Triangle Groups: Automorphic Forms and Nonlinear Differential Equations
We study the relations governing the ring of quasiautomorphic forms associated with triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated with these triangle groups are shown to satisfy Ramanujan-like identities. These identities,...
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| description | We study the relations governing the ring of quasiautomorphic forms associated with triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated with these triangle groups are shown to satisfy Ramanujan-like identities. These identities, in turn, allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated with the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 102, 13 pages
Triangle Groups: Automorphic Forms
and Nonlinear Differential 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 April 21, 2020, in final form October 05, 2020; Published online October 11, 2020
https://doi.org/10.3842/SIGMA.2020.102
Abstract. We study the relations governing the ring of quasiautomorphic forms associated
to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups.
The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-
like identities. These identities in turn allow us to associate a nonlinear differential equation
to each triangle group. We show that they are solved by the quasiautomorphic weight-2
Eisenstein series associated to the triangle group and its orbit under the group action. We
conclude by discussing the Painlevé property of these nonlinear differential equations.
Key words: triangle groups; Chazy equations; Painlevé analysis
2020 Mathematics Subject Classification: 34M55; 11F12; 33E30
1 Introduction and discussion
Triangle groups are an infinite family of discrete subgroups of PSL(2,R), the group of orienta-
tion-preserving isometries of the upper-half plane. Abstractly, these groups are characterised
by an ordered triple of positive integers {mi}3i=1 and generated by the letters {gi}3i=1, and
circumscribed by the relations
gmi
i = 1 for i ∈ {1, 2, 3} and g1g2g3 = 1.
The theory of automorphic functions and forms associated to triangle groups has been detailed
in [11]. Triangle groups with arithmetic properties have appeared in the study of string theo-
ries [7, 14, 19], supersymmetric gauge theories [3], the study of knots [17, 25, 26], and simple
quantum mechanical systems [5, 23] and further motivates our investigations. The present work
can be thought of as an extension of our results for Hecke groups obtained in [4] to all triangle
group with a single cusp.
Differential equations satisfied by modular [13, 16] and quasi-modular [1, 6, 21] objects have
been the subject of sustained interest. The primary objects of study in this paper are an infinite
class of nonlinear differential equations naturally associated to triangle groups. In Section 2,
we review the construction of quasi-automorphic forms associated to arbitrary triangle groups
as solutions to the generalised Halphen system, a system of first-order differential equations,
following [11]. The automorphic forms associated to these triangle groups are then found to
mailto:sashok@imsc.res.in
mailto:dileep@hri.res.in
mailto:madhur@theory.tifr.res.in
https://doi.org/10.3842/SIGMA.2020.102
2 S.K. Ashok, D.P. Jatkar and M. Raman
satisfy a system of first-order differential equations whose structure is strongly reminiscent of
the Ramanujan identities governing the ring of (quasi-)modular forms of PSL(2,Z).
The construction of automorphic forms also exhibits the existence of certain ring relations –
that is, a set of relations that imply that the ring of automorphic forms associated to a triangle
group is not freely generated. These ring relations play a central role in this note. Indeed,
in Section 3 we use both the Ramanujan identities and the ring relations to derive a non-linear
differential equation previously derived in [15]. Notably, our derivation does not rely on any
explicit Fourier expansions or require the use of computer algebra.
In the final section, we present the derivation of higher-order differential equations associated
to each triangle group via differential elimination on the Ramanujan identities. These equations
are natural analogues of the Chazy equation associated to PSL(2,Z). We then prove that
these differential equations possess the Painlevé property, which is to say that the only movable
singularities of these differential equations are poles.
In his analysis of second-order ordinary differential equations, Painlevé restricted his attention
to only those equations whose movable singularities were poles. All these equations turned out
to be integrable. Based on this, the authors of [2] devised the Painlevé test, a set of criteria that,
if satisfied by a differential equation, establishes the Painlevé property. This test works perfectly
for all soliton equations but requires a more careful stability analysis for those equations which
possess negative resonances, along the lines of [9]. The nonlinear differential equations we study
in this note generically possess negative resonances. In some cases, as we will show, this is no
obstruction to the demonstration of the Painlevé property.
A proof of the Painlevé property is a prerequisite to the demonstration of integrability.
Indeed, the higher-order Chazy equations uncovered in [4] already possess tantalising hints that
these equations may be integrable. Consider as an example the Chazy equation corresponding to
the Hecke group H(4) in [4, Theorem 4.3], written in terms of the Ramanujan–Serre derivative D
and a weight-4 modular form Z:
D3Z + 6ZDZ = 0.
