# Uniformization of punctured spheres and modular forms

Gabriele Bogo

###### Abstract

We investigate some aspects of the classical accessory parameters problem from
the point of view of modular forms. We show that if $X\simeq\mathbb{H}/\Gamma$
is an $n$-punctured sphere, the whole ring of modular forms $M_{*}(\Gamma)$
can be constructed from a Frobenius basis of solutions of the uniformizing
differential equation of $X.$ The argument is based on the more general
construction of certain rings of modular forms of rational weight from the
uniformizing differential equation of $X$. In the second part we present an
algorithm for the computation of the uniformizing value of the accessory
parameters for certain punctured spheres. It is based on the modularity of the
holomorphic solution of the uniformizing differential equation. The algorithm
works uniformly in the case of four-punctured spheres, and it can be used to
compute the uniformizing accessory parameters for spheres with $n\geq 4$
punctures and enough automorphisms.

## Introduction

The uniformization theorem states that a simply connected Riemann surface is
biholomorphic to (only) one of the following three: the Riemann sphere
$\widehat{\mathbb{C}}$, the complex plane $\mathbb{C}$, the upper half-plane
$\mathbb{H}$. In the classical works of Klein and Poincaré (1883/1884), this
statement was related to the study of second order linear differential
equations. Let $X$ be a Riemann surface whose universal cover $\widetilde{X}$
is not isomorphic to the Riemann sphere nor to $\mathbb{C}.$ Every local
biholomorphism $\widetilde{X}\to\mathbb{C}$ arises from the analytic
continuation of the ratio of independent solutions of certain second order
linear ODEs (projective connections) defined on $X.$ In particular, if a
global biholmorphism between $\widetilde{X}$ and a subdomain of $\mathbb{C}$
exists, it must come from a differential equation on $X$. Klein, Poincaré, and
others tried to show the existence of a global biholomorphism
$\widetilde{X}\overset{\sim}{\rightarrow}\mathbb{H}$ by determining a
“special” differential equation on $X,$ called _uniformizing differential
equation_ , but eventually could not. This approach was later abandoned, and
the uniformization theorem was proved by Koebe and Poincaré in 1907 using
different methods.

When $X$ is of genus zero, i.e., a $n$-punctured sphere with $n\geq 3$, the
uniformizing differential equation is a Fuchsian differential equation on
$\mathbb{C}$ with rational coefficients. It can be determined up to $n-3$
complex parameters called _accessory parameters_. The _accessory parameters
problem_ consisted in finding the unique value of these parameters, called the
_Fuchsian value_ , such that the associated differential equation induces the
global biholomorphism $\widetilde{X}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ \mathbb{H}.$ The projective monodromy group of the
uniformizing equation, being the Deck group of the covering $\mathbb{H}\to X,$
is a discrete subgroup $\bar{\Gamma}$ of
$\mathrm{Aut}(\mathbb{H})\simeq\mathbb{P}\mathrm{SL}_{2}(\mathbb{R}).$

The accessory parameters problem turned out to be very difficult; a direct
proof of the existence of the Fuchsian value, guaranteed by the uniformization
theorem, and its determination are still, after 140 years, open problems. For
an introduction to different aspects of the accessory parameter problem and
later developements, see the papers by by Chudnovsky and Chudnovsky [4] Hejhal
[7],[8], Hempel [9], Kra [15], Smirnov [21], Takhtajan and Zograf [22],[23],
and Thompson [24].

This paper is based on the following observation: a holomorphic solution of
the uniformizing differential equation is a (parametrized) modular form $f$
with respect to the monodromy group $\Gamma$. This observation is not new: it
was indeed at the heart of Poincaré’s study of uniformization; nevertheless,
it seems that nobody studied the accessory parameter problem from this point
of view.

The modular form $f$ can be constructed from a Frobenius basis of solutions at
a regular singular point of the uniformizing differential equation. A first
result is that, under some assumptions on the group $\Gamma,$ the whole ring
of quasimodular forms can be constructed from a Frobenius basis.

######  Theorem.

Let $X$ be a $n$-punctured sphere, $n\geq 3$, and let
$\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$ be the monodromy group of the
unifomizing differential equation. If $\Gamma$ has at most one irregular cusp,
the ring of quasimodular forms $\widetilde{M}_{*}(\Gamma)$ can be constructed
form a Frobenius basis of solutions of the uniformizing differential equation.

Among the groups that satisfy the hypothesis of the theorem are, for instance,
the genus zero congruence subgroups $\Gamma(N),\Gamma_{0}(N),$ and
$\Gamma_{1}(N)$.

The theorem is an immediate consequence of a more general result on _modular
forms of rational weight_. Let $J\colon\Gamma\times\mathbb{H}\to\mathbb{C}$ be
an automorphy factor of rational weight $r$ for the group $\Gamma,$ and let
$M_{J^{k}}(\Gamma)$ denote, for every $k\geq 0,$ the space of modular forms
with respect to the automorphy factor $J^{k}$ (of weight $rk$). The space
$M_{J^{*}}(\Gamma):=\bigoplus_{k\geq 0}M_{J^{k}}(\Gamma)$ is a graded ring.
The result is the following.

######  Theorem.

Let $\Gamma$ be a Fuchsian group such that $\mathbb{H}/\Gamma$ is isomorphic
to a punctured sphere, and assume that $\Gamma$ has at most one irregular
cusp. Then:

  1. 1.

There exists an automorphy factor $J$ on $\Gamma$ such that the ring
$M_{J^{*}}(\Gamma)$ is freely generated by two elements, and the ring of
ordinary modular forms $M_{*}(\Gamma)$ is contained in $M_{J^{*}}(\Gamma);$

  2. 2.

The automorphy factor $J$ and the generators of $M_{J^{*}}(\Gamma)$ can be
constructed from a Frobenius basis of the uniformizing differential equation
of any punctured sphere isomorphic to $\mathbb{H}/\Gamma.$

The last part of the paper is of algorithmic nature. The interpretation of the
classical accessory parameter problem in terms of modular forms gives a way to
compute the Fuchsian value for the uniformization of certain punctured
spheres. Since this is not formally a theorem, we call it “Result”.

######  Result.

The Fuchsian value for the uniformization of a punctured sphere $X$ with
“enough automorphisms” can be computed algorithmically from the location of
the punctures and the explicit description of the automorphisms of $X$. The
algorithm can be easily implemented and permits to compute the Fuchsian value
numerically to high precision.

If $X$ has $n$ punctures, then one needs $|\mathrm{Aut}(X)|\geq n-2$ to run
the algorithm (recall that $|\mathrm{Aut}(X)|\leq 6(n-2)$); if all the
punctures are real, then $|\mathrm{Aut}(X)|\geq n/2-1$ suffices. In
particular, since a generic four-punctured sphere has an automorphisms group
of order four, the algorithm can always be used to compute the uniformization
of spheres with four punctures.

The basic idea of the algorithm is very simple. From the Frobenius solutions
of a “potential uniformizing differential equation”, where the accessory
parmeters are considered formal parameters, one constructs a new series
expansion. This expansion is the Fourier expansion of a modular form at a cusp
only when the accessory parameters take the Fuchsian value. This gives a
concrete way to test if a given value is the Fuchsian value. The monodromy
group $\Gamma$ of the differential equation, which is needed to test the
modularity, is determined with the help of the automorphisms of the punctured
sphere $X.$

To give an example of how the algorithm works, we compute numerically the
local expansion of the function that to a four-punctured sphere associates its
Fuchsian value for the uniformization; such function is real-analytic by a
result of Kra. Some observations on the coefficients of this local expansion
are explained using a result of Takhtajan and Zograf and some special
symmetry.

Other methods, not based on modularity, for the numerical computation of the
Fuchsian value can be found in the literature. The reader may want to consult
the works of Chudnovsky and Chudnovsky [4], Hoffman [10], Keen, Rauch, and
Vasquez [16] and Komori and Sugawa [14]. A new approach, based on the theory
of Painlevé VI equation and isomonodromy deformations is presented in [1].

#### Acknowledgments

The results of this paper are based on a part of my Ph.D. thesis, written
while I was a student at SISSA (International School for Advanced Studies) in
Trieste. I am deeply indebited with my advisors Don Zagier and Fernando
Rodriguez Villegas. I want to thank the Max Planck Institut for Mathematics in
Bonn, where part of these results were discovered while I was a visiting
member of the IMPRS graduate school.

##  1 Uniformization, modular forms, and differential equations

###  1.1 Uniformization and differential equations

In their first approach to the uniformization of Riemann surfaces, Klein [13]
and Poincaré [18] related the study of the universal covering of a Riemann
surfaces $X$ to certain linear differential equations defined on $X$. In the
following we will briefly explain their basic ideas in the case of genus zero
hyperbolic surfaces. A good reference for the general theory is the book [20].

