another consistent choice of boundary conditions is
reminiscent of relations stemming from the theory of microdifferential
operators Ohta1988 . Additionally, the demonstration that the higher-order
Chazy equations possess the Painlevé property hints at an underlying
integrable system.

##### Supersymmetric Gauge Theories

We already observed in the introduction that Hecke groups appeared as duality
groups acting on the low-energy effective couplings in certain supersymmetric
gauge theories and on a special loci of the Coulomb moduli space. We expect
the low energy effective action and other calculable quantities to be
expressible in terms of the automorphic forms discussed in this work. While
this has been done for arithmetic Hecke groups in Ashok:2015cba ;
Ashok:2016oyh  it would be interesting to carry it out for the other Hecke
groups. A more ambitious goal would be to understand the manner in which the
resummation program may be carried out _away_ from these special loci; in this
case, the results of these studies would serve as consistency checks.

##### BPS State Counting

Arithmetic Hecke groups arise in the context of BPS state counting in four
dimensional $N=4$ supersymmetric string theories Dabholkar:2005dt . This class
of string theories is obtained via toroidal compactifications of heterotic
string theory or of the CHL models, the latter being $\mathbb{Z}_{N}$
asymmetric orbifolds of heterotic string theory or equivalently K3
compactifications of type IIA string theory with asymmetric orbifold
Chaudhuri:1995dj . The arithmetic Hecke groups appear for $\mathbb{Z}_{N}$
orbifolds with $N=1,\ 2,\ 3,$ and $4$, which correspond in turn to the Hecke
groups with heights $3,\ 4,\ 6,$ and $\infty$ respectively. In fact, there
exists a wider class of CHL models whose S-duality group contains the Fricke
involution $\tau\to-\frac{1}{N\tau}$, in addition to the subgroup of
SL$(2,\mathbb{Z})$ Persson:2015jka . Among them, those which are called self-
dual are invariant under the Fricke involution. For these models the
$\frac{1}{2}$-BPS state counting formula is given by $\eta$-products which are
balanced cycle shapes Dummit:1985 . The cycle shapes $1^{24}$ and $1^{8}2^{8}$
appear as discriminants in the solution to the Chazy equation and its
generalization. It would nice to explore this relation further. When the CHL
model is not self-dual under the Fricke involution it gives us a map between
different CHL models. In this case the relevant modular forms are not
$\eta$-products but $\eta$-quotients. It would be interesting to explore the
role of the Hecke groups in this context. The cycle shapes and the
$\eta$-quotients are known to be related to certain sporadic groups. The
relation between the automorphic forms of the Hecke groups and the cycle
shapes dictated by the sporadic groups may unravel new connections between
these two fields.

##  Appendix A Generalized Halphen System and Fourier Expansions

The generalized Halphen system was introduced in (3). In this section, we
provide explicit solutions to this system of differential equations following
(Doran:2013npa, , Theorem 3). In terms of the parameters $(a,b,c)$ introduced
in (4), we have the solution

$\displaystyle\begin{split}t_{1}(\tau)&=\frac{1}{\alpha_{m}}\,(a-1)\,z\,Q(z)\,F(1-a,b,1;z)\,F(2-a,b,2;z)\
,\\\ t_{2}(\tau)-t_{1}(\tau)&=\frac{1}{\alpha_{m}}\,Q(z)\,F^{2}(1-a,b,1;z)\
,\\\ t_{3}(\tau)-t_{1}(\tau)&=\frac{1}{\alpha_{m}}\,z\,Q(z)\,F^{2}(1-a,b,1;z)\
.\end{split}$ (149)

The function $F$ is the Gauss hypergeometric function ${}_{2}F_{1}$ while the
function $Q(z)$ is given by

$Q(z)=\frac{\text{i}\pi(1-b)}{2\sin(\pi b)\,\sin(\pi a)}\,(1-z)^{b-a}\,.$
(150)

The parameter $z$ is related to the hauptmodul $j$ of the Hecke group H$(m)$
by the formula:

$z=\frac{1}{1-J}\ .$ (151)

Here we have rescaled the solutions in Doran:2013npa  by an overall constant
$\alpha_{m}$, which we will fix momentarily. For the Hecke group H$(m)$ we now
work out the Fourier expansions of the solutions of the generalized Halphen
equations $t_{k}$ near the cusp at $\text{i}\infty$. These, in turn, determine
the $q$-expansions for the Eisenstein series $E_{2k}$ of the Hecke group via
(8).

