The quasi-periods of the Weierstrass zeta-function

We study the ratio $p=\eta_1/\eta_2$ of the pseudo-periods of the Weierstrass $\zeta$-function in dependence of the ratio $\tau=\omega_1/\omega_2$ of the generators of the underlying rank-2 lattice. We will give an explicit geometric description of the map $\tau\mapsto p(\tau)$. As a consequence, we obtain an explanation of a theorem by Heins who showed that $p$ attains every value in the Riemann sphere infinitely often. Our main result is implicit in the classical literature, but it seems not to be very well known. Essentially, this is an expository paper. We hope that it is easily accessible and may serve as an introduction to these classical themes.


Introduction
Throughout this paper, we assume that ω 1 , ω 2 ‰ 0 are two numbers in the complex plane C that are linearly independent over the field of real numbers R. We define (1.1) τ :" ω 1 {ω 2 , and assume that (1.2) Impτ q ą 0.
generated by ω 1 and ω 2 .The Weierstrass ζ-function associated with this lattice Γ is given by the series We simply write ζpuq if the underlying lattice Γ is understood.It is well-known that the series (1.4) converges absolutely and locally uniformly for u P CzΓ.Moreover, ζ is an odd meromorphic function with poles of first order precisely at the lattice points.It has the periodicity property (1.5) ζpu `ωk q " ζpuq `ηk for some constants η k P C, k " 1, 2 (for a discussion of all these facts, see [Ch85,Chapter 4]).Throughout this paper, we assume that u represents a variable in C and so (1.5) and similar formulas are identities valid for all u P C. We call η 1 and η 2 the quasi-periods of ζ associated with the given generators of ω 1 and ω 2 of Γ. Evaluating (1.5) at u " ´ωk {2 and using that ζ is an odd function, we see that (1.6) η k " 2ζp 1 2 ω k q for k " 1, 2. It also follows from (1.5) that (1.7) ζpu `kω 1 `nω 2 q " ζpuq `kη 1 `nη 2 for k, n P Z.
This implies that the function p is homogeneous of degree 0 and hence we can consider p " ppτ q as function of τ " ω 1 {ω 2 .Let H :" tτ P C : Impτ q ą 0u be the open upper halfplane.The function τ P H Þ Ñ ppτ q P p C can be described explicitly, as the main result of this paper shows.In order to state this, we first introduce some terminology.
A closed Jordan region X in the Riemann sphere p C is a compact set homeomorphic to a closed disk.Then its boundary BX is a Jordan curve and the set of interiors points intepXq :" XzBX a simply connected open region.
A circular arc triangle T is a closed Jordan region in p C whose boundary is decomposed into three non-overlapping circular arcs considered as the sides of T .The three endpoints of these arcs are the vertices of T .
We say that f is a conformal map between two circular arc triangles X and Y in p C if f is a homeomorphism between X and Y that sends the vertices of X to the vertices of Y and is a biholomorphism between intepXq and intepY q.We say that f is an anti-conformal map between X and Y if z P X Þ Ñ f pzq P Y is a conformal map between X and the complex conjugate Y :" tw : w P Y u of Y .
We can now state our result.
Of course, the last relation has to be understood in a limiting sense: if τ P T 0 X H Ñ 8, then ppτ q P T 1 X H Ñ 8.In the following, we always think of p (and similar functions) as extended to the point 8 in this way.
In order to give a more intuitive description of the map p based on Theorem 1.1, we proceed as follows.Note that the triangle T 0 has angles 0, π{2, π{3 at its vertices 8, i, ρ :" 1 2 p1 `i? 3q, respectively.Let T Ď p C be an arbitrary circular arc triangle in the Riemann sphere with angles equal to 0, π{2, π{3.Each such triangle is Möbius equivalent to T 0 or its complex conjugate T 0 (see A circular arc triangle T with angles 0, π{2, π{3, and its complementary triangle T 1 .
The circular arc triangle T 0 with its complement T 1 .
Lemma 3.6 below).This implies that the triangle T is contained in the intersection D 1 X D 2 of two closed disks D 1 and D 2 in p C each bounded by a circle in p C containing one of the sides of T and such that these boundaries touch at the vertex of T with angle 0. Then the closure T 1 of the set pD 1 X D 2 qzT is a circular arc triangle that we call complementary to T , since D 1 X D 2 " T Y T 1 .Note that T 1 has the same vertices as T , is bounded by arcs of the same three circles as T , and has angles 0, π{2, 2π{3 (see Figure 1).This relation of T and its complementary triangle T 1 is invariant under Möbius transformations and complex conjugation.
