Geom DedicataDOI 10.1007/s10711-011-9688-7

ORIGINAL PAPER

Equivariant functions and integrals of elliptic functions

Abdellah Sebbar · Ahmed Sebbar

Received: 8 August 2011 / Accepted: 3 December 2011© Springer Science+Business Media B.V. 2011

Abstract In this paper, we introduce the theory of equivariant functions by studying theiranalytic, geometric and algebraic properties. We also determine the necessary and sufficientconditions under which an equivariant form arises from modular forms. This study wasmotivated by observing examples of functions for which the Schwarzian derivative is amodular form on a discrete group. We also investigate the Fourier expansions of normalizedequivariant functions, and a strong emphasis is made on the connections to elliptic functionsand their integrals.

Keywords Equivariant functions · Schwarz derivative · Cross-ratio · Modular forms ·Platonic solids · Integrals of elliptic functions

Mathematics Subject Classification (2000) 11F03 · 33E05

1 Introduction

Though the problem we are studying is analytic and geometric in its nature, it can be givena general algebraic formulation as follows: Let G be a group acting on two sets X, Y and letS be a set of functions from X to Y on which G acts in the following way

g. f (x) = g. f (g−1.x) , for all g ∈ G , f ∈ S , x ∈ X.

An equivariant function f : X → Y (also sometimes called concomitant) is a function suchthat g. f = f , that is

A. SebbarDepartment of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canadae-mail: asebbar@uottawa.ca

A. Sebbar (B)Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 cours de la Libération,33405 Talence cedex, Francee-mail: ahmed.sebbar@math.u-bordeaux1.fr

123

Geom Dedicata

f (g.x) = g. f (x), ∀g ∈ G,∀x ∈ X.

If G acts trivially on Y , then f is called an invariant. We propose to illustrate this problemin the following setting which encompasses arithmetic, analytic and geometric flavors.

A subgroup � of the modular group SL2(Z) acts on the upper half-plane

H = {z ∈ C, �z > 0},by linear fractional transformation

γ · z = az + b

cz + d, z ∈ H , γ =

(a b

c d

)∈ �.

We would like to investigate the class of meromorphic functions h on H which commutewith this action. In other words, h satisfies the equivariance relation

h

(az + b

cz + d

)= ah(z)+ b

ch(z)+ d, z ∈ H ,

(a b

c d

)∈ �. (1.1)

These functions first appeared in the works [5] and [34] in the 1960s. Our interest in thesefunctions began with the study of the modular properties of the Schwarz derivative. If f is amodular function on a Fuchsian group G of the first kind, then its Schwarz derivative { f, z}is a weight 4 modular form on a group which is larger than G. In particular if G has genus0, then this larger group is the normalizer of G in PSL2(R). One formulates the converseto this statement as follows: If a meromorphic function f on the upper half-plane H, withsome favorable conditions at the cusps, is such that its Schwarz derivative { f, z} is a weight4 modular form on a certain group G, what can we say about f itself in terms of invariance?Is f a modular function under a nontrivial subgroup of G? How is the size of this subgroupdependent on the analytic properties of f ?

This problem is equivalent to the following: Let f be a meromorphic function on H. LetG be a discrete subgroup of PSL2(R) such that for all z ∈ H and for all γ ∈ G we have

f (γ · z) = �γ · f (z),

where �γ is a certain matrix in PGL2(C). It is clear that � is a group homomorphism.The problem is to determine the kernel of � depending on analytic properties of f and/orgeometric properties of G. When the homomorphism � is the identity, then the function fsatisfies the relation (1.1) for the group G.

If a function h satisfies the relation (1.1) and such that h(z) − z is meromorphic at thecusps, then h will be called an equivariant form. The group � can be taken to be any dis-crete subgroup of SL2(R), but for the time being, only the modular group will be under ourconsideration.

It turns out that to each meromorphic modular form f of weight k for �, one can associatean equivariant form h as follows

h(z) = z + kf (z)

f ′(z). (1.2)

In fact, we determine the necessary and sufficient conditions for an equivariant form to ariseas in (1.2) and this will be called a rational equivariant form. We will also exhibit equivariantforms that are not rational.

Further investigations of an equivariant function h is carried out by looking at the Fourier

coefficients of the periodic function h(z)− z, the Lambert series of1

h(z)− zor at the infinite

123

Geom Dedicata

product of the modular form attached to h in the sense of Borcherds. This will be madeexplicit for equivariant functions attached to classical modular forms such as the Eisensteinseries. The same examples are used in studying the Fourier coefficients of the reciprocal ofthe classical Eisenstein series as was carried out by Hardy, Ramanujan, and more recentlyby Berndt and Bialek among others.

Beside the modular aspect in the structure of equivariant functions, there is also a fasci-nating elliptic aspect to them. Indeed, the fundamental example of the equivariant functionattached to the weight 12 cusp form � given by

h1(z) = z + 12�

�′ = z + 6

iπE2(z),

where E2 is the weight 2 Eisenstein series, is closely related to the Weierstrass ζ -functionwhere ζ ′ = ℘ and ℘ is the classical Weierstrass elliptic function. In fact, as was observedby Heins [15], if L = Zω1 + Zω2 is a lattice with τ = ω2/ω1 ∈ H, and if η1 and η2 are thepseudo-periods of ζ , then ω1η2, as a function of τ , is an equivariant function for SL2(Z) byway of the Legendre relation. It turns out that this equivariant function is nothing else butthe fundamental example h1 above. We will show that, with two exceptions, the integral ofeach ℘n, n ∈ Z, is an equivariant function.

From the differential algebra point of view, we have the important feature that each equi-variant form satisfies a differential equation of degree at most 6; a fact that one expects froma function that satisfies sufficiently many functional equations. To justify this property, onegoes back to the differential ring of modular forms and their derivatives, also known as thering of quasi-modular forms, which is simply C[E2, E4, E6], and thus has transcendencedegree 3. One more time, when we specify to explicit examples of equivariant forms com-ing from the Eisenstein series, we find important differential properties of the reciprocal of

E2, E4 and E6. Namely, using a theorem of Maillet, we show that1

E2,

1

E4and

1

E6satisfy

algebraic differential equations over Q. The fact that the equivariant functions are differ-entially algebraic enables us to use theorems, well known in transcendence theory, such asthose of Maillet and Popken, to control gaps or growth coefficients, in the expansion of thesefunctions in q-series. The same argument is also valid for the reciprocal of E2, E4 and E6

completing, in some sense, the previous work of Hardy, Ramanujan and more recently ofBerndt and Bialek.

Beside the modular, the elliptic and the differential aspects of equivariant functions, thebulk of this work basically articulates on three main axes: the link between the cross-ratioand the Schwarz derivative, then between the equivariance and the Schwarz derivative andfinally between equivariance and the cross-ratio. This scheme is a consequence of the fol-lowing intriguing facts:

• The Schwarz derivative is simply the infinitesimal counterpart of the cross-ratio.• The Schwarz derivative of an equivariant function is a weight four modular form.• The Riccati equation is closely related to the Schwarz differential equation.• The cross-ratio of four solutions to the Riccati equation is a constant in the field C.• The cross-ration of four equivariant functions is a modular function, that is in the function

field of a compact Riemann surface.

All these connections make the equivariant functions extremely rich objects to study.Though the study of equivariant functions was undertaken since the 1960s by

M. Heins and M. Brady in the framework of elliptic functions, we learned only recently fromD. Zagier that the fundamental example h1 was known by W. Nahm in connection with some

123

Geom Dedicata

physical problems. We cite in this context, in Sect. 12, some examples of a special kind ofequivariance, named platonic, that appeared in the Physics literature.

2 SL2(R), the Riccati equation, the Schwarz derivative and the cross-ratio

The Riccati equation is naturally related to the Schwarz differential equation through a changeof function. In this section, we will exhibit similar relations between the Riccati equation andthe functional equations satisfied by equivariant functions. Many properties of these nonlin-ear equations have their origins in the projective differential geometry of the special lineargroup SL2(R) that we recall now. The associated Lie algebra sl(2) is three dimensional withbasis

A1 =(

0 1

0 0

), A2 =

( 12 00 − 1

2

), A3 =

(0 0

1 0

).

The associated infinitesimal generators are ∂x , x∂x , −x2∂x , which have the same commu-tation rules as A1, A2, A3. The group SL2(R) acts on the real line as the projective group

x ∈ R → ax + b

cx + d∈ R,

(a b

c d

)∈ SL2(R).

To give a general idea about the link between the group SL2(R) and nonlinear differentialequations, we first observe that if ft : R → R is a smooth function with f0(x) = x thenft (x) = x + t X (x)+ O(t2) so that X is the infinitesimal generator vector field associated toft . In the particular case where ft is the projective transformation associated with

M(t) =(

1 + ta tbtc 1 + td

)∈ SL2(R) ,

we have

ft (x) = (1 + ta)x + tb

tcx + (1 + td)= x + t

(b + (a − d)x − cx2) + O(t2)

and the associated vector field is X = (b + (a − d)x − cx2) ∂

∂x. We note that for the linear

differential system (u(t)v(t)

)=

(a(t) b(t)c(t) d(t)

) (u(t)v(t)

)

the quotient x = u

vsatisfies the Riccati equation

x(t) = b(t)+ (a(t)− d(t))x(t)− c(t)x(t)2

which can be considered as a differential equation for the integral curve of the vector field

X = (b + (a − d)x − cx2) ∂

∂x. It is important to notice here that the Riccati equation is

the only first order nonlinear ordinary differential equation which possesses the Painlevéproperty, that is of not having removable singularities. Its link with the cross-ratio, and henceprojective geometry, was shown by Lie [21]. We recall that the cross-ratio of four differentcomplex numbers is defined as

[z1, z2, z3, z4] = (z2 − z1)(z4 − z3)

(z3 − z1)(z4 − z2).

123

Geom Dedicata

If u1, u2, u3, u are four solutions of the Riccati equation, then their cross-ratio is constant

(u − u1)(u2 − u3)

(u1 − u2)(u3 − u)= k ,

leading to what is called the superposition formula that will be considered later:

u = u3(u2 − u1)+ k(u1 − u3)u2

u2 − u1 + k(u1 − u3). (2.1)

The Riccati equation is also the only ordinary nonlinear differential equation of first orderwhich possesses such nonlinear superposition formula. As we will see later, this superpo-sition formula is similar to one for the equivariant functions once we substitute the field ofcomplex numbers by the function field of a Riemann surface.

We now introduce the Schwartz derivative (or Schwarzian). This is a differential oper-ator introduced by Schwarz in [31] in the context of differential equations and quadraticdifferentials. It is defined for meromorphic functions over regions of the complex plane by

{ f, z} = 2

(f ′′

f ′

)′−

(f ′′

f ′

)2

= 1

f ′2 (2 f ′ f ′′′ − 3 f ′′2). (2.2)

If w is a function of z, then the Schwarzian satisfies the chain rule:

{ f, z} = { f, w}(dw/dz)2 + {w, z}.Moreover, If f is a linear fractional transformation of z, then { f, z} = 0. As a consequence,if w′(z0) �= 0 for some point z0, then in a neighborhood of this point, the inverse functionz(w) satisfies

{z, w} = −{w, z}(dz/dw)2.

More significant properties of the Schwarzian follow from its close relationship with secondorder differential equations. Indeed, let y1 and y2 be two linearly independent solutions to

y′′ + 1

4R(z)y = 0, (2.3)

where R(z) is a meromorphic function on a certain domain. Then the quotient f = y1/y2 sat-isfies { f, z} = R(z). Conversely, if f is a locally univalent function satisfying { f, z} = R(z),then y1 = f/

√f ′ and y2 = 1/

√f ′ are two linearly independent solutions to (2.3). As a

consequence,

Proposition 2.1 We have

(1) { f, z} = 0 if and only if f is a linear fractional transformation.(2) { f, z} = {g, z} if and only if each function is a linear fraction of the other.

As a corollary of the above, we have

{ f, z} ={

f,az + b

cz + d

}(ad − bc)2

(cz + d)4. (2.4)

Finally, from the definition of the Schwarzian, we have

Proposition 2.2 If f is a meromorphic function, then { f, z} has double poles at the criticalpoints of f and is holomorphic elsewhere.

123

Geom Dedicata

The cross-ratio is projectively invariant, that is if f : z → az + b

cz + d, a, b, c, d ∈ C, ad −bc �=

0 is a Möbius transformation, then

[ f (z1), f (z2), f (z3), f (z4)] = [z1, z2, z3, z4].For a general smooth function f , the Schwarz derivative measures the distortion of the cross-ratio in a certain sense. The following result is well known and is fundamental in Cartan [8].

Proposition 2.3 Let a, b, c and d be four distinct complex numbers and f a twice-differen-tiable function whose second derivative is continuous in an open set D ⊂ C. For z ∈ D andsmall t we define the cross-ratio

S f (z, t) = [ f (z + ta), f (z + tb), f (z + tc), f (z + td)],then

S f (z, t) = [a, b, c, d](

1 + 1

6(a − b)(c − d)S f (z)t2 + o(t2)

).