Let
$X:=\mathbb{P}^{1}\smallsetminus\\{\alpha_{1},\alpha_{2},\dots,\alpha_{n-2},\alpha_{n-1}=0,\alpha_{n}=\infty\\},$
where $\alpha_{i}\in\mathbb{C}\smallsetminus\\{0\\}$ and
$\alpha_{i}\neq\alpha_{j}$ if $i\neq j,$ be an $n$-punctured sphere. Let

$\frac{d^{2}y(t)}{dt^{2}}+p(t)\frac{dy(t)}{dt}+q(t)y(t)=0$ (1)

be a linear differential equation on $X,$ where $p,q\colon X\to\mathbb{C}$ are
holomorphic functions, and let $y_{1}$ and $y_{2}$ be linearly independent
local solutions of (1). The ratio $y_{2}/y_{1}$ can be analitically continued
to the whole $X$ and defines a non-constant complex-valued function
$\widetilde{\tfrac{y_{2}}{y_{1}}}$ on the universal covering $\widetilde{X}$
of $X.$ It is easy to verify that this function is a local biholomorphism.
Conversely, every local biholomorphism $\widetilde{X}\to\mathbb{C}$ arises in
this way. In particular, every global biholomorphism between $\widetilde{X}$
and a subdomain of $\mathbb{C}$, if any, arises from independent solutions of
certain equations (1). It is well known, [6],[20], that such global
biholomorphisms can only arise from differential equations of the following
type

$\frac{d^{2}y(t)}{dt^{2}}+\Biggl{(}\frac{1}{4}\sum_{j=1}^{n-1}{\frac{1}{(t-\alpha_{j})^{2}}}+\frac{1}{2}\sum_{j=1}^{n-1}{\frac{m_{j}}{(t-\alpha_{j})}}\Biggr{)}y(t)=0\,,$
(2)

where $m_{0},\dots,m_{n-1}$ are complex parameters, called _accessory
parameters_ , subject to the following relations111In the literature often
appears another parameter $m_{n}$ associated to the puncture at $\infty;$ it
is defined from the asymptotic expansion of the rational function in (2) as
$t\to\infty.$ It turns out that $m_{n}$ can be expressed in terms of
$m_{1},\dots,m_{n}$ and of the punctures $\alpha_{1},\dots,\alpha_{n-1}$ as
$m_{n}=\sum_{j=1}^{n-1}{\alpha_{j}(1+m_{j}\alpha_{j})}$.:

$\sum_{j=1}^{n-1}{m_{j}}=0,\quad\sum_{j=1}^{n-1}{\alpha_{j}m_{j}}=1-\frac{n}{2}.$
(3)

The name “accessory parameters” comes from the fact that the choice of
$m_{1},\dots,m_{n-1}$ does not influence the local behaviour of solutions of
(2) near the singular points (but of course influences the global behaviour of
the solutions).

Different values of the accessory parameters define different ODEs (2) and
different functions $\widetilde{X}\to\mathbb{C}$. The (unique) value of the
accessory parameters that induces a global biholomorphism
$\widetilde{X}\to\mathbb{H}$, whose existence is guaranteed by the
uniformization theorem, is called the _Fuchsian value_. The corresponding
differential equation (2) is called the _uniformizing differential equation_ ;
its projective monodromy group, which is the Deck group of the covering
$\mathbb{H}\to X,$ is then a discrete subgroup of
$\mathbb{P}\mathrm{SL}_{2}(\mathbb{R}).$ The problem, given $X,$ of
determining the Fuchsian value of the accessory parameters is known as the
_accessory parameters problem_.

###  1.2 Rankin-Cohen brackets, modular forms, and differential equations

We recall the definitions and the main results about Rankin-Cohen (RC for
short) brackets and Rankin-Cohen structures. In the following, we denote by
$\mathrm{D}:=(2\pi i)^{-1}d/d\tau,\tau\in\mathbb{H}$ the differentiation on
$\mathbb{H}.$ It is well known that if $f$ is a modular form then
$\mathrm{D}f$ is not a modular form.

Rankin-Cohen brackets, introduced by Cohen in [5], are bilinear operators
$[\;,\,]_{n},\,n\geq 0,$ that can be defined in general on spaces of
holomorphic functions with some weight; they are of particular relevance in
the theory of modular forms, since they provide combinations of derivatives of
modular forms that are again modular.

Let $f,g\colon\mathbb{H}\to\mathbb{C}$ be holomorphic functions of weight
$k,l\in\mathbb{R}$ respectively. For every integer $n\geq 0,$ the $n$-th
_Rankin-Cohen bracket_ is the bilinear form defined by

$[f,g]_{n}:=\sum_{\overset{r,s>0}{r+s=n}}{(-1)^{r}\binom{k+n-1}{s}\binom{l+n-1}{r}\mathrm{D}^{r}f\mathrm{D}^{s}g}\,.$

When $f,g$ are modular forms of (integral) weight $k,l$ respectively, the
holomorphic function $\bigl{[}f,g\bigr{]}_{n}$ is a modular form of weight
$k+l+2n.$

In the general case, Rankin-Cohen brackets verify a number of algebraic
relations; for instance

$\bigl{[}[f,g]_{1},h\bigr{]}_{1}+\bigl{[}[g,h]_{1},f\bigr{]}_{1}+\bigl{[}[h,f]_{1},g\bigr{]}_{1}=0\quad\text{
(Jacobi identity)},$

and, if $f,g,h$ have weight $k,l,m$ respectively,

$\displaystyle m[f,g]_{1}h+k[g,h]_{1}f+l[h,f]_{1}g=0\,,$ (4) $\displaystyle
k^{2}(k+1)f^{2}[g,g]_{2}=l^{2}(l+1)g^{2}[f,f]_{2}-(k+1)(l+1)[f,g]_{1}^{2}+l(l+1)g[[f,g]_{1},f]_{1}\,.$
(5)

The study of these algebraic relations lead Zagier [25] to the definition of a
new algebraic structure called Rankin-Cohen algebra. A _Rankin-Cohen algebra_
(or _RC algebra_) over a field $K$ is a graded $K$-vector space
$R=\bigoplus_{i\geq 0}R_{i},$ where $R_{i}$ has finite dimension for every
$i\geq\leavevmode\nobreak\ 0,$ equipped with a family of operators (brackets)
$[\cdot,\cdot]_{R,n}\colon R_{k}\otimes R_{l}\to\leavevmode\nobreak\
R_{k+l+2n},\,n\geq\leavevmode\nobreak\ 0,$ that satisfy all the algebraic
identities satisfied by the Rankin-Cohen brackets. The rings of modular and
quasimodular forms are natural examples of Rankin-Cohen algebras.

######  Theorem (Zagier).

Let $R$ be a $RC$ algebra with brackets $[\;,\,]_{R,n}$ and assume it contains
a non-zero divisor $F$ of weight $N>0$ such that

$[F,R]_{R,1}\subset RF,\quad[F,F]_{R,2}\in RF^{2}.$

Then every bracket $[\cdot,\cdot]_{R,n}$ can be computed in terms of:

  1. 1.

the zero bracket $[f,g]_{R,0}$ of arbitrary $f,g\in R;$

  2. 2.

the first bracket $[g,F]_{R,1}$ of an arbitrary $g\in R$ with the fixed
element $F,$ which is a derivation;

  3. 3.

the second bracket $[F,F]_{R,2}$ of $F$ with itself.

Rankin-Cohen brackets also appear in the theory of modular forms in relation
with linear differential equations. To explain this, we start by recalling the
following basic theorem (see [3]).

######  Theorem.

Let $\Gamma$ be a Fuchsian group of finite covolume. Let $f(\tau)\in
M^{!}_{k}(\Gamma)$ be a meromorphic modular form of positive weight $k$, and
let $t(\tau)\in M^{!}_{0}(\Gamma)$ be a modular function. Express $f$ locally
as a function of $t,$ $\Phi(t)=f(\tau).$ Then $\Phi(t)$ satisfies a complex
linear differential equation of order $k+1$ with algebraic coefficients (with
rational coefficients if $\Gamma$ is of genus zero and $t$ is an Hauptmodul,
i.e. $t\colon\overline{\mathbb{H}}/\Gamma\to\mathbb{P}^{1}(\mathbb{C})$ is an
isomorphism).

Much more is true: the above theorem holds if we consider, instead of a
modular form $f$, a function defined locally as a root of a modular form. For
example, and this is the most relevant case in what follows, given a
meromorphic modular form $f$ of weight $k>0,$ we can consider locally the
function $f^{1/k},$ which is formally of weight one. If we write
$\Psi(t)\leavevmode\nobreak\ =\leavevmode\nobreak\ f^{1/k}(\tau),$ then
$\Psi(t)$ satisfies a second-order linear ODE with algebraic coefficients.