Note that T 1 is the complementary triangle to T 0 in this sense (see Figure 2).In particular, the complex conjugate τ Þ Ñ ppτ q :" ppτ q of p maps T 0 anticonformally onto its complementary triangle T 1 0 " T 1 " t8u Y tτ P C : 0 ď Repτ q ď 1{2uztτ P C : Impτ q ą 0, |τ | ą 1u such that the common vertices i, ρ, 8 are fixed under p.Now it is well-known that by successive reflections in the sides of T 0 we generate a tessellation T of the set H ˚:" H Y Q Y t8u by circular arc triangles with angles 0, π{2, π{3 (see [Sch74, Section 2.2]) and Figure 3 for an illustration).
Let G be the group of conformal or anti-conformal homeomorphisms of p C generated by the reflections in the sides of T 0 .By applying the Schwarz Reflection Principle repeatedly and the invariance of the relation between a triangle and its complementary triangle under this procedure, we immediately see that p ˝S " S ˝p for all S P G.This gives the following consequence.
Corollary 1.2.Let T be any triangle in the tessellation T of H ˚obtained by successive reflections in the sides of T 0 , and let T 1 be the complementary triangle of T as defined above.Then p is an anti-conformal map of T onto T 1 that fixes the vertices of T .
, where a, b, c, d P Z with ad ´bc " 1 form the modular group PSL 2 pZq.This is a subgroup of index 2 of the group G defined above.
It follows from our previous considerations that (1.14) p ˝S " S ˝p for all S P PSL 2 pZq.Meromorphic functions f with this type of transformation behavior (f ˝S " r S ˝f , where S runs through a Fuchsian group Γ and r S is a Möbius transformation associated with S) are called polymorphic in the classical literature.We will refer to property (1.14) of p as its PSL 2 pZqequivariance.
We will soon see that this equivariant behavior of p can easily be derived analytically (see Proposition 2.2).Theorem 1.1 and Corollary 1.2 essentially explain this behavior from a geometric perspective.
The circular arc triangle has all its angles equal to 0. It consists of a union of six triangles from the tessellation T (see Figure 4).It is easy to see from Corollary 1.2 that the continuous extension of p to V 0 attains every value in p C once, twice, or thrice.This is illustrated in Figure 4, where the level of darkness indicates how often the regions are covered (the darker the gray, the more often the region is covered; note that some relevant circles are drawn in black which does not correspond to how often the points in these circles are attained).In particular, ppV 0 X Hq " p C " ppV 0 X Hq.Now if we successively reflect in the sides of V 0 , we obtain another tessellation of T 1 of H ˚by circular triangles whose angles are all equal to 0. Each of the triangles V in this tessellation is a union of six triangles in T .Then Corollary 1.2 and the previous discussion imply that ppV X Hq " p C. We obtain the following consequence.
Corollary 1.3.The map p attains every value in p C infinitely often on H.
This was the main result in the papers [He66,He67] by Heins.He proved it by a purely function-theoretic argument without an explicit description of the map p.
We will give two proofs of Theorem 1.1.The first one is along classical lines.We suitably normalize the quasi-periods η 1 and η 2 and show that these normalized periods form a fundamental system of solutions of a hypergeometric differential equation with a certain elliptic modular function, the absolute invariant J, as the variable (see (3.21)).This will show that p " ppJq is a Schwarz triangle map sending the closed lower J-halfplane considered as a circular arc triangle with vertices 0, 1, 8 to a circular arc triangle with angles 2π{3, π{2, 0. On the other hand, it is known that J is a conformal map of the circular arc triangle T 0 onto the lower J-halfplane.This will lead to the explicit description of the map τ Þ Ñ ppτ q.
Our second approach to Theorem 1.1 is more direct: we will show that p is a homeomorphism from BT 0 onto BT 1 .The statement then essentially follows from the Argument Principle (see Proposition 4.1).
We should emphasize that none of our results are really new.The hypergeometric ODE (3.21) for the suitably normalized quasi-periods η 1 and η 2 goes back to at least as far as Fricke-Klein.The quasi-periods of ζ are really the periods of the elliptic integral of the second kind in Weierstrass normalization (see [Fri,p. 152 ff. and p. 198 ff.] for a thorough discussion).The ODE valid for them (in suitable normalization) is an example of what is now called a Picard-Fuchs equation.They are satisfied by periods of other elliptic and more general integrals (see [Za18] for a gentle introduction).Fricke-Klein were also well aware of the connection to Schwarz triangle maps, but they seem not to have stated Theorem 1.1 explicitly even though it easily follows from their considerations.