The proof of Proposition 2.3 basically uses the following expansion, valid for all x and yclose to z with x = z + h, y = z + k,

logf (x)− f (y)

x − y= log f ′(z)+ 1

2

f ′′(z)f ′(z)

(h + k)

+(

1

6

f ′′′(z)f ′(z)

− 1

8

(f ′′(z)f ′(z)

)2)(h2 + k2)−

(1

6

f ′′′(z)f ′(z)

−1

4

(f ′′(z)f ′(z)

)2)

hk + · · · .

Remark 2.4 Taking the second derivative with respect to x and y gives another interpretationof the Schwarzian derivative, as a bi-differential

d f (x)d f (y)

( f (x)− f (y))2= dxdy

(x − y)2+

(1

6

f ′′′(z)f ′(z)

− 1

4

(f ′′(z)f ′(z)

)2)

dxdy + ψ(h, k)dxdy,

where the term ψ(h, k) vanishes for x = y = z.

3 The case of modular functions

Let H = {z ∈ C, �z > 0} be the upper half of the complex plane. The group SL2(R)

of Möbius transformations acts on H in the usual manner. We restrict our attention to themodular group SL2(Z) and its subgroups, but the general picture involves all the discretesubgroups of SL2(R). Let G be such a group, and let f be a modular form on G of weight k(k ≥ 0), that is, a meromorphic function on H satisfying

f

(az + b

cz + d

)= (cz + d)k f (z) , z ∈ H ,

(a b

c d

)∈ G, (3.1)

with some growth conditions at the cusps. When k = 0, f is called a modular function. If fis holomorphic, it will be mentioned explicitly.

The effect of the Schwarzian on modular functions is illustrated as follows. Using (2.4)and the fact that the derivative of a modular function is a modular form of weight 2, we have

123

Geom Dedicata

Proposition 3.1 If f is a modular function for G, then {z, f } is a modular function and{ f, z} is a modular form of weight 4 on G.

Suppose now that G is a discrete group of genus 0, that is the compactification of the quotientG\H is a Riemann surface of genus 0. Any analytic embedding of this surface in the extendedcomplex plane induces a modular function for G defined on H. It is called a Hauptmodul, andit generates the function field of the Riemann surface. Let f be a Hauptmodul for G. Thenthe modular function {z, f } is a rational function of f . As for { f, z}, we have a deeper result:

Proposition 3.2 [25] Let G be a genus 0 discrete group and f a Hauptmodul for G. Then{ f, z} is a weight 4 modular form on the normalizer of G in SL2(R) and this normalizer isthe maximal group with this property.

To illustrate this proposition, Let �(2) be the principal congruence group of level 2. A Hau-ptmodul is given by the classical Klein λ function. Since �(2) has no elliptic elements, andthus λ has no critical point, we see that {λ, z} is a holomorphic weight 4 modular form on thenormalizer of �(2) in SL2(R)which is SL2(Z). However, the space of weight 4 holomorphicmodular forms for SL2(Z) is one-dimensional and is generated by the weight 4 Eisensteinseries E4. In fact we have

{λ, z} = π2 E4(z). (3.2)

A deeper study of this type of relations can be found in [25]. Here

λ(z) =(η(z)

η(4z)

)8

,

where the eta function is defined by

η(z) = q1

24∏n≥1

(1 − qn) , q = e2π i z,

and the Eisenstein series E4 is defined by

E4(z) = 1 + 240∑n≥1

σ3(n)qn , q = e2π i z,

where σ3(n) is the sum of the cubes of the positive divisors on n.Because the normalizer of congruence subgroups is often not a subgroup of the modulargroup, our focus should be on a more general class of discrete groups. Namely all the discretesubgroups that are commensurable with the modular group. These groups have finite indexinside their normalizers. The next section deals with the converse to the previous proposition.

4 Modular Schwarzians

In this section we look at the following question: Assume that f is a meromorphic functionon H such that F(z) = { f, z} is a modular form of weight 4 on a certain group G F . Theinvariance group of f (on which f is a modular function) is a subgroup G f of G F . What is thesize of G f inside G F ? Keeping in mind the relationship between G f and G F for a modularfunction f as it was seen in the previous section. We will make explicit some instances wherewe have complete answers. It turns out that these answers depend on the analytic propertiesof f and on the structure of G F . So far, for all kind of plausible conditions, it seems thatthere are always examples that provide different answers.

123

Geom Dedicata

We shall rephrase the problem differently. Since F(z) = { f, z} is a modular form ofweight 4, then for every γ = (a b

c d

) ∈ G F we have

F(γ · z) = (cz + d)4 F(z) = (cz + d)4 { f, z}.On the other hand, using (2.4), we have

F(γ · z) = { f (γ · z), γ · z} = (cz + d)4{ f (γ · z), z}.It follows that

{ f, z} = { f (γ · z), z}.Therefore, using Proposition 2.1, there exists �γ ∈ GL2(C) such that

f (γ · z) = �γ · f (z). (4.1)

This defines a group homomorphism

� : G F −→ GL2(C)

γ −→ �γ

The invariance group G f of f is simply the kernel of �. It is worth mentioning that M.Kaneko and M. Yoshida have considered a similar problem in [18] where � is an epimor-phism r : G −→ G ′ between two Fuchsian groups of the first kind. In particular, the authorswere interested in the case G and G ′, the kernel and the co-kernel of r , are infinite groups.They have constructed the Kappa function defined by j (κ(z)) = λ(z) as an answer to thisproblem. We are mainly interested in the case where the co-kernel is rather finite.

We now look at a case where there is a precise answer to the above question.

Proposition 4.1 Let f be a meromorphic function on H such that f (z +1) = f (z). If { f, z}is a weight 4 modular form on SL2(Z), then f is a modular function for SL2(Z).

Proof Let U = (1 11 0

). The condition f (z + 1) = f (z) means that U ∈ Ker�, and that

〈U 〉 = {U n, n ∈ Z} ⊆ Ker�. Since Ker� is normal in SL2(Z), it contains {L−1U L , L ∈SL2(Z)}, the normal closure of 〈U 〉. Now if we set V = (0 −1

1 0

), then one can show that

V −1 = U V −1U V U . Hence V −1 belongs to the normal closure of 〈U 〉, and so does V . SinceU and V generate SL2(Z), it follows that Ker� = SL2(Z). ��Remark 4.2 The condition f (z + 1) = f (z) means that f has a Fourier expansion in q =exp(2π i z).

The above theorem provides an example where f is a modular function on the same group onwhich its Schwarzian is a modular form. This is equivalent to saying that the homomorphism� is constant. This proposition can be generalized as follows.

Theorem 4.3 Let n be an integer such that 1 ≤ n ≤ 5. Suppose that f is a meromorphicfunction on H such that f (z +n) = f (z) and suppose that { f, z} is a modular form of weight4 for SL2(Z). Then f is a modular function for a finite index normal subgroup of SL2(Z).

Proof The case n = 1 is settled in the above proposition. For 2 ≤ n ≤ 5, the group �(n)is of genus 0 and has no elliptic elements. Hence it is generated by parabolic elements only.Using this fact, one can show that the normal closure �(n) of 〈U n〉 is simply �(n). Thus fis a modular function for a finite index subgroup of SL2(Z). ��

123

Geom Dedicata

As for n ≥ 6, the argument above is no longer valid. Indeed the normal closure�(n) of 〈U n〉has infinite index in SL2(Z).

We now provide an example where the invariance group of f is not larger than 〈U 〉,although the function f has the same conditions as in the theorem. Let G be a genus 0 grouphaving at least two cusps. Let g be a Hauptmodul for G normalized to have the value 1 at ∞and 0 at the other cusp. Let g(z) = log f (z). Then f (z + 1) = f (z) and for γ ∈ SL2(Z) notin 〈U 〉 we have

f (γ · z) = f (z)+ 2π in� , n� ∈ Z, n� �= 0.

However, { f, z} is a weight 4 modular form on G.

5 Equivariant forms, a first example

The principal motivation behind equivariant forms was to look for examples of meromorphicfunctions f where { f, z} is a modular form of weight 4 for a discrete group G but f is notinvariant under any nontrivial matrix. That is, the kernel of� defined in the previous sectionis trivial. This will be the case if, for instance, � =Id. In other word, f would satisfy

f

(az + b

cz + d

)= a f (z)+ b

c f (z)+ d, for all z ∈ H and

(a b

c d

)∈ G.

On the other hand, in the paper [32], we studied and solved a family of Riccati equations ofthe form

k

iπu ′ + u2 = E4 , 1 ≤ k ≤ 6.

While the cases 2 ≤ k ≤ 6 were given in terms of modular forms and functions, the solutionin the case k = 1 turns out to satisfy the above functional equations for G = SL2(Z).

This paper is devoted to construct and study these very special functions with surprisingproperties arising along the way. We will focus on the case of G = SL2(Z) for most of thepaper.

Definition 5.1 A meromorphic function h(z) on H is called an equivariant form for SL2(Z)

if

• For all z ∈ H and for all γ ∈ SL2(Z), we have

h(γ · z) = γ · h(z). (5.1)

• The function h(z)− z is meromorphic at the cusps.

Since h(z + 1) = h(z) + 1, the function h(z) − z is 1-periodic and hence has a Fourierexpansion in q = exp(2π i z), the local parameter of SL2(Z) at ∞. To say that h(z) − zis meromorphic at ∞ means that the Fourier expansion has finitely many negative powersof q .

The trivial example is h0(z) = z which is equivariant for any group. This example is veryparticular in two ways: First, it is the only Möbius transformation that is equivariant, and sec-ond, as was shown by Heins [15], h0 is the only equivariant function that maps H into itself.In particular, this means that the composition of maps does not provide the set of equivariantholomorphic functions with a group structure. This result has a certain connection with theiteration of holomorphic maps from H into H. We give a quick proof using the followingtheorem of Denjoy and Wolff [7].

123

Geom Dedicata

Theorem 5.2 Let f : D → D be a holomorphic map. We assume that f is neither an ellipticMöbius transformation nor the identity, then the successive iterates f on converge, uniformlyon compact subsets of D, to a constant function z → c0 ∈ D.

We recall that the elliptic Möbius transformations in H are all of the following form

z → cos θ2 z + sin θ2

− sin θ2 z + cos θ2

, θ ∈ (0, 2π) ,

and they are not equivariant, for any θ ∈ (0, 2π). Thus, if h is an equivariant holomorphicfunction sending H into itself, so does any iterate hon, n ∈ N which is also equivariant and ho-lomorphic. By the Wolff-Denjoy theorem, hon should tend to c ∈ H which, by equivariance,verifies the contradictory conditions

c = −1

c, c = c + 1.

Thus, the only equivariant holomorphic function in H which maps H into H is the identitymap.

We now provide the first nontrivial example of equivariant functions. Let E2(z) be theclassical Eisenstein series

E2(z) = 1 − 24∞∑

n=1

σ1(n)qn = 1 − 24

∞∑n=1

nqn

1 − qn, (5.2)

where q = exp(2π i z), z ∈ H and σ1(n) is the sum of the positive divisors of n. The seriesE2(z) is a holomorphic function on H, and if γ = (a b

c d

) ∈ SL2(Z), we have

E2(γ · z) = (cz + d)2 E2(z) + 6c

iπ(cz + d), (5.3)

that is to say that E2 is a quasi-modular form of weight 2. Meanwhile, if we define

E∗2 (z) = E2(z)− 3

πy, y = �z, (5.4)

then E∗2 is a non-holomorphic modular form of weight 2 for SL2(Z). Moreover, E2 is the

logarithmic derivative of the discriminant function �(z), the classical cusp form of weight12 for SL2(Z)

�(z) = η(z)24 = q∞∏

n=1

(1 − qn)24 , q = e2π i z .

In fact, we have

E2(z) = 1

2iπ

�′(z)�(z)

. (5.5)

Theorem 5.3 The function h1 given by

h1(z) = z + 6

iπE2(z)(5.6)

is an equivariant form for SL2(Z).

123

Geom Dedicata

Proof It suffices to show that h1 is equivariant under the transformations z → z + 1 andz → −1/z which generate SL2(Z). Since E2(z + 1) = E2(z), it is clear that h1(z + 1) =h1(z)+ 1. On the other hand, using (5.3), one has:

h1

(−1

z

)= −1

z− 6i/π

z2 E2(z)− 6i z/π

= E2(z)

zE2(z)− 6i/π

= −1

h1(z).

Furthermore, one can see from (5.2) that h1(z) − z has a holomorphic q-expansion, andtherefore h1 is an equivariant function for SL2(Z). ��

The equivariant function h1(z) has no fixed points in H and also at ∞ in the sense that

limz→i∞(h1(z)− z) = 6

iπ�= 0.

In the following, we will show that h1(z) is the unique equivariant function for SL2(Z) withthe property that it has no fixed points in H ∪ {∞} and that this example actually fits in ageneral construction of equivariant functions.

Theorem 5.4 Let h(z) be an equivariant function for SL2(Z)with no fixed points in H∪{∞}.Then

h(z) = h1(z) = z + 6

iπE2(z).

