Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations
We study various relations governing quasi-automorphic forms associated with discrete subgroups of SL(2, ℝ), namely the Hecke groups. We show that 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-...
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| description | We study various relations governing quasi-automorphic forms associated with discrete subgroups of SL(2, ℝ), namely the Hecke groups. We show that 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 quasi-modular forms of SL(2, ℤ). Each Hecke group is then associated with a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the m = 3 and 4 cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear 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⁽ᵐ⁾₂ associated to H(m) and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 001, 26 pages
Aspects of Hecke Symmetry:
Anomalies, Curves, and Chazy Equations
Sujay K. ASHOK †, Dileep P. JATKAR ‡ and Madhusudhan RAMAN §
† Institute of Mathematical Sciences, Homi Bhabha National Institute (HBNI),
IV Cross Road, C. I. T. Campus, Taramani, Chennai 600 113, India
E-mail: sashok@imsc.res.in
‡ Harish-Chandra Research Institute, Homi Bhabha National Institute (HBNI),
Chhatnag Road, Jhunsi, Allahabad 211 019, India
E-mail: dileep@hri.res.in
§ Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400 005, India
E-mail: madhur@theory.tifr.res.in
Received May 06, 2019, in final form December 29, 2019; Published online January 01, 2020
https://doi.org/10.3842/SIGMA.2020.001
Abstract. We study various relations governing quasi-automorphic forms associated to
discrete subgroups of SL(2,R) called Hecke groups. We show that 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 quasi-modular forms
of SL(2,Z). Each Hecke group is then associated to a (hyper-)elliptic curve, whose coeffi-
cients are determined by an anomaly equation. For the m = 3 and 4 cases, the Ramanujan
identities admit a natural geometric interpretation as a Gauss–Manin connection on the
parameter space of the elliptic curve. The Ramanujan identities also allow us to associate
a nonlinear 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
(m)
2 associated to H(m) and its orbit under the Hecke
group. We conclude by demonstrating that these nonlinear equations possess the Painlevé
property.
Key words: Hecke groups; Chazy equations; Painlevé analysis
2010 Mathematics Subject Classification: 34M55; 11F12; 33E30
1 Introduction
In this work, we study properties of automorphic forms for Hecke groups. An element of the
Hecke group, which we denote H(m), is any word made up of the letters S and T that is further
circumscribed by the relations
S2 = 1, (ST )m = 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 τ that takes values in the
upper-half plane H, the generators of the Hecke group act as1
T : τ → τ + 1, S : τ → − 1
λmτ
,
1Hecke groups are usually defined in the literature as generated by the symbols T and S that act as
T : τ → τ +
√
λm and S : τ → − 1
τ
,
mailto:sashok@imsc.res.in
mailto:dileep@hri.res.in
mailto:madhur@theory.tifr.res.in
https://doi.org/10.3842/SIGMA.2020.001
2 S.K. Ashok, D.P. Jatkar, and M. Raman
where
λm = 4 cos2
( π
m
)
. (1.1)
The Hecke groups are indexed by an integer m ≥ 3 that will be called its height. For m ∈
{3, 4, 6,∞} the corresponding λm are integers. These groups are called arithmetic Hecke groups,
while the rest (for which λm is not an integer) will be referred to as non-arithmetic Hecke
groups [32]. 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 [15] on (quasi-)automorphic forms for Hecke
groups H(m). The ring of quasi-automorphic forms for the Hecke group H(m), which we de-
note Rm, is generated by the Eisenstein series E
(m)
2k 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 H(3)
1
2πi
d
dτ
E
(3)
2 =
1
12
((
E
(3)
2
)2 − E(3)
4
)
,
1
2πi
d
dτ
E
(3)
4 =
1
3
(
E
(3)
2 E
(3)
4 − E
(3)
6
)
,
1
2πi
d
dτ
E
(3)
6 =
1
2
(
E
(3)
2 E
(3)
6 −
(
E
(3)
4
)2)
.
In Section 3, we associate an algebraic curve to the Hecke group H(m), whose coefficients A
(m)
k
are quasi-automorphic forms.2 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 [5, 6, 20] in that it fixes the dependence of these coefficients on
the quasi-automorphic Eisenstein series E
(m)
2 . A complete solution to these anomaly equations
requires the specification of boundary conditions that fix the dependence of A
(m)
k on purely
automorphic pieces; this is done by insisting on certain fall-offs near the cusp τ = i∞ which we
term ‘cuspidal’ boundary conditions.
The discussion of curves is developed with the goal of interpreting the Ramanujan identities
as corresponding to some natural geometric object on the parameter space of these curves. This
has already been done for the modular group in [16, 23, 26, 27]. 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. We also show that the discriminant of the curve is
closely related to the automorphic discriminant defined in [15].
Another motivation for our studies comes from [33], where it was shown that for the mod-
ular group, the quasi-modular form E
(3)
2 and its SL(2,Z)-orbits satisfy the well-known Chazy
equation and further, that the weight 12 modular discriminant plays the role of a Hirota τ -
function for the Chazy equation. We use the integrable systems nomenclature here and by
Hirota τ -function, we mean a solution to (possibly a collection of) nonlinear PDEs [22, 21, 33].
In Section 4, we observe that for every Hecke group the corresponding Eisenstein series E
(m)
2
satisfies an ordinary differential equation of order m that can be systematically constructed using
the Ramanujan identities. We go on to show that the H(m)-orbit of E2 also satisfies the same
differential equation, and an analogue of the modular discriminant plays the role of a Hirota
τ -function for the Hecke group, thereby generalizing the results of [33] to all Hecke groups.
where λm is defined as in (1.1). It is easily verified that under the rescaling τ →
√
λmτ , we recover the definition
provided in the main text.
2As we will see, for m = 3, 4, this curve is elliptic, and for m > 4 it is hyperelliptic.
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 3
One of the important differences between the modular group and other Hecke groups is that
there are non-trivial relations among the Eisenstein series of H(m) for m > 3. These relations
along with the Ramanujan identities naturally lead us to a third-order nonlinear differential
equation, allowing us to make contact with earlier work on the subject by Maier [25].
Finally, in Section 5, we perform a generalized Painlevé analysis of the higher-order Chazy
equations and Maier’s equation and show that all these equations possess the Painlevé property.
Some technical material is collected in Appendices A and B.
2 Ramanujan identities
The automorphic forms we study in this paper are Eisenstein series corresponding to the Hecke
group, denoted H(m). They will be built out of solutions to the generalized Halphen system,
following [15]. Once constructed, we prove analogues of the Ramanujan identities – first discussed
in [31] – for all Hecke groups.
2.1 Generalized Halphen systems and a proof
The generalized Halphen system is a set of coupled ordinary differential equations for three
variables {t(m)
k (τ)}3k=1 that satisfy
t
(m)′
1 = (a− 1)
(
t
(m)
1 t
(m)
2 + t
(m)
1 t
(m)
3 − t(m)
2 t
(m)
3
)
+ (b+ c− 1)
(
t
(m)
1
)2
,
t
(m)′
2 = (b− 1)
(
t
(m)
2 t
(m)
3 + t
(m)
2 t
(m)
1 − t(m)
1 t
(m)
3
)
+ (a+ c− 1)
(
t
(m)
2
)2
,
t
(m)′
3 = (c− 1)
(
t
(m)
3 t
(m)
1 + t
(m)
3 t
(m)
2 − t(m)
1 t
(m)
2
)
+ (a+ b− 1)
(
t
(m)
3
)2
, (2.1)
where the parameters (a, b, c) are specified by the height m of the Hecke group H(m) as
a =
1
2
(
1
2
+
1
m
)
, b =
1
2
(
1
2
− 1
m
)
, c =
1
2
(
3
2
− 1
m
)
, (2.2)
and the accent ′ denotes the following derivative
f ′ :=
1
2πi
d
dτ
f(τ). (2.3)
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 = e2πiτ , and when working with q-series, the
accent ′ is
1
2πi
d
dτ
= q
d
dq
.