In the more recent literature, the function p (in the form (2.10) below) was considered by various authors (see, for example, [SS12, IJT14, WY14]).In [SS12] the connection to hypergeometric ODEs was realized and an explicit formula for the Schwarzian derivative tp, τ u was obtained (see Section 5 for more discussion), but the explicit geometric description of p was not pointed out.In [WY14] some mapping properties of p were studied without the full realization of the statement of Theorem 1.1.Actually, the authors of [WY14] explicitly ask whether p is a conformal map on the fundamental region of the modular group given by the union of T 0 and its reflection in the imaginary axis.This immediately follows from Corollary 1.2.In [IJT14] the authors do prove a biholomorphism property of p that is equivalent to Theorem 1.1.Their method is somewhat ad hoc, based on similar considerations as our second argument for the proof of Theorem 1.1.
This paper is of an expository nature giving an introduction to this classical subject.An interesting topic for further investigation is whether mapping properties similar to Theorem 1.1 can also be obtained for ratios of periods of other naturally appearing elliptic integrals.It is to be hoped that this paper is a starting point for further studies along these lines.

First facts about the quasi-periods
We first compute some special values of p and then derive a representation of p in terms of a Fourier series.All of this is well known from the classical literature (see [Fri], for example).A modern account is given in [Ch85].
Now let ω 1 and ω 2 again be arbitrary, Γ be the associated rank-2 lattice as in (1.3), and ζpuq " ζpu; Γq.If we integrate ζ over the contour given by the parallelogram Q with vertices ˘ω1 {2 ˘ω2 {2, in positive orientation, then the residue theorem gives us the Legendre relation See [Ch85, pp.50-51] for more details.
We will now prove a much deeper fact that connects the quasi-periods with the theory of modular forms which we will review in Section 3. We first set up some notation.As before, τ P C with Impτ q ą 0 will denote a variable in the open upper halfplane H. Throughout this paper, we set (2.5) q :" e 2πiτ .
Note that |q| ă 1 for τ P H.We define for k, n P N.So σ k pnq is the sum of all kth-powers m k of natural numbers m that divide n.Finally, we define (2.7) It is immediate that this series (as a function of τ P Hq converges locally uniformly on H. Hence E 2 is a holomorphic function on H.
Formulas (2.8) and (2.9) are well-known; see, for example, [Fri,p. 311].Since they are the basis for our further considerations, we will give a proof for the sake of completeness.We will follow an argument outlined in [Fri,.
Our starting point is formula (1.4).For ease of notation we use the abbreviation pk, nq " kω 1 `nω 2 and set Σ k puq :" Since the series (1.4) converges absolutely and uniformly in u P CzΓ, we can rearrange the order of summation; so ζpuq " The idea now is to express η 2 " ζpu `ω2 q ´ζpuq in terms of the differences of the functions Σ k .First note that Σ 0 pu `ω2 q ´Σ0 puq " For the other differences, we use the standard fact that (2.12) π 2 sin 2 pπuq " for u P CzZ.By considering partial sums, we see that for k P Zzt0u, we have Σ k pu `ω2 q ´Σk puq " Putting this all together, we see that as desired.□ We now give an analytic argument for the equivariant nature of p under the modular group and a closely related transformation behavior of the function Proof.We define ω 1 1 " aω 1 `bω 2 and ω 1 2 " cω 1 `dω 2 .Then ω 1 1 , ω 1 2 generate the same lattice Γ as ω 1 , ω 2 , and so give rise to the same ζ-function.The first quasi-period η 1 1 associated with the pair ω 1 1 , ω 1 2 can be computed from the following identity in u: η 1 1 " ζpu `ω1 1 q ´ζpuq " ζpu `aω 1 `bω 2 q ´ζpuq " aη 1 `bη 2 .Similarly, for the second quasi-period η 1 2 associated with ω 1 1 , ω 1 2 , we have η 1 2 " cη 1 `dη 2 .Equation (2.13) then immediately follows by passing to quotients.
Note that (2.13) implies that η 1 " 0 for a pair pω 1 , ω 2 q if and only if η 2 " 0 for the pair p´ω 2 , ω 1 q.On the other hand, by (2.9) we have η 2 " 0 if and only if E 2 pτ q " 0. So studying values pω 1 , ω 2 q where η 1 or η 2 vanishes amounts to the same as finding all zeros of E 2 .As Heins alluded to in [He67], it is easy to see that E 2 has a zero on the positive imaginary axis.Watson [Wa52] had actually computed this zero with great accuracy, The location of all the zeros of E 2 in H was studied in depth in [WY14,IJT14].