Proof Since h(z)− z does not have zeros on H ∪ {∞}, then the function

g(z) = 1

h(z)− z− iπ

6E2(z)

is holomorphic in H ∪ {∞}. Moreover, we have g(z + 1) = g(z) and using the equivarianceof h and (5.3) we get

g(−1/z) = zh(z)

h(z)− z− iπ

6z2 E2(z)− z

= z(h(z)− z)+ z2

h(z)− z− iπ

6z2 E2(z)− z

= z2g(z).

Therefore, g(z) is a weight 2 holomorphic modular form and thus g(z) = 0 since the spaceof weight 2 holomorphic modular forms for SL2(Z) is trivial. The theorem follows. ��

Remark 5.5 An alternative way to prove this theorem is to notice that under the assumptionsof the theorem, the integral

z∫i

dw

h1(w)− w

123

Geom Dedicata

does not depend on the path of integration Wz from i to z in H since h1(z)− z is holomorphicand non-vanishing. Therefore, the function

f (z) = exp

z∫i

12dw

h1(w)− w

is a well defined function that is holomorphic and non-vanishing on H. Using the equivari-ance of h1, one can show that f (z) is a non-vanishing modular form of weight 12 and thusis a scalar multiple of �(z). Taking the logarithmic derivative of f and using (5.5) yield thetheorem.

Proposition 5.6 If z0 is an elliptic fixed point of SL2(Z), then

h1(z0) = z0.

Proof If z0 ∈ H ∪Q is fixed by an element γ of SL2(Z), then h1(z0) is also fixed by γ . Thusif z0 is an elliptic fixed point, and since h1 does not have fixed points in H, we must haveh1(z0) = z0. ��

6 Analytic properties of h1 and an application

In this section we show that there are infinitely many poles of h1 and that they are all simple.

Proposition 6.1 We have

(1) The poles of h1(z) are located at the zeros of E2 and they are simple.(2) The critical points of h1 are located at the zeros of E4.

Proof It is clear that the poles of h1(z) are exactly the zeros of E2. Now recall the followingdifferential relation between E2 and E4 due to Ramanujan, [27,28]

1

2π i

d

dzE2(z) = 1

12(E2

2 − E4). (6.1)

It follows that if a zero of E2 is not simple, then it is also a zero of E4 and such a zero lies inthe SL2(Z)-orbit of ρ, the cubic root of unity. This is impossible since Proposition 5.4 statesthat E2 does not vanish at the elliptic fixed points. Therefore, the poles of h1(z) are simple.Furthermore,

h′1(z) = 1 − 6

iπ

E ′2

E22

= 1 − E22 − E4

E22

= E4

E22

.

Hence h′1(z) vanishes exactly at the zeros of E4, ��

As a consequence, and using Proposition 2.2, E24{h, z} is a weight 12 holomorphic modular

form and thus it is a linear combination of � and E34 which constitute a basis of the space

of weight 12 holomorphic forms for SL2(Z). Investigation of the first two coefficients of theFourier expansion yields

Proposition 6.2 We have

{h1, z} = −2632π2 �

E24

. (6.2)

123

Geom Dedicata

Remark 6.3 The above proposition can be established by direct computation using similaridentities to (6.1), also known as the Ramanujan identities [27], namely

1

2π i

d

dzE4(z) = 1

3(E2 E4 − E6) (6.3)

1

2π i

d

dzE6(z) = 1

2(E2 E6 − E2

4), (6.4)

together with the identity

E34 − E2

6 = 2732�.

Here, E6 is the weight 6 Eisenstein series

E6(z) = 1 − 504∑n≥1

σ5(n) qn,

where σ5(n) is the sum of the fifth powers of the positive divisors of n.

If z0 is a pole of h1 then all the translates of z0 by integers are also poles of h1. It turns outthat the only other poles of h1 that are SL2(Z)-equivalent to z0 are its translates by integers.Indeed,

Lemma 6.4 If z0 and z1 are two poles of h1, and if there exists γ ∈ SL2(Z) such thatz1 = γ · z0 then γ = (1 n

0 1

)for some integer n.

Proof If γ = (a bc d

)and if z0 and z1 = γ · z0 are poles of h1, then necessarily c = 0 and

a = d = ±1. Thus z1 = z0 + n where n is an integer. ��Using this property, we can restrict ourselves to the half strip

D ={

z = x + iy : y > 0, −1

2< x ≤ 1

2

}.

We will denote by T and S the transformations

T z = z + 1 , Sz = −1

z.

Proposition 6.5 There exists a pole z0 of h1 on the purely imaginary axis {z = iy : y > 0}.Proof Recall that E2(z) = 1 − 24

∑n≥1 σ1(n)qn . We see that E2(iy), y > 0 is real and

strictly decreasing for y ∈ (0,∞). It takes the value 1 at ∞ and limy→0 E(iy) = −∞.Therefore, there exists y0 > 0 such that E2(iy0) = 0 and thus iy0 is a pole of h1. Moreover,y0 < 1 since by (5.3), we have E2(i) = 3/π . ��Recall that the fundamental domain for SL2(Z) is

F = {z ∈ H : |z| > 1, −1/2 < �z ≤ 1/2}.Since Im(z0) < 1 we see that

z0 ∈ SF , Sz0 ∈ F . (6.5)

Theorem 6.6 There are infinitely many poles for h1 in D.

123

Geom Dedicata

Proof Let z1 = Sz0 = −1/z0 where z0 is as in Proposition 6.5. Then h1(z1) = 0 and z1 isnot an elliptic fixed point by Proposition 5.6. Choose an open neighborhood U ⊂ F of z1 onwhich h1 is holomorphic, not containing an elliptic fixed point and such that no two pointsof U are SL2(Z)−equivalent. Then U is mapped by h1 onto an open neighborhood V of 0.There are infinitely many rational numbers in the open set V . If x = h1(zx ), zx ∈ U , is such arational number, let γx be such that γx ·x = ∞. Then h1(γx · zx ) = γx ·h1(zx ) = γx ·x = ∞.Therefore, γx · zx is a pole of h1. In the meantime, no two such poles γx · zx and γy · zy areSL2(Z)−equivalent because zx and zy , which are in U , are not. ��Corollary 6.7 The Eisenstein series E2 has infinitely many non-equivalent zeros.

Remark 6.8 As a corollary, the Eisenstein series E2 has infinitely many non-equivalent zeros;a result that has been established without the notion of equivariant functions in [12].

7 Rational equivariant functions, the general case

In this section, from each modular form we construct an equivariant function. Using (5.5),one can rewrite the equivariant function h1 as

h1(z) = z + 12�

�′ .

It turns out that this expression can be generalized as follows

Theorem 7.1 [34] Let f be a modular form on SL2(Z) of weight k. Then the function

h(z) = z + kf (z)

f ′(z)(7.1)

is equivariant for SL2(Z).

Proof Let γ = (a bc d

) ∈ SL2(Z). We have f (γ · z) = (cz + d)k f (z), hence

f ′(γ · z) = kc(cz + d)k+1 f (z)+ (cz + d)k+2 f ′(z).

Therefore,

h(γ · z) = az + b

cz + d+ k(cz + d)k f (z)

kc(cz + d)k+1 f (z)+ (cz + d)k+2 f ′(z)

= k f (z)(abz + bc + 1)+ (az + b)(cz + d) f ′(z)(cz + d)(kc f (z)+ (cz + d) f ′(z))

.

On the other hand, we have

γ · h(z) = (az + b) f ′(z)+ ak f (z)

(cz + d) f ′(z)+ kc f (z).

Since ad − bc = 1, we have (az + b)c + 1 = a(cz + d). The identity h(γ · z) = γ · h(z)follows. ��Proposition 7.2 If f is a modular form of weight k, then scalar multiples of f and integralpowers of f give rise to the same equivariant function h. The modular functions correspondto the trivial equivariant function h0(z) = z.

123

Geom Dedicata

Proof This is straightforward keeping in mind that, for an integer m, the weight of f m iskm. ��In what follows, we will find sufficient conditions for an equivariant function to arise froma modular form.

Proposition 7.3 If h is equivariant for SL2(Z), then the set of residues of the meromorphicfunction 1/(h(z)− z) at the simple poles is finite.

Proof Let γ =(

a b

c d

)∈ SL2(Z). Differentiating h(γ z) = γ h(z) yields

h′(γ z)

(cz + d)2= h′(z)

(ch(z)+ d)2.

Thus, if h(z0)= z0, then h′(γ z0)= h′(z0), that is, h′ takes the same value at the orbit of afixed point of h. Hence, the set of values h′(z0)when z0 describes the set of fixed points of h iscompletely determined if we restrict ourselves to the fundamental domain F . Moreover, sinceh(z)− z is meromorphic at i∞, there is a neighborhood of i∞ of the form {z ∈ H : �z > y0}on which h(z)−z does not vanish (except possibly at i∞). Therefore, all the zeros of h(z)−zin F are within the closure of {z ∈ F : �z ≤ y0} which is compact and thus we have onlyfinitely many zeros. In the meantime, the residue of 1/(h(z)− z) at a simple z0 of h is simply1/(h′(z0)− 1). The proposition follows. ��Theorem 7.4 Let h be an equivariant function satisfying the following conditions:

(1) The poles of 1/(h(z)− z) in H are simple and their residues are rational numbers.(2) At ∞ we have:

limz−→∞

1

h(z)− z∈ 2π iQ.

Then there exists a modular form f of integer weight k for SL2(Z) such that

h(z) = z + kf (z)

f ′(z).

Proof Define the function f (z) by

f (z) = exp

z∫i

kdz

h(z)− z,

where k is a positive integer to be chosen conveniently. The path of integration �z is chosento lie in H \ S, where S is the set of (simple) zeros of h(z)− z. By assumption, the residuesof 1/(h(z)− z) are rational numbers, and using Proposition 7.3, these rational numbers havebounded denominator. Therefore, there exists k ∈ Z

+ such that for each such residue r, kris an integer and

limz−→∞

k

h(z)− z∈ 2π iZ ,

and we also suppose k ≡ 0 mod 4. If we choose a different path of integration�′z from i to

z lying in H \ S, then∫�z

kdz

h(z)− z−

∫�′

z

kdz

h(z)− z= 2π ik

∑Residues ∈ 2π iZ,

123

Geom Dedicata

where the sum of the residues is taken over the finite number of poles within the closed path�z −�′

z . Therefore, f (z) is well defined on H \ S. We extend f to a meromorphic functionon S in the following way. Let m (an integer) be the residue of k/(h(z) − z) at z0. If r > 0we define f (z0) = 0 to make f holomorphic at z0 and the order of f at z0 is r . If r < 0then z0 is a pole of f of order −r . Thus f is a well-defined meromorphic function of H.Furthermore,

f (z + 1) = exp

z+1∫i

kdz

h(z)− z= f (z)g(z)

where

g(z) = exp

z+1∫z

kdz

h(z)− z.

Since h is an equivariant function, it is clear that g′(z) = 0 and hence g is constant. Takingthe limit z → i∞ yields g(z) = 1 since by assumption,

k

h(z)− z= ka0 +

∑n≥1

anqn , q = e2π i z,

and ka0 ∈ 2π iZ. Therefore,

f (z + 1) = f (z).

On the other hand,

f (−1/z) = exp

−1/z∫i

kdw

h(w)− w

= exp

z∫i

kh(t)dt

t (h(t)− t)

= f (z) exp

z∫i

kdt

t

= zk f (z) since k ≡ 0 mod 4.

Thus f is a meromorphic modular form of weight k for SL2(Z). ��

Motivated by the above theorem, we have

Definition 7.1 An equivariant function that arises from a modular form as in (7.1) is calleda rational equivariant function.

Remark 7.5 If f is a weight k modular form, then the corresponding equivariant functionh(z) = z + k f (z)/ f ′(z) satisfies the conditions of the above theorem; that is to say that thetwo conditions are necessary and sufficient conditions for h to be of that form. Moreover the

123

Geom Dedicata

conditions are optimal as we will see that there are indeed examples of equivariant functionsthat do not satisfy them and thus they are not rational. Indeed, if we take

h1(z) = z + 6E22 E6

iπE2 E24 E6 +�

,

then one can show that h1 is equivariant but h1(z)− z has a double pole at the cubic root ofunity ρ. Also,

h2(z) = z + 6E4

iπ(E2 E4 + E6)

is equivariant having poles only at the zeros of E4 but the residue of 1/(h2(z) − z) at ρ isirrational. Finally,

h3(z) = z + 6�

iπE2�+ E14

is equivariant, but limz→i∞(h3(z) − z) = 0. These three examples show that one cannotremove the conditions of the converse theorem above.

8 Lambert series and Borcherds products

For a rational equivariant function h(z) = z + k f (z)/ f ′(z), we will investigate the Fourier

coefficients of the periodic function h(z)− z, the Lambert series of1

h(z)− z. We also point

out a possible link with a theorem of Borcherds at least by considering some examples. Webegin by recalling some classical analytic facts [36], p. 147.