The solution to the generalized Halphen system can be obtained explicitly in terms of hyper-
geometric functions whose arguments depend on the standard hauptmodul of the Hecke group
[15, Theorem 3(i)]. We have included a few details about these solutions and their Fourier
expansions in Appendix A.3
The Eisenstein series
{
E
(m)
2k
}m
k=2
are holomorphic automorphic forms of weight 2k under the
Hecke group H(m), and they have simple expressions in terms of the solutions to the generalized
Halphen system [15, see p. 707 and Theorem 4(iv)]. In order to simplify expressions, we define
the linear combinations
x(m) = t
(m)
1 − t(m)
2 and y(m) = t
(m)
3 − t(m)
2 . (2.4)
3It will be important to keep in mind that some of our normalizations differ from those of [15].
4 S.K. Ashok, D.P. Jatkar, and M. Raman
The automorphic forms E
(m)
2k in these variables are
E
(m)
2k =
(
x(m)
)k−1
y(m). (2.5)
It is clear from this resolution of the Eisenstein series into combinations of generalised Halphen
variables that for m > 3, the algebra of automorphic forms is not freely generated by the{
E
(m)
2k
}m
k=2
, as
E
(m)
2p E
(m)
2(k−p+1) =
(
x(m)
)k−1(
y(m)
)2
= E
(m)
2p′ E
(m)
2(k−p′+1) (2.6)
for any integers 2 ≤ p ≤ k − 1 and 2 ≤ p′ ≤ k − 1.
The Hecke groups also come equipped with a quasi-automorphic weight 2 Eisenstein se-
ries E
(m)
2 , which is a linear combination of solutions to the generalized Halphen system [15,
Theorem 4(iii)]. In terms of the variables
(
x(m), y(m)
)
the Eisenstein series E
(m)
2 can be written as
E
(m)
2 = − 1
m− 2
[
4x(m) + 2my(m) + (3m+ 2)t
(m)
2
]
, (2.7)
and using the conventions made explicit in Appendix A, we can check that as τ → i∞,
E
(m)
2 = 1 +O(q).
Lemma 2.1. The Ramanujan identities for Eisenstein series E
(m)
2k corresponding to Hecke
groups H(m) take the form
E
(m)′
2 =
m− 2
4m
((
E
(m)
2
)2 − E(m)
4
)
,
E
(m)′
2k =
k
2
(
m− 2
m
)
E
(m)
2 E
(m)
2k −
(
k − 2
2
)
E
(m)
2k′ E
(m)
2(k−k′+1) −
(
m− k
m
)
E
(m)
2k+2, (2.8)
for any k′ such that 2 ≤ k′ ≤ k − 1.
Proof. Perform a linear transformation from the generalized Halphen variables
(
t
(m)
1 , t
(m)
2 , t
(m)
3
)
to the variables
(
x(m), y(m), E
(m)
2
)
using (2.4) and (2.7). The generalized Halphen system then
takes the form
E
(m)′
2 =
m− 2
4m
((
E
(m)
2
)2 − x(m)y(m)
)
,
x(m)′ =
(
m− 2
2m
)
E
(m)
2 x(m) +
(
x(m)
)2
m
− x(m)y(m)
2
,
y(m)′ =
(
m− 2
2m
)
E
(m)
2 y(m) −
(
m− 1
m
)
x(m)y(m) +
(
y(m)
)2
2
. (2.9)
The first equation in (2.9) proves the first of the Ramanujan identities. For the higher weight
forms, we simply differentiate (2.5) and use (2.9) to find
E
(m)′
2k = (k − 1)
(
x(m)
)k−2
y(m)
[(
m− 2
2
)
E
(m)
2 x(m) +
(
x(m)
)2
m
− x(m)y(m)
2
]
+
(
x(m)
)k−1 [(m− 2
2
)
E
(m)
2 y(m) +
(
m− 1
m
)
x(m)y(m) +
(
y(m)
)2
2
]
=
k
2
(
m− 2
m
)
E
(m)
2k E
(m)
2 −
(
k − 2
2
)
E
(m)
4 E
(m)
2k−2 −
(
m− k
m
)
E
(m)
2k+2.
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 5
The second term can be resolved in multiple equivalent ways, courtesy of (2.6). We have thus
derived the analogues of the Ramanujan identities for Hecke groups, which for fixed m and
k ≤ m take the stated form for any k′ such that 2 ≤ k′ ≤ k − 1. �
It is straightforward to verify that the case m = 3 reproduces the well known Ramanujan
identities corresponding to the modular group. The above identities for m = 4 and 6 have
appeared in [36]. For Hecke groups H(m), we get analogous relations; systems of this form will
be collectively referred to as Ramanujan identities.
In earlier work [30], similar Ramanujan identities were derived for cusp forms associated to
the Hecke group H(m). In Appendix B we translate between the two sets of automorphic forms –
the Eisenstein series E
(m)
2k and the cusp forms f
(m)
2k – and relate (2.8) to the identities obtained
in [30].
3 Hyperelliptic curves, anomaly equations,
and boundary conditions
In this section we propose to give a geometric interpretation to the Ramanujan identities for the
arithmetic Hecke groups following earlier work on the subject by [16, 23, 26, 27].
3.1 Curves and anomalies
We begin by assigning to each of the Hecke groups, an algebraic curve defined by the equation
y2 = pm(x),
where pm(x) is a polynomial of degree m defined as
pm(x) = xm +
m∑
k=1
xm−kA
(m)
k . (3.1)
Definition 3.1. The coefficients A
(m)
k are chosen recursively via an anomaly equation
∂A
(m)
k
∂E
(m)
2
=
c(m− k + 1)
m
A
(m)
k−1, (3.2)
supplemented by boundary conditions which demand that, as τ → i∞,
A
(m)
k = O
(
qbk/2c
)
. (3.3)
We will set c = 1/4 for definiteness in (3.2), and conventionally set A
(m)
0 = 1. The boundary
condition (3.3) will be referred to as a cuspidal boundary condition.
We now supply a derivation of the anomaly equation. First, assign to the coefficient A
(m)
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
(m)
1 has weight 2,
and can thus be chosen to be proportional to E
(m)
2
A
(m)
1 = cE
(m)
2 . (3.4)
The A
(m)
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
6 S.K. Ashok, D.P. Jatkar, and M. Raman
appear in supersymmetric gauge theories [2, 13, 14], we note that it is often desirable to have al-
gebraic curves whose coefficients are purely automorphic forms. This requirement is imposed by
demanding that pm(x) with shifted argument is such that the coefficients are purely automorphic
p̃m(x) := pm
(
x− c
m
E
(m)
2
)
= xm +
m∑
k=2
xm−kÃ
(m)
k , (3.5)
where under the action of γ =
(
a b
c d
)
∈ H(m), the Ã
(m)
k transform as
Ã
(m)
k (γ · τ) = (cτ + d)2kÃ
(m)
k (τ), with γ · τ =
aτ + b
cτ + d
,
i.e., they are automorphic forms of weight 2k. This would in turn imply that the coefficients Ã
(m)
k
for k > 2 do not depend on the quasi-automorphic Eisenstein series E
(m)
2 . An interesting outcome
of this requirement can be derived by rewriting pm(x) as
pm(x) =
(
x+
c
m
E
(m)
2
)m
+
m∑
k=2
(
x+
c
m
E
(m)
2
)m−k
Ã
(m)
k ,
which in turn allows us to translate our requirement into a constraint of the form[
∂
∂E
(m)
2
− c
m
∂
∂x
]
pm(x) = 0,
where we now write the polynomial pm(x) as in (3.1). This constraint along with the rela-
tion (3.4) yields
m∑
k=1
[
∂A
(m)
k
∂E
(m)
2
− c
m
(m− k + 1)A
(m)
k−1
]
xm−k = 0.