Modular forms
In order to proceed, we need some facts about modular forms.This is all standard material going back to Fricke-Klein [FK] and beyond.A modern account can be found in [Sch74].The survey by Zagier [Za08] gives a fresh perspective with connections to many other areas.All we need are some basic fundamentals of the theory.Many of them are discussed in [Ch85] and all of them in [Sch74, Chapters 1-3], for example.We start with a definition.Definition 3.1.A holomorphic function f on H is called a (holomorphic and inhomogeneous) modular form of weight k P N if it satisfies for all τ P H and a, b, c, d P Z with ad ´bc " 1.Moreover, we require that (3.2) f pτ q " Op1q as Impτ q Ñ `8.
Note that a modular form is 1-periodic: f pτ `1q " f pτ q.So it will have a Fourier series expansion that converges for all q P C with |q| ă 1. Condition (3.1) ensures that no terms with negative powers of q occur in this series and so If f is a function on H, k P N, and S P PSL 2 pZq is as in (1.13), then we can define a operation f | k S by setting pf | k Sqpτ q " f pSpτ qqpcτ `dq ´k.Condition (3.1) then simply becomes pf | k Sqpτ q " f pτ q.
One can easily check that if S, T P PSL 2 pZq, then This implies that in order to verify (3.1), it it enough to do this for generators of PSL 2 pZq, for example for τ Þ Ñ τ `1 and τ Þ Ñ ´1{τ .We see that condition (3.1) is equivalent to the identities f pτ `1q " f pτ q and f p´1{τ q " τ k f pτ q.
We say that f is a modular function if it is meromorphic on H and satisfies f ˝S " f for all S P PSL 2 pZq.Moreover, we require that that for some N P N 0 we have (3.5)f pτ q " Opq ´N q as Impτ q Ñ `8.
A modular function f is 1-periodic and has a Fourier expansion as in (3.3) converging if |q| ą 0 is small enough.Condition (3.5) ensures that in this Fourier expansion only finitely many non-zero terms with negative n occur.Note that if g and h ‰ 0 are modular forms of the same weight, then f " g{h is a modular function.If g and h are modular forms of weight k and l, respectively, then gh is a modular form of weight k `l.
Let f be a modular form of weight k.Then we can pass to an associated homogeneous modular form r f depending on two variables ω 1 and ω 2 (satisfying our standing assumptions) by setting The incorporation of the factor 2π here has some advantages.Then r f is a homogeneous function of degree ´k in the sense that r f ptω 1 , tω 2 q " t ´k r f pω 1 , ω 2 q for t P C ˚.Moreover, (3.1) is equivalent to So the transformation behavior of a homogeneous modular form becomes more transparent at the cost of having to deal with a function of two variables.
The (inhomogeneous and holomorphic) modular forms of a given weight k P N obviously form a vector space M k over C. It immediately follows from the definitions that there are no non-trivial modular forms of odd weight (indeed, change the signs of a, b, c, d in (3.1)).If k is even, then the following well-known fact gives the dimension of M k for even k (see [Sch74, Theorem 18, p. 47]).The fact that M k is finite-dimensional is probably the single most important fact in the theory of modular forms.
In particular, there are no non-trivial modular forms of weight 2.Moreover, the spaces of modular forms of weight 4, 6, and 8 are all one-dimensional and hence spanned by any non-trivial form in these spaces.The latter fact is all we need in the following.
To obtain non-trivial modular forms in M 4 and M 6 , we consider the Eisenstein series G k defined for even k ě 4 as Here the prime on the sum means that the term with m " n " 0 should be omitted from the summation.
The series representing the function G k , k ě 4, converges absolutely and locally uniformly for τ P H; so G k is a holomorphic function of τ P H. Actually, G k is a modular form of weight k.Indeed, it is immediate to see that G k has the right transformation behavior as in (3.1).Moreover, one can explicitly obtain a Fourier expansion of G k as in (3.4).This can be seen by a computation similar to the proof of Proposition 2.1 where one uses an identity obtained from (2.12) by differentiating pk ´2q-times.This is standard and the details can be found in [Sch74, Chapter 3], for example.We only need the result for k " 4 and k " 6. Namely, we have where In particular, G 4 , or equivalently E 4 , is a modular form spanning M 4 and G 6 or E 6 are modular forms each spanning M 6 .The space M 8 is spanned by While E 2 (as defined in (2.7)) is not a modular form, it has a transformation behavior that is closely related.Indeed, each function E k , k " 2, 4, 6 is 1periodic, that is, for τ P H it satisfies E k pτ `1q " E k pτ q; moreover, we have E 4 p´1{τ q " τ 4 E 4 pτ q, (3.7) E 6 p´1{τ q " τ 6 E 6 pτ q. (3.8) The first equation follows from (2.14), while the last two equations follow from the fact E 4 and E 6 are modular forms of weight 4 and weight 6, respectively.Due to its simple transformation behavior, E 2 is called a quasi-modular form of weight 2.