Lemma 8.1 Given any sequence (an)n≥0, a0 �= 0, we formally have

1∑∞n=0 anqn

= 1

a0+

∞∑1

(−1)nGn

n!an+10

qn

with

Gn =

∣∣∣∣∣∣∣∣∣∣∣∣

2a1 a0 0 0 · · · 04a2 3a1 2a0 0 · · · 06a3 5a2 4a1 3a0 · · · 0. . . . · · · .

(2n − 2)an−1 . . . · · · (n − 1)a0

nan (n − 1)an−1 . . . a1

∣∣∣∣∣∣∣∣∣∣∣∣Corollary 8.2 Let E2(q) be the weight 2 Eisenstein series (5.2) seen as a function of q andset

1

E2(q)=

∞∑n=0

αnqn, (8.1)

123

Geom Dedicata

then α0 = 1 and

n!(24)−nαn =

∣∣∣∣∣∣∣∣∣∣∣∣

2σ1(1) 1 0 0 · · · 04σ1(2) 3σ1(1) 2 0 · · · 06σ1(3) 5σ1(2) 4σ1(1) 3 · · · 0. . . . · · · .

(2n − 2)σ1(n − 1) . . . · · · n − 1nσ1(n) (n − 1)σ1(n − 1) . . . σ1(1)

∣∣∣∣∣∣∣∣∣∣∣∣.

Lemma 8.3 Consider a formal power series∑

n≥1 bn xn with complex coefficients, then asequence (an)n≥1 can be found such that the following expansion in Lambert series holds

∑n≥1

bn xn =∑n≥1

anxn

1 − xn(8.2)

with

bn =∑m|n

am, am =∑d|m

μ(m

d

)bd =

∑d|m

μ(d)b md, (8.3)

μ being the Möbius function defined as usual by the inverse of the Riemann zeta function

ζ(s) =∑n≥1

1

ns,

1

ζ(s)=

∑n≥1

μ(n)

ns, �s > 1.

We now consider an equivariant function h and the associated function g, defined by

k

2iπg(z) = 1

h(z)− z=

∑n≥0

bne2iπnz .

It is a meromorphic periodic function on the upper half plane H. Let, with the notation ofLemma 8.3,

a(n) = − 1

n

∑d|n

μ(d)b nd

and f (z) = e2iπb0z∏n≥1

(1 − e2iπnz)a(n). (8.4)

Applying the theta differential operator θ = 1

2iπ

d

dzand using the Möbius inversion formula,

we have

θ f

f= b0 −

∞∑n=1

∑d|n

da(d)e2iπnz = k

2iπg.

We define the slash operator of weight k as usual: Let γ =(

a bc d

)∈ SL(2,Z), then

F |k[γ ](z) = (cz + d)−k f (γ z).

According to [19] we have the following

Definition 8.1 Let � ⊂ �(1) containing −I . A meromorphic function F : H → C is saida generalized modular form of weight k and a character ν if

123

Geom Dedicata

(1) F satisfies the modular transformation law

F |k[M](z) = ν(M)F(z)

for all M ∈ �, with ν(M) independent of z ∈ H.

(2) F has a left-finite Fourier expansion at each parabolic cusp in a fundamental region Rof �.

The generalized modular forms differ from the classical modular forms with a character bythe fact the multiplier system ν need not be unitary.

Proposition 8.4 For h equivariant, the infinite product f (z), given in (8.4), is a generalizedmodular form

Proof We have

θ f |0[γ ]f |0[γ ] = θ f

f|2[γ ] = k

2iπg|2[γ ] = k

2iπg + k

2iπ

c

cz + d

where the last equality is a consequence of the equivariance of h. Therefore,

θ f |0[γ ]f |0[γ ] − θ f

f= k

2iπ

c

cz + d, (8.5)

which is a fundamental equation to study the modular properties of f . It follows that

θ(

f |0[γ ]f

)f |0[γ ]

f

= θ f |0[γ ]f |0[γ ] − θ f

f= k

2iπ

c

cz + d= θ(cz + d)k

(cz + d)k.

Hence,f |k[γ ]f (z)

= f |0[γ ]f (z)(cz + d)k

is a non-zero constant ν(γ ) on H. Using the cocycle rela-

tion of the slash operator, it is easily seen that ν is actually a character of SL2(Z) yieldingthe multiplier system of the generalized modular form f of weight k. ��Let us give some examples. For the basic equivariant function h1 given by (5.6), the associatedinfinite product is given by the discriminant function

�(z) = q∞∏

n=1

(1 − qn)24 , q = e2π i z .

According to (5.5)

E2(z) = 1

2iπ

�′(z)�(z)

,

and thus the fundamental Eq. (8.5) reduces to (5.3).We can also consider, as in [4], the following infinite products

j (τ ) = q−1∏n>0

(1 − qn)3c0(n2)

= q−1(1 − q)−744(1 − q2)80256(1 − q3)−12288744 · · · ,E6(τ ) =

∏n>0

(1 − qn)a(n2)

= (1 − q)504(1 − q2)143388(1 − q3)51180024 · · · .

123

Geom Dedicata

and define the corresponding equivariant functions and associated Lambert series expansions.More generally, in [4], Borcherds gives a striking description of the exponents in the infiniteproduct expansion of several modular forms in terms of the Fourier coefficients of some halfinteger meromorphic modular forms.

The q-expansion of j starts as

J (z) = q−1 + 744 + 196884q + 21493760q2 + · · · , q = e2iπ z (8.6)

and we introduce a sequence of modular functions

J0(z) = 1, J1(z) = J (z)− 744

and for m ≥ 2, we define

jm(z) = J1(z)|T0(m)

where T0(m) is the normalized mth Hecke operator defined by

g(z)|T0(m) =∑

d|m, ad=m

d−1∑b=0

g

(az + b

d

)

Then we have [4]

E4(z) = 1 + 240∞∑

n=1

σ3(n)qn =

∞∏n=1

(1 − qn)c(n2) (8.7)

where the c(n)s denote the coefficients of a weight 1/2-modular form on �0(4) whoseexpansion starts as

f (z) = q−3 + 4 − 240q + 26760q4 − 85995q5 + 1707264q8 + · · ·or more explicitly,

c(n) = 8 + 1

3n

∑d|n

μ(n

d

)jd(ω) , ω = 1 + √−3

2;

a formula which can be compared with the identity J = E34

�. Borcherds theorem shows

at once that the knowledge of the sequence of exponents c(n) with the use of Lemma 8.1gives, in principal, the full Fourier coefficients of the periodic part h(z) − z of an equivari-ant function h. However, this not an easy task in practice, even for basic example such as

h1.Indeed, due to the lack of modularity, the Fourier series of1

E2has not been investigated

before, contrary to1

E4and

1

E6. The Fourier series of

1

E6has been studied by Hardy and

Romanian [14], and very recently the Fourier series of1

E4has been studied by Bernie, Bialys

[3]. The results are deep and for comparison and later discussion we quote them and give animmediate consequence using Lemma 8.1.

Theorem 8.5 (Hardy–Ramanujan) Let C = 3π2

4�8( 34 )

. Define the coefficients pn by

1

E6(q)=

∞∑n=0

pnqn .

123

Geom Dedicata

Then, for n ≥ 0,

pn =∑μ

Tμ(n),

where μ runs over all integers of the form

μ = 2ar∏

j=1

pa jj , (8.8)

where a = 0 or 1, p j is a prime of the form 4m + 1, and a j is a nonnegative integer andwhere

T1(n) = 2

C2 enπ , T2(n) = 2

C2

(−1)n

24 enπ ,

and for for μ > 2,

Tμ(n) = 2

C2

e2nπμ

μ4

∑c,d

2 cos

((ac + bd)

2nπ

μ+ 8 tan−1 c

d

),

where the sum is over all pairs (c, d) satisfying μ = c2 + d2 and (a, b) is any solution toad − bc = 1.

Comparing with Lemma 8.1, we obtain at once, with σ5(k) =∑d|n

d5

n!(504)−n pn =

∣∣∣∣∣∣∣∣∣∣∣∣

2σ5(1) 1 0 0 · · · 04σ5(2) 3σ5(1) 2 0 · · · 06σ5(3) 5σ5(2) 4σ5(1) 3 · · · 0. . . . · · · .

(2n − 2)σ5(n − 1) . . . · · · n − 1nσ5(n) (n − 1)σ5(n − 1) . . . σ5(1)

∣∣∣∣∣∣∣∣∣∣∣∣.

In a similar way, we have for E4

Theorem 8.6 (Berndt–Bialek) Let ρ = −1

2+ i

√3

2and set

1

E4(q)=

∞∑n=0

βnqn

and

G = E6(ρ),

then

βn = (−1)n3

G

∑(λ)

hλ(n)

λ3 enπ√

3λ .

Here λ runs over the integers of the form (8.8),

h1(n) = 1, h3(n) = −1,

123

Geom Dedicata

and, for λ ≥ 1

hλ(n) = 2∑c,d

cos

((ad + bc − 2ac − 2bd + λ)

nπ

λ− 6 tan−1

(c√

3

2d − c

)),

where the sum is over all pairs (c, d) satisfying λ = c2 − cd + d2 and (a, b) is any solutionto ad − bc = 1.

Comparing with the Lemma 8.1 we obtain once again, with σ3(k) =∑d|n

d3,

n!(240)−nβn =

∣∣∣∣∣∣∣∣∣∣∣∣

2σ3(1) 1 0 0 · · · 04σ3(2) 3σ3(1) 2 0 · · · 06σ3(3) 5σ3(2) 4σ3(1) 3 · · · 0. . . . · · · .

(2n − 2)σ3(n − 1) . . . · · · n − 1nσ3(n) (n − 1)σ3(n − 1) . . . σ3(1)

∣∣∣∣∣∣∣∣∣∣∣∣.

Remark 8.7 It would be interesting to have another formulation for the determinant in (8.1)

similar to those given for the coefficients of1

E4(q)and

1

E6(q)despite of the lack of the

modularity for E2.

9 ℘n as an elliptic function and a differential algebra

In this section, we give some properties of the meromorphic elliptic function ℘n, n ∈ Z

and with two exceptions, we will associate to each ℘n an equivariant function. As wasobserved by Heins, the basic example of equivariant functions is related to the Weierstrassfunctions ζ ′ = −℘ by way of the Legendre relation. Our main task is to find integrals of℘n, n ∈ Z. For later use, let us recall the essential idea. If ω1, ω2 are two complex numberswith �(ω2/ω1) > 0 and � = 2ω1Z + 2ω2Z, the Weierstrass functions are

℘(z;ω1, ω2) = ℘(z) = 1

z2 +∑

ω∈�\{0}

(1

(z − ω)2− 1

ω2

),

ζ(z;ω1, ω2) = ζ(z) = 1

z+

∑ω∈�\{0}

(1

z − ω+ 1

ω+ z

ω2

),

σ (z;ω1, ω2) = σ(z) =∏

ω∈�\{0}

(1 − z

ω

)exp

(z

ω+ z2

2ω2

).

Hence ℘ = −ζ ′, σ′

σ= ζ . In some situations, it is better to use the odd Jacobi theta function

θ1(z|τ) = i∑

n∈ 12 +Z

(−1)n− 12 e2iπnzeiπn2τ .

The zero divisors of this function form the lattice Z + τZ. The general principal that wefollow and which goes back to Liouville and Hermite is that if the principal part of an ellipticmeromorphic function at each of its poles is known, then this function is determined up toan additive constant. More precisely we have the decomposition theorem [36].

123

Geom Dedicata

Theorem 9.1 If ak, 1 ≤ k ≤ n, is the set of poles of an elliptic function f , of periods2ω1, 2ω2 and if at ak the principal part is

rk∑s=1

cks(z − ak)−s,

then there exists constant A such that

f (z) = A +n∑

k=1

rk∑s=1

(−1)s−1

(s − 1)! cksζ(s−1)(z − ak).

Consequently, a primitive of f is given by

∫f (z)dz = Az + B +

n∑k=1

[ck1 log σ +

rk∑s=2

(−1)(s−1)

(s − 1)! cksζ(s−2)(z − ak)

],

with B being an arbitrary constant. Moreover, with Jacobi theta function , we have

f (z) = C +n∑

k=1

rk∑s=1

(−1)(s−1)

(s − 1)! cksds

dzslog θ1

(π z − πak

2ω1|2ω2

2ω1

)

and

∫f (z)dz = Cz + D +

n∑k=1

rk∑s=1

(−1)(s−1)

(s − 1)! cksds−1

dzs−1 log θ1

(π z − πak

2ω1|2ω2

2ω1

).

The coefficient ck1 is the residue of f at the pole ak , hence

n∑i=1

ck1 = 0.

In particular c11 = 0 if there is only one pole. This theorem is very deep in the sense thatit gives all the differential relations that will be considered below and also all the knownrelations between Weierstrass elliptic functions and Jacobi elliptic functions. In addition, it

essentially says that if D = C

[d

dz

]is the ring of differential operators with constant coef-

ficients and M the C-vector space of elliptic meromorphic functions with a pole at 0, thenM is a left D-module, that is (M, d

dz ) is a differential graded algebra

M = C ⊕ C℘ ⊕ C℘′ ⊕ · · · C℘(n) ⊕ · · · .