This presentation manifests a striking resemblance to the stationary Korteweg–de Vries equation:
u′′′ + 6uu′ = 0.
Given the uniform structure of the Ramanujan identities for triangle groups we write down, and
the higher-order Chazy and Maier equations that are consequent upon them, one is tempted to
look for a framework, perhaps even a hierarchy, within which these differential equations may
be organised. The results of this note, in particular investigations into the Painlevé property,
are an important step in this direction.
2 Triangle groups and Ramanujan identities
We begin with a brief review of triangle groups and the associated generalised Halphen system.
We will then review the main theorems of [11] regarding the construction of Eisenstein series
in terms of the solutions to the Halphen system. Starting from these definitions, we then derive
the Ramanujan relations that are valid for any triangle group with a single cusp.
2.1 Triangle groups
A triangle group is identified by its type, denoted by t and consisting of a triple of integers
2 ≤ m1 ≤ m2 ≤ m3 ≤ ∞. The integers that make up a triple define a triple of angles
(π/m1, π/m2, π/m3), which are taken to be the angles subtended by the edges of a triangle
Triangle Groups: Automorphic Forms and Nonlinear Differential Equations 3
in the hyperbolic plane. The group of reflections of this triangle across its sides generates
a tessellation of the hyperbolic plane, and is called a triangle group.
Triangle groups are discrete subgroups of PSL(2,R), the group of isometries of the upper-
half plane H, and we study those whose fundamental domains have finite hyperbolic area.1 We
will further restrict our attention to those triangle groups with a single cusp – meaning the
angle subtended by one pair of edges of the aforementioned triangle is zero – and without loss
of generality this choice will be implemented by setting m3 = ∞.2 In what follows, when we
refer to triangle groups we will always mean triangle groups with a single cusp, characterised by
a pair of integers so its type t = (m1,m2,∞) such that 2 ≤ m1 ≤ m2 < ∞. For example, the
modular group PSL(2,Z) is a triangle group of type (2, 3,∞), and the Hecke groups H(m) are
of type (2,m,∞). Finally, throughout this paper, the subscript t will indicate that the object
in question is associated to any specific choice of triangle group.
We now move on to discuss the construction of Eisenstein series associated to these triangle
groups. This will be done following [11], who construct these automorphic forms out of solutions
to a generalised Halphen system.
2.2 Generalized Halphen systems and Eisenstein series
The generalized Halphen system is a set of coupled ordinary first-order differential equations for
three variables {tk,t(τ)}3k=1 given as follows:
t′1,t = (a− 1)(t1,tt2,t + t1,tt3,t − t2,tt3,t) + (b+ c− 1)t21,t,
t′2,t = (b− 1)(t2,tt3,t + t2,tt1,t − t1,tt3,t) + (a+ c− 1)t22,t,
t′3,t = (c− 1)(t3,tt1,t + t3,tt2,t − t1,tt2,t) + (a+ b− 1)t23,t. (2.1)
Here, the parameters {a, b, c} are specified by the type t of the triangle group as
a =
1
2
(
1− 1
m1
+
1
m2
)
, b =
1
2
(
1− 1
m1
− 1
m2
)
, c =
1
2
(
1 +
1
m1
− 1
m2
)
, (2.2)
and the accent ′ denotes the following derivative:
h′ :=
1
2πi
d
dτ
h(τ),
where τ is a coordinate on the upper-half plane H.
The solutions to the generalized Halphen system can be obtained explicitly in terms of
hypergeometric functions [11, Theorem 3(i)]. We relegate a discussion of these solutions to
Appendix A – notably, our conventions differ slightly from theirs – and instead proceed to the
construction of Eisenstein series. We start with the following theorem due to [11, see p. 707
and Theorem 4(iv)]:
Theorem 2.1. The algebra of holomorphic automorphic forms of a triangle group t is generated
by two branches of Eisenstein series:
{
E
(1)
2k,t
}m1
k=2
and
{
E
(2)
2k,t
}m2
k=2
. For each admissible k, the
corresponding Eisenstein series has an automorphic weight 2k under the triangle group. These
Eisenstein series are constructed out of solutions to the generalised Halphen system as follows:
define the linear combinations
xt = t1,t − t2,t and yt = t3,t − t2,t, (2.3)
1As a result, the following types are not considered: (i) (2, 2,m) for all m ≤ ∞, (ii) (2, 3, n) for n ≤ 6, and
(iii) the types (2, 4, 4) and (3, 3, 3).
2For a more algebraic characterisation, these triangle groups are isomorphic to the free product group Zm1∗Zm2 ,
where the notation Zk denotes a cyclic group of order k.