In [3] three different proofs of the above theorem are given, each one giving
different insights into the statement. We collect here two important facts
from these proofs. First, if $f\in M_{k}(\Gamma)$, then a basis of solutions
of the differential equation solved by $f$ is
$\\{f(\tau),\tau{f(\tau)},\dots,\tau^{k}{f(\tau)}\\},$ and the monodromy group
of this differential equation is the $k$-th symmetric power of $\Gamma.$ The
second fact is that we can write the differential equation associated to $f$
and $t$ by computing certain Rankin-Cohen brackets of $f$ and $t.$ If $f$ is a
modular form of weight $k,$ one first considers the differential equation
solved by $f^{1/k}$ in the sense explained above. This is of the form
$L\Psi(t)=0$, where $L$ is the following differential operator

$L=L_{f,t}:=\;\frac{d^{2}}{dt^{2}}+\frac{[f,t^{\prime}]_{1}}{kft^{\prime
2}}\frac{d}{dt}-\frac{[f,f]_{2}}{k^{2}(k+1)f^{2}t^{\prime 2}}.$ (6)

The coefficients of $L$ are algebraic functions of $t$ because both
$\tfrac{[f,t^{\prime}]_{1}}{kft^{\prime 2}}$ and
$\tfrac{[f,f]_{2}}{k^{2}(k+1)f^{2}t^{\prime 2}}$ are weight zero modular
forms. They are in particular rational functions of $t$ if $\Gamma$ is of
genus zero and $t$ is an Hauptmodul. The differential equation satisfied by
$f$ can then be obtained in a standard way from (6) by considering the
differential operator $\mathrm{Sym}^{k}(L),$ whose solutions are the $k$-fold
products of solutions of $L\Psi=0.$

###  1.3 The modular nature of the uniformizing differential equations

Let $\Gamma$ be a torsion-free Fuchsian group of genus zero with $n\geq 3$
cusps, and let $t$ be an Hauptmodul; then $X_{\Gamma}:=t(\mathbb{H}/\Gamma)$
is an $n$-punctured sphere, and every punctured sphere can be obtained in this
way. The differential equation satisfied by any modular form (or a root of a
modular form) on $\Gamma$ with respect to $t$ is defined over $X,$ and has
holomorphic coefficients. Assume that this differential equation has order
two: a basis of solutions is given by $\Psi(t),\hat{\Psi}(t),$ where
$\Psi(t)=f(\tau),\,\hat{\Psi}(t)=\tau{f(\tau)}$ locally on $\mathbb{H}.$ It
follows that the ratio $\hat{\Psi}(t)/\Psi(t)$ corresponds to the function
$\tau$ on $\mathbb{H},$ so that $\hat{\Psi}(t)/\Psi(t)$ is the (multivalued)
inverse of $t(\tau).$ This means that the equation solved by $\Psi(t)$ is the
uniformizing differential equation for $X_{\Gamma}.$ We record this
observation as a corollary of the above theorem.

######  Corollary.

Let $\Gamma$ be of genus zero and torsion-free, let $t\in M_{0}^{!}(\Gamma)$
be a Hauptmodul, and let $f\in M_{k}^{!}(\Gamma)$. The differential operator
that annihilates $\Phi(t)=f(\tau)$ is the $k$-th symmetric power of the
differential operator associated to the uniformization of the Riemann surface
$X_{\Gamma}.$

##  2 The uniformizing differential equation from the RC structure

In this section, we denote by $\Gamma\subset SL_{2}(\mathbb{R})$ a genus zero
Fuchsian group with no torsion and with $n\geq 3$ inequivalent cusps. We can
assume that one of its cusps is at $\infty,$ and that this cusp has width one.
Let $t$ be an Hauptmodul, and assume it has its unique pole at a cusp $c_{0},$
and its unique zero at the cusp $\infty$ of $\Gamma.$ Finally, let
$X_{\Gamma}$ be the $n$-punctured sphere isomorphic to $\mathbb{H}/\Gamma$ via
$t$

$t\colon\mathbb{H}/\Gamma\;\overset{\sim}{\rightarrow}\;X_{\Gamma}\;=\;\mathbb{P}^{1}\\!\smallsetminus\\!\\{\alpha_{1},\alpha_{2},\dots,\alpha_{n-1}=0,\infty\\},$
(7)

where $\alpha_{i}\neq\alpha_{j}$ if $i\neq j.$

In this section we compute explicitly the differential equation satisfied by a
certain modular form $f$ with respect to $t$ using Rankin-Cohen brackets. We
then relate it to the family of differential equations (2) associated to the
uniformization of $X_{\Gamma}.$ Since these differential equations has order
two, it would be natural to consider a weight one modular form on $\Gamma,$
which gives a second order differential equation. However, not every group
$\Gamma$ admits weight one modular forms (this depends on the number of
irregular cusps); it is true instead that $\dim{M_{2}(\Gamma)}\;=\;n-1$
independently of the nature of the cusps. It makes sense then, in order to
treat uniformely all the possible cases, to consider the square root of a
modular form of weight two. To carry out this construction, we choose a weight
two form whose zeros are concentrated in a certain cusp. As the next lemma
shows, this choice is always possible.

######  Lemma 1.

Let $\Gamma$ be torsion free and of genus zero, let $t$ be an Hauptmodul, and
denote by $c_{0}$ the cusp of $\Gamma$ where $t$ has its unique pole. There
exists a modular form $f\in M_{2}(\Gamma),$ unique up to scalar
multiplication, with all its zeros in $c_{0}.$ In particular, $f$ has no zeros
in $\mathbb{H}.$

###### Proof.

Let $g\in M_{2}(\Gamma)$ and let $\sigma\in\mathrm{SL}(2,\mathbb{R})$ be such
that $\sigma{c_{0}}=\infty.$ Let

$\bigl{(}g\bigr{|}_{2}\sigma^{-1}\bigr{)}(\tau)=\sum_{m\geq 0}{g_{m}q^{m}}$

denote the Fourier expansion of $g$ at $c_{0},$ where $q=e^{2\pi
i\tau/h},\tau\in\mathbb{H},$ is a local parameter. It is known that the degree
of the divisor associated to any $g\in M_{2}(\Gamma)$ is $d=n-2.$ Let $\phi$
be the map

$\phi\colon M_{2}(\Gamma)\to\mathbb{C}^{d},\quad
g\mapsto(g_{0},g_{1},\dots,g_{d-1}).$

that sends a modular form of weight $2$ to the vector defined by its first $d$
Fourier coefficients at the cusp $c_{0}.$ This map is of course linear.

The dimension of $M_{2}(\Gamma)$ is $n-1=d+1$, so the map $\phi$ has a non-
trivial kernel of dimension $\geq 1.$ Let $f\in\mathrm{Ker}(\phi).$ Such $f$
can have at most $d$ zeros in ${\mathbb{H}}\cup\\{\text{cusps}\\},$ and they
are all in $c_{0}$ by construction. Finally, let $f,g\in\mathrm{Ker}(\phi)$ be
linearly independent. The ratio $f/g$ is a weight zero modular form
holomorphic in $\mathbb{H}$ and in all the cusps, since $f$ and $g$ have all
their zeros at the same cusp $c_{0}.$ This implies that $f/g$ is a constant,
i.e., $\dim\mathrm{Ker}(\phi)=1.$ ∎

Given $f$ and $t$ as in Lemma 1 we can construct all the even weight modular
forms on $\Gamma$.

######  Lemma 2.

Let $k\geq 0$ be an integer, and let $f$ and $t$ be as in Lemma 1. The
functions

$f^{k}t^{i},\quad i=0,\dots,k(n-2),$

form a basis of the space $M_{2k}(\Gamma).$

###### Proof.

By construction, the weight $2k$ modular form $f^{k}$ has $k(n-2)$ zeros at
the cusp $c_{0}$ where $t$ has a simple pole, and these are the only zeros of
$f^{k}.$ It follows that $f^{k}t^{i}$ is a holomorphic modular form for every
$i=0,\dots,k(n-2),$ and meromorphic for every other value of $i.$ By looking
at the location of the zeros, we can prove that the holomorphic functions in
the statement are linearly independent. From the dimension formula for
$M_{2k}(\Gamma)$ (see for example [17], Chapter 2) we conclude that they form
a basis. ∎

When the number of regular cusps of $\Gamma$ is maximal ($n$ if $n$ is even,
or $n-1$ if $n$ is odd) a similar statement holds also for odd weights. We
will discuss this in the next section.

Now we are ready to compute the second order linear differential operator
associated to a square root of the modular form $f\in M_{2}(\Gamma)$ and to
the Hauptmodul $t$ in Lemma 1.

######  Proposition 1.

Let $\Gamma$ be a genus zero torsion-free Fuchsian group with $n\geq 3$ cusps,
and let $t$ be an Hauptmodul such that
$t\colon\mathbb{H}/\Gamma\overset{\sim}{\rightarrow}X=\mathbb{P}^{1}\smallsetminus\\{\alpha_{1},\dots,\alpha_{n-2},\alpha_{n-1}=0,\alpha_{n}=\infty\\}.$
Denote by $c_{0}$ the cusp of $\Gamma$ where $t$ has its unique pole, and let
$f\in M_{2}(\Gamma)$ be such that all its zeros are at the cusp $c_{0}.$ Then
the differential operator $L$ associated to a square root of $f$ and to $t$ is
given by

$L=\frac{d}{dt}\Bigl{(}P(t)\frac{d}{dt}\Bigr{)}+\sum_{i=0}^{n-3}\rho_{i}t^{i},$
(8)

where $P(t)=\prod_{j=1}^{n-1}{(t-\alpha_{j})},\;\rho_{n-3}=(n/2-1)^{2},$ and
$\rho_{0},\dots,\rho_{n-3}\in\mathbb{C}$ are uniquely determined by $f,t.$

###### Proof.

Recall from section 1.2 that $L$ can be computed from $f$ and $t$ via

$L=\;\frac{d^{2}}{dt^{2}}+\frac{[f,t^{\prime}]_{1}}{2ft^{\prime
2}}\frac{d}{dt}-\frac{[f,f]_{2}}{12f^{2}t^{\prime 2}}.$ (9)