As a D-module, M is generated by two elements 1, ℘, hence we a have the free resolution

0 −→ D φ−→ D2 ψ−→ M −→ 0

where ψ(D1, D2) = D1.1 + D2℘ and φ(D) = (D,−D.1).

123

Geom Dedicata

Perhaps the most fascinating examples of applications of this theorem are the two follow-ing identities of Frobenius and Stickelberger

(−1)n(n−1)

2

n∏k=1

k!σ(z0 + z1 + · · · + zn)∏

0≤i< j≤n(zi − z j )

σ n+1(z0) · · · σ n+1(zn)

=

∣∣∣∣∣∣∣∣∣∣

1 ℘(z0) ℘′(z0) · · · ℘(n−1)(z0)

1 ℘(z1) ℘′(z1) · · · ℘(n−1)(z1)

· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·1 ℘(zn) ℘′(zn) · · · ℘(n−1)(zn)

∣∣∣∣∣∣∣∣∣∣and of Kiepert

(−1)(n−1)

(n−1∏k=1

k!)2

σ(nu)

σ n2(z)

=

∣∣∣∣∣∣∣∣∣∣

℘′(z) ℘′′(z) · · · ℘(n−1)(z)℘′′(z) ℘′′′(z) · · · ℘(n)(z)· · · · · · · · · · · ·· · · · · · · · · · · ·

℘(n−1)(z) ℘(n)(z) · · · ℘(2n−3)(z)

∣∣∣∣∣∣∣∣∣∣.

In the identity of Frobenius and Stickelberger, the left hand side considered as function of z0,is an elliptic function having z0 = 0 as a pole of order at most n. Its decomposition accordingto the theorem (9.1) is given by the right hand side. The coefficients of the decompositionare obtained by developing the determinant with respect to the elements of the first row. Theidentity of Kiepert can be obtained from the one of Frobenius and Stickelberger by a limitingprocess.

The Weierstrass function ℘ is homogeneous of degree −2 and is a generating function ofthe classical Eisenstein series

℘(z;ω1, ω2) = 1

z2 +∞∑

k=1

(2k + 1)G2k+2z2k, G2k+2 =∑

ω∈�\{0}ω−2k .

For τ = ω2ω1

fixed, ℘ and its derivative ℘′ are elliptic functions for Z+ τZ. The zeros of ℘′ in

C/(Z + τZ) occur at the points of order 2, namely1

2,τ

2and

1 + τ

2. On the other hand, the

zeros of ℘ were described in [11] and more recently in [10]. Since ℘ assumes every value inC ∪ {∞} exactly twice in C/(Z + τZ), it follows that ℘ has two zeros therein which can bewritten as ±z0 since ℘ is even.

Proposition 9.2 The zeros of the ℘-function are given by ±z0 where, by the Eichler–Zagierformula,

z0 = m + 1

2+ nτ ±

⎛⎝log

5 + 2√

6

2iπ+ 144iπ

√6

i∞∫τ

(σ − τ)�(σ)

E6(σ )32

d σ

⎞⎠

for all m, n ∈ Z, where the integral is to be taken over the vertical line σ = τ + iR+ or bythe Duke-Imamoglu formula

z0 = 1 + τ

2+

c2x14 F

(13 ,

23 , 1; 3

4 ,54 |x

)

F(

112 ,

512 |1 − x

) , c2 = −i√

6

3π

123

Geom Dedicata

where x = 1 − 1728

jand where the generalized hypergeometric series defined for |x | < 1

by

F(a1, . . . , am; b1, . . . , bm−1|x) =∞∑

n=0

(a1)n . . . (am)n

(b1)n · · · (bm−1)n

xn

n! .

and (a)n = a(a + 1) · · · (a + n − 1).

The pseudo-periods η1, η2 of the Weierstrass ζ -function are defined by

ζ(u + 2ωα) = ζ(u)+ 2ηα, ζ(ωα) = ηα, α = 1, 2.

The Legendre relation is η1ω2 − η2ω1 = 12 iπ . The periods and pseudo-periods of the Wei-

erstrass ℘-function are related to Ramanujan Eisenstein series by

E2(z) = 12

π2 η1ω1, E4(z) = 12(ω1

π

)4g2, E6(z) = 216

(ω1

π

)6g3.

In particular if 2ω1 = 1, 2ω2 = τ, τ = ω2

ω1, then

h1(τ ) = τ + 6

iπE2(τ )= η2

η1. (9.1)

Thus the basic equivariant function is a quotient of pseudo-periods. This interpretation willreveal important differential properties. We would like to extend this construction to thepowers ℘n, n ∈ Z with two exceptions.

Let L be the set of lattices in the R-vector space C and

M = {(ω1, ω2) ∈ C∗2 : τ = �(ω2/ω1) > 0}.

Then L can be identified with the quotient M/SL2(Z). Moreover, C∗ acts on L and on M

yielding two more identifications

M/C∗ ≈ H, R/C∗ ≈ H/PSL2(Z).

Following Brady [5], we introduce

Definition 9.1 A function f : C × L → P is called pseudo-periodic if it is meromorphic inz and for each ω ∈ � = 2ω1Z + 2ω2Z, (ω1, ω2) ∈ M there is a constant η(ω|�) such thatfor each z ∈ C

f (z + ω,ω) = f (z, ω)+ η(ω|�).In [5], Brady observed first that if f is a homogeneous pseudo-periodic function with pseudo-

periods η(ω|�), then the map τ → η(τ |�τ )η(1|�τ ) is an equivariant function, provided that

τ → η(1|�τ ), τ → η(τ |�τ ) are meromorphic and η(1|�τ ) does not vanish identically.In this paper, we are studying a similar question, namely the zeta function associated to

the elliptic function℘n, (n ∈ Z∗). This function is an even elliptic function, homogeneous of

degree −2n, of periods ω1, ω2. We look for a primitive of ℘n giving rise to pseudo-periodicfunction. We recall the following definition [11]

Definition 9.2 Let k and m be two fixed integers. A function φ : H × C → P is called ameromorphic Jacobi form of weight k and index m if

123

Geom Dedicata

(i) φ is meromorphic on C × H,

(ii) φ satisfies

φ

(z

γ τ + δ,ατ + β

γ τ + δ

)= (γ τ + δ)k exp

(2iπm

γ z2

γ τ + δ

)φ(z, τ ),

for every

(α β

γ δ

)∈ SL2(Z).

(iii) φ has a meromorphic q-expansion of the form

φ(z, τ ) =∑n≥h

cn(z)qn, 0 < |ξ | < A, 0 < |q| < B|ξ |N ,

ξ = e2iπ z, A > 0, B > 0, N ∈ N, ,

where the coefficients cn(z) are in the function field C(ξ).

The function ℘ is an example of a Jacobi form of weight 2 and index 0, that is a mero-morphic function that satisfies

℘

(z

γ τ + δ; ατ + β

γ τ + δ

)= (γ τ + δ)2 ℘(z; τ).

Thus℘n, n ∈ N∗ is a Jacobi form of weight 2n. The origin is the unique pole in a fundamental

domain, of order 2n.In general, let �(z, τ ) be a meromorphic periodic function in z with respect to the lattice

�τ = Z + τZ. Assume that, as function of τ , it satisfies

�

(z

γ τ + δ; ατ + β

γ τ + δ

)= (γ τ + δ)m�(z, τ )

for every

M =(α β

γ δ

)∈ SL2(Z).

Let gp(τ ) be the p-th coefficient of the Taylor expansion of�(z; τ) at x0 = xτ + y for somex, y ∈ R. Then for any M ∈ SL2(Z) such that (x ′, y′) = (x, y)M ≡ (x, y)∈Z

2, we have

gp

(ατ + β

γ τ + δ

)= (γ τ + δ)m+pgp(τ ).

This means that for a fixed integer n �= 0, the Taylor coefficient gnp, p ∈ N at the origin of

the function ℘n is a modular form on SL2(Z) of weight p + 2n.As is well known, many analytic properties of the ℘-function come from the differential

equation

℘′2 = 4℘3 − g2℘ − g3, g2 = 4π4

3E4, g3 = 8π6

27E6. (9.2)

From the decomposition theorem (9.1) or from (9.2) we obtain

℘2 = 1

6℘′′ + 1

12g2, ℘℘′ = 1

12℘′′′,

℘3 = 1

10

(1

12℘(4) + 3

2g2℘ + g3

), . . . ,

123

Geom Dedicata

and more generally, we have

Proposition 9.3 For each positive integer,

℘n = B(n)0

(2n − 1)!℘(2n−2) + B(n)1

(2n − 3)!℘(2n−4) + · · · + B(n)r

(2n − 2r − 1)!℘(2n−2r−2)

+ · · · + B(n)n−2℘′′ + B(n)n−1℘ + B(n)n , (9.3)

where, for 0 ≤ r ≤ n + 1

B(n+1)r = (2n−2r)(2n−2r+1)

2n(2n+1) B(n)r + 2n−14(2n+1) B(n−1)

r−2 g2 + n−12(2n+1) B(n−2)

r−3 g3 (9.4)

and B(n)r = 0 for r < 0 and r > n.

As a consequence of (9.3), we obtain

℘n℘′ = 1

n + 1

B(n+1)0

(2n + 1)!℘(2n+1) + · · · + 1

n + 1

B(n+1)r

(2n − 2r + 1)!℘(2n−2r+1) (9.5)

+ · · · + 1

n + 1B(n+1)

n−1 ℘′′′ + 1

n + 1B(n+1)

n ℘′.

Bnn−1 are constants and Bn

1 , Bn2 , . . . , Bn

n−1 are homogeneous polynomials in g2 and g3. Inparticular Bn

n−2 = 0.

Lemma 9.4 For every n ∈ N \ {0}, we have:

℘n+1 = 1

2n(2n + 1)(℘n)′′ + (2n − 1)

4(2n + 1)g2℘

n−1 + n − 1

2(2n + 1)g3℘

n−2 ,

with ℘−1 taken as 0.

The proof of this lemma is a straightforward computation from (9.2) and (9.3). We con-clude from (9.3) that lower order coefficients and pseudo-periods are given by the four-termrelations

B(n+1)n+1 = 2n − 1

4(2n + 1)g2 B(n−1)

n−1 + n − 1

2(2n + 1)g3 B(n−2)

n−2 , (9.6)

B(n+1)n = 2n − 1

4(2n + 1)g2 B(n−1)

n−2 + n − 1

2(2n + 1)g3 B(n−2)

n−3 , (9.7)

ηn+1 = 2n − 1

4(2n + 1)g2ηn−1 + n − 1

2(2n + 1)g3ηn−2. (9.8)

By inversion of these relations or by taking successive derivations of (9.2) we obtain

℘′′ = 6℘2 − 1

2g2,

℘′′′ = 12℘℘′, (9.9)

℘(4) = 12

(10℘3 − 3

2g2℘ − g3

),

etc.

More generally, we have for each n ∈ N

℘(2n) = Pn+1(℘), ℘(2n+1) = Qn(℘)℘′ , (9.10)

123

Geom Dedicata

where Pn+1, Qn are polynomials in ℘ of degree n + 1 and n, respectively and their coeffi-cients are polynomials in g2 and g3 with rational coefficients. For l < n + 1, the coefficientof ℘l is a modular form of weight 2n + 2 − 2l. Looking at one of these polynomials

A℘m + A1℘m−1 + · · · ,

we notice that it must be a homogeneous polynomial of degree −2m in u, ω1, ω2 so that itscoefficients must be of the following form

A1 = 0, A2 = ag2, A3 = bg3,

A4 = cg22, A5 = dg2g3, A6 = eg3

2 + f g23, · · ·

where a, b, . . . , f are numerical constants. A precise analysis of the polynomials Pn, Qn

can also be done using the following results due to Feldman [2] and which are very useful intranscendence theory.

Lemma 9.5 The jth derivative ℘( j)(z) of ℘(z) can be expressed in the form

∑u(t, t ′, t ′′)(℘ (z))t (℘′(z))t ′(℘′′(z))t ′′

where the summation is over all non-negative integers t, t ′, t ′′ with 2t + 3t ′ + 4t ′′ = j + 2and u(t, t ′, t ′′, j, k) denotes rational integers with absolute values at most 3 j ( j + 7)!.

Lemma 9.6 For any positive integer k, the j th derivative of (℘ (z))k can be expressed inthe form

∑u(t, t ′, t ′′, j, k)(℘ (z))t (℘′(z))t ′(℘′′(z))t ′′

where the summation is over all non-negative integers t, t ′, t ′′ with 2t + 3t ′ + 4t ′′ = j + 2kand u(t, t ′, t ′′, j, k) denote rational integers with absolute values at most j !48 j (7!28)k .

We recall that for a field k and a non-zero polynomial P ∈ k[X1, X2, · · · , Xn], the weightw(P) is defined by

w(P) = degt P(t X1, t2 X2, . . . , tn Xn

).