4 S.K. Ashok, D.P. Jatkar and M. Raman
in terms of which the Eisenstein series are
E
(1)
2k,t = xty
k−1
t for 2 ≤ k ≤ m1, (2.4)
E
(2)
2k,t = xk−1t yt for 2 ≤ k ≤ m2. (2.5)
The above construction of the Eisenstein series in terms of solutions to the generalised
Halphen system straightforwardly implies the following results.
Corollary 2.2. The algebra of automorphic forms is not freely generated. For both branches of
Eisenstein series, i.e., for ` ∈ {1, 2} we have the following relations:
E
(`)
2p,tE
(`)
2(k−p+1),t = E
(`)
2p′,tE
(`)
2(k−p′+1),t, (2.6)
for any integers 2 ≤ p ≤ k− 1 and 2 ≤ p′ ≤ k− 1. Additionally, we have the following relations
binding the two branches of Eisenstein series:
E
(1)
2k,tE
(2)
2k,t = (E4,t)
k. (2.7)
Collectively, we will refer to the above identities as ring relations. They will be used exten-
sively in the following sections.
Remark 2.3. Associated to every triangle group, there exists a unique weight-4 Eisenstein
series, obtained as the case k = 2 of (2.5). Since this automorphic form may be thought of
as belonging to either branch of Eisenstein series, we denote it without any reference to these
branches, as E4,t.
The triangle groups also come equipped with a quasi-automorphic weight 2 Eisenstein se-
ries E2,t, which is also given in terms of a linear combination of the generalised Halphen vari-
ables ti,t [11, Theorem 4(iii)].
Theorem 2.4. For any triangle group with type t, we can associate a holomorphic, quasi-
automorphic weight-2 Eisenstein series E2,t. In terms of solutions to a generalised Halphen
system, it is given by
E2,t =
1
m1 +m2 −m1m2
(
2m1xt + 2m2yt + (m1 +m2 +m1m2)t2,t
)
. (2.8)
This choice of linear combination is tied to the choice of normalisation for solutions to the
generalised Halphen system and ensures that at i∞, E2,t is unity – see Appendix A.
With the introduction of the Eisenstein series that generate the ring of holomorphic, quasi-
automophic forms of triangle groups t now complete, we are in a position to state our first
result. It generalizes our previous results on Ramanujan identities for Hecke groups [4, 22] to
an arbitrary triangle group with cusp.
Lemma 2.5. The Eisenstein series associated to a triangle group t, as defined in Theorems 2.1
and 2.4, and for k > 2, obey the following identities:
E′2,t =
(m1m2 −m1 −m2)
2m1m2
(
E2
2,t − E4,t
)
,
E′4,t = 2
(m1m2 −m1 −m2)
m1m2
E2,tE4,t −
m1 − 2
m1
E
(1)
6,t −
m2 − 2
m2
E
(2)
6,t ,
E
(1)′
2k,t =
k(m1m2−m1−m2)
m1m2
E2,tE
(1)
2k,t−
(km2−k−m2)
m2
E4,tE
(1)
2k−2,t−
(m1−k)
m1
E
(1)
2k+2,t,
E
(2)′
2k,t =
k(m1m2−m1−m2)
m1m2
E2,tE
(2)
2k,t−
(km1−k−m1)
m1
E4,tE
(2)
2k−2,t−
(m2−k)
m2
E
(2)
2k+2,t. (2.9)
Triangle Groups: Automorphic Forms and Nonlinear Differential Equations 5
Proof. We perform a linear transformation from the generalized Halphen variables {ti,t}3i=1
to the variables (E2,t, xt, yt) using (2.3) and (2.8). The generalised Halphen system is then
equivalent to the following system of ordinary first-order differential equations:
E′2,t =
(m1m2 −m1 −m2)
2m1m2
(
E2
2,t − E4,t
)
,
x′t =
(m1m2 −m1 −m2)
m1m2
E2,txt +
x2t
m2
−
(
1− 1
m1
)
E4,t,
y′t =
(m1m2 −m1 −m2)
m1m2
E2,tyt +
y2t
m1
−
(
1− 1
m2
)
E4,t. (2.10)
The first equation in (2.10) proves the first of the Ramanujan identities. For E4,t, we consider
the derivative of (2.5) for k = 2 to get
E′4,t =
[
(m1m2 −m1 −m2)
m1m2
E2,txt +
x2t
m2
−
(
1− 1
m1
)
xtyt
]
yt
+ xt
[
(m1m2 −m1 −m2)
m1m2
E2,tyt +
y2t
m1
−
(
1− 1
m2
)
xtyt
]
= 2
(m1m2 −m1 −m2)
m1m2
E2,tE4,t −
m1 − 2
m1
E
(1)
6,t −
m2 − 2
m2
E
(2)
6,t .