We have to write the coefficients of $L$ as rational functions of $t.$ First
we prove that

$(-1)^{n}\Biggr{(}\prod_{j=1}^{n-2}{\alpha_{j}}\Biggl{)}t^{\prime}=fP(t).$

The ratio $t^{\prime}/f$ is a meromorphic modular function, so it is a
rational function of $t$. From the assumption on the zeros of $f$ it follws
that the modular function $t^{\prime}/f$ has a simple zero at every cusp
different from $c_{0}$, i.e. $n-1$ simple zeros (since these are the zeros of
$t^{\prime}$). It has also a unique pole of order $n-1$ at $c_{0}$, since $f$
has $n-2$ zeros there and $t^{\prime}$ a simple pole. The rational functions
of $t$ with this zeros and poles are given by the polynomials
$\kappa^{-1}P(t),\,\kappa\in\leavevmode\nobreak\ \mathbb{C}^{*},$ where $P(t)$
is as in the statement. Looking at the coefficient of the $q$-expansion of
$t^{\prime}/f$ at $\infty,$ we find the correct factor
$\kappa=(-1)^{n}\prod_{j=1}^{n-2}{\alpha_{j}}.$

Next, we compute the brackets $[f,t^{\prime}],$ and $[f,f]_{2}.$ The first one
is very easy:

$[f,t^{\prime}]=2ft^{\prime\prime}-2f^{\prime}t^{\prime}=f(fP(t)\kappa)^{\prime}-2f^{\prime}t^{\prime}=2f^{\prime}t^{\prime}+2\kappa
f^{2}P^{\prime}(t)t^{\prime}-2f^{\prime}t^{\prime}=2\kappa
f^{2}t^{\prime}P^{\prime}(t).$

Dividing then by $2ft^{\prime 2}=2\kappa f^{2}P(t)t^{\prime}$ we finally get
the rational function $P^{\prime}(t)/P(t)$ as in the statement.

The computation of the bracket $[f,f]_{2}$ needs a little more work. From the
definition of RC brackets, we see that $[f,f,]_{2}$ is a cusp form of weight
eight. Moreover, it has a zero of order $2n-4$ where $f$ is zero, so it is
necessarly divisible by $f^{2}.$ There exists then an element $h_{4}\in
M_{4}(\Gamma)$ such that $[f,f]_{2}=f^{2}h_{4}.$ By Lemma 2 we know that
$h_{4}$ is of the form

$h_{4}=f^{2}Q(t),$

where $Q(t)$ is a polynomial in $t$ of degree $\dim{M_{4}(\Gamma)}=2n-3.$
Since $[f,f]_{2}$ is a cusp form, then $[f,f]_{2}/f^{2}$ has a zero in every
cusp different from $c_{0},$ and these zeros are simple. This means that the
polynomial $Q(t)$ is divisible by $P(t).$ We have then

$h_{4}=f^{2}P(t)\bigl{(}\hat{\rho}_{n-3}t^{n-3}+\hat{\rho}_{n-2}t^{n-2}+\dots+\hat{\rho}_{0}\bigr{)},$

for some $\hat{\rho}_{0},\dots,\hat{\rho}_{n-3}\in\mathbb{C}.$ We can
determine $\hat{\rho}_{n-3}$ by considering the expansion of $f$ at the cusp
$c_{0}.$ Let $q_{0}$ denote the local parameter at $c_{0}$. We have then in
this parameter

$f=cq_{0}^{n-2}+\cdots,\quad t=sq_{0}^{-1}+s_{0}+\cdots,\ $

for some non-zero $c,s\in\mathbb{C}$ and $s_{0}\in\mathbb{C}.$ The bracket
$[f,f]_{2}$ has the expansion

$[f,f]_{2}=6ff^{\prime\prime}-9f^{\prime
2}=3c^{2}(n-2)^{2}q_{0}^{2n-4}+\cdots,$

while the expansion of $h_{4}$ is

$h_{4}=\bigl{(}cq_{0}^{n-2}+\cdots\bigr{)}^{2}\bigl{(}\hat{\rho}_{n-3}s^{n-3}q_{0}^{3-n}+\cdots\bigr{)}\bigl{(}s^{n-1}q_{0}^{1-n}+\cdots\bigr{)}=\hat{\rho}_{n-3}c^{2}s^{2n-4}q_{0}^{0}+\cdots.$

Combining all this information we get

$\hat{\rho}_{n-3}c^{2}s^{2n-4}=(n-2)^{2}.$

From the relation $t^{\prime}=\kappa^{-1}P(t)f$ we can compute the constant
$\kappa$ in terms of the coefficients appearing in the expansions at $c_{0}$,
obtaining $\kappa=-cs^{n-2}.$ This implies that

$\hat{\rho}_{n-3}\;=\;3\kappa^{-2}(n-2)^{2}.$

It finally follows that the ratio $[f,f]_{2}/(12f^{2}t^{\prime 2})$ is equal
to

$\displaystyle\frac{[f,f]_{2}}{12f^{2}t^{\prime 2}}$
$\displaystyle\;=\;\frac{f^{4}P(t)(\kappa^{-2}(n-2)^{2}t^{n-3}+\cdots+\hat{\rho}_{0})}{12\kappa^{-2}f^{4}P(t)^{2}}$
(10)
$\displaystyle\;=\;\frac{(n/2-1)^{2}t^{n-3}+\rho_{n-4}t^{n-4}+\cdots+\rho_{0}}{P(t)},$
(11)

where $\rho_{i}=\hat{\rho_{i}}\kappa^{2}/12.$ ∎

######  Definition 1.

The elements $\rho_{1},\dots,\rho_{n-4}$ in (8), considered as free
parameters, are called modular accessory parameters associated to the group
$\Gamma.$

The unique value $\rho=(\rho_{0},\dots,\rho_{n-3})$ of the modular accessory
parameters that make the identity (10) true in the Rankin-Cohen algebra
$M_{*}(\Gamma)$ is called the modular Fuchsian value, and it is denoted by
$\rho_{F}.$

The modular accessory parameters and the accessory parameters defined in (2)
are different objects. The modular ones are defined from a Fuchsian group, and
depend on the choice of the Hauptmodul $t$ and the modular form $f.$ The
accessory parameters in (2) are defined from a Riemann surface $X$, and depend
on the choice of a basis of quadratic differentials on $X.$ Nevertheless, they
play the same role from the point of view of differential equations, and in
fact these two sets of parameters are very much related. In general we will
refer to the elements $\rho_{0},\dots,\rho_{n-4}$ simply by accessory
parameters.

A set of algebraic relations between the elements of the two sets of accessory
parameters can be determined from the theory of differential equations. It
follows in fact from the corollary in section 1.2 that when $\rho=\rho_{F}$ in
(8) and the differential equation in (2) is the uniformizing one, then these
two equations are _projectively equivalent_ , i.e. the ratios of independent
solutions of (2) and of (8) induce the same function on the universal cover
$\widetilde{X}$ of $X.$

The differential equation (2) is in _canonical form_ (or _reduced_ form): it
simply means that the coefficient of $dY/dt$ is zero. It general, given a
family of projectively equivalent second order equations, there is a unique
one in canonical form. There exists a standard transformation which brings any
second order Fuchsian equation to its canonical form. Given such an equation

$\frac{d^{2}}{dt^{2}}Y(t)+p(t)\frac{d}{dt}Y(t)+u(t)Y(t)=0,$

where $p(t),u(t)$ are rational functions, the equation in canonical form is

$\frac{d^{2}}{dt^{2}}\widetilde{Y}(t)+\Bigl{(}q-\frac{p^{\prime}}{2}-\frac{p^{2}}{4}\Bigr{)}\widetilde{Y}(t)=0.$
(12)

Applying this transformation to equation (8), we obtain the relations between
the Fuchsian value of the modular accessory parameters
$\rho_{i},\,i=1,\dots,n-3$ and of the accessory parameters
$m_{j},\,j=1,\dots,n-1.$

######  Lemma 3.

Let $X$ and $P(t)$ be as in Proposition 1. Then the differential equations (2)
and (8) are projectively equivalent if and only if the accessory parameters
$m_{1},\dots,m_{n-1}$ and the modular ones $\rho_{0},\dots,\rho_{n-3}$ satisfy
the following relations

$m_{j}=\mathrm{Res}_{t=\alpha_{j}}\Biggl{(}\frac{4\sum_{i=0}^{n-3}{\rho_{i}t^{i}}+P^{\prime\prime}(t)}{2P(t)}\Biggr{)},\quad
j=1,\dots,n-1.$

###### Proof.

We explained above that projectively equivalent differential equations are
related by the transformation (12). In the case of the differential equations
(2) and (8) this leads to the following relation

$\frac{1}{4}\sum_{i=j}^{n-1}{\frac{1}{(t-\alpha_{j})^{2}}}+\frac{1}{2}\sum_{i=j}^{n-1}{\frac{m_{j}}{(t-\alpha_{j})}}=\frac{\sum_{i=0}^{n-3}\rho_{i}t^{i}}{P(t)}-\frac{1}{2}\biggl{(}\frac{P^{\prime}(t)}{P(t)}\biggr{)}^{\prime}-\biggl{(}\frac{P^{\prime}(t)}{2P(t)}\biggr{)}^{2},$
(13)