It follows from the valence formula for modular forms (see [Sch74, Theorem 13, p. 41]) that ρ is the only zero of E 4 on H ˚" H Y Q Y t8u up to equivalence under PSL 2 pZq.More precisely, we have (3.10)E 4 pτ q " 0 for τ P H ˚if and only if τ " Spρq for some S P PSL 2 pZq.
Again i is the only zero of E 6 up to equivalence under PSL 2 pZq, but we will not need this fact.We will derive expressions for the derivates E 1 k " dE k {dτ .These formulas are often attributed to Ramanujan, but they can be traced back (in different notation) to Fricke-Klein and beyond.To absorb a factor 2πi that appears in these formulas, we introduce the abbreviation DF " 1 2πi dF pτ q dτ for ease of notation.
The other identities follow from similar considerations: one shows that DE 4 ´1 3 E 2 E 4 is a modular form of weight ´6 and DE 6 ´1 2 E 2 E 6 is a modular form of weight ´8.Moreover, the spaces of these forms are spanned by E 6 and E 2 4 , respectively.Again one determines the proportionality constants by considering q-expansions.□ To put the preceding argument into a more general perspective: if f is a modular form of weight k " 4, 6, . . ., then one can show by computations similar to the ones in the proof of Proposition 3.3 that its Serre derivative (see [Za08,p. 48]) We need two more auxiliary functions.The first one is This is a modular form of weight 12 (note that the notation in [Sch74,Ch85] is different from ours).From Proposition 3.3 it follows that Based on this and the expansion of E 2 as a q-series, it is easy to derive the well-known formula (3.15) ∆ " q 8 ź n"1 p1 ´qn q 24 Indeed, both sides in this identity are holomorphic functions on H with the same logarithmic derivatives, namely 2πiE 2 ; so they must represent the same function up to a multiplicative constant.This constant is equal to 1 as a comparison of the leading terms of the q-expansions shows.It immediately follows from (3.15) that ∆ has no zeros on H and that it takes positive real values for τ on the positive imaginary axis.In the following, we will consider various roots ∆ 1{k of ∆ for k P N. Since ∆ has no zeros on H, these are holomorphic functions on H.A priori, ∆ 1{k is only defined up to multiplication by a k-th root of unity.We fix this ambiguity so that ∆ 1{k attains positive real values on the positive imaginary axis.
At least the correspondence of vertices is easy to see: the definition of J immediately gives Jpτ q " 1 1728 q ´1 `Op1q as Impτ q Ñ `8.Hence Jp8q " 8 (understood in a limiting sense).Moreover, the definition of J in combination with (3.9) and (3.11) gives Jpρq " 0 and Jpiq " 1.By (3.16) we obtain a uniquely determined holomorphic third root of J by setting Similarly, by (3.16) we have J ´1 " 1 1728 E 2 6 {∆, which allows us to define the holomorphic function pJ ´1q 1{2 :" 1
Note that these definitions fix the ambiguity of J 1{3 and pJ ´1q 1{2 in such a way that these functions take positive real values with τ on the positive imaginary axis with Impτ q large.The previous formulas can be rewritten as follows We also have We want to introduce J as a new variable instead of τ for some functions of τ .The map τ P H Þ Ñ Jpτ q is locally injective at all points not equivalent to i or ρ under the modular group.The latter points are exactly those where J takes on the values 1 or 0.Moreover, Jpτ q attains all values in p C except 8.This means that τ " τ pJq is locally well-defined away from the points 0, 1, 8 in the J-plane.Once we fix such a local branch τ " τ pJq, we can analytically continue it along any path in p Czt0, 1, 8u.In this way, we obtain a multivalued function on p Czt0, 1, 8u, whose branches differ by postcomposition with elements in the modular group PSL 2 pZq.
We will now fix such a local branch τ " τ pJq and introduce the following functions, considered as depending on the variable J: Note that by (2.10) we have (3.20) H 1 pτ q{H 2 pτ q " τ ´6i πE 2 pτ q " ppτ q.
The meaning of the expressions (3.18) and (3.19) becomes clearer if one transitions to homogeneous functions of the variables ω 1 and ω 2 .See Section 5 for more details.