Similarly, for the relations governing either branch of Eisenstein series E
(`)
2k,t of weight 2k > 4,
differentiate (2.5) and use (2.10), then resolve all products of xt and yt into Eisenstein series
according to (2.5). �
Owing to their similarities with relations satisfied by the Eisenstein series associated to
PSL(2,Z), these identities will henceforth be referred to as the Ramanujan identities associ-
ated to the triangle group t.
3 Maier equations for triangle groups
The Ramanujan identities are a set of coupled first order differential equations that involve
all the Eisenstein series. However as we have seen, there are non-trivial ring relations that
relate the higher weight Eisenstein series. We now show how it is possible to use the ring
relations and deduce from the Ramanujan relations a higher order non-linear differential equation
for every triangle group. We show that the equation coincides precisely with the one obtained
by Maier [15]. Our derivation has the advantage of being completely elementary without the
need for advanced computer algebra or explicit Fourier expansions of the automorphic forms and
turns out to follow simply from the existence of the Ramanujan identities and the ring relations.
Before we proceed further we first introduce the notion of a covariant derivative. In fact,
the structure of the Ramanujan identities lends itself to a natural choice of derivation on ring
of automorphic forms. Let the space of weight-k automorphic forms associated to a triangle
group t be denoted mk.
3 We define the operator D such that
D: mk −→ mk+2,
Explicitly, this Ramanujan–Serre derivative is defined as
D =
1
2πi
d
dτ
− k
2
(
1− 1
m1
− 1
m2
)
E2,t.
3The dependence on the choice of triangle group is left implicit to avoid clutter.
6 S.K. Ashok, D.P. Jatkar and M. Raman
With this definition, the first few Ramanujan identities may be written in terms of the Rama-
nujan–Serre derivative as
DE4,t =
2−m1
m1
E
(1)
6,t +
2−m2
m2
E
(2)
6,t , (3.1)
DE
(1)
6,t =
3−m1
m1
E
(1)
8,t +
3− 2m2
m2
E2
4,t, (3.2)
DE
(2)
6,t =
3−m2
m2
E
(2)
8,t +
3− 2m1
m1
E2
4,t. (3.3)
We now act with the Serre derivative on (3.1), and use equations (3.2) and (3.3) along with the
ring relation (2.6), i.e., we use
E
(`)
4,tE
(`)
8,t =
(
E
(`)
6,t
)2
for ` = 1, 2,
to obtain the following differential equation relating Eisenstein series’ of weights four and six:
E4,tD
2E4,t =
(3− 2m1)(2−m2) + (3− 2m2)(2−m1)
m1m2
E3
4,t
+
(m1 − 2)(m1 − 3)
m2
1
(
E
(1)
6,t
)2
+
(m2 − 2)(m2 − 3)
m2
2
(
E
(2)
6,t
)2
. (3.4)
Note that each term in the above equation is automorphic with weight 12, and consequently
that the equation is invariant under triangle group action.
Before turning to the main goal of this section, i.e., deriving the Maier equations for arbitrary
triangle groups [15], we first consider two special cases. These cases are distinguished by simpler
Maier equations with lower weight than one might expect, chiefly due to the simplifications that
arise when the integers specifying the type t of the triangle group are tuned to special values.
Theorem 3.1 (isosceles triangle groups). The Maier equations corresponding to isosceles tri-
angle groups with type tM = (M,M,∞) are identical to the Maier equations for the Hecke
group H(M).
Proof. Equations (3.1) and (3.4) simplify considerably for isosceles triangle groups with
m1 = m2 = M . Using the square of (3.1) and the ring relation (2.7), i.e.,
E
(1)
6,tM
E
(2)
6,tM
= E3
4,tM
, (3.5)
we can write (3.4) as
E4,tM D2E4,tM =
M − 3
M − 2
(DE4,tM )2 +
2(M − 2)
M
E3
4,tM
.
Up to an unimportant rescaling of E4,tM , this is the same equation that was derived for the
Hecke group H(M) in [4]. �
This fact was already observed in [15] and we see that it is a simple consequence of the
Ramanujan relations and the ring relations. There exists a simple geometrical basis for this
fact: an isosceles hyperbolic triangle of type (M,M,∞) can be bisected into two hyperbolic
triangles of type (2,M,∞) [18]. We now turn to triangles groups of type t3,M = (3,M,∞). We
present this example because it is instructive and will help build intuition for the most general
case.