The right-hand side is equal to
$\Bigl{(}4\sum_{i=0}^{n-3}{\rho_{i}t^{i}}-2P^{\prime\prime}(t)P(t)+P^{\prime}(t)^{2}\Bigr{)}/4P(t)^{2},$
and we have

$\frac{1}{4}\sum_{j=1}^{n-1}{\frac{1}{(t-\alpha_{j})^{2}}}+\frac{P^{\prime\prime}(t)}{2P(t)}-\frac{P^{\prime}(t)^{2}}{4P(t)^{2}}=-\frac{P^{\prime\prime}(t)}{4P(t)}.$

Identity (13) then becomes

$\sum_{j=1}^{n-1}{\frac{m_{j}}{(t-\alpha_{j})}}=\frac{4\sum_{i=0}^{n-3}{\rho_{i}t^{i}}+P^{\prime\prime}(t)}{2P(t)},$

from which the statement follows. ∎

As an example, when $n=4$ we have the following relations for the punctured
sphere $\mathbb{P}^{1}\smallsetminus\\{\infty,1,0,\alpha^{-1}\\}:$

$m_{0}=\alpha+1-2\rho,\;m_{1}=\frac{1-2\rho}{\alpha-1},\;m_{\alpha}=\frac{\alpha(2\rho-\alpha)}{\alpha-1},\;m_{\infty}=\frac{1+\alpha-2\rho}{\alpha}.$
(14)

##  3 The Rankin-Cohen structure from the uniformizing equation

In the last section we computed the uniformizing differential equation using
Rankin-Cohen brackets. It is interesting to notice that the accessory
parameters appear only in the computation of the second bracket of $f$ with
itself:

$[f,f]_{2}=12f^{4}P(t)\bigl{(}(n/2-1)^{2}t^{n-3}+\rho_{n-4}t^{n-2}+\cdots+\rho_{0}\bigr{)}.$

In particular, the ability of expressing the bracket $[f,f]_{2}$ in terms of
$f$ and $t$ leads to the knowledge of the Fuchsian value of the accessory
parameters.

In this section we will show that, under certain assumptions, the converse is
also true. Starting with an $n$-punctured sphere $X,$ we will show that the
knowledge of the Fuchsian value of the accessory parameters gives not only the
uniformization of $X\simeq\mathbb{H}/\Gamma,$ but also the complete ring of
modular forms $M_{*}(\Gamma)$ and even its Rankin-Cohen structure. Our result
is the following.

######  Theorem 1.

Let $X$ be a punctured sphere, and suppose that the Fuchsian value of the
uniformization of $X$ is known. Let $\Gamma$ be such that
$X\simeq\mathbb{H}/\Gamma$, and, if $-1\notin\Gamma,$ assume that it has at
most one irregular cusp. Then, from a Frobenius basis of solutions of the
uniformizing differential equation, one can determine the ring of modular
forms $M_{*}(\Gamma)$ and the Rankin-Cohen structure on $M_{*}(\Gamma).$

We will prove a more general result, of which the above theorem is a
corollary. To this end, we introduce and discuss modular forms of rational
weight.

###  3.1 Modular forms of rational weight

We are interested in rings of modular forms of rational weight, with respect
to certain multiplier systems, that contains as a subring the ring of ordinary
modular forms. In [2] it was noticed that the ring of modular forms of weight
$k/5$ on $\Gamma(5)$ is freely generated by two elements of weight $1/5.$
Ibukiyama [11],[12], inspired by this work, found more examples and elaborated
a more general theory for principal congruence subgroups $\Gamma(N)$ with
$N>3$ odd. We will prove similar results for rings of rational weight modular
forms on certain genus zero groups $\Gamma$. In particular, we define an
automorphy factor $J$ of rational weight from the Frobenius solutions of the
uniformizing differential equation of certain punctured spheres, and construct
a ring of modular forms of rational weight $M_{J^{*}}(\Gamma)$ whose integral
subring (the subring of forms of integral weight) is $M_{*}(\Gamma).$ As a
special case we will recover the $\Gamma(5)$ example above. We start with the
definitions.

######  Definition 2.

Let $\Gamma\subset SL_{2}(\mathbb{R})$ be a cofinite Fuchsian group. An
automorphy factor of weight $r\in\mathbb{Q}$ on $\Gamma$ is a function
$J(\gamma,\tau)\colon\Gamma\times\mathbb{H}\to\mathbb{C}$ holomorphic in
$\tau$ and such that:

(1)

$J(\gamma_{1}\gamma_{2},\tau)=J(\gamma_{1},\gamma_{2}\tau)J(\gamma_{2},\tau),$
for every $\gamma_{1},\gamma_{2}\in\Gamma;$

(2)

$|J(\gamma\tau)|=|c\tau+d|^{r}$ for every $\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma;$ equivalently

$\frac{J^{\prime}(\gamma,\tau)}{J(\gamma,\tau)}=\frac{rc}{c\tau+d},\quad
J^{\prime}(\gamma\tau)=\frac{dJ(\gamma,\tau)}{d\tau}.$

Let $k\geq 0$ be an integer. A holomorphic function
$f\colon\mathbb{H}\to\mathbb{C}$ is a modular form of rational weight $kr$
with respect to $J(\gamma,\tau)$ if

$f(\gamma\tau)=J(\gamma,\tau)^{k}f(\tau)$

for every $\gamma\in\Gamma,$ and if $f$ is holomorphic at every cusp of
$\Gamma$.

In some books (for instance [19]), there is an extra assumption on
$J(\gamma,\tau)$ when $-1\in\Gamma,$ that $J(-1,\tau)=1.$ This will be
automatic in the cases we study, where $J(\gamma,\tau)$ will be constructed as
$f(\gamma\tau)/f(\tau)$ for $f$ a modular form of weight $r\in\mathbb{Q}.$

For a fixed $J=J(\gamma,\tau)$ denote by $\mathrm{M}_{J^{k}}(\Gamma)$ the
linear space of modular forms of weight $kr,$ for every integer $k\geq 1.$ The
direct sum of these spaces

$M_{J^{*}}(\Gamma):=\bigoplus_{k\geq 0}{M_{J^{k}}(\Gamma)}$

is a graded ring. Our result is the following

######  Proposition 2.

Let $\Gamma$ be a genus zero Fuchsian group with $n$ cusps and no torsion. If
$\Gamma$ has at most one irregular cusp, then there exist an automorphy factor
$J$ of rational weight $r$ such that $M_{*}(\Gamma)\subset M_{J^{*}}(\Gamma),$
and $M_{J^{*}}(\Gamma)$ is freely generated by two elements. In particular:

  1. 1.

If $-1\in\Gamma,$ then $r=2/(n-2)$ and $M_{J^{*}}(\Gamma)$ is freely generated
by two elements of weight $r.$

  2. 2.

If $-1\notin\Gamma$ and $\Gamma$ has only regular cusps, then
$r\leavevmode\nobreak\ =\leavevmode\nobreak\ 1/(n/2-1),$ and
$M_{J^{*}}(\Gamma)$ is freely generated by two elements of weight $r.$

  3. 3.

If $-1\notin\Gamma$ and $\Gamma$ has exactly one irregular cusp, then
$r=1/(n-2),$ and $M_{J^{*}}(\Gamma)$ is freely generated by an element of
weight ${r}$ and an element of weight $2{r}.$

Note that the statement holds, for instance, for all the genus zero congruence
groups of the form $\Gamma(N),\Gamma_{0}(N),\Gamma_{1}(N).$ An analogous
statement seems to be false in higher genus and in the case where $\Gamma$ is
of genus zero and has more than one irregular cusp.

###### Proof.

We first prove $\mathrm{(2)}$. Let $\Gamma$ be such that $-1\in\Gamma$ and has
only regular cusps. We showed in Lemma 1 the existence of a weight two modular
form with all its zeros concentrated at a cusp $c_{0}.$ When $\Gamma$ has only
regular cusps the same argument, together with the fact that
$\dim{M_{1}(\Gamma)}=n,$ (see [17], Chapter 2) proves that we can find $f\in
M_{1}(\Gamma)$ with all its $n/2-1$ zeros at a given cusp $c_{0}.$

If we set $r=1/(n/2-1),$ the above discussion proves the existence of a
holomorphic function $g\colon\mathbb{H}\to\mathbb{C}$ such that $g$ is non
zero on $\mathbb{H},$ is holomorphic at all cusps, has a simple zero at
$c_{0},$ and $g^{r}=f\in M_{1}(\Gamma).$ Then if we define

$J(\gamma,\tau):=\frac{g(\gamma,\tau)}{g(\tau)},$

we see that $J(\gamma,\tau)$ is an automorphy factor of weight $r.$ It is
holomorphic everywhere since $g$ has a unique zero at $c_{0},$ and, using
$g^{r}=f,$ it is easy to see that also $(1),(2)$ in Definition 2 are
satisfied.