Lemma 3.4.For k " 1, 2 we have the following identities: For the second identity note that Similarly, The functions H 1 and H 2 considered as functions of J form a fundamental system of solutions of the hypergeometric differential equation As mentioned in the introduction, this is not new.It follows from the methods of Fricke-Klein and was explicitly stated in [Fri,formula (9), p. 326].To derive (3.21), Fricke used homogeneous modular forms and an associated differentiation process in contrast to our approach based on inhomogeneous modular forms (see Section 5 for more discussion).
Before we proceed, we record a simple geometric fact.
Here we say that two circular arc triangles X and Y are Möbius equivalent if there exists a Möbius transformation S (i.e., a biholomorphism on p C) that gives a conformal map between X and Y (as defined in the introduction).Recall that T 0 and T 1 are defined in (1.11) and (1.12), respectively.
Proof.The first part is true in even greater generality and its proof is fairly standard (see, for example, [Ca54, Section 394 and Figure 83]); so we will only give an outline of the argument.
Suppose T Ď p C has angles 0, π{2, π{3 at its vertices a, b, c P BT , respectively.By applying an auxiliary Möbius transformation to T , we may assume that a " 8. Then the side of T containing b and c must be an arc on a circle C and cannot be a segment on a line (in the latter case the angles at b and c would have to add up to π which they do not).By applying another Möbius transformation (namely, a Euclidean similarity), we may also assume that C is the unit circle and that the two sides of T containing a " 8, b and a, c, respectively, are rays parallel to the positive imaginary axis.It is then easy to see that the only possibilities are b " i, c " ρ, in which case T " T 0 , or b " i, c " ´ρ in which case T is equal to the reflection image of T 0 in the imaginary axis, and hence Möbius equivalent to T 0 .
If T Ď p C has angles 0, π{2, 2π{3 at its vertices a, b, c P BT , respectively, then the quickest way to verify the statement is to reduce it to the first part.For this we consider the unique circular arc triangle r T with the same vertices a, b, c and angles 0, π{2, π{3, respectively, so that T is complementary to r T in the sense discussed in the introduction.Then r T Y T forms a lune as in Figure 1.By what we have seen, we can find a Möbius transformation S that gives a conformal map of r T onto T 0 or T 0 .Then S is a conformal map of T onto T 1 or T 1 and the statement follows. □ The different cases in the previous lemma can easily be distinguished by taking boundary orientation into account.Let us assume the circular arc triangle T has angles 0, π{2, π{3 at its vertices a, b, c P BT , respectively.Then we orient BT so that we traverse a, b, c in this cyclic order if we run through BT according to the orientation.Now suppose T lies on the left of BT with this orientation.Since Möbius transformations are orientation-preserving, it then follows that T cannot be Möbius equivalent to T 0 and so it must be Möbius equivalent to T 0 .If T lies on the right of BT , then T is Möbius equivalent to T 0 .A similar analysis applies to circular arc triangles that are Möbius equivalent to T 1 and T 1 .
The proof of our main result is now easy.
Proof of Theorem 1.1.We know that τ Þ Ñ Jpτ q sends T 0 conformally to the circular arc triangle given by the closed lower halfplane.Here the vertex correspondence under the map is On the other hand, we know that p " H 1 {H 2 , now considered as a function of J, sends the lower J-halfplane conformally onto some circular arc triangle T with vertices a, b, c P p C where the angles are 0, π{2, 2π{3, respectively.Here the vertex correspondence under If we compose τ Þ Ñ J with J Þ Ñ H 1 {H 2 , then we obtain the map τ Þ Ñ H 1 pτ q{H 2 pτ q " ppτ q.In particular, p is a conformal map between the circular arc triangles T 0 and T such that pp8q " a, ppiq " b, ppρq " c.On the other hand, we know the values that p obtains at these locations (see (2.1), (2.3), and (2.11)).It follows that a " pp8q " 8, b " ppiq " ´i, c " ppρq " ρ.
If we orient BT 0 so that T 0 lies on the left of BT 0 , then the vertices of T 0 are in cyclic order 8, i, ρ.Since the conformal map p of T 0 onto T preserves orientation, T lies on the left of BT if BT carries the induced orientation under p.This corresponds to the cyclic order pp8q " 8, ppiq " ´i, ppρq " ρ of the vertices of T .
Lemma 3.6 and the subsequent discussion after this lemma now imply that T is Möbius equivalent to T 1 (and not to T 1 ).Since T and T 1 have the same vertices, it follows that T " T 1 and the statement follows.□

An alternative approach
Our proof of Theorem 1.1 is in a sense the "classical" proof.It may be interesting to point out a different and simple direct argument that avoids the theory of hypergeometric ODEs.We require some preparation.