The polynomial P is called isobaric of weight w(P) if for any monomial Xi11 . . . Xin

n ofP(X1, . . . , Xn), we have

w(P) =n∑

r=1

rir .

It is a natural problem to investigate the polynomial in three indeterminate variables

123

Geom Dedicata

Q(X, Y, Z) =∑

u(t, t ′, t ′′)Xt Y t ′ Zt ′′ .

It is isobaric if one attributes to X, Y and Z the weights 2, 3 and 4, respectively.To give a similar formula for ℘n for a negative integer n, we use the precise analysis of

Zagier and Eichler concerning the zeros of the Weierstrass ℘-function, quoted in proposition(9.2). They gave two proofs of their result which reveal that the zero z0(τ ) of ℘(z, τ ) is,as function of τ , a (multi-valued) modular form of weight −1 and, in addition, its secondderivative is given by

z′′0(τ ) = ±144iπ

√6�(τ)

E6(τ )

and hence

z0(τ ) = m + 1

2± 1

2iπlog ε + nτ ± 144iπ

√6

(e2iπτ) + 183e4iπτ) + · · ·

).

We will consider two cases:The case τ �= i : The function ℘(z) = ℘(z; 1, τ ) has two simple zeros in L , the funda-

mental parallelogram of the periods 1 and τ , of opposite residues. Now for a zero z0 ∈ L of℘

℘−1(z) =[℘′(z0)(z − z0)+ ℘′′(z0)

2(z − z0)

2 + · · ·]−1

= 1

℘′(z0)(z − z0)− ℘′′(z0)

2℘′2(z0)+ · · · .

Since ℘ is even, the other zero in L is z1 = −z0 + 1 + τ . From the decomposition theorem(9.1) we obtain

1

℘(z)= A + 1

℘′(z0)[ζ(z − z0)− ζ(z + z0 − 1 − τ)] . (9.11)

The case τ = i (Lemniscatic case [30]): We first recall some known facts about the lem-niscatic case. The lattice � = Z + iZ admits a complex multiplication in the sense that themultiplication is an automorphism of �. Hence all the Eisenstein series G p vanish exceptwhen p is divisible by 4. In this situation G4k are real because � is also invariant undercomplex conjugation. The ℘-function verifies in this case

℘′2(z) = 4℘(z)3 − g2℘(z),

℘ (z) = 1

z2 +∞∑

k=1

(4k − 1)G4k z4k−2

from which we obtain the classical formulas g2 = 60G4, g3 = 140G6. We set g2 = 4w4,

w > 0 and we obtain for the Lattice L = w∧:

30G4 = 2, G8 = 6

7G2

4, G12 = 7

22G4G8,

and more generally the following recursion formula

((4k − 1)(4k − 2)(4k − 3)− 12(4k − 1))G4k = 6Gl+m=k(4l − 1)(4m − 1)G4l G4m

which shows that all G4p, p = 1, 2, . . ., are positive. For our purpose, we retain that forτ = i , we have from the theory of complex multiplication (Chowla-Selberg formula)

123

Geom Dedicata

g2(i) = 60G4(i) = 60∑

(m,n)∈Z\(0,0)

1

(m + ni)4= 60

�( 14 )

8

263.5.π2 .

The number�( 14 ) has very interesting arithmetical properties. It was shown to be transcenden-

tal by Chudnovsky, and Nesterenko showed that �( 14 ), π, eπ are algebraically independent.

Moreover, several of its infinite product expansions are known, from which we conclude thatsome relations exist between the exponents in (8.7) and the Catalan constant for example,or the Glaisher–Kinkelin constant as well [1]. More precisely, the Fourier series expansion

shows that for the lattice Z + iZ we have G4(i) = π4

45 E4(i), and hence

E4(i) = 3�( 1

4 )8

(2π)6,

and from (8.7), we obtain the infinite product expansion

�

(1

4

)=

(1

3

) 18

(2π)34

∞∏n=1

(1 − e−2πn)c(n)

8 .

In the meantime, τ = i, z0 = (1 + i)/2 is a double zero for ℘ and by (9.9) we have

1

℘(z)= 1

℘′′(z0)(z − z0)2− 2

3

℘′′′(z0)

℘′′(z0)(z − z0)+ · · · = 1

℘′′(z0)(z − z0)2+ · · · .

Finally, from the decomposition theorem (9.1) we obtain

1

℘(z)= −2

1

℘′′( 1+i2 )

ζ ′(

z − 1 + i

2

)= 4

g2(i)℘

(z − 1 + i

2

). (9.12)

10 Algebraic differential equations, a special Lie algebra and hypergeometricequations

In this section we study various differential equations related to equivariant functions. Wefirst prove the following result

Theorem 10.1 All equivariant functions on H satisfy (nonlinear) differential relations oforder at most 6.

This result will emerge from the full functional Eq. (1.1). We recall that, in general, onefunctional equation is not enough to insure the existence of a differential equation. The Eulergamma function� verifies essentially one functional equation and a theorem of Hölder assertsthat it is differentially transcendental over the field of rational functions R(X), which meansthat there is no polynomial P(X1, X2, . . . , Xn) ∈ R(x)[X1, X2, . . . , Xn] such that

P(�,�′, �′′, . . . , �(n)

)= 0,

or, in other words, the transcendence degree

Trdeg

(x, �′(x), �′′(x), . . . , �(n)(x), . . .

)= ∞.

We have at least two specific methods to check the asserted statement for the basic example

h1(z) = z + 6

iπE2(z). One has been given in Sect. 9 and the second one follows from the

fact that the Eisenstein series y = E2(z) = 6

iπ

1

h1(z)− zsatisfies the Chazy equation [37]

123

Geom Dedicata

D3 y − y D2 y + 3

2(Dy)2 = 0, Dy = 1

2iπ

dy

dz. (10.1)

This Chazy equation for E2 is converted, for y = 1

E2, into a differential equation as follows

Lemma 10.2 We have

−y2 D3 y + 6y Dy D2 y − 6(Dy)3 + y D2 y − 1

2(Dy)2 = 0.

As a consequence, we obtain, after a lengthy but elementary computation,

Proposition 10.3 The derivative h′ of h satisfies the following differential equation

3h′4h(4)2 − 24h′3h′′h(3)h(4) + 8h′3h(3)

3 − 6h′3h(4)2

+18h′2h(2)3h(4) + 12h′2h(2)

2h(3)

2 − 12h′2h′′h(3)h(4)

+32h′2h(3)3 + 3h′2h(4)

2 − 18h′h′′4h(3) + 54h′h′′3h(4) − 48h′h′′2h(3)2

−36h′h(2)h(3)h(4) + 32h′h(3)3 − 36h′′4h(3) + 36h′′3h(4) − 36h(2)2h(3)

2 = 0.

To prove Theorem 10.1 in its full generality we use the fundamental fact that the Schwarzderivative h of an equivariant holomorphic function f is given by a differential expression oforder 3 and is a modular form of weight 4. It is a classical fact that the graded ring of modularforms for SL2(Z) and their derivatives is generated over C by the three basic Eisenstein seriesE2, E4, E6. In the meantime, C(E2, E4, E6) has, according to [22], a transcendence degree3 over C. Hence, there exists a polynomial P(X1, X2, X3, X4) ∈ C[X1, X2, X3, X4] suchthat P(h, h′, h′′, h′′′) = 0. This establishes Theorem 10.1.

As we already pointed out, one needs to know the Fourier coefficients of1

E2in order to

determine the periodic part of the fundamental example h1. Though no elegant descriptionseems to emerge (except what we said in Lemma 8.1), these coefficients can also be givenby induction from Lemma 10.2. However, some arithmetical properties are guaranteed bythe following general result of Hurwitz [17] concerning the analytic solutions of algebraicdifferential equations. Before we proceed further, let us recall the general context mentionedearlier in this paper regarding the algebraic setting.

Theorem 10.4 (Eisenstein [26]) A series∞∑

n=0

anzn, an ∈ Q ,

of radius of convergence 1, has the property that if the sum is algebraic, then there exists aninteger N such that the quantities an N n are integers.

The converse is not true, for in any series with a finite radius of convergence, it suffices tochange the signs of an infinite number of coefficients so that the circle of convergence of thenew series shall be reduced to the extent that the new series cannot be algebraic although itscoefficients have the required property. Taking into account that an algebraic equation is azero order differential equation, Hurwitz, in [17], used an argument similar to Heine’s proofof Eisenstein’s theorem to prove the following result

Theorem 10.5 Suppose that a holomorphic function f around the origin is given by a powerseries expansion

f (z) = a0 + a1z + a2z2 + · · · , ak = pk

qk∈ Q

123

Geom Dedicata

and is a solution of an algebraic differential equation

P(x, f, f ′, . . . , f (l)) = 0, P ∈ Z(x)[X1, X2, . . . , Xl ].Then there exists a polynomial T (X) = γ0 + γ1 X + · · · + γl Xl ∈ Z[X ] and an integerN such that every prime which divides qN , qN+1, qN+2 . . . must divide T (N ), T (N )T (N +1), T (N )T (N + 1)T (N + 2), . . . respectively.

This result knew several refinements and extensions, including results of Popken and Os-trowski [23], p. 206. They have been used in transcendence problems [23,33]. They show, inparticular, that if pm denotes the largest prime in the denominator of the coefficient am , then

lim supm→∞

pm

log m< ∞.

Moreover, the following result of Maillet [24] gives an idea on the gaps in power seriessolutions of an algebraic differential equation:

Theorem 10.6 (Maillet) Let∑∞

m=0 bm zm be a given formal solution to a differential equa-tion of order k and degree μ, we can find a fixed number τ , independent of m such that forlarge m, if bm is non-zero, the previous non-zero coefficient has an index greater than or

equal to(m + τ)

μ.

This will enable us to establish a different approach to study the coefficients of1

E2,

1

E4and

1

E6as power series in q = e2iπ z solutions to algebraic differential equations.

The three Eisenstein series E2, E4, E6 all satisfy algebraic differential equations; a fact

that is well known [29,37]. As we had already observed in Lemma 10.2, E2 and1

E2are

solutions to algebraic differential equations, but we could not find any reference giving theneeded explicit algebraic differential equation for E6. We will provide it here after giving adetailed summary of the necessary ideas following [29,37].

Let Mk(�, ν) be the space of all automorphic forms f (z) of weight k with respect to aFuchsian group of the first kind � with a multiplier system ν, and set Mk(�, 1) = Mk(�).The classical Bol operator [29] is the differential operator D such that for each integer m > 1,

Dm : Mk(�, ν) → Mm(k+2)(�, ν)

and is defined as follows: Let f ∈ Mk(�, ν), k �= 0 and recall the differential operator

θ = 1

(2iπ)

d

dz= q

d

dq, q = e2iπ z , then

θm(( f )((1−m)/k

)= ( f )((1−(k+1)m)/k Dm( f )

where Dm( f ) is a polynomial in f and its derivatives f, f ′, . . . , f (m). On the other hand,the Rankin-Cohen product is defined [37] for f ∈ Mk(�) and g ∈ Ml(�) as follows

Definition 10.1

[ f, g]n =n∑

r=0

(−1)r(

k + n − 1

n − r

)(l + n − 1

r

)f (r)g(n−r),

where f (r) stands for Dr f = 1

(2iπ)rdr f

dzr.

123

Geom Dedicata

In this way [ f, g]0 = f g and

[ f, g]1 = k f g′ − l f ′g, [ f, g]2 = k(k + 1)

2f g′′ − (k + 1)(l + 1) f ′g′ + l(l + 1)

2f ′′g.

The Rankin–Cohen product sends Mk(�)× Ml(�) into Mk+l+2n(�). Furthermore, the iter-ated Rankin-Cohen products are related in a remarkable way to the Bol operator. In fact it iseasily seen that if f is of weight k, then

D2( f ) = − 1

k2(k + 1)[ f, f ]2,

D3( f ) = 2

k3(k + 1)[ f, [ f, f ]1]1.

On the other hand and very generally, if � is a discrete group acting on H such that asuitably chosen fundamental domain for � in H is finite with respect to the invariant metric,then the quotient H/� can be given a structure of a compact Riemann surface R. As a con-sequence, any two meromorphic functions on R are algebraically dependent over C. This isthe main reason for a reasonably well behaved automorphic function to satisfy an algebraicdifferential equation over C. More precisely [29].

Theorem 10.7 Let � be a discrete group such that the quotient space H/� is of finite vol-ume. Let f ∈ Mk(�, ν) be an analytic automorphic form with a multiplier system ν satisfyingνs ≡ 1 for some positive integer s. Then if the product ks is a positive integer, there exists apolynomial P ∈ C[X1, X2, X3] such that on H

P(

f, D2( f ), D3( f )) = 0.

In [29] (Proposition 4), the following differential equation for E4 is given. We give it heretogether with the corresponding differential equation for E6.