This provides a set of (m− 1) equations, each constraining the dependence of the A
(m)
k on the
quasi-automorphic Eisenstein series E
(m)
2 , and each of the form
∂A
(m)
k
∂E
(m)
2
=
c(m− k + 1)
m
A
(m)
k−1. (3.6)
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 topological string theory
and superconformal gauge theories [3, 4, 5, 6, 20] and are referred to as modular anomaly equa-
tions. Since the above equation constrains the dependence of the A
(m)
k on the quasi-automorphic
Eisenstein series E
(m)
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 case m = 3, we fix this constant to be c = 1/4 by
appealing to a well-known factorisation of the elliptic curve associated to the modular group,
which we discuss in Section 3.2.
Integrating the anomaly equations fixes the dependence of A
(m)
k on the quasi-automorphic
form E
(m)
2 . In order to determine A
(m)
k completely, we must supply a boundary condition that
fixes the purely automorphic pieces. We now highlight one possible choice of boundary condition
that will be useful in later sections.
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 7
Starting with the A
(m)
k of the lowest weight, we solve the anomaly equation, thereby “inte-
grating in” any E
(m)
2 dependence. For example, on using (3.4), the anomaly equation for A
(m)
2
reads
∂A
(m)
2
∂E
(m)
2
=
c2(m− 1)
m
E
(m)
2 ,
which is solved by
A
(m)
2 =
c2(m− 1)
2m
(
E
(m)
2
)2
+ (automorphic form).
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
(m)
2 , this means
A
(m)
2 =
c2(m− 1)
2m
(
E
(m)
2
)2
+ aE
(m)
4 ,
as for all H(m), the dimension of the space of weight 4 forms is unity. For general A
(m)
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 m2k, which from (B.4) is bk/2c.
We propose to fix these coefficients by demanding that near the cusp at i∞, the A
(m)
k has
a Fourier expansion that starts at qdimm2k . That is, as τ → i∞
A
(m)
k = O
(
qbk/2c
)
.
This boundary condition provides as many equations as the number of coefficients to be deter-
mined, and is consequently an unambiguous prescription. Further, by construction these A
(m)
k
satisfy the anomaly equation. Additionally, we will see in Section 3.4 that the choice of cuspidal
boundary conditions allow us to relate the ‘curve’ discriminant ∆(m) constructed out of the
(hyper-)elliptic curve to the ‘automorphic’ discriminant ∆m defined solely with reference to the
ring of quasi-automorphic forms for H(m) in an interesting manner.
3.2 Examples
3.2.1 H(3)
In this section we show explicitly that the solutions to the modular anomaly equations along
with the cuspidal boundary conditions are completely consistent with the usual parametrisation
of the elliptic curves. The elliptic curve associated to the modular group is
y2 = x3 +A
(3)
1 x2 +A
(3)
2 x+A
(3)
3 . (3.7)
Notice that the assignment of weights here implies that the coefficients A
(3)
2 and A
(3)
3 have
weights 4 and 6 respectively. On solving the anomaly equations and using the cuspidal boundary
conditions, we find
A
(3)
1 = cE
(3)
2 , A
(3)
2 =
c2
3
((
E
(3)
2
)2 − E(3)
4
)
,
A
(3)
3 =
c3
27
((
E
(3)
2
)3 − 3E
(3)
2 E
(3)
4 + 2E
(3)
6
)
.
8 S.K. Ashok, D.P. Jatkar, and M. Raman
Factorization of elliptic curves. The polynomial defining the elliptic curve admits a well-
known factorisation
y2 =
3∏
k=1
(x+ sk), (3.8)
where, following [19], we define the sk to be logarithmic derivatives of Jacobi θ-constants4
s1 =
1
iπ
d
dτ
log θ2(τ), s2 =
1
iπ
d
dτ
log θ3(τ), s3 =
1
iπ
d
dτ
log θ4(τ).
The A
(3)
k are symmetric polynomials in the sk, i.e.,
A
(3)
1 = s1 + s2 + s3, A
(3)
2 = s1s2 + s2s3 + s3s1, A
(3)
3 = s1s2s3,
and the requirement of consistent assignment of weights leads us to conclude that the roots sk
have weight 2. Let us focus on the first of the anomaly equations, which yields A
(3)
1 = cE
(3)
2 in
terms of a sum of three sk. As we know, a τ -derivative raises the weight of a modular (function
or form) by two units. Using this explicit factorized solution, we can solve for A
(3)
1 as
A
(3)
1 =
1
iπ
d
dτ
log θ2(τ)θ3(τ)θ4(τ), (3.9)
=
1
iπ
d
dτ
log η(τ)3 =
1
4
E
(3)
2 . (3.10)
In going from equation (3.9) to equation (3.10) we have made use of the following identity that
relates the product of the three θ-constants to the Dedekind η-function5
θ2(τ)θ3(τ)θ4(τ) = 2η3(τ).
In the last equality in equation (3.10) we have used the fact that the modular discriminant
(which equals the curve discriminant in this case) is related to the η-function by the relation
∆ = (2π)12η24(τ),
and that the Eisenstein series is related to the modular discriminant by the relation
E
(3)
2 (τ) =
1
2πi
d
dτ
log ∆.
This fixes the constant c = 1
4 , and motivates the simplifying assumption that c = 1
4 for the
anomaly equations associated to all H(m). The other A
(3)
k can similarly be checked to be
consistent with those that arise from the factorized curve.
4The Jacobi θ-function is defined as follows
θ [ ab ] (v|τ) =
∑
n∈Z
q(n−
a
2 )2e2πi(n−
a
2 )(v− b
2 ).
The θ-constants are defined by setting v = 0 and θ2 ≡ θ [ 10 ] , θ3 ≡ θ [ 00 ] , θ4 ≡ θ [ 01 ].
5The Dedekind η-function is defined as the following product η(τ) = q
1
24
∞∏
n=1
(1− qn).
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 9
3.2.2 H(4)
The case of H(4) works out in much the same way. We begin with the quartic curve
y2 = x4 +
4∑
k=1
x4−kA
(4)
k .
The Ak satisfy our anomaly equation (3.6) with m = 4
∂A
(4)
k
∂E
(4)
2
=
5− k
16
A
(4)
k−1.
Together with the cuspidal boundary conditions, we find the following solutions to the anomaly
equations
A
(4)
1 =
1
4
E
(4)
2 , A
(4)
2 =
3
128
((
E
(4)
2
)2 − E(4)
4
)
,
A
(4)
3 =
1
1024
((
E
(4)
2
)3 − 3E
(4)
2 E
(4)
4 + 2E
(4)
6
)
,
A
(4)
4 =
1
65536
((
E
(4)
2
)4 − 6
(
E
(4)
2
)2
E
(4)
4 + 8E
(4)
2 E
(4)
6 +
(
E
(4)
4
)2 − 4E
(4)
8
)
.