The function $g$ is a weight one modular form with respect to $J,$ i.e., $g\in
M_{J}(\Gamma).$ We can easily construct another element in $M_{J}(\Gamma)$.
Let $t$ be an Hauptmodul for $\Gamma$ with its pole in the cusp $c_{0}.$
Define

$g_{1}:=gt.$

This function is holomorphic in $\mathbb{H}$ and in every cusp, since the pole
of $t$ cancels with the zero of $g$ at $c_{0}.$ Moreover, $t$ being of weight
zero, $g_{1}$ transforms with respect to $\Gamma$ like $g,$ so $g_{1}\in
M_{J}(\Gamma).$ It follows by looking at the location of the zeros that $g$
and $g_{1}$ are linearly independent; they are also algebraically independent.
This is a standard fact, and holds in general for holomorphic modular forms of
the same weight which are linearly independent; for a proof see [3].

From this discussion it follows that, for every integer $k\geq 0,$ the space
of homogeneous polynomials of degree $k,$ $V_{kr}:=\mathbb{C}[g,g_{1}]_{k}$ is
a vector space of modular forms with respect to the automorphy factor $J^{kr}$
of weight $kr.$ Its dimension is easy to compute:

$\dim{V_{kr}(\Gamma)}=\dim_{\mathbb{C}}{\mathrm{Sym}^{k}(\mathbb{C}\oplus\mathbb{C})}=k+1.$

We see in particular that, when $k=k^{\prime}/r\in\mathbb{Z},$ for some
$k^{\prime}\in\mathbb{Z}_{\geq 0},$

$\dim{V_{kr}}=\frac{k^{\prime}}{r}+1=k^{\prime}\Bigl{(}\frac{n}{2}-1\Bigr{)}+1=\dim{M_{k^{\prime}}(\Gamma)},$
(15)

where $M_{k^{\prime}}(\Gamma)$ is a space of modular forms of integral weight
$k^{\prime}.$

Note that from this it follows that $M_{J^{*}}(\Gamma)$ is freely generated by
$g,g_{1}.$ Let $h\in M_{J^{k}}(\Gamma)$ for some $k\in\mathbb{Z}_{\geq 0},$
and suppose that $h$ is not a polynomial combination of $g,g_{1}.$ Let $w$ be
the mimimum positive integer such that $hg^{w}$ has integral weight. From (15)
it follows that $hg^{w}$ is an homogeneous polynomial in $g,g_{1}$ of degree
$d=k+w:$

$hg^{w}=a_{d,0}g^{d}+a_{d-1,1}g^{d-1}g_{1}+\cdots+a_{0,d}g_{1}^{d},\quad
a_{i}\in\mathbb{C}.$

Since $h$ is by definition holomorphic, it follows that
$(a_{d,0}g^{d}+\cdots+a_{0,d}g_{1}^{d})/g^{w}$ is holomorphic. This, by the
location of zeros of $g,g_{1},$ is possible only if

$h=a_{d,0}g^{k}+\cdots+a_{w,k}g_{1}^{k},$

which means that $h$ is a polynomial combination of $g,g_{1}$ as an element of
$M_{J^{k}}(\Gamma).$ This concludes the proof of $\mathrm{(2)}$.

To prove $\mathrm{(1)},$ we notice that the existence of a function $f\in
M_{2}(\Gamma)$ as in Lemma 1 guarantees the existence of a holomorphic
function $g$ on $\mathbb{H}$ such that $g^{n-2}=f.$ The function $g$ can be
used to construct an automorphy factor of weight $r=2/(n-2)$ on $\Gamma,$ and
we can show as above that $g$ and $gt$ freely generate $M_{J^{*}}(\Gamma).$

Part $\mathrm{(3)}$ of the proposition is proved in a similar way, but there
are some differences. First we fix $r=1/(n-2).$ As in the previous case, we
can construct an element $\hat{f}\in M_{1}(\Gamma)$ with all its zeros
concentrated in a cusp, but this cusp $\hat{c}_{0}$ is necessarily the
irregular one. This fact is proved using the same argument of Lemma 1 and the
fact that $\dim{M_{1}(\Gamma)}=(n-1)/2,$ together with the observation that a
weight one form has always a zero at the irregular cusp. Note that
$(n-1)/2\in\mathbb{Z}$ since $\Gamma$ can have exactly one irregular cusp only
if $n$ is odd.

We can find, as in the regular case, a holomorphic function $\hat{g}$ such
that $\hat{g}^{r}=\hat{f}$ and $g$ has its unique zero in the cusp
$\hat{c}_{0}$ Then, we can define an automorphy factor
$J:=\hat{g}(\gamma\tau)/\hat{g}$ of weight $r$ as in the regular case; we have
by construction that $\hat{g}\in M_{J}(\Gamma).$

Let $\hat{t}$ be an Hauptmodul on $\Gamma$ with a pole where $\hat{g}$ has its
zero. The product $\hat{g}_{2}:=\hat{g}^{2}\hat{t}^{2}$ is holomorphic and
modular of weight $2r,$ so $\hat{g}_{2}\in M_{J^{2}}(\Gamma).$ From the fact
that $\hat{g}_{2}$ has a zero of order two at $\hat{c}_{0}$, it follows that
the functions $\hat{g}^{2}$ and $\hat{g}_{2}$ are linearly independent, and
hence algebraically independent. It follows than that also $\hat{g}$ and
$\hat{g}_{2}$ are algebraically independent.

For every $k\geq 0,$ the space of weighted homogeneous polynomials of degree
$k,$ denoted by $W_{kr}\leavevmode\nobreak\ :=\leavevmode\nobreak\
\mathbb{C}[\hat{g},\hat{g}_{2}]_{k},$ is a vector space of modular forms with
respect to the automorphy factor $J^{kr}$ of weight $kr.$ Its dimension is

$\dim{(W_{kr})}=\begin{cases}(k+1)/2&\text{ $k$ odd}\\\ (k+2)/2&\text{ $k$
even}.\end{cases}$

In particular, if $k=k^{\prime}/r,\,k^{\prime}\in\mathbb{Z}_{\geq 0},$ we have

$\dim{(W_{kr})}=\begin{cases}((n-2)k^{\prime}+1)/2&\text{ $k^{\prime}$ odd}\\\
((n-2)k^{\prime}+2)/2&\text{ $k^{\prime}$ even}.\end{cases}$

These are precisely the dimensions of the spaces $M_{k^{\prime}}(\Gamma)$ when
$k^{\prime}$ is odd or even. As before, this implies that $M_{J^{*}}(\Gamma)$
is freely generated by $\hat{g},\hat{g}_{2}.$ ∎

######  Example 1.

From the proposition follows, as a special case, Bannai’s example on
$\Gamma(5).$ This group has genus zero and $12$ regular cusps. From point
$\mathrm{(2)}$ of the above proposition, we find that $r=1/5$ and that the
ring $M_{J^{*}}(\Gamma)$ is freely generated by two elements of weight $1/5.$

In the proof of Proposition 2 we constructed in each case the automorphy
factor $J$ and the ring $M_{J^{*}}(\Gamma)$ from special modular forms of low
weight with all their zeros concentrated in the cusp where the Hauptmodul $t$
has its pole. As follows form Proposition 1, these functions are exactly the
one that solve the uniformizing differential equation of $X_{\Gamma}.$ Since
we can reconstruct these modular forms form a Frobenius basis of this
uniformizing differential equation (this will be explained in detail in
Section 4.2), it follows that we can construct $M_{J^{*}}(\Gamma)$ from a
Frobenius basis of solutions of the unifomizing differential equation of
$X_{\Gamma}.$ We proved the following corollary.

######  Corollary 1.

Let $X$ be a punctured sphere, and suppose that the uniformizing differential
equation of $X$ is known. If the uniformizing group $\Gamma$ has at most one
irregular cusp, then we can construct the ring $M_{J^{*}}(\Gamma)$ in
Proposition 2 from a Frobenius basis of solutions of the uniformizing
differential equation of $X.$

###  3.2 RC structure on the space of modular forms of rational weight

The ring of modular forms of rational weight $M_{J^{*}}(\Gamma)$ can be
endowed with a Rankin-Cohen structure using the Rankin-Cohen brackets defined
in Section 1.2. If $J$ is an automorphy factor of rational weight $r$ and such
that $M_{*}(\Gamma)\subset M_{J^{*}}(\Gamma)$ as in Proposition 2, then the RC
algebra $M_{*}(\Gamma)$ is canonically a sub-RC algebra of
$M_{J^{*}}(\Gamma).$ We show that, if the Fuchsian value for the
uniformization of the surface $X_{\Gamma}$ is known, we can completely
describe the Rankin-Cohen structure on $M_{J^{*}}(\Gamma),$ hence on
$M_{*}(\Gamma),$ in terms of its multiplicative generators. We work this out
only in the case $\mathrm{(2)}$ of Proposition 2, the other cases being
similar.

As explained in Section 1.2, to describe the Rankin-Cohen structure on
$M_{J^{*}}(\Gamma)$ we only need the multiplicative structure of
$M_{J^{*}}(\Gamma),$ and, for a fixed element $0\neq h\in M_{J^{*}}(\Gamma),$
its first bracket with every element of the ring, and its second bracket with
itself. Let $J$ be the automorphy factor of weight $r$ described in
Proposition 2. There we showed that the multiplicative structure of
$M_{J^{*}}(\Gamma)$ is completely determined by two generators $g$ and
$g_{1}=gt,$ where $g^{r}\in M_{1}(\Gamma)$ and
$t\colon\mathbb{H}/\Gamma\overset{\sim}{\rightarrow}X_{\Gamma}=\mathbb{P}^{1}\smallsetminus\\{\alpha_{1},\dots,\alpha_{n-2},\alpha_{n-1}=0,\alpha_{n}=\infty\\}$
is an Hauptmodul.