Theorem 10.8 The three Eisenstein series E2, E4 and E6, as q−series, are all solutions todifferential equations over Q. The equation for E2 is the Chazy equation, given in proposition(10.1) and for E4 and E6, we have

5 (D3(E4))2 − 576D2(E4)

3 + 20E34 D2(E4)

2 = 0,

2744D2(E6)3 E12

6 − 343D3(E6)3 E8

6 + 226492416D2(E6)6 E4

6

−3096576D3(E6)2 D2(E6)

3 E46 + 10584D3(E6)

4 E46 = 0.

Corollary 10.9 The three inverses1

E2,

1

E4,

1

E6, as q-series, are all solutions to algebraic

differential equations over Q. Their coefficients, as given in Sect. 8, satisfy the conclusionsgiven in Hurwitz’s theorem (10.5) and Maillet’s theorem (10.6).

On the other hand, we can show that the function h1 is also related to a monodromy problemby considering it as a function of the modular invariant J rather than the variable z ∈ H. Thiswill be undertaken later in this section.

Another aim of the present section is to consider a special Lie algebra related to ellipticfunctions, well suited to study the powers ℘n . We also show here that the basic equivariantfunction h1 verifies a third order differential equation, with respect to the coordinate j , arisingfrom a hypergeometric equation of Picard-Fuchs type. This shows that h1 is also intimately

123

Geom Dedicata

related to uniformization theory. We first recall some classical facts on periods associated toelliptic curves. We begin with the two differential forms

dx√4x3 − g2x − g3

,xdx√

4x3 − g2x − g3

and introduce the periods

ω =∫

dx√4x3 − g2x − g3

, η =∫

xdx√4x3 − g2x − g3

, (10.2)

where the integrals are taken over suitable cycles on the elliptic curve y2 = 4x3 − g2x − g3.We normalize the periods in the following way

� =√

g3

g2ω , H =

√g3

g2η.

As usual, we introduce� = g32 −27g2

3 that we suppose is non-vanishing and we let J = g32

�.

We thus have the connexion formula⎛⎝

d�d J

d Hd J

⎞⎠ =

⎛⎜⎝

J+212J (J−1)

118J

−18(J−1) − J+2

12J (J−1)

⎞⎟⎠

⎛⎝�

H

⎞⎠ . (10.3)

A short calculation shows that a general differential system (whatever the variable J and thefunctions �, H ) of the form

d

d J

⎛⎝�

H

⎞⎠ =

⎛⎝a(J ) b(J )

c(J ) d(J )

⎞⎠

⎛⎝�

H

⎞⎠

leads to the second order differential equation

d2�

d J 2 −(

a + d + b′

b

)d�

d J+ (ad − bc − a′).

Moreover, the periods satisfy the Legendre relation⎛⎝ω1 ω2

η1 η2

⎞⎠

⎛⎝ 0 1

−1 0

⎞⎠

⎛⎝ω1 η1

ω2 η2

⎞⎠ = −iπ

2

⎛⎝ 0 1

−1 0

⎞⎠ .

Recall that the J−function given by (8.6) realizes a complex analytic isomorphism of Rie-mann surfaces

J : H/SL2(Z) → C, τ → J (τ ).

If H∗ = H ∪ P1(Q), then the function J extends as a complex analytic isomorphism of

compact Riemann surfaces

J : H∗/SL2(Z) → P1(C).

It is also related to Klein’s absolute invariant λ and to Eisenstein series E4, E6 by

J = 4

27

(λ2 − λ+ 1)3

λ2(λ− 1)2= E3

4

E34 − E2

6

.

123

Geom Dedicata

The link to uniformization theory is illustrated by the hypergeometric equation

J (J − 1)d2 y

d J 2 +(

2

3− 7

6J

)dy

d J− 1

144y = 0,

in the sense that J is the inverse of the (multi-valued) function obtained by taking the quotientof two linearly independent solutions to this differential equation. A general fact, shown in[13], concerning elliptic functions is that if φ is an elliptic function of periods ω1, ω2, thenthe following functions

4g2∂φ

∂g2+ 6g3

∂φ

∂g3− u

∂φ

∂u, 18g3

∂φ

∂g2+ g2

2∂φ

∂g3− 3ζ(u)

∂φ

∂u(10.4)

are also elliptic with the same periods. This leads, for the function℘(u;ω1, ω2), to the preciseidentities

4g2∂℘

∂g2+ 6g3

∂℘

∂g3− u

∂℘

∂u= 2℘, (10.5)

18g3∂℘

∂g2+ g2

2∂℘

∂g3− 3ζ(u)

∂℘

∂u= 6℘2 − g2.

Following [13] or the more recent treatment [6], we introduce the differential operators

L0 = 4g2∂

∂g2+ 6g3

∂

∂g3− u

∂

∂u,

L1 = ∂

∂u, (10.6)

L2 = 6g3∂

∂g2+ 1

3g2

2∂

∂g3− ζ(u; g2, g3)

∂

∂u.

The Lie brakets are

[L0, Lk] = kLk, [L1, L2] = ℘(u, g2, g3)L1.

These operators generate the ring Der F of the derivations of the ring F of elliptic functions.We introduce the fundamental operator

D = 12g3∂

∂g2+ 2

3g2

2∂

∂g3= 2L2 + 2ζ(u; g2, g3)

∂

∂u, (10.7)

then [L0, D] = 0 and

L0℘ = 2℘, L0℘n = 2n℘n, n ∈ N. (10.8)

This eigenfunction character of ℘n could be used to compute the coefficients in the relations(9.3). A more important property is that the operator operator D connects the periods topseudo-periods by the relation

Dωi = −2ηi , i = 1, 2

and the discriminant function � belongs to KerD as D� = 0. A consequence of thisdescription is that the fundamental operator gives a correspondence between different typesof hypergeometric differential equations

J (J − 1)d2ωi

d J 2 +(

7

6J − 2

3

)dωi

d J+ 1

144ωi = 0 i = 1, 2 (10.9)

123

Geom Dedicata

and

J (J − 1)d2ηi

d J 2 +(

5

6J − 1

3

)dηi

d J+ 1

144ηi = 0 i = 1, 2. (10.10)

A variant of (2.3) is the fact that if f = y1

y2, with two independent solutions y1, y2 to the

differential equation

d2 y

dx2 + 2Q1dy

dx+ Q2 y = 0,

then the Schwarz derivative { f, x} satisfies

{ f, x} = 2

(Q2 − Q2

1 − d Q1

dx

),

and hence, from (9.1) and (10.10) we obtain, for the fundamental equivariant function, athird order differential equation, as was pointed out in the proof of the theorem (10.1).

Corollary 10.10 The basic equivariant function h1 satisfies

{h1, J } = −3

8

1

(J − 1)2+ 4

9

1

J 2 − 23

72

1

J (J − 1).

This equation is the same as that of Proposition 6.2 except that it is now given in connection toa monodromy problem for a linear second order differential equation. That the two equationsare the same is due to the following equations

J = g32

�, J − 1 = 27

g23

�, � = g3

2 − 27g23 .

11 Equivariant functions and Binary forms

In this section we revisit the notion of cross-ratio with more details. Let� be the diagonal of(P1(C))4, the set of 4-tuples with at least two equal coordinates and set X = (P1(C))4 \�.The group G = GL(2,C) acts naturally on X by g.(z1, z2, z3, z4) = (g.z1, g.z2, g.z3, g.z4).The map

R : (z1, z2, z3, z4) → (z2 − z1)(z4 − z3)

(z3 − z1)(z4 − z2)

is an invariant function under this action inducing a bijection on the orbit space

R : X/GL(2,C) → C.

This is strongly connected with the theory of invariants of binary quartics

F(x, y) = f0 y4 + 4 f1xy3 + 6 f2x2 y2 + 4 f3x3 y + f4x4.

The algebra of invariants A = C[ f0, . . . , f4]SL2(C) is freely generated by two invariants ofweight 4 and 6, respectively

g2 =∣∣∣∣ f0 f2

f2 f4

∣∣∣∣ − 4

∣∣∣∣ f1 f2

f2 f3

∣∣∣∣ , g3 =∣∣∣∣∣∣

f0 f1 f2

f1 f2 f3

f2 f3 f4

∣∣∣∣∣∣ . (11.1)

123

Geom Dedicata

We can identify the binary quartic F with its roots z1, . . . , z4 ∈ P1(C) and consider the

discriminant � = g32 − 27g2

3 of weight 12 and the absolute rational invariant J , related tothe cross-ratio

λ = (z1 − z2)(z3 − z4)

(z1 − z3)(z2 − z4), J = g3

2

�= 4(λ2 − λ+ 1)3

27λ2(λ− 1)2.

The symmetric group S4 acts on rational functions of the roots zi by permuting the indices.The alternating group A4 has the Klein 4-group V as a proper normal subgroup consist-ing in double transpositions Id, (12)(34), (13)(24), (14)(23). The Klein 4-group V fixes thecross-ratio λ and induces an action of S3 = S4/V under which the orbit of λ is

λ,1

λ, 1 − λ,

1

1 − λ,

λ

λ− 1,λ− 1

λ.

This means that the permutation group S3 acts on the Riemann sphere P1(C) by Möbius trans-

formations and the orbit space P1(C)/S3 is a 2-sphere [20]. The map λ → J is a rational

map on the Riemann sphere. It commutes with the action of S3. The orbit space of S3 may beidentified with the Riemann sphere, and the quotient map with λ → J too. The inverse mapis exactly the Kappa function considered by Kaneko and Yoshida in [18]. There are threeexceptional values of J whose

(i) J = 1; λ = −1, 12 , 2. Here g3 = 0 and the roots lines from a harmonic range (Lem-

niscatic case).

(ii) J = ∞, λ = 0, 1,∞. The form is degenerate, and � = 0.

(iii) J = 0; λ = −ω,ω2, ω = e2iπ/3. Here g2 = 0 and the roots lines from a equi-anhar-monic range.

In [5], Brady gives a parametrization of equivariant functions by using the projective invari-ance of the cross-ratio:

[ f (z1), f (z2), f (z3), f (z4)] = [z1, z2, z3, z4] ,where f (z) is a Möbius transformation.

Let A the class of all meromorphic equivariant functions for SL2(Z) on H and let M thefield of all meromorphic modular functions on H.

Theorem 11.1 [5] Let F1, F2 and F3 be three mutually distinct elements in A. The function

φ : A → M, φ(F) = F1 − F2

F1 − F3.F3 − F

F2 − F= [F1, F2, F3, F] (11.2)

is a (1, 1)-map of A \ {F2} onto M, whose inverse map, as in (2.1), is

G → F2(F1 − F3)G − (F1 − F2)F3

G(F1 − F3)− (F1 − F2).

We give an example: let hi , 1 ≤ i ≤ 4 four rational equivariant functions as in theorem (7.4),

hi = z + kif ′i

fi, fi being a modular form of weight ki , then

[h1, h2, h3, h4] = [ f1, f2, f3, f4]1

123

Geom Dedicata

where

[h1, h2, h3, h4] = (h1 − h2)(h4 − h3)

(h1 − h3)(h4 − h2)

is the usual cross-ratio and

[ f1, f2, f3, f4]1 = [ f1, f2]1[ f4, f3]1

[ f1, f3]1[ f4, f2]1

with [ fi , f j ]1 is the first Rankin-Cohen bracket (10.1) of fi , f j .This result is similar to what we have encountered in Sect. 2: the constants in the Riccati

equation are replaced by modular functions. The (1, 1)-map φ can serve to provide the classA with many properties. We give one example here. The functions that are meromorphic onthe upper half plane H can be seen as functions taking their values in the Riemann sphere,or the extended complex plane, by attributing the value ∞ to the poles. Usually a topologyon the set of meromorphic functions is defined as follows. We consider a sequence of com-pacts sets (Kn)n∈N such that Kn ⊂ K o

n+1, ∪n≥0 Kn = H. Let σ the spherical metric onthe Riemann sphere. For each n ∈ N and two meromorphic functions f, g on H, we defineρn( f, g) = supz∈Kn

σ( f (z), g(z)). In this way the set of meromorphic functions on H isendowed with a structure of metric space (not complete) with the distance

d( f, g) =∑n≥0

ρn( f, g)

1 + ρn( f, g).

As a consequence of (11.2), we obtain that the pace A is connected. It is possible to definepaths in A by defining paths on the space of modular functions and considering their imagesby the map (11.2). In addition, it would be worthwhile to transfer to equivariant functionsthe following stability properties of modular functions:

(1) Action of Hecke operators.(2) If g is modular for � so is g′−2{g, z} and {z, g}.(3) If k is a positive integer and F(τ ) is a modular form of weight 2k with respect to �,

then F(τ )

(d J (τ )

dτ

)−k

is a modular function with respect to �.

These operations induce corresponding operations on the set of equivariant functions.

12 Special cases: Platonic equivariance

We would like to give some examples, borrowed from geometry and physics [16], wheresome special equivariant functions are considered. We emphasize the examples of equivari-ant functions which are rational and associated to platonic solids and with lattice maps. Theseexamples appeared recently in some physical literature in connection with monopoles andNahm equations or in dynamical systems. We provide few examples:

• Example 1: Tetrahedral symmetry. The rational function

R(z) = i√

3z2 − 1

z(z2 − i√

3)

verifies

R(−z) = −R(z), R

(1

z

)= 1

R(z), R

(i z + 1

−i z + 1

)= i R(z)+ 1

−i R(z)+ 1.