An important ingredient in the Fourier expansion of the solutions to the
generalized Halphen system is the Fourier expansion of the hauptmodul
corresponding to the Hecke group in question. This is obtained by solving a
Schwarzian differential equation order-by-order about the point
$\tau=\text{i}\infty$ following the prescriptions in Doran:2013npa ;
Raman:2018owg . This yields:

$J=\frac{1}{dq}+\frac{4+3m^{2}}{8m^{2}}+\frac{dq}{1024m^{2}}(69m^{4}-8m^{2}-48)+\frac{d^{2}\left(27m^{6}-116m^{4}+16m^{2}+64\right)q^{2}}{3456m^{6}}+\ldots$
(152)

where $d$ is defined following Doran:2013npa . First define the integers
$a^{\prime},b^{\prime},c^{\prime},d^{\prime}$ such that

$\frac{a^{\prime}}{b^{\prime}}=\frac{3m-2}{4m}\qquad\frac{c^{\prime}}{d^{\prime}}=\frac{3m+2}{4m}\
.$ (153)

Then $d$ is defined to be the following product:

$d=b^{\prime}\,d^{\prime}\prod_{k=1}^{b^{\prime}-1}\left(2-2\cos\frac{2\pi
k}{b^{\prime}}\right)^{-\frac{1}{2}\cos\frac{2\pi
a^{\prime}}{b^{\prime}}}\prod_{\ell=1}^{d^{\prime}-1}\
\left(2-2\cos\frac{2\pi\ell}{d^{\prime}}\right)^{-\frac{1}{2}\cos\frac{2\pi
c^{\prime}}{d^{\prime}}}\,.$ (154)

For the arithmetic cases of $m=3,4,$ and $6$, $d$ takes integer values and is
equal to $1728,256,$ and $108$ respectively. We now demand that the expansion
for $t_{2}(\tau)$ begins with unit coefficient; this uniquely fixes the
coefficient $x_{m}$ to be

$\alpha_{m}=-\frac{(3m+2)\sec\frac{\pi}{m}}{8m}\,.$ (155)

With our conventions now made fully explicit, it is easy to check that (149)
solves the generalized Halphen system. We can now use (8) and (9) to construct
Fourier expansions of the Eisenstein series we use in this paper. For example,
a few light Eisenstein series have the following Fourier expansions:

$\displaystyle\begin{split}E_{2}&=1-\frac{(m-2)^{2}}{8m^{2}}(qd)-\frac{(m(m+12)-44)(m-2)^{2}}{512m^{4}}(qd)^{2}+\cdots\
,\\\
E_{4}&=1+\frac{\left(m^{2}-4\right)}{4m^{2}}(qd)+\frac{\left(7m^{4}-88m^{2}+240\right)}{256m^{4}}(qd)^{2}+\cdots\
,\\\
E_{6}&=1-\frac{\left(m^{2}+12\right)}{8m^{2}}(qd)+\frac{\left(-31m^{4}+152m^{2}+912\right)}{512m^{4}}(qd)^{2}+\cdots\
,\\\
E_{8}&=1-\frac{\left(m^{2}+4\right)}{2m^{2}}(qd)-\frac{\left(m^{4}-168m^{2}-368\right)}{128m^{4}}(qd)^{2}+\cdots\
.\end{split}$ (156)

##  Appendix B Sato Boundary Conditions

In this section we present an alternative choice of boundary conditions that
is fully compatible with the modular anomaly equations, in addition to also
possessing some interesting connections to the theory of integrable systems
Ohta1988 . We begin with the hyperelliptic curve associated to H$(m)$:

$y^{2}=x^{m}+\sum_{k=2}^{m}A_{k}x^{m-k}\ ,$ (157)

and recall that the anomaly equation naturally arose due to the structure of
$E_{2}$ dependence of the $A_{k}$, governed by the equation:

$\frac{\partial A_{k}}{\partial E_{2}}=\frac{(m-k+1)}{4m}\,A_{k-1}\ .$ (158)

We now introduce new quasi-modular forms $A_{\ell}$ of definite weight
${\ell}$ under the Hecke group H$(m)$ that automatically satisfy a set of
generalized modular anomaly equations. These are constructed from the
Eisenstein series $E_{2k}$ via the following generating function:

$G(y)=1+\sum_{\ell=1}^{\infty}(-1)^{\ell}\,c_{\ell}\,A_{\ell}\,y^{\ell}=\text{exp}\left\\{-\sum_{k=1}^{\infty}\frac{y^{k}}{k}E_{2k}\right\\}\,,$
(159)

with coefficients $c_{\ell}$ to be determined by the anomaly equations. By
using the definition of $G(y)$ in terms of the Eisenstein series, we find that

$\frac{\partial G}{\partial E_{2}}=-\,y\,G\,.$ (160)

Substituting the definition of $G$ in terms of the $A_{\ell}$ and equating
powers of $y$ on both sides, we find that

$\sum_{\ell=1}^{\infty}(-1)^{\ell}c_{\ell}\left(\frac{\partial
A_{\ell}}{\partial E_{2}}-\frac{c_{\ell-1}}{c_{\ell}}A_{\ell-1}\right)=0\,.$
(161)

Here we use the convention $c_{0}=A_{0}=1$. Demanding that this coincides with
our modular anomaly equation in (158) leads to the following result for the
coefficients:

$c_{k}=\prod_{j=1}^{k}\frac{4m}{(m+1-j)}\,.$ (162)

The key point in this approach is that the exponential map also fixes the
dependence of the $A_{\ell}$ on the “higher times” $E_{2k}$ in much the same
way thanks to a hierarchy of equations:

$\frac{\partial G}{\partial E_{2k}}=-\,\frac{y^{k}}{k}\,G\,.$ (163)

This leads to the following relations: for $\ell<k$ we have

$\frac{\partial A_{\ell}}{\partial E_{2k}}=0\ ,$ (164)

while for $\ell\geq k$ we find

$\frac{\partial A_{\ell}}{\partial
E_{2k}}=\frac{(-1)^{k+1}}{k}\prod_{j=1}^{k}\left(\frac{m-\ell+j}{4m}\right)\,A_{\ell-k}\
.$ (165)

This completely fixes the dependence of the $A_{\ell}$ on the $E_{2k}$ for
$k>1$ as well. We can consequently interpret the exponential map as specifying
the boundary condition for the modular anomaly equation.

From the form of the coefficients $c_{k}$ one can check that, for the Hecke
group H$(m)$, there are precisely $m$ non-zero forms $A_{\ell}$ while the
higher ones are automatically set to zero. The first few of these take the
following form:

$\displaystyle\begin{split}A_{1}&\propto\left(E_{2}\right)\ ,\\\
A_{2}&\propto\left(E_{2}^{2}-E_{4}\right)\ ,\\\
A_{3}&\propto\left(E_{2}^{3}-3E_{2}E_{4}+2E_{6}\right)\ ,\\\
A_{4}&\propto\left(E_{2}^{4}-6E_{2}^{2}E_{4}+8E_{2}E_{6}+3E_{4}^{2}-6E_{8}\right)\
,\\\
A_{5}&\propto\left(E_{2}^{5}-10E_{2}^{3}E_{4}+20E_{2}^{2}E_{6}+15E_{2}E_{4}^{2}-20E_{4}E_{6}-30E_{2}E_{8}+24E_{10}\right)\
,\end{split}$ (166)

where as before, the constant of proportionality is given by (44). The
$A_{\ell}$ as defined via the exponential map are a unique solution to a
_system_ of anomaly equations. It is enlightening to compare these expressions
for $A_{k}$ with (43). While the $E_{2}$-dependence of the $A_{\ell}$
calculated this way is identical to the ones we found via the cuspidal
boundary conditions, the coefficients accompanying the automorphic pieces are
different.

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