Now let $h=g.$ The first bracket of $g$ with every other element of
$M_{J^{*}}(\Gamma)$ is determined by the bracket $[g,g_{1}]$ since $g,g_{1}$
generate the whole ring and $[g,g]=0.$ It can be written only in terms of $g$
and $t$ in the follwing way

$[g,g_{1}]=\frac{1}{r}(gg_{1}^{\prime}-g^{\prime}g_{1})=\frac{1}{r}(g(gt)^{\prime}-g^{\prime}gt)=\frac{1}{\kappa
r}(g^{2r+2}P(t)),\quad P(t)=\prod_{j=1}^{n-1}(t-\alpha_{j}),$

where $\kappa^{-1}=(-1)^{n}\prod_{j=1}^{n-2}\alpha_{j},$ and we have used the
identity $t^{\prime}=g^{2r}P(t)$ proved in Proposition 1.

To express the second bracket $[g,g]_{2}$ in terms of $g$ and $t$ we need to
know the Fuchsian value of the accessory parameters. First notice that, since
$g$ is such that $g^{r}=f\in M_{1}(\Gamma),$

$[f,f]_{2}=\frac{2r^{3}}{r+1}g^{2r-2}[g,g]_{2}.$

Then, if the uniformizing equation is known, we can read $[f,f]_{2}$ from it,
as proven in Theorem 1:

$[f,f]_{2}=g^{6r}P(t)\bigl{(}\hat{\rho}_{n-3}t^{n-3}+\hat{\rho}_{n-2}t^{n-2}+\dots+\hat{\rho}_{0}\bigr{)},$

where $\hat{\rho}_{0},\dots,\hat{\rho}_{n-3}$ are the known Fuchsian values.
It follows that

$[g,g]_{2}=\frac{r+1}{2r^{3}}g^{4r+2}P(t)\bigl{(}\hat{\rho}_{n-3}t^{n-3}+\hat{\rho}_{n-2}t^{n-2}+\dots+\hat{\rho}_{0}\bigr{)}.$

We have proved

######  Theorem 2.

Let $X$ be a punctured sphere, and suppose that the Fuchsian value for the
uniformization of $X$ is known. Let $\Gamma$ be such that
$X=\mathbb{H}/\Gamma$, and, if $-1\notin\Gamma,$ assume it has at most one
irregular cusp. Finally let $J(\gamma,\tau)$ be as in Proposition 2. Then,
from a Frobenius basis of solutions of the uniformizing differential equation
of $X$, one can determine the full ring of modular forms $M_{J^{*}}(\Gamma)$
and the Rankin-Cohen structure on $M_{J^{*}}(\Gamma).$

Theorem 1 follows then from the above result by considering the canonical sub-
RC algebra $M_{*}(\Gamma)$ of integral weight modular forms of
$M_{J^{*}}(\Gamma).$

##  4 Finding the Fuchsian value

In this section we present an algorithm to compute the Fuchsian value of the
uniformization of a generic four-punctured sphere. The algorithm depends on
the geometry of these objects, in particular on the existence of non-trivial
automorphisms. However, the basic idea is quite general, and can be applied
also to spheres with more then four punctures with sufficiently many
automorphisms. We sketch below the general idea of the algorithm in order to
explain its functioning.

1\. Construction of potential modular forms

Let $X$ be an $n$-punctured sphere and consider the differential equation (8),
where the accessory parameters are considered as formal parameters. We
construct, from a Frobenius basis of solutions, a new power series $F(\rho,Q)$
that depends on the accessory parameters. This series will become the Fourier
expansions at $\infty$ of a modular form $f$ of formal weight one when the
accessory parameters take the Fuchsian value, i.e. $f=F(\rho_{F},Q).$ This
gives a criterion to decide whether a value of the accessory parameters is the
Fuchsian value or not. To be able to run this criterion we need to find the
group $\Gamma$ for which the constructed function $f$ is modular.

2\. The uniformizing group

We fix a fundamental domain on $\mathbb{H}$ for the action of a torsion-free
Fuchsian group of genus zero with $n$ cusps. We show that a set of generators
of this group, that eventually will be the uniformizing group, can be
expressed as matrix-valued functions of certain unknown cusp representatives.
Our goal is to compute these cusp representatives in terms of the punctures of
$X$ and of the accessory parameters.

3\. Cusp representatives

We show that the cusp representatives can be computed as functions of the
accessory parameters. This step exploits the existence of a number of
automorphisms of the punctured sphere $X$ and their fixed points on $X$. This
permits to express a set of generators of $\Gamma=\Gamma(\rho)$ as functions
of the accessory parameters.

4\. Determination of the Fuchsian value

Finally, we can set up the modular transformation equations for the function
$F(\rho,Q)$ with respect to various elements $\gamma(\rho)$ of the group
$\Gamma(\rho):$

$F(\rho,\widetilde{Q})-F(\rho,Q)\bigl{(}c(\rho)\tau_{0}+d(\rho)\bigr{)}=0,$
(16)

where $\tau_{0}\in\mathbb{H}$ is fixed and

$Q=re^{2\pi i\tau_{0}},\quad\widetilde{Q}=re^{2\pi
i\gamma(\rho)\tau_{0}},\quad\gamma(\rho)=\begin{pmatrix}a(\rho)&b(\rho)\\\
c(\rho)&d(\rho)\end{pmatrix}\in\Gamma(\rho).$

The modular equations (16) are zero only if the value of the accessory
parameters is the Fuchsian value. The problem is then reduced to the
computation of the unique common zero of a set of equations. This last step
can be performed, for instance, by implementing a version of Newton’s method.

###  4.1 Preliminaries

Let $\alpha\neq 0\,,1$ be a complex number and consider the four-punctured
sphere

$X_{\alpha}=\mathbb{P}^{1}\\!\smallsetminus\\!\\{\infty,1,0,\alpha^{-1}\\}.$

By the uniformization theorem there exist a Fuchsian group
$\Gamma=\Gamma_{\alpha}\leavevmode\nobreak\ \subset\leavevmode\nobreak\
SL_{2}(\mathbb{R})$ and an unbranched holomorphic covering
$t\colon\mathbb{H}\to X_{\alpha}$ such that

$t\colon{\mathbb{H}}/\Gamma\to{X_{\alpha}}$

is a biholomorphism.

The group $\Gamma$ is torsion free and has four inequivalent cusps; we denote
the equivalence classes of cusps by
$c_{1},c_{2},c_{3},c_{4}\in\overline{\mathbb{R}}.$ Later we will fix
$c_{3}=[\infty],\,c_{4}=[0].$ The Hauptmodul $t$ sends these classes to
distinct punctures of $X_{\alpha}.$

An example of a fundamental domain for the action on $\mathbb{H}$ of such
groups is represented in Figure 1.

Figure 1: Fundamental domain of $\Gamma_{1}(5).$ The cusp representatives are
$0,1/3,2/5$ and $\infty.$ The colors indicate how to identify the boundary
geodesics.

In the following we will discuss the explicit uniformization of $X$ by
determining a specific uniformizing group and a specific Hauptmodul. Neither
the group nor the Hauptmodul are detemined uniquely form the uniformizing
differential equation: the uniformizing group is found as the monodromy of the
equation, so it is determined only up to conjugacy in $SL_{2}(\mathbb{C}),$
while the Hauptmodul can be composed with any automorphism of $X.$ It follows
that we have different choices of $\Gamma$ and $t.$ To discuss exlpicit
uniformizations, we should make this choice unique. For this reason, we fix
the following normalization.

######  Lemma 4.

Let
$X_{\alpha}=\mathbb{P}^{1}\\!\smallsetminus\\!\\{\infty,1,0,\alpha^{-1}\\}$ be
as above. We can choose the uniformizing group $\Gamma$ and the Hauptmodul $t$
such that

$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\in\Gamma$

generates the stabilizer $\Gamma_{\infty}\subset\Gamma,$ and the values of $t$
at the inequivalent cusps $\infty,0$ are

$t(\infty)=0,\quad t(0)=\alpha^{-1}.$

These choices uniquely determine $\Gamma$ and $t.$

###### Proof.

Let $\hat{\Gamma}$ and $\hat{t}$ give a uniformization of $X_{\alpha}.$ We can
compose $\hat{t}$ with an automorphism $\phi$ of $X_{\alpha}$ in such a way
that $t:=\phi\circ\hat{t}$ maps $\infty$ to the puncture $0$ (such
automorphism of $X_{\alpha}$ always exists, and will be discussed in detail in
the next section.)