Triangle Groups: Automorphic Forms and Nonlinear Differential Equations 7
Example 3.2 (type (3,M,∞) triangle groups). In this case as well the goal is to write an equa-
tion purely in terms of the modular covariant derivatives DkE4,t3,M . Start with (3.1) and solve
for E
(1)
6,t3,M
as
E
(1)
6,t3,M
= −3DE4,t3,M −
3(M − 2)
M
E
(2)
6,t3,M
.
By combining this with the ring relation (3.5), we obtain the relation:
DE4,t3,ME
(2)
6,t3,M
= −M − 2
M
(
E
(2)
6,t3,M
)2 − 1
3
E3
4,t3,M
.
One can write an independent expression for the square of E
(2)
6,t3,M
in terms of the E4,t3,M and
its modular covariant derivatives courtesy of (3.4):
(
E
(2)
6,t3,M
)2
=
M2
(M − 3)(M − 2)
E4,t3,M D2E4,t3,M +
M(5M − 9)
3(M − 3)(M − 2)
(E4,t3,M )3. (3.6)
This leads to the following relation:
DE4,t3,ME
(2)
6,t3,M
=
2
3
(2M − 3)
(M − 3)
(E4,t3,M )3 − M
M − 3
E4,t3,M D2E4,t3,M .
Squaring both sides of this equation and using the ring relation (3.6) once again leads to the
following equation, with each term of weight 20:
9M2(M − 3)D2E4,t3,M (DE4,t3,M )2 − 9M2(M − 2)(D2E4,t3,M )2E4,t3,M
+ 3M(M − 3)(9− 5M)(DE4,t3,M )2(E4,t3,M )2
+ 12M(M − 2)(2M − 3)D2E4,t3,M (E4,t3,M )3 − 4(M − 2)(2M − 3)2(E4,t3,M )5 = 0.
We now bring this strategy to bear on arbitrary triangle groups. This theorem corresponds
directly to [15, Theorem 6.4] and we use the same notation for the coefficients for ease of
comparison.
Theorem 3.3. For an arbitrary triangle group of type t = (m1,m2,∞), the weight four auto-
morphic form E4,t satisfies the following weight-24 differential equation:
C88E
2
4,t
(
D2E4,t
)2
+ C86E4,t
(
DE4,t
)2
D2E4,t + C84E
4
4,t
(
D2E4,t
)
+ C66(DE4,t)
4
+ C64E
3
4,t(DE4,t)
2 + C44E
6
4,t = 0, (3.7)
where the coefficients are given by
C88 = m2
1m
2
2(m1 − 2)(m2 − 2),
C86 = −m2
1m
2
2((m1 − 2)(m2 − 3) + (m2 − 2)(m1 − 3)),
C84 = −4m1m2(m1 − 2)(m2 − 2)(m1m2 −m1 −m2),
C66 = m2
1m
2
2(m1 − 3)(m2 − 3),
C64 = m1m2
(
3(m1 −m2)
2 + 2m2(m2 − 3)(m1 − 2)2 + 2m1(m1 − 3)(m2 − 2)2
)
,
C44 = 4(m1 − 2)(m2 − 2)(m1m2 −m1 −m2)
2. (3.8)
Proof. First, solve for E
(1)
6,t using (3.1):
E
(1)
6,t = −DE4,tm1
m1 − 2
− m1 (m2 − 2)
m2 (m1 − 2)
E
(2)
6,t .
8 S.K. Ashok, D.P. Jatkar and M. Raman
Then substitute this into the ring relation (3.5) to obtain the following equation:
m2(m1 − 2)
m1(m2 − 2)
E3
4,t +
m2
m2 − 2
E
(2)
6,t DE4,t +
(
E
(2)
6,t
)2
= 0. (3.9)
A second independent weight-12 equation is arrived at by acting with E4,tD on (3.1) and in the
resulting equation, we substitute the equations (3.2) and (3.3). After this, on taking suitable
linear combinations we eliminate
(
E
(2)
6,t
)2
and obtain the following equation:
DE4,tE
(2)
6,t = −(m1 − 2)m2
m1 −m2
E4,tD
2E4,t +
(m1 − 3)m2
m1 −m2
(DE4,t)
2
+
2(m1 − 2)(m1m2 −m1 −m2)
m1(m1 −m2)
E3
4,t. (3.10)
We substitute this into (3.9) to obtain the ring relation for
(
E
(2)
6,t
)2
:
(
E
(2)
6,t
)2
= − (m1 − 2)m2
2
(m1 −m2)(m2 − 2)
E4,tD
2E4,t +
(m1 − 3)m2
2
(m1 −m2)(m2 − 2)
DE2
4,t
+
m2(m1 − 2)(2m1m2 − 3m1 −m2)
m1(m1 −m2)(m2 − 2)
E3
4,t. (3.11)
We now have all the ingredients in place to derive the Maier equation for the general triangle
group. Consider the square of (3.10) and substitute the expression for the square of E
(2)
6,t derived
in (3.11) above. This leads to the weight-24 equation written purely in terms of E4,t and its
modular covariant derivatives, given in (3.7), with the coefficients given in (3.8). Finally, that
the orbit of E4,t under the corresponding triangle group also solves the nonlinear differential
equation under consideration follows from its manifest covariance. �
Up to a normalization factor, this precisely matches the Maier equation for the triangle
group of type (m1,m2,∞) that is presented in [15, Theorem 6.4]. We see that this differential
equation is a direct consequence of the Ramanujan identities satisfied by the Eisenstein series
in combination with the algebraic ring relations that bind them.