123

Geom Dedicata

Hence it is equivariant with respect to the sub-group of PSL2(C) generated by

z → z, z → 1

z, z → i z + 1

−i z + 1.

The Wronskian of R(z) = p

q,W = p′q − pq ′, is given by

W (z) = −i√

3(z4 + 2i√

3z2 + 1)

which is proportional to a tetrahedral Klein polynomial.• Example 2: Octahedral symmetry. The rational function

R(z) = z(z4 − 5)

−5z4 + 1

verifies

R

(i z + 1

−i z + 1

)= i R(z)+ 1

−i R(z)+ 1

and the Wronskian is W (z) = −5(z8 + 14z4 + 1) which is proportional to a octahedralKlein polynomial.

• Example 3: Icosahedral symmetry [9]. We identify the Riemann sphere with a sphere inR

3 so that 0 and ∞ are poles and the circle |z| = 1 is the equator. We inscribe a regularicosahedron in the sphere normalized so that one vertex is at 0 and the adjacent vertex lieson the positive real axis in the Riemann sphere P

1(C). The isometries of the icosahedronact on P

1(C) via a group � ⊂ PSL2(C). More precisely,

� ≈ PSL(4) ≈ PSL(5) ≈ A5,

the latter being the alternating group. We have [9]

Proposition 12.1 There are exactly four rational maps of degree< 31 which commute withthe icosahedral group. These four maps, of degree 1, 11, 19 and 29 are given respectivelyby:

f1(z) = z,

f11(z) = z11 + 66z6 − 11z

−11z10 − 66z5 + 1,

f19(z) = −57z15 + 247z10 + 171z5 + 1

−z19 + 171z14 − 247z9 − 57z4 ,

f29(z) = −87z25 − 3335z20 − 6670z10 − 435z5 + 1

−z29 − 435z24 + 6670z19 + 3335z9 + 87z4 .

These facts can be checked by direct computations. The generators of the icosahedron are[20], p. 47

z → εμz, z → −ε4μ

z, z → εν

−(ε − ε4)εμz + (ε2 − ε3)

(ε2 − ε3)εμz + (ε − ε4),

z → −ε4ν (ε2 − ε3)εμz + (ε − ε4)

−(ε − ε4)εμz + (ε2 − ε3); ε = e

2iπ5 , μ, ν = 0, 1, 2, 3, 4.

The following table, from Tannery and Molk [35] (volume IV, pp. 108–109) gives the integrals∫℘n(u)du; ℘(u) = ℘(u; 1, z), ζ(u) = ζ(u; 1, z), �z > 0:

123

Geom Dedicata

Jn

J1 −ζ(u)

J213!℘

′(u)+ g2223

u

J315!℘

′′′(u)− 3g2225ζ(u)+ g3

2.5 u

J4℘(5)(u)

37! + g25℘′(u)

3! − g37 ζ(u)+ 5g2

224.3.7

u

J5℘(7)(u)

9! + g222℘′′′(u)

5! + 5g322.7

℘′(u)3! − 7g2

224.3.5

ζ(u)+ g2g32.3.5 u

J6℘(9)(u)

11! + 3g22.5

℘(5)(u)7! + 3g3

2.7℘′′′(u)

5! + 17g22

24.52℘′(u)

3! − 3.29g2g322.5.7.11

ζ(u)+(

3.5g32

26.7.11+ g2

35.11

)u

J7℘(11)(u)

13! + 7g222.5

℘(7)(u)9! + g3

22℘(5)(u)

7! + 7g22

23.3.5℘′′′(u)

5! + 3.23.g2g324.5.11

℘′(u)3!

−(

7.11g32

26.3.5.13+ 5g2

32.7.13

)ζ(u)+ 433g2

2 g3

25.3.5.7.13u

J8℘(13)(u)

15! + 2g25℘(9)(u)

11! + 2g37℘(7)(u)

9! + 23g22

22.3.52℘(5)(u)

7! + 23g2g37.11

℘′′′(u)5!

(61g3

222.53.13

+ 3.31g23

22.72.13

)℘′(u)

3! − 167g22 g3

23.3.5.7.11ζ(u)+

(13g4

228.7.11

+ 7g2g23

22.3.5.11

)u

J9℘(15)(u)

17! + 32g222.5

℘(11)(u)13! + 32g3

22.7℘(9)(u)

11! + 3.13g23

24.52℘(7)(u)

9! + 32.13g2g324.5.11

℘(5)(u)7!

+(

3.47g32

25.52.13+ 32.53g2

324.72.13

)℘′′′(u)

5! + 32.181.g22 g3

25.52.7.11℘′(u)

3!

−(

7.11g42

28.13.17+ 33.223g2.g2

322.5.7.11.13.17

)ζ(u)+

(7g3

32.5.7.11.17 + 383g3

2 g3

23.7.11.13.17

)u

J10℘(17)(u)

19! + g22℘(13)(u)

15! + 5g32.7

℘(11)(u)13! + 29g2

224.3.5

℘(9)(u)11! + 3.17g2g3

22.7.11℘(7)(u)

9!

+(

587g32

25.3.52.13+ 5.17g2

324.7.13

)℘(5)(u)

7! + 137g22 g3

2.3.7.11℘′′′(u)

5!

+(

31.1453g42

28.3.53.13.17+ 3.15817g2g2

324.72.11.13.17

)℘′(u)

3!

−(

3251g32 g3

25.7.11.13.19+ 2.5.g3

37.13.19

)ζ(u)+

(1357g2

2 g23

24.7.11.13.19+ 13.17g5

2210.7.11.19

)u

The equivariant functions hn(z)From the previous table, we deduce the first ten equivariant functions,

123

Geom Dedicata

hn

h1η2η1

h2 z

h3− 3g2

225η2+ g3

2.5 z

− 3g2225

η1+ g32.5

h4− g3

7 η2+ 5g22

24 .3.7z

− g37 η1+ 5g2

224 .3.7

h5− 7g2

224 .3.5

η2+ g2g32.3.5 z

− 7g22

24 .3.5η1+ g2g3

2.3.5

h6

− 3.29g2g322 .5.7.11

η2+(

3.5g32

26 .7.11+ g2

35.11

)z

− 3.29g2g322 .5.7.11

η1+(

3.5g32

26 .7.11+ g2

35.11

)

h7

−(

7.11g32

26 .3.5.13+ 5g2

32.7.13

)η2+ 433g2

2 g325 .3.5.7.13

z

−(

7.11g32

26 .3.5.13+ 5g2

32.7.13

)η1+ 433g2

2 g325 .3.5.7.13

h8

− 167g22 g3

23 .3.5.7.11η2+

(13g4

228 .7.11

+ 7g2g23

22 .3.5.11

)z

− 167g22 g3

23 .3.5.7.11η1+

(13g4

228 .7.11

+ 7g2g23

22 .3.5.11

)

h9

−(

7.11g42

28 .13.17+ 33 .223g2 .g

23

22 .5.7.11.13.17

)η2+

(7g3

32.5.7.11.17 + 383g3

2 g323 .7.11.13.17

)z

−(

7.11g42

28 .13.17+ 33 .223g2 .g

23

22 .5.7.11.13.17

)η1+

(7g3

32.5.7.11.17 + 383g3

2 g323 .7.11.13.17

)

h10

−(

3251g32 g3

25 .7.11.13.19+ 2.5.g3

37.13.19

)η2+

(1357g2

2 g23

24 .7.11.13.19+ 13.17g5

2210 .7.11.19

)z

−(

3251g32 g3

25 .7.11.13.19+ 2.5.g3

37.13.19

)η1+

(1357g2

2 g23

24 .7.11.13.19+ 13.17g5

2210 .7.11.19

)

The equivariant functions hn can be given in terms of E2, E4, E6 by means of the relations(9.2).

References

1. Abramowitz, M., Stegun, I.A. (ed.): Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. Dover, New York (1972)

2. Baker, A.: On the Periods of the Weierstrass℘-Function Symposia Mathematica, vol. IV (INDAM, Rome,1968/69). pp. 155–174. Academic Press, New York (1970)

3. Berndt, B.C., Bialek, P.R.: On the power series coefficients of certain quotients of Eisenstein series. Trans.Am. Math. Soc. 357, 4379–4412 (2005)

4. Borcherds, R.E.: Automorphic forms on Os+2,2(R) and infinite products. Invent. Math. 120, 161–213 (1995)

5. Brady, M.: Meromorphic solutions of a system of functional equations involving the modular group. Proc.Am. Math. Soc. 30, 271–277 (1971)

123

Geom Dedicata

6. Buchstaber, V.M., Leykin, D.V.: Solution of the problem of differentiation of Abelian functions overparameters for families of (n, s)-curves. Funct. Anal. Appl. 42(4), 268–278 (2008)

7. Carleson, L., Gamelin, T.: Complex Dynamics. Springer, New York (1993)8. Cartan, E.: Leçons sur la théorie des espaces à connexion projective. Gauthier-Villars, Paris (1937)9. Doyle, P., McMullen, C.: Solving the quintic by iteration. Acta. Math. 163, 151–180 (1989)

10. Duke, W., Imamoglu, Ö.: The zeros of the Weierstrass ℘-function and hypergeometric series. Math.Ann. 340, 897–905 (2008)

11. Eichler, M., Zagier, D.: On the zeros of the Weierstrass ℘-function. Math. Ann. 258, 399–407 (1981)12. Elbasraoui, A., Sebbar, A.: The zeros of the Eisenstein series E2. Proc. Am. Math. Soc. 138(7),

2289–2299 (2010)13. Frobenius, F.G., Stickelberger, L.: Über die differentiation der ellipyischen functionen nach den perioden

und invarianten. J. Reine Angew. Math. 92, 311–327 (1882)14. Hardy, G.H., Ramanujan, S.: On the coefficients in the expansions of certain modular functions. Proc. R.

Soc. A 95, 144–155 (1918)15. Heins, M.: On the pseudo-periods of the Weirstrass zeta-function. SIAM J. Numer. Anal. 3,

266–268 (1966)16. Houghton, C.J., Manton, N.S., Sutcliffe, P.M.: Rational maps, monopoles and skyrmions. Nucl. Phys. B

510 [PM], 507–537 (1998)17. Hurwitz, A.: Sur le développement des fonctions satisfaisant à une équation différentielle algébrique. Ann.

Sci. École Norm. Sup. 3(6), 327–332 (1889)18. Kaneko, M., Yoshida, M.: The kappa function. Int. J. Math. 14–9, 1003–1013 (2003)19. Knopp, M., Mason, G.: Generalized modular forms. J. Number Theory. 99, 1–28 (2003)20. Klein, F.: Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. Dover, New

York (1956)21. Lie, S.: Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen.

Chelsea Publishing Co., Bronx, New York (1971)22. Mahler, K.: On algebraic differential equations satisfied by automorphic functions. J. Aust. Math.

Soc. 10, 445–450 (1969)23. Mahler, K.: Lectures on transcendental numbers. Lecture Notes in Math., vol. 546, Springer, Berlin (1976)24. Maillet, E.: Introduction à la théorie des nombres transcendants et des proprités arithmétiques des fonc-

tions. Gauthier-Villars, Paris (1906)25. McKay, J., Sebbar, A.: Fuchsian groups, automorphic functions and Schwarzians. Math. Ann. 318,

255–275 (2000)26. Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. I. Springer, Berlin (1976)27. Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916)28. Rankin, R.A.: Modular Forms and Functions. Cambridge University Press, Cambridge (1977)29. Resnikoff, H.L.: On differential operators and automorphic forms. Trans. Am. Math. Soc. 124,

334–346 (1966)30. Schappacher, N.: Some milestones of Lemniscatomy. Lecture Notes in Pure and Applied Mathematics

Series 193. M. Dekker, New York, 257–290 (1997)31. Schwarz, H.A.: Gesammelte Mathematische Abhandlungen, vol. 2. Springer, Berlin (1880)32. Sebbar, A. Sebbar, A.: Eisenstein series and modular differential equations. Can. Math. Bull. doi:10.4153/

CMB-2011-091-333. Sibuya, Y., Sperber, S.: Arithmetic properties of power series solutions of algebraic differential equa-

tions. Ann. Math. (2) 113(1), 111–157 (1981)34. Smart, J.: On meromorphic functions commuting with elements of a function group. Proc. Am. Math.

Soc. 33, 343–348 (1972)35. Tannery, J., Molk, J.: Eléments de la théorie des fonctions elliptiques. pp. 1893–1902. Gauthier-Vil-

lars, Paris (1990)36. Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, 4th edn. Cambridge University Press, Cam-

bridge (1990)37. Zagier, D.: Elliptic Modular Forms and Their Applications in: The 1-2-3 of Modular Forms Universi-

text. Springer, Berlin (2008)

123