As in the case of H(3), it is possible to present a factorized form for the quartic curve
x4 +
4∑
k=1
A
(4)
k x4−k =
4∏
k=1
(x+ sk), (3.11)
from which it follows that A
(4)
k are elementary symmetric polynomials
A
(4)
1 = s1 + s2 + s3 + s4,
A
(4)
2 = s1s2 + s3s2 + s4s2 + s1s3 + s1s4 + s3s4,
A
(4)
3 = s1s2s3 + s1s4s3 + s2s4s3 + s1s2s4,
A
(4)
4 = s1s2s3s4. (3.12)
The sk that factorize the quartic polynomial in (3.11) are given as follows
s1 =
1
2πi
d
dτ
log θ2(2τ), s2 =
1
2πi
d
dτ
log θ3(2τ),
s3 =
1
2πi
d
dτ
log θ3(τ), s4 =
1
2πi
d
dτ
log θ4(τ). (3.13)
This is verified by explicitly comparing the Fourier expansions on both sides of (3.12).6
3.3 General solution to the anomaly equation
To recapitulate, we started by associating to each Hecke group a curve that was elliptic for
m ≤ 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
(m)
k on the quasi-automorphic
form E
(m)
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
6The expressions for the sk in (3.13) were independently derived by [18].
10 S.K. Ashok, D.P. Jatkar, and M. Raman
are obtained after “integrating in” the E
(m)
2 dependence. Apart from an overall constant that
varies with the height m of the Hecke group, the A
(m)
k are found to be proportional to the same
combinations of the Eisenstein series. The first few A
(m)
k are given below
A
(m)
1 ∝ E(m)
2 ,
A
(m)
2 ∝
[(
E
(m)
2
)2 − E(m)
4
]
,
A
(m)
3 ∝
[(
E
(m)
2
)3 − 3E
(m)
2 E
(m)
4 + 2E
(m)
6
]
,
A
(m)
4 ∝
[(
E
(m)
2
)4 − 6
(
E
(m)
2
)2
E
(m)
4 + 8E
(m)
2 E
(m)
6 +
(
E
(m)
4
)2 − 4E
(m)
8
]
,
A
(m)
5 ∝
[(
E
(m)
2
)5 − 10
(
E
(m)
2
)3
E
(m)
4 + 20
(
E
(m)
2
)2
E
(m)
6
+ 5E
(m)
2
((
E
(m)
4
)2 − 4E
(m)
8
)
− 4E
(m)
4 E
(m)
6 + 8E
(m)
10
]
,
and so on. One can show that
A
(m)
k =
(
m− 1
k − 1
)
1
k4kmk−1
[(
E
(m)
2
)k
+ · · ·
]
,
where the constant of proportionality is a simple consequence of the anomaly equation.
3.4 Discriminants
We have defined a set of (hyper-)elliptic curves with coefficients determined by an anomaly
equation together with cuspidal boundary conditions. Since these curves y2 = pm(x) are specified
by the polynomials pm(x), it is natural to consider the discriminant of the polynomial, defined
in the usual way as the resultant of the polynomial pm(x) and its derivative p′m(x). It is
a polynomial in the A
(m)
k with integer coefficients, which we will denote by ∆(m) and refer to as
the curve discriminant.
Since the coefficients A
(m)
k transform as quasimodular forms of weight (2k), we then ask what
the weight of the curve discriminant might be. Let us suppose that the polynomial admits
a factorisation
pm(x) =
m∏
k=1
(x+ sk),
in which case the A
(m)
k are easily seen to be symmetric polynomials in the sk. For the weights
to be consistently defined, the sk must transform as weight 2 quasimodular forms. In terms of
the factorised representation of the polynomial, the discriminant is defined as
∆(m) =
∏
i 6=j
(si − sj),
which makes it clear that the weight of the algebraic discriminant is w = 2m(m− 1).
Now, we must also use the information that the ring of quasi-modular forms for Hecke
groups H(m) with m > 3 is not freely generated, and that there are non-trivial relations between
the Eisenstein series of higher weight. It is easily verified that for E
(m)
2k (with k > 3) we have
E
(m)
2k
(
E
(m)
4
)k−3
=
(
E
(m)
6
)k−2
,
which we can plug into the definition of the curve discriminant in terms of resultants. This
motivates the following conjecture:
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 11
Conjecture 3.2. The curve discriminant for the (hyper-)elliptic curves associated to the Hecke
group H(m) is given by
∆(m) = am
((
E
(m)
4
)3 − (E(m)
6
)2)(m−1)(m−2)/2(
E
(m)
4
)(m−1)(m−3) , with am =
22−m(m+1)
mm(m−2) .
Let us try and massage the above expression into something more familiar. We rewrite the
automorphic forms in terms of the standard hauptmodul J(m) and its derivatives using the
formulae in Appendix B (see equation (B.2))
∆(m) = am
(
J ′(m)
)m
Jm−1(m)
(
J(m) − 1
)m/2
(m−1)
=
{
am
(
f
(m)
2m
)(m−1)
for m ∈ 2Z,
am
(
f
(m)
4m
)(m−1)/2
for m ∈ 2Z + 1.
(3.14)
Finally, the automorphic discriminant (which we will encounter later as well) is defined in
[15, Theorem 2(ii)] as ∆m := f
(m)
2L with L = lcm(2,m), where f
(m)
2k are cusp forms defined
in (B.1). It is natural to wonder what the relationship between ∆(m), the curve discriminant of
the (hyper-)elliptic curve which we have determined earlier to have weight w = 2m(m− 1), and
the automorphic discriminant ∆m is. From [15, Theorem 2(ii)] we see that the weight of ∆m is
wm =
{
2m for m ∈ 2Z,
4m for m ∈ 2Z + 1.
From (3.14) we find that the algebraic and automorphic discriminants are related as(
∆(m)
am
)wm
= (∆m)w,
which serves as a strong, non-trivial consistency check on the web of relationships we have
uncovered. In particular, it confirms the correctness the (hyper-)elliptic curve and the anomaly
equation, and justifies the use of cuspidal boundary conditions.
3.5 Gauss–Manin connections
Here we discuss a geometric interpretation of the Ramanujan identities following [16, 23, 26, 27].
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 parameter space of the elliptic curve in (3.8). We
then do the same for H(4). For this, we use the notion of a Gauss–Manin connection, which
formalizes the following observation: the variation of an elliptic integral – defined using some
basis of differentials – with respect to a parameter t can be written as a linear combination of
period integrals
d
(∮
Πa
)
=
∑
b
Aab
(∮
Πb
)
.
The coefficients A form a 2× 2 matrix, and for multiple such parameters {tk}mk=1 we define the
differential form
A =
m∑
k=1
A(k)dtk,
12 S.K. Ashok, D.P. Jatkar, and M. Raman
so the variation of the period integrals with respect to these parameters is captured in the
equations
∇Π =
m∑
k=1
(
A(k)Π
)
dtk =: A ·Π.
More properly, A should be viewed as a differential 1-form on the parameter space of the elliptic
curve. The matrix A is referred to as a Gauss–Manin connection, and in [27, Proposition 6] this
Gauss–Manin connection was computed explicitly for the m = 3 case.
We now study what happens to this basis of differential 1-forms as we vary τ , the modular
parameter of the underlying torus. This leads us to define the Ramanujan vector field
R :=
1
2πi
d
dτ
.
In terms of variations of the parameters that appear in the curve (which are the Eisenstein
series), we have
R =
m∑
k=1
(
1
2πi
d
dτ
E
(m)
2k
)
∂
∂E
(m)
2k
. (3.15)
The portion in parentheses above may be replaced in accordance with the Ramanujan identi-
ties (2.8).
We can contract this differential 1-form on the parameter space of the elliptic curve with
a vector from the same space. The Ramanujan identities define the vector field (3.15) on the
parameter space of the elliptic curve, and we define its contraction with the connection as
∇R = ιR∇,
using the rule
∂
∂tk
(dt`) = δk`.