4 Painlevé analysis
We now turn to a study of the nonlinear differential equations that we introduced in the previous
section. In particular, we will show that these equations satisfy the Painlevé property, i.e., that
the only movable singularities of these differential equations are poles. For a brief review of the
stability of differential equations and the Painlevé property, we refer the reader to [4].
4.1 Chazy equations
In this section, we outline a process of differential elimination that may be used to derive
higher-order nonlinear differential equations satisfied by E2,t. These differential equations are of
order m2, and are natural analogues of the higher-order Chazy equations studied in [4].4
The process of differential elimination is complicated by presence of two distinct branches
of Eisenstein series. In order to sidestep this difficulty, we effect a change of basis in the space
of automorphic forms that is “diagonal” in the sense that only specific linear combinations of
4We remind the reader that without loss of generality, we choose the integers that specify the type t of the
triangle group such that m1 ≤ m2.
Triangle Groups: Automorphic Forms and Nonlinear Differential Equations 9
the two branches of Eisenstein series appear together in the Ramanujan identities. For triangle
group t and k ≥ 3, define the following combinations of the two branches of Eisenstein series:
G2k,t =
k−1∏
p=2
(
m1 − p
m1
)
E
(1)
2k,t +
k−1∏
p=2
(
m2 − p
m2
)
E
(2)
2k,t. (4.1)
Since the Eisenstein series’ of weights two and four are unique we identify G4,t = E4,t and
G2,t = E2,t for uniformity of notation. Once this change of basis is effected, we may proceed with
straightforward differential elimination. Note that the G2k,t are not algebraically independent.
This is once again a consequence of the ring relations. The utility of this change of basis is
presented in the following simple example.
Example 4.1 (type (4, 4,∞) triangle group). In terms of the linear combinations defined
in (4.1), the Ramanujan identities (2.9) associated to the triangle group (4, 4,∞) take the
following form:
G′2,t =
1
4
(
G2
2,t −G4,t
)
,
G′4,t = G2,tG4,t −G6,t,
G′6,t =
1
4
(
6G2,tG6,t − 5G2
4,t − 4G8,t
)
,
G′8,t =
1
2
(4G2,tG8,t −G4,tG6,t).
Using a procedure outlined in [4, Section 4], we may use these equations to arrive at a differ-
ential equation C(4,4) satisfied by y = G2,t(τ):
C(4,4) : y(4) − 5yy(3) − 9yy′2 + (6y′ + 6y2)y′′ = 0. (4.2)
Remark 4.2. The differential equation (4.2), as we will see shortly, has Bureau symbol P1,
meaning the differential equation admits solutions having a Laurent expansion with leading
divergence ∼(τ − τ0)−1. This equation is known to possess the Painlevé property, and appears
in [10, equation (2.3)].
These differential equations are natural analogues of the Chazy equation associated to the
quasimodular form of PSL(2,Z) [8, 24], and also the higher-order Chazy eqautions associated
to Hecke groups [4]. The above systematic procedure for constructing higher-order nonlinear
differential equations motivates the following proposition.
Proposition 4.3. Each triangle group is associated to a nonlinear differential equation of or-
der m2, which we shall denote by Ct. Further, each term in this differential equation is of order
2m2 + 2.