Proposition 4.1.Suppose U Ď p C is a closed Jordan region and f : intepU q Ñ C is a holomorphic function that has a continuous extension (as a map into p C) to the boundary BU Ď p C. Suppose this extension maps BU homeomorphically to the boundary BV Ď p C of a closed Jordan region V Ď p C with intepV q Ď C. If 8 P BV we assume in addition that if the Jordan curve BU is oriented so that U lies on the left of BU and if BV carries the orientation induced by f , then V lies on the left of BV .Under these assumptions, f is a homeomorphism of U onto V that is a biholomorphism between intepU q and intepV q.
In our application of this statement, 8 will be on the boundary of U and V .This is the reason why the formulation is somewhat technical and we cannot simply assume U, V Ď C.
Proof.The Argument Principle implies that on intepU q the function f attains each value in intepV q once and no other values.The statement easily follows.As this reasoning is fairly standard, we skip the details (see [Bu79, for some background and [Bu79, Exercise 9.17 (i)] for a closely related statement).□ We need the following estimate for E 2 (see [IJT14, Lemma 2.3]).

The statement follows. □
Lemma 4.3.The function p is holomorphic near each point τ P T 0 X H.Moreover, p 1 pτ q " 0 for τ P T 0 X H if and only if τ " ρ.
So the only critical point of p on T 0 X H is at τ " ρ.
Proof.The first part follows from (2.10), because E 2 does not take the value 0 on T 0 X H by Lemma 4.2.By (2.10) and (3.12) we also have It follows directly from the definition (see (2.7)) that E 2 pτ q P R when τ P H with Repτ q " 0 or Repτ q " 1{2.Here E 2 pτ q ą 0 if in addition Impτ q is large enough.
As a consequence, Repppτ qq " 0 if Repτ q " 0 (see (2.10)).In particular, p sends Azt8u into the imaginary axis.Moreover, since p 1 pτ q ‰ 0 on A, we move strictly monotonically from ppiq " ´i to pp8q " 8 as τ P A travels from i to 8.Here Impppτ qq must be strictly increasing (and not decreasing) with increasing Impτ q for τ P A as Impppτ qq is positive for τ P A with large Impτ q.This implies that p sends A homeomorphically onto A 1 .
Similarly, Repppτ qq " 1{2 if Repτ q " 1{2.Hence p sends Czt8u into the line L :" tτ P C : Repτ q " 1{2u.Since p 1 pτ q ‰ 0 for interior points of C, we move strictly monotonically from ppρq " ρ to pp8q " 8 on the line L as τ P C moves from ρ to 8 on C. Again Impppτ qq ą 0 for τ P C with large Impτ q, and so we must have a strict increase here.This implies that p sends C homeomorphically onto C 1 .
Since p sends A into A 1 , by the Schwarz Reflection Principle we must have pp´τ q " ´ppτ q for all τ P H.If τ P B, then τ lies on the unit circle and so τ " 1{τ .Then the transformation behavior of p implies ppτ q " ´pp´τ q " ´pp´1{τ q " 1{ppτ q, and so |ppτ q| " 1.We see that p sends B into the unit circle.Since p 1 does not vanish at any interior point of B, we move strictly monotonically along the unit circle from ppiq " ´i to ppρq " ρ as τ moves from i to ρ along B. This implies that p sends B homeomorphically onto B 1 or the complementary arc B 2 of B 1 on the unit circle with the same endpoints as B 1 .
Here we actually have must have ppBq " B 1 as follows by considering orientations.Indeed, p 1 piq ‰ 0, and so p is a conformal and hence orientationpreserving map near i.We know that p sends A oriented in a positive direction from i to 8 into the arc A 1 oriented from ´i to 8. Since B lies on the right of the oriented arc A locally near i, the image ppBq must lie on the right of A 1 near ppiq " ´i.This only leaves the possibility that p send B homeomorphically onto B 1 and not onto B 2 .
We conclude that p sends the three sides A, B, C of T 0 homeomorphically onto the three sides A 1 , B 1 , C 1 of T 1 .It follows that p is a homeomorphism of BT 0 onto BT 1 .Since p is holomorphic on the interior of T 0 , Proposition 4.1 now implies that p is a conformal map of the circular arc triangle T 0 onto the circular arc triangle T 1 .Here the vertex correspondence is as in the statement and the claim follows. □ We want to point out a consequence of our considerations.
Corollary 4.4.The critical points of p on H are precisely the points that are equivalent to ρ under the action of PSL 2 pZq.