We now ask how ∇R acts on a basis of differential 1-forms associated to our elliptic curve,
and in [27, Proposition 7] it was demonstrated that
∇RΠ = ARΠ,
with AR determined by explicit calculation. Let us quickly review the manner in which this
computation was 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→ x− 1
3
A
(3)
1 .
Then, the curve (3.7) in terms of the Eisenstein series takes its Weierstrass normal form
y2 = p̃3(x) = x3 − E
(3)
4
48
x+
E
(3)
6
864
.
The coefficients of the polynomial p̃3 are precisely the automorphic forms Ã
(3)
k that we encoun-
tered in (3.5). Our goal will be to determine the action of the Ramanujan vector fields on the
period integrals; this computation is built out of constituents that have the general form
∂
∂E
(3)
2k
∮
x`dx
y
=
∮
dx
p̃3y
(
−x
`
2
∂p̃3
∂E
(3)
2k
)
. (3.16)
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 13
At this stage, a happy consequence of using the shifted polynomial p̃3 is that we can conclude
by construction
∂
∂E
(3)
2
∮
x`dx
y
= 0,
i.e., the period integrals do not vary in the direction ∂
E
(3)
2
as the curve y2 = p̃3(x) carries
no dependence on E
(3)
2 . For the other components of the Ramanujan vector field ∂
E
(3)
2k
(for
k ∈ {2, 3}) and each independent differential form x`dx
y (for ` ∈ {0, 1}), we follow a technique
of [27] and look for polynomials α and β that satisfy the constraint
−x
`
2
∂
∂E
(3)
2k
p̃3 = α
dp̃3
dx
+ βp̃3.
Once determined, we plug this into the variation in question (3.16) and after some elementary
manipulations, it is then easy to see that the result of the variation with respect to E
(3)
2k is simply
∂
∂E
(3)
2k
∮
x`dx
y
=
∮
dx
y
(
2
dα
dx
+ β
)
.
Earlier work has used this technique to establish the following theorem:
Theorem 3.3 (Movasati [27]). The Gauss–Manin connection corresponding to the Ramanujan
vector field associated to the modular group rotates the canonical basis of differential 1-forms on
the elliptic curve as
AR =
−
E
(3)
2
12
1
−E
(3)
4
144
E
(3)
2
12
.
Remark 3.4. This computation may also be performed in the original (unshifted) basis, and
in that case the result is
AR =
(
0 1
0 0
)
,
which is precisely the result of [27] up to a sign.
3.5.1 Height four
While the holomorphic differential is still dx
y – and this is true for all hyperelliptic curves –
the differential xdx
y is no longer a good candidate as it has a simple pole at infinity, making it
a differential of the third kind [7]. Instead, a valid differential of the second kind is given by
Π2 =
x2dx
y
.
This choice of differential does not have a pole at infinity. We use the following convenient basis
of differential 1-forms for an elliptic curve defined by a quartic polynomial
Π =
dx
y
x2dx
y
.
14 S.K. Ashok, D.P. Jatkar, and M. Raman
Theorem 3.5. The Gauss–Manin connection corresponding to the Ramanujan vector field as-
sociated to H(4) rotates the above basis of differential 1-forms on the elliptic curve as
AR =
−
E
(4)
2
4
0
−E
(4)
6
512
E
(4)
2
4
.
Proof. The proof proceeds by explicit computation. The Ramanujan vector field is given by
R =
1
2πi
d
dτ
=
4∑
k=1
(
1
2πi
d
dτ
E
(4)
2k
)
∂
∂E
(4)
2k
,
and as before, the expression in parentheses may be replaced in accordance with the Ramanujan
identities, giving
R =
1
8
((
E
(4)
2
)2 − E(4)
4
) ∂
∂E
(4)
2
+
1
2
(
E
(4)
2 E
(4)
4 − E
(4)
6
) ∂
∂E
(4)
4
+
1
4
(
3E
(4)
2 E
(4)
6 − 2
(
E
(4)
4
)2 − E(4)
8
) ∂
∂E
(4)
6
+
(
E
(4)
2 E
(4)
8 − E
(4)
4 E
(4)
6
) ∂
∂E
(4)
8
.
The basis of differential 1-forms will rotate into itself under the action of the Ramanujan vector
field. After a shift of the form
x→ x− 1
4
A
(4)
1
the curve (in terms of Eisenstein series) takes the form
y2 = p̃4(x) = x4 − 3E
(4)
4
128
x2 +
E
(4)
6
512
x+
(
E
(4)
4
)2 − 4E
(4)
8
65536
.
Employing the technique of solving for polynomials α and β allows us to determine the effect of
the Ramanujan vector field on the basis of differential 1-forms. We find that
∇RΠ1 = −E
(4)
2
4
Π1, ∇RΠ2 = −E
(4)
6
512
Π1 +
E
(4)
2
4
Π2.
Alternatively, we may write down the following connection equation
∇RΠ = ARΠ,
where AR is as claimed. �
4 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.
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 15
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
E
(3)′
2 =
1
12
((
E
(3)
2
)2 − E(3)
4
)
,
E
(3)′
4 =
1
3
(
E
(3)
2 E
(3)
4 − E
(3)
6
)
,
E
(3)′
6 =
1
2
(
E
(3)
2 E
(3)
6 −
(
E
(3)
4
)2)
.
A well-known strategy (outlined for example in [8]) consists of using the above equations to find
a differential equation satisfied by the weight 2 Eisenstein series E
(3)
2 , which we will denote by y.
This is done straightforwardly by elimination and we find the following equation:
2y(3) − 2yy′′ + 3y′2 = 0, (4.1)
where y(m) is m-th derivative of y with respect to τ (see, e.g., equation (2.3)). Up to rescalings,
this nonlinear third-order differential equation is known as the Chazy equation [9]. It originally
arose in the study of third-order ordinary differential equations having the Painlevé property. We
shall return to a detailed study of this property in Section 5. Before this, we derive higher-order
Chazy equations that are derived straightforwardly from the Ramanujan identities.
4.1 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 Cm look like?
An example: height four. Let us consider the Ramanujan identities (2.8) for m = 4.
Using the symbol y to once again denote E
(4)
2 in analogy with the case of the modular group,
and following the procedure outlined in the previous section yields a new, fourth-order analogue
of the Chazy equation7
C4 : 4y(4) − 10y(3)y + 6y2y′′ − 9y(y′)2 + 12y′y′′ = 0.
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, 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 for H(m). Below, we list the next two members of this family,
for future reference
C5 : 300y(5) − 1350y(4)y + 1932y(3)y2 − 882y3y′′ + 920(y′′)2
+ 1323y2(y′)2 − 168(y′)3 + y′
(
620y(3) − 2772yy′′
)
= 0,
C6 : 9y(6) − 63y(5)y + 158y(4)y2 − 168y(3)y3 − 156y(y′′)2
+ 48y(y′)3 +
(
78y(3) + 64y4
)
y′′ − (y′)2
(
48y′′ + 96y3
)
− y′
(
3y(4) + 68y(3)y − 240y2y′′
)
= 0. (4.2)
Thus, we find constructively that:
Proposition 4.1. Each Hecke group H(m) is associated to an order m nonlinear ordinary
differential equation. Further, each term in the Chazy equation corresponding to H(m) has
weight 2m+ 2.
7This equation was independently derived in [18].
16 S.K. Ashok, D.P. Jatkar, and M. Raman
Proof. Consider the Ramanujan identities corresponding to H(m), a set of m first-order differ-
ential equations in m variables. The above proposition follows by differential elimination. �
4.2 H(m) orbits
In the previous section, we have demonstrated constructively that y = E
(m)
2 satisfies a higher-
order Chazy equation, which for Hecke group H(m) is a nonlinear ordinary differential equation
of order m. This allows us to generalise [33, equation (8)] in the following way:
Proposition 4.2. The automorphic discriminant ∆m is the Hirota τ -function for the Chazy
equation Cm.