Proof. The Ramanujan identities in terms of the combinations G2k,t are a system of m2 first
order differential equations in m2 variables. The proposition follows by differential elimina-
tion. �
4.2 Painlevé analysis
We now briefly recall the main steps of the generalized Painlevé analysis, following the work
of [9]. Consider a differential equation represented by
M [y] = 0. (4.3)
10 S.K. Ashok, D.P. Jatkar and M. Raman
We first propose a local solution yc (around a simple pole singularity) to the nonlinear differential
equation in Frobenius form. One then linearises the equation around the solution
d
dε
M [yc + εw]
∣∣∣
ε=0
= 0, (4.4)
and look for solutions w = w(τ) also in the Frobenius form. In this case, as in the original Chazy
equation and in the higher order generalizations in [4], we find that the linearised solution has
poles of higher order than the originally proposed solution to the differential equation. At first
glance these so-called “negative resonances” appear to make the leading order solution unstable
to linear perturbations. However, as shown in [9, 12], it is possible to perform the stability
analysis in such a situation by writing the solution as a functional Frobenius series:
y =
∞∑
i=0
yiχ
i−a.
Here, χ = χ(τ) satisfies the Riccati equation of the form (see [4] for more details):
χ′ = 1− S
2
χ2,
and yi are the undetermined coefficients in the Frobenius ansatz. We find that leading order
ansatz satisfies the differential equation provided a is an integer solution to an indicial equation.5
Once a particular branch of the indicial equation is chosen, we can solve for the coefficients yi.
We then propose the following ansatz for the next to leading order solution:
w =
∞∑
i=0
wiχ
i−b.
We then substitute this into the linearized equation in (4.4) and solve for b. The phenomenon of
“negative resonances” is when b is a positive integer larger than a . This appears to destabilize
the leading order solution as it appears to have a more singular behaviour near χ = 0. However
on setting the coefficients of the leading order solution to be the yi that we have solved for,
we find that the coefficients of the higher order negative resonance vanishes and this implies
that the negative resonance does not destabilize the solution of the differential equation. Thus,
following the methods of [9, 12] we can claim that the Chazy equation for the triangle groups of
type (M,M,∞) satisfies the Painlevé property and this leads us to the following proposition.
Proposition 4.4. For a triangle group of type t = (M,M,∞), the associated Chazy equa-
tion Ct possesses the Painlevé property. The Eisenstein series y = E2,t(τ) that satisfies both the
differential equations has the following leading order expansion:
y =
M
M − 2
(
− 2
χ
+
S
3
χ+
2S2
45
χ3 +
11S3
945
χ5 + · · ·
)
.
The indicial equation for the resonances has the roots b ∈ {2, . . . ,m+ 1}.
This extends the results obtained for the Hecke groups in [4] to all isosceles triangle groups
with a cusp. Before concluding, we briefly discuss the status of the Chazy equations for triangle
groups of type (m1,m2,∞) with m1 6= m2, and the Maier equation presented in Theorem 3.7.
In the former case, some of the resonance numbers b are negative rational numbers. We have
5Integrality of solutions to an indicial equation is one of the criteria laid out in [9], and ensures single-valuedness
of solutions.
Triangle Groups: Automorphic Forms and Nonlinear Differential Equations 11
observed that triangle groups of type (m,m + k,∞) have k such negative rational resonances.
For the Maier equation, the analysis is more intricate [20].6 In both these cases, the criteria
of [9, 12] that test for the Painlevé property are not straightforwardly applicable. Work on
extending these techniques is currently underway.
A Generalized Halphen system
In this section we collect a few results about the Halphen system and the Fourier expansions of
the automorphic forms E2k,t associated to triangle group of type t = (m1,m2,∞). In terms of
the parameters (a, b, c) introduced in (2.2), the explicit solution of the Halphen system in (2.1)
is given by (see [11, Theorem 3]):
t1,t(τ) =
1
αt
(a− 1)zQ(a, b; z) 2F1(1− a, b; 1; z)2F1(2− a, b; 2; z),
t2,t(τ)− t1,t(τ) =
1
αt
Q(a, b; z) 2F1(1− a, b; 1; z)2,
t3,t(τ)− t1,t(τ) =
1
αt
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.
The parameter z is related to the standard hauptmodul Jt of the triangle group:
z =
1
1− Jt
.
The solutions in [11] differ from those in (A.1) by an overall constant αt that we are free to
choose. We choose this coefficient such that the expansion for t2,t begins with unit coefficient;
this uniquely fixes the coefficient αt to be
αt = −m1 +m2 +m1m2
8m1m2
csc
[
π
2
(
1− 1
m1
+
1
m2
)]
sec
[
π(m1 +m2)
2m1m2
]
.
With this choice of normalisation, the quasi-automorphic weight-2 Eisenstein series E2,t has the
following behavior as τ → i∞:
E2,t(τ) = 1 +O(q).