Proof.It follows from Corollary 1.2 and the Schwarz Reflection Principle that if T is a triangle from the tessellation T (as defined in the introduction), then p is a conformal map near each point τ P T X H distinct from the vertex τ T of V where the angle is π{3.There the angle is doubled by p and so τ T is a critical point (of first order).Therefore, p has precisely the critical points τ V , V P T .These are precisely the points that are equivalent to ρ under the action of PSL 2 pZq.□ As the proof shows, all critical points of p are of first order, meaning that the local degree of the map is 2 at these points.Corollary 4.4 can also be derived from (4.3) and (3.10).

Remarks
1. Our route to derive Proposition 3.5 is essentially due to Fricke-Klein.In contrast to our approach, Fricke-Klein usually prefer to work with homogeneous modular forms.If r f pω 1 , ω 2 q is such a form, then they use an associated form If r f is a homogeneous modular form of degree ´k, then it is straightforward to see that r g " D η r f is a homogeneous modular form of degree ´k ´2.The differentiation process D η occurs implicitly in Fricke-Klein [FK] and was systematically used by Fricke in [Fri] (see also [FS]).
Let us consider the associated inhomogeneous forms f and g so that So passing to the inhomogeneous version g of r g, we have gpτ q " 1 p2πq k`2 r gpτ, 1q " 1 2πi f 1 pτ q ´k 12 E 2 pτ qf pτ q, and so This shows that the Serre derivative ϑ k f of a modular form f of weight k corresponds to the homogenous form D η r f .So Serre derivative and the differentiation process D η are essentially the same, but only differ whether one considers inhomogeneous or homogeneous modular forms.
So Ω 1 and Ω 2 are essentially just the periods ω 1 and ω 2 renormalized to make them homogeneous of degree 0. Similarly, using (2.8) and (2.9), one can see that Again here H 1 and H 2 are homogeneous of degree 0 as functions of the pair pω 1 , ω 2 q and hence functions of τ " ω 1 {ω 2 alone; so the transition from pη 1 , η 2 q to pH 1 , H 2 q is also a normalization procedure.
Indeed, this is in agreement with the fact, pointed out earlier, that τ P T 0 Þ Ñ Jpτ q is a conformal map of T 0 onto the closed lower halfplane.4. The Schwarzian derivative of a meromorphic function f pzq depending on a complex variable z is defined as tf, zu :" d 2 dz 2 log f 1 pzq ´1 2 ˆd dz log f 1 pzq ˙2 " 2f 1 pzqf 3 pzq ´3f 2 pzq 2 2f 1 pzq 2 .
The Schwarzian derivative of a function f is invariant under post-composition with a Möbius transformations S, namely, tS ˝f, zu " tf, zu.
Moreover, we have the following chain rule for the Schwarzian derivative tf ˝g, zu " tf, gu ˆdg dz ˙2 `tg, zu.
If S P PSL 2 pZq is arbitrary, then by PSL 2 pZq-equivariance of our function p we have S ˝p " p ˝S.It follows that f pτ q :" tp, τ u " tS ˝p, τ u " tp ˝S, τ u " f pSpτ qqS 1 pτ q 2 .This implies that f has the same transformation behavior as a modular form of weight 4.
This can be seen more explicitly as follows: tp, τ u " tp, JuJ 1 pτ q 2 `tJ, τ u " ptp, Ju ´tτ, JuqJ 1 pτ q 2 " ´J12 6 ˆ1 J 2 `1 Jp1 ´Jq ˙" J 12 6J 2 pJ ´1q " ´8π 2 ∆ 1{3 {J 2{3 " ´1152π 2 ∆{E 2 4 .This last expression for tp, τ u was also recorded in [SS12, Proposition 6.2].Since ∆ has weight 12 and E 4 has weight 4, it clearly transforms as a modular form of weight 4. Note that tp, τ u is not a modular form according to our definition as this function has poles, namely exactly at the points equivalent to ρ under PSL 2 pZq.These points are precisely the critical points of p as we know from Corollary 4.4.

Figure 3 .
Figure 3.Some circular arc triangles in the tessellation T .

Figure 4 .
Figure 4. Sets covered by the map p restricted to V 0 .
It follows that H 1 and H 2 are solutions of the above differential equation.Since the ratio p " H 1 {H 2 of these functions is non-constant by (2.10), they are linearly independent and form a fundamental system of solutions.□ by (3.10) all zeros of E 4 are given by Spρq, S P PSL 2 pZq.No point τ P T 0 ztρu is equivalent to ρ under the action of PSL 2 pZq on H ˚. It follows that τ " ρ is the only zero of E 4 and hence of p 1 on T 0 .□