This follows rather trivially following the representation of y = E
(m)
2 in terms of the ‘auto-
morphic’ discriminant ∆m as given in [15, Theorem 2(ii)]
y =
1
2πi
d
dτ
log ∆m.
We now characterise general solutions to the Chazy equation Cm. This is inspired by [33,
Lemma 3] and proceeds by rewriting Cm as a linear combination of automorphic forms of a given
weight. Let us see how this works in the case of the usual Chazy equation corresponding to H(3),
thereby reviewing [33, Theorem 2]. The expression
Z = E
(3)′
2 − 1
12
(
E
(3)
2
)2
is a weight 4 modular form. We want to generate forms of higher weight, and a well-known
procedure to do this is to use the Ramanujan–Serre derivatives [35] that send
D: mk → mk+2.
Explicitly, this derivative takes the form (with our normalizations)
D =
1
2πi
d
dτ
− k
12
E
(3)
2 ,
and one can check that with this definition DE
(3)
4 = −1
3E
(3)
6 and DE
(3)
6 = −1
2
(
E
(3)
4
)2
. 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
aD2Z + bZ2,
which is guaranteed to be an automorphic form of weight 8. Finally, we can check that the
Chazy equation can be written as
C3 : D2Z + 2Z2 = 0.
We have thus demonstrated that C3 can be written as a linear combination of weight 8 automor-
phic forms; thus, under the action of a γ =
(
a b
c d
)
∈ H(3), each term x in C3 will transform as
x 7→ (cτ + d)8x.
With this, we can conclude that all H(3) ∼= SL(2,Z)-orbits of y solve the Chazy equation.
Further, the space of weight 8 forms is 1-dimensional while 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. We now
generalise these statements to all Hecke groups H(m).
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 17
Theorem 4.3. All H(m) orbits of E
(m)
2 are solutions to Cm. That is, if y = E
(m)
2 is a solution
to Cm, then so is ỹ = E
(m)
2 (γ · τ), where γ ∈ H(m).
Proof. The strategy of our proof is similar: for H(m) we will define the weight 4 form
Z = E
(m)′
2 −
(
m− 2
4m
)(
E
(m)
2
)2
,
and invoke a result of [30] that defines an appropriate analogue of the Ramanujan–Serre deriva-
tives for the Hecke group H(m). In our present normalization this takes the form
D =
1
2πi
d
dτ
− k
2
(
m− 2
2m
)
E
(m)
2 . (4.3)
It will turn out that under the action of a γ =
(
a b
c d
)
∈ H(m), each term x that appears in Cm
will transform as
x 7→ (cτ + d)2m+2x.
This can be explicitly verified for Hecke groups with small values of m. For example, we find
C4 : D3Z + 6ZDZ = 0,
C5 : 3D4Z + 26ZD2Z + 20(DZ)2 + 12Z3 = 0,
C6 : D5Z + 12ZD3Z + 24D2ZDZ + 32Z2DZ = 0,
and so on. In each of these cases, the proof goes through as before. �
Similarly, it is easy to understand why it is reasonable to expect that Cm is zero.
Remark 4.4. It is known that for the Hecke group H(m) there are m of generators of the ring
of quasi-automorphic forms Rm [15]. Thus, at weight (2m+ 2), all forms that span this vector
space will be products of forms of lower weight 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.
4.3 Relation to Maier’s equation
An important point to keep in mind is that the quasi-automorphic forms do not generate a free
polynomial algebra for m > 3. This is immediately obvious from the definition of the E
(m)
2k
in (2.5); for instance it is true for all m > 3 that
E
(m)
4 E
(m)
8 =
(
E
(m)
6
)2
. (4.4)
Similar relations hold for the higher weight forms as well. Given the way we derived the gen-
eralized Chazy equation, this immediately suggests the existence of a lower-order differential
equation satisfied by y = E
(m)
2 for all m by using such identities; in this section we show that
this is indeed the case and in fact the differential equation we obtain is identical to the one
obtained in [25].
Let us see this in detail. Consider the action of the Ramanujan–Serre derivative (4.3) on E4.
The following identity is straightforwardly verified
(m− 2)E
(m)
4 D2E
(m)
4 = (m− 3)
(
DE
(m)
4
)2
+
(m− 2)2
2m
(
E
(m)
4
)3
.
18 S.K. Ashok, D.P. Jatkar, and M. Raman
Upon defining Y = (m−2)2
16m2 E
(m)
4 , we obtain the following relation of Maier [25]
(m− 2)YD2Y − (m− 3)(DY )2 − 8mY 3 = 0.
This shows the complete equivalence of the Ramanujan identities for H(m) and Maier’s equation.
In particular, this implies the following theorem.
Theorem 4.5. For all H(m), the weight 2 quasi-automorphic form y = E
(m)
2 satisfies the
following third-order nonlinear differential equation
y′′′
(
2y2 − 8my′
(m− 2)
)
+ y′′
(
24yy′
(m− 2)
− 2y3
)
+
8m(m− 3)(y′′)2
(m− 2)2
+ 3y2(y′)2 − 4(m+ 6)(y′)3
(m− 2)
= 0. (4.5)
Proof. With the identity (4.4), the set of Ramanujan identities (2.8) for k = {1, 2, 3} becomes
a closed system of first-order ordinary differential equations. By elimination, we recover the
above differential equation for y = E
(m)
2 . �
This form of the Maier equation will prove to be very useful when we perform a Painlevé
analysis on the solutions to the differential equation.
5 Painlevé analysis of the Chazy equations
We now test the Chazy equations for the Painlevé property, which roughly corresponds to
a statement that the only movable singularities of these differential equations are poles. We
begin with a brief primer on stability analysis and go on to apply the methods of [10, 11] to the
higher-order Chazy equations as well as the Maier equation.
5.1 A primer on stability
We begin with a brief review of the stability analysis of nonlinear differential equations. Consider
a nonlinear ordinary differential equation of order n. The analysis due to Painlevé [28] involves
expanding the independent variable near the singular point – this “nearness” is parametrized
by a small parameter α – and expanding the dependent variable as a formal power series in α.
The method due to Kowalevskaya [24] 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 [1], 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 Frobenius form
y(τ) =
∞∑
i=0
yi(τ − τ0)i−a. (5.1)
In [1] 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
[
y
]
= 0,
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 19
where K[y] is a polynomial function of y, y′, etc. up to the nth derivative, then the linearized
equation is obtained as
d
dε
K[y + εw]
∣∣∣∣
ε=0
= 0. (5.2)
Substituting the Frobenius form (5.1) into the above equation, one can equivalently write the
linearized equation as
KL
[
d
dτ
]
w(τ) = 0,
for some polynomial KL whose coefficients are given by the yi. We assume the following ansatz
for the linearized solution:
w(τ) = w0(τ − τ0)−a +
∞∑
i=1
wi(τ − τ0)i−a. (5.3)
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 (τ − τ0)i−a−n vanishes for each i. This condition gives a recursion relation,
which determines wi in terms of wj with j < i, and the parameters appearing in the linear
differential equation. We write this equation as
P (i)wi − f(τ0, a, yk;w1, . . . , wi−a) = 0.