B Chazy equations
In this appendix we give examples of nonlinear differential equations naturally associated to
triangle groups and solved by their quasiautomorphic weight-2 Eisenstein series. These diffe-
rential equations were discussed in Proposition 4.4 and are arrived at by differential elimination
on the system of equations governing the G2k,t defined in (4.1). For a triangle group of type
6On revisiting our previous work, it appears that the our analysis of the Maier equation [4, equation (4.5)]
using the methods of Conte–Fordy–Pickering [9] are insufficient to prove the Painlevé property. The conclusions
of [4, Proposition 5.1] pertaining to the higher-order Chazy equations Cm [4, Proposition 4.1], however, remain
unchanged. A more comprehensive analysis of the Maier equation is in preparation.
12 S.K. Ashok, D.P. Jatkar and M. Raman
t = (m1,m2,∞), they are denoted Ct ≡ C(m1,m2). The following equations all possess the
Painlevé property:
C(3,3) : y(3) − 2yy′′ + 3y′2 = 0,
C(4,4) : y(4) − 5yy(3) − 9yy′2 +
(
6y′ + 6y2
)
y′′ = 0,
C(5,5) : 75y(5) − 675yy(4) + 520y′′2 − 366y′3 + 2727y2y′2
+ y(3)
(
220y′ + 1959y2
)
+
(
−2664yy′ − 1818y3
)
y′′ = 0,
C(6,6) : 3y(6) − 42y(5)y − 228yy′′2 − 552y3y′2 + 228yy′3 + y(4)
(
214y2 − 12y′
)
+
(
600y2y′ − 114y′2 + 368y4
)
y′′ + y(3)
(
57y′′ − 34yy′ − 468y3
)
= 0.
Acknowledgments
We thank Robert Conte for helpful correspondence, and Hossein Movasati for valuable discus-
sions. We would also like to thank the anonymous referees for valuable comments and feedback.
DPJ and MR are grateful to IMSc, Chennai for hospitality. MR acknowledges support from the
Infosys Endowment for Research into the Quantum Structure of Spacetime. DPJ acknowledges
support from SERB grant CRG/2018/002835.
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1 Introduction and discussion
2 Triangle groups and Ramanujan identities
2.1 Triangle groups
2.2 Generalized Halphen systems and Eisenstein series
3 Maier equations for triangle groups
4 Painlevé analysis
4.1 Chazy equations
4.2 Painlevé analysis
A Generalized Halphen system
B Chazy equations
References
|
| id | nasplib_isofts_kiev_ua-123456789-211018 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T16:57:05Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan 2025-12-22T09:30:37Z 2020 Triangle Groups: Automorphic Forms and Nonlinear Differential Equations. Sujay K. Ashok, Dileep P. Jatkar and Madhusudhan Raman. SIGMA 16 (2020), 102, 13 pages 1815-0659 2020 Mathematics Subject Classification: 34M55; 11F12; 33E30 arXiv:2004.06035 https://nasplib.isofts.kiev.ua/handle/123456789/211018 https://doi.org/10.3842/SIGMA.2020.102 We study the relations governing the ring of quasiautomorphic forms associated with triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated with these triangle groups are shown to satisfy Ramanujan-like identities. These identities, in turn, allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated with the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations. We thank Robert Conte for helpful correspondence and Hossein Movasati for valuable discussions. We would also like to thank the anonymous referees for their valuable comments and feedback. DPJ and MR are grateful to IMSc, Chennai, for hospitality. MR acknowledges support from the Infosys Endowment for Research into the Quantum Structure of Spacetime. DPJ acknowledges support from SERB grant CRG/2018/002835. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Triangle Groups: Automorphic Forms and Nonlinear Differential Equations Article published earlier |
| spellingShingle | Triangle Groups: Automorphic Forms and Nonlinear Differential Equations Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan |
| title | Triangle Groups: Automorphic Forms and Nonlinear Differential Equations |
| title_full | Triangle Groups: Automorphic Forms and Nonlinear Differential Equations |
| title_fullStr | Triangle Groups: Automorphic Forms and Nonlinear Differential Equations |
| title_full_unstemmed | Triangle Groups: Automorphic Forms and Nonlinear Differential Equations |
| title_short | Triangle Groups: Automorphic Forms and Nonlinear Differential Equations |
| title_sort | triangle groups: automorphic forms and nonlinear differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211018 |
| work_keys_str_mv | AT ashoksujayk trianglegroupsautomorphicformsandnonlineardifferentialequations AT jatkardileepp trianglegroupsautomorphicformsandnonlineardifferentialequations AT ramanmadhusudhan trianglegroupsautomorphicformsandnonlineardifferentialequations |