Let r1 ≤ r2 ≤ · · · ≤ rn be roots of the polynomial P (i); then we can determine wi in terms of wj
(with j < i) as long as i 6= r1. When i = r1 then we have P (i) = 0 and we end up with a condition
f(τ0, a, yk, w1, . . . , wi−a) = 0. If this condition is satisfied then wr1 is indeterminate and the
linearized equation is said to pass the Painlevé test. If, however, f(τ0, a, yk, w1, . . . , wi−a) 6= 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 = r1, we continue the test for i = rk
for k = {2, . . . , 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 [1]. If the linearized equation satisfies the
ARS stability conditions then we say that the original nonlinear 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 [1] are insufficient. Luckily, the analysis of negative resonances has
been carried out in [10, 11] and has the added advantage of being applicable to nonlinear partial
differential equations, thereby subsuming the analysis of [34].
We now turn to the notion of stability for systems with negative resonances. In a nutshell,
the argument of [10, 11] is that if the coefficients of the negative resonances vanish identically
when the zeroth-order nonlinear equation is satisfied, we are permitted to conclude that the
equation possesses the Painlevé property.
20 S.K. Ashok, D.P. Jatkar, and M. Raman
5.2 The Painlevé property
We will now consider the Painlevé analysis of the Chazy equations Cm, examples of which are
presented in equations (4.1) to (4.2), as well as the Maier equation in (4.5). These equations
are nonlinear ordinary differential equations and 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 C3 in (4.1), following [17].
In order to illustrate the procedure, we first seek a solution to the equation in the Frobenius
form, i.e., (5.1). 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 τ → λτ and y → λay.
The integrality of a ensures that the Chazy equation passes the first criterion of ARS. We then
proceed to determine the coefficients yi by recursively solving the equation (4.1). Some low-order
coefficients are
y0 = −12, y1 = y2 = y3 = 0.
To see if this solution is stable against perturbation we substitute y(τ) = ys(τ) + εw(τ) in the
Chazy equation, where ys(τ) is the solution to the Chazy equation. The linearized equation is
obtained by picking up terms linear in ε, as in (5.2), and we find
3y′(τ)w′(τ)− y′′(τ)w(τ)− y(τ)w′′(τ) + w′′′(τ) = 0. (5.4)
Substituting the ansatz (5.3) into this equation, one finds that the linearized solutions of equa-
tion (5.4) 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 vio-
lates the criterion of [1], 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 [17]. 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 φ(τ), which parametrizes the singular manifold when φ(τ) = 0.
The solution, however, is written in terms of a Frobenius series in another function χ(τ), related
to φ(τ) as follows
χ(τ) =
2φφ′
2(φ′)2 − φφ′′
.
This provides us with a germ which is formally independent of the function defining the singular
manifold. The function χ(τ) satisfies a Riccati type equation
χ′(τ) = 1− 1
2
Sχ2(τ),
where the Schwarzian S is defined to be
S =
φ′′′(τ)
φ(τ)
− 3
2
(
φ′′(τ)
φ′(τ)
)2
.
We now make the following ansatz for the leading order solution
y(τ) =
∞∑
i=0
yi (χ(τ))i−a , (5.5)
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 21
and find that a = 1 as before but now find the following solutions for the coefficients
y0 = −12, y1 = 0, y2 = 2S, y3 = 0, (5.6)
and so on. Note that we now get a non-zero value for y2, unlike the earlier (naive) Laurent
expansion. For the next-to-leading order we now use the following ansatz
w(τ) =
∞∑
i=0
wi(χ(τ))i−b. (5.7)
Substituting this in the linearized Chazy equation (5.4) 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 yi in (5.6), 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 now carry out a similar analysis on the general third-order differential equation that is
equivalent to the Maier equation and that we obtained in (4.5). Substituting the ansatz (5.5)
and solving for the exponent, we find a = 1. For the leading coefficients, we now find
y0 = − 4m
m− 2
, y1 = 0, y2 =
2m
3(m− 2)
S, y3 = 0. (5.8)
At the next to leading order analysis we find b = 2, which shows the presence of higher-order
poles for the linearized equations compared to the solution of the Maier equation itself. But
exactly as for the Chazy equation, we find that when we substitute the coefficients yi in (5.8),
the coefficients of the higher resonance vanish identically and implies that the resonances do not
destabilize the solution of the Maier equation.
We thus conclude that the Maier equation possesses the Painlevé property. Further, from
the discussion in Section 4.3, it is clear that the higher-order Chazy equations Cm are entirely
equivalent to the Maier equation when the algebraic dependence of the forms
{
E
(m)
2k
}m
k=2
(see
equation (2.6)) is taken into account. We therefore conclude that each Cm also possesses the
Painlevé property. We have explicitly verified this for 3 ≤ m ≤ 10 and the analysis of [10, 11]
leads us to the following proposition.
Proposition 5.1. The Maier equation in (4.5) as well as the higher-order Chazy equation Cm
possess the Painlevé property. The leading order ansatz y(τ) defined in (5.5) satisfies both sets
of differential equations for the following values of the coefficients
y(τ) =
2m
m− 2
(
− 2
χ(τ)
+
S
3
χ(τ) +
2S2
45
χ(τ)3 +
11S3
945
χ(τ)5 + · · ·
)
. (5.9)
The analysis of the next-to-leading order ansatz (5.7) shows that the Chazy equation has
negative resonances, with b ∈ {1, 2, . . . ,m} for Cm. 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 C3 as well as the Maier equation, we find that when evaluated on the
leading order solution, the coefficients of these resonances vanish identically, guaranteeing that
all these equations have the Painlevé property. It is interesting to observe that while the Maier
equation fixes the first two coefficients in the expansion in (5.9), it is the Painleve analysis of
the higher-order Chazy equations that allows one to fix the sub-leading terms in the ansatz.
22 S.K. Ashok, D.P. Jatkar, and M. Raman
6 Discussion
We have studied several interesting properties of automorphic forms and associated nonlinear
differential equations associated to the Hecke group H(m). We now discuss some future directions
for research motivated by these developments, with a special emphasis on applications relevant
to gauge theories, integrability, and string theories.
Hecke groups have appeared as strong-weak duality groups acting on coupling constants in
certain supersymmetric gauge theories [3]. These coupling constants appear in what is called
the low energy effective action of the gauge theory. Given this, we expect the 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 [3, 4] it would be interesting
to carry it out for the other Hecke groups. From the gauge theory perspective what it amounts
to is a resummation of the observables that allows one to probe the theory at all values of the
coupling, which is a desirable feature in any quantum field theory.
The differential equations Cm we have uncovered merit further investigation from the point of
view of integrability, since we have demonstrated that each Cm possesses the Painlevé property,
a necessary condition for a differential equation to be integrable.
The congruence subgroups of SL(2,Z) also arise in the context of state counting in four
dimensional N = 4 supersymmetric string theories [12]. Of particular interest is the generating
function that counts states which preserve half of the N = 4 supersymmetry; this generating
function is given in terms of a denominator formula which contains either η-products or η-
quotients.8 It is not difficult to check that the solution to the Chazy equation for m = 3, 4
can be written in terms of logarithmic derivatives of the very same η-products that appear in
these generating functions.9 It would be interesting to understand this relation further, and in
particular understand how the Hecke groups organise the degeneracies of the supersymmetric
states in these string theories.
A Generalized Halphen system and Fourier expansions
In this section we collect a few results about the Halphen system and the automorphic forms E
(m)
2k
associated to H(m). The generalized Halphen system was introduced in (2.1). In this section,
we provide explicit solutions to this system of differential equations following [15, Theorem 3].
In terms of the parameters (a, b, c) introduced in (2.2), we have the solution
t
(m)
1 (τ) =
1
αm
(a− 1)zQ(a, b; z)2F1(1− a, b; 1; z)2F1(2− a, b; 2; z),
t
(m)
2 (τ)− t1(τ) =
1
αm
Q(a, b; z)2F1(1− a, b; 1; z)2,
t
(m)
3 (τ)− t1(τ) =
1
αm
zQ(a, b; z)2F1(1− a, b; 1; z)2. (A.1)
The function 2F1 is the Gauss hypergeometric function while the function Q(a, b; z) is given by
Q(a, b; z) =
iπ(1− b)
2 sin(πb) sin(πa)
(1− z)b−a.
8In fact, there exists a larger class of N = 4 superstring models whose duality group contains the Fricke
involution τ → − 1
Nτ
, in addition to the subgroup of SL(2,Z) [29].
9Since E
(m)
2 , which solves the Chazy equation, is proportional to the function A
(m)
1 that appears in the curve
associated to the Hecke group, this can be seen from the factorized form of the curve in (3.8) and (3.11), and
some well known θ-function identities.
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 23
The parameter z is related to the standard hauptmodul J(m) of the Hecke group H(m) by the
formula
z =
1
1− J(m)
.
Here we have rescaled the solutions in [15] by an overall constant αm, which we will fix mo-
mentarily. For the Hecke group H(m) we now work out the Fourier expansions of the solutions
of the generalized Halphen equations t
(m)
k near the cusp at i∞. These, in turn, determine the
q-expansions for the Eisenstein series E
(m)
2k of the Hecke group via (2.5).
An important ingredient in the Fourier expansion of the solutions to the generalized Halphen
system is the Fourier expansion of the standard hauptmodul corresponding to the Hecke group
in question. This is obtained by solving a Schwarzian differential equation order-by-order about
the point τ = i∞ following the prescriptions in [15, 30]. This yields
J(m) =
1
dq
+
4 + 3m2
8m2
+
dq
1024m2
(
69m4 − 8m2 − 48
)
+
d2
(
27m6 − 116m4 + 16m2 + 64
)
q2
3456m6
+ · · · ,
where d is defined following [15]. First define the integers a′, b′, c′, d′ such that
a′
b′
=
3m− 2
4m
,
c′
d′
=
3m+ 2
4m
.
Then d is defined to be the following product
d = b′d′
b′−1∏
k=1
(
2− 2 cos
2πk
b′
)− 1
2
cos 2πa′
b′
d′−1∏
`=1
(
2− 2 cos
2π`
d′
)− 1
2
cos 2πc′
d′
.
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
(m)
2 begins with unit coefficient; this
uniquely fixes the coefficient αm to be
αm = −
(3m+ 2) sec π
m
8m
.
With our conventions now made fully explicit, it is easy to check that (A.1) solves the generalized
Halphen system. We can now use (2.5) and (2.7) to construct Fourier expansions of the Eisenstein
series we use in this paper. For example, the first few Eisenstein series have the following Fourier
expansions
E
(m)
2 = 1− (m− 2)2
8m2
(qd)− (m(m+ 12)− 44)(m− 2)2
512m4
(qd)2 + · · · ,
E
(m)
4 = 1 +
(
m2 − 4
)
4m2
(qd) +
(
7m4 − 88m2 + 240
)
256m4
(qd)2 + · · · ,
E
(m)
6 = 1−
(
m2 + 12
)
8m2
(qd) +
(
−31m4 + 152m2 + 912
)
512m4
(qd)2 + · · · ,
E
(m)
8 = 1−
(
m2 + 4
)
2m2
(qd)−
(
m4 − 168m2 − 368
)
128m4
(qd)2 + · · · .
These expansions prove useful in implementing the cuspidal boundary conditions and finding
the solutions to the anomaly equations satisfied by the A
(m)
k .
24 S.K. Ashok, D.P. Jatkar, and M. Raman
B Ramanujan identities for cusp forms
Recall the definition [15, Theorem 2(i)] of the cusp form f
(m)
2k of a Hecke group H(m)
f
(m)
2k = (−1)k
(
J ′(m)
)k
J1−k
(m)
(
J(m) − 1
)d k
2
e−k
. (B.1)
Here J(m) is the standard 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
(m)
2k in (2.5), we recall
from [15, Theorem 4(iii)] that the solutions to the generalized Halphen system can be written
down in terms of the standard hauptmodul of the corresponding Hecke group
x =
J ′(m)
J(m)
and y =
J ′(m)
J(m) − 1
.
Substituting these into the definition of E
(m)
2k , we obtain
E
(m)
2k =
(
J ′(m)
)k
J1−k
(m) (J(m) − 1)−1. (B.2)
This yields the following simple relation between the Eisenstein series and the cusp form
(−1)kE
(m)
2k = f
(m)
2k
(
J(m) − 1
)d2k , (B.3)
where d2k is defined as [15, Theorem 2(i)]
d2k = k −
⌈
k
2
⌉
−
⌈
k
m
⌉
.
It is related to the dimension of the space of weight (2k) automorphic forms m2k (for k ≤ m) as
dimm2k = d2k + 1 = k −
⌈
k
2
⌉
=
⌊
k
2
⌋
. (B.4)
In (B.4), we have used the fact that k ≤ m, and the floor and ceiling functions are defined as
bxc = max{n ∈ Z : n ≤ x}, dxe = min{n ∈ Z : n ≥ x}.
From these definitions it is clear that
dxe − bxc =
{
0 for x ∈ Z,
1 for x /∈ Z,
and in particular, for n ∈ Z, we have bnc = dne = n. On using (B.3) the Ramanujan identi-
ties conjectured in [30, Section 3] may be directly related to the ones derived in the previous
subsection.
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, and the anonymous referees for valuable comments and feedback. 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).
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations 25
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1 Introduction
2 Ramanujan identities
2.1 Generalized Halphen systems and a proof
3 Hyperelliptic curves, anomaly equations, and boundary conditions
3.1 Curves and anomalies
3.2 Examples
3.2.1 H(3)
3.2.2 H(4)
3.3 General solution to the anomaly equation
3.4 Discriminants
3.5 Gauss–Manin connections
3.5.1 Height four
4 Chazy equations
4.1 Higher-order Chazy equations
4.2 H(m) orbits
4.3 Relation to Maier's equation
5 Painlevé analysis of the Chazy equations
5.1 A primer on stability
5.2 The Painlevé property
6 Discussion
A Generalized Halphen system and Fourier expansions
B Ramanujan identities for cusp forms
References
|
| id | nasplib_isofts_kiev_ua-123456789-210609 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:20Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan 2025-12-12T10:41:57Z 2020 Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations. Sujay K. Ashok, Dileep P. Jatkar and Madhusudhan Raman. SIGMA 16 (2020), 001, 26 pages 1815-0659 2010 Mathematics Subject Classification: 34M55; 11F12; 33E30 arXiv:1810.07919 https://nasplib.isofts.kiev.ua/handle/123456789/210609 https://doi.org/10.3842/SIGMA.2020.001 We study various relations governing quasi-automorphic forms associated with discrete subgroups of SL(2, ℝ), namely the Hecke groups. We show that 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 quasi-modular forms of SL(2, ℤ). Each Hecke group is then associated with a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the m = 3 and 4 cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear 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⁽ᵐ⁾₂ associated to H(m) and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property. We would like to thank Suresh Govindarajan for the discussions and for the collaboration during an early stage of the project. We also thank Renjan John for helpful comments on an earlier version of the manuscript, and the anonymous referees for valuable comments and feedback. 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 to participate in the program Quantum Fields, Geometry and Representation Theory (Code: ICTS/qftgrt/2018/07). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations Article published earlier |
| spellingShingle | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan |
| title | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_full | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_fullStr | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_full_unstemmed | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_short | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_sort | aspects of hecke symmetry: anomalies, curves, and chazy equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210609 |
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