This operation amounts to determine the coordinate $\tau$ on $\mathbb{H}$
given by the uniformizing differential equation up to a linear map
$\tau\mapsto a\tau+b,$ where $a,b\in\mathbb{R}$ (in principle the variable on
$\mathbb{H}$ is determined up to a fractional linear transformation). We see
that there are two free real constants $a,b$. To fix this linear map is
equivalent to choosing a matrix $\sigma\in SL_{2}(\mathbb{R})$ which sends
$\infty$ to $\infty$ and to consider, instead of $\hat{\Gamma},$ the
conjugated group

$\Gamma:=\sigma\hat{\Gamma}\sigma^{-1},\quad\sigma=\begin{pmatrix}a&b\\\
0&1/a\end{pmatrix}.$

From this perspective, we can see that to fix $a$ is equivalent to fixing the
width of the cusp at $\infty$ of $\Gamma.$ Since we are interested in Fourier
expansions of modular forms on $\Gamma$ at $\infty,$ we choose $a$ such that
the width at $\infty$ is one.

Now only $b$ has to be chosen. For different values of $b$ we would have
different coordinates $\tau$ on $\mathbb{H}$ and, in turn, different values
(among the possible ones) of $t$ at the cusps. Because of this we can fix $b$
by choosing the cusp where $t$ takes a determined value, for example
$\alpha^{-1}.$ A natural idea is to consider a cusp for which we can determine
a convenient representative, and we choose $0.$ Finally, we can set that $t$
maps $0$ to the puncture $\alpha^{-1}.$ ∎

In the following, while discussing the uniformization of four-punctured
spheres, we will always consider $\Gamma$ and $t$ normalized as in the above
proposition. Notice that, under this normalization, the cusp at $\infty$ has
width one. Together with $t(\infty)=0,$ it implies that the Fourier expansion
of $t$ at $\infty$ reads

$t=rq+\cdots,\qquad q=e^{2\pi i\tau},\tau\in\mathbb{H},$ (17)

for some non-zero $r\in\mathbb{C}.$

###  4.2 Construction of modular forms

The uniformizing differential equation for $X_{\alpha}$ can be written in the
following form

$P(t)\frac{d^{2}Y(t)}{dt^{2}}+P^{\prime}(t)\frac{dY(t)}{dt}+(t-\hat{\rho})Y=0,$
(18)

where $\hat{\rho}$ is the accessory parameter. In the following, we will
consider $\hat{\rho}$ and the rescaled parameter $\rho:=\alpha\hat{\rho}$ as
formal parameters.

In this paragraph, we make concrete the construction of modular forms from the
uniformizing equation using power series solutions. In order to do this, we
explicitly construct, starting from a Frobenius basis of solutions, some new
power series. These will eventually be the Fourier expansions of certain
modular forms on $\Gamma$ if we specialize $\rho$ to the Fuchsian value.

Using the standard Frobenius method, we can find in a neighborhood of the
regular singular point $t=0$ of (18) a basis of solutions
$\\{y(t),\hat{y}(t)\\}$ where $y(t)$ is holomorphic, and $\hat{y}(t)$ has a
logarithmic singularity in $t=0.$ We normalize the Frobenius basis by
assigning, in $t=0,$ the values $1$ and $0$ to $y(t)$ and to the holomorphic
part of $\hat{y}(t)$ respectively. The chosen basis is then given by the
following power series:

$\displaystyle y(\rho,t)=\sum_{n\geq
0}{a_{n}(\rho)t^{n}}=1+\rho{t}+\frac{1}{4}(\rho^{2}-2\rho(\alpha+1)-\alpha)t^{2}+\cdots,$
$\displaystyle\hat{y}(\rho,t)=\log(t)y(t)+\sum_{n\geq
0}{b_{n}(\rho)t^{n}}=\log(t)y(t)+(-2\rho+\alpha+1)t+\cdots.$

The coefficients $a_{n}=a_{n}(\rho),\,b_{n}=b_{n}(\rho)$ of the series
expansions of $y,\hat{y}$ are computed from the following linear recursions
(Frobenius method):

$\alpha n^{2}a_{n-1}-((\alpha+1)(n^{2}+n)+\rho)a_{n}+(n+1)^{2}a_{n+1}=0,$
$\alpha n^{2}b_{n-1}-((\alpha+1)(n^{2}+n)+\rho)b_{n}+(n+1)^{2}b_{n+1}\\\
+2\alpha na_{n-1}-(2n+1)(\alpha+1)a_{n}+2(n-1)a_{n+1}=0.$

The relevant function for the uniformization is the ratio of the two solutions
$y,\hat{y}.$ However, due to the logarithmic term, using power series it is
more appropriate to work with the exponential of this ratio

$Q(\rho,t)=\exp(\hat{y}/y)=\sum_{n\geq
0}{Q_{n}(\rho)t^{n}}=t+(-2\rho+\alpha+1)t^{2}+\cdots.$ (19)

The function $Q(\rho,t)$ is a local biholomorphism as a function of $t$;
inverting the series (19) we find a new $Q$-expansion $T(\rho,Q)$ around
$Q=0:$

$T(\rho,Q)=\sum_{n\geq
0}{\hat{t}_{n}(\rho)Q^{n}}=Q+(2\rho-\alpha-1)Q^{2}+\cdots.$ (20)

Finally substitute the above series $T(\rho,Q)$ into the holomorphic function
$y(\rho,t)$ to get a holomorphic function in $Q:$

$F(\rho,Q):=y(\rho,T(\rho,Q))=\sum_{n\geq
0}{\hat{f}_{n}(\rho)Q^{n}}=1+\rho{Q}+\frac{1}{4}\bigl{(}9\rho^{2}-2\rho(\alpha+1)-\alpha\bigr{)}Q^{2}+\cdots.$
(21)

When the accessory parameter specializes to the Fuchsian value $\rho_{F}$, the
ratio $\hat{y}(\rho_{F},t)/y(\rho_{F},t)$ gives a coordinate on the universal
covering $\mathbb{H}$ of $X_{\alpha}.$ From our normalization it follows that
the $Q$-expansions of $f$ and $t$ are, respectively, expansions at $\infty$ of
the weight one modular form $f=y\circ t$ and of the Hauptmodul $t.$222When
$\Gamma$ does not admit weight one modular forms, $f$ is a root of some higher
weight modular form as explained in the previous sections. In this case in
fact, a comparison between tha expressions (20) and (17) gives

$Q=rq,\quad q=e^{2\pi i\tau},\tau\in\mathbb{H},$ (22)

for some non-zero $r\\!\in\\!\mathbb{C}.$ It follows that the $Q$-expansions
(20),(21) of $T(\rho_{F},Q)$ and $F(\rho_{F},Q)$ can be converted into
$q$-expansions, which eventually turn $T(\rho_{F},Q)$ and $F(\rho_{F},Q)$ into
functions $t(\tau)$ and $f(\tau)$ on $\mathbb{H}:$

$t(\tau)=\sum_{n\geq 0}{t_{n}q^{n}},\quad
t_{n}:=\hat{t}_{n}(\rho_{F})r^{n},\quad f(\tau)=\sum_{n\geq
0}{f_{n}q^{n}},\quad f_{n}:=\hat{f}_{n}(\rho_{F})r^{n}.$

These constructions have been carried out in the four-punctured sphere case,
but clearly they generalize to every $n$-punctured sphere (this does not apply
to the normalization introduced in Proposition 4, which is tailored to the
four-puncured sphere case).

###  4.3 The uniformizing group

The goal of this section is to write a set of parabolic generators of $\Gamma$
whose coefficients are functions of cusp representatives. Recall that a set of
_parabolic generators_ of a torsion free Fuchsian group with $n$ cusps is a
collection of $n$ matrices $M_{1},\dots,M_{n},$ with
$\bigl{(}\mathrm{trace}\,M_{i}\bigr{)}^{2}=4,$ which generate $\Gamma$ with
the relation

$\prod_{i=1}^{n}{M_{i}}=\mathrm{Id}.$ (23)

To be a generating set, each $M_{i}$ should be the generator of the stabilizer
of a cusp $c_{i};$ two cusps $c_{i},c_{j}$ are inequivalent if $i\neq j.$

Let $c\neq\infty$ be a regular cusp of $\Gamma.$ An element in the stabilizer
$\Gamma_{c}$ of $c$ in $\Gamma$ has the form

$S_{c}=\begin{pmatrix}1+cD_{c}&-c^{2}D_{c}\\\ D_{c}&1-cD_{c}\end{pmatrix}$
(24)

for some positive $D_{c}\in\mathbb{R}.$

In the case of four-punctured spheres, we have three finite cusps
$[0],[c_{1}],[c_{2}]$ and the cusp $[\infty].$ Choose $0$ and $\infty$ as
representatives of the cusps $[0]$ and $[\infty]$ respectively, and let
$0\leavevmode\nobreak\ <\leavevmode\nobreak\ c_{1}\leavevmode\nobreak\
<\leavevmode\nobreak\ c_{2}\leavevmode\nobreak\ <\leavevmode\nobreak\ 1$ be
representatives of the other finite cusps. Then the generators of the
stabilizers of $\Gamma_{0},\Gamma_{c_{1}},\Gamma_{c_{2}},$ are of the form

$S_{0}=\begin{pmatrix}1&0\\\
D_{0}&1\end{pmatrix},\;S_{c_{1}}=\begin{pmatrix}1+c_{1}D_{1}&-c_{1}^{2}D_{1}\\\
D_{1}&1-c_{1}D_{1}\end{pmatrix},\;S_{c_{2}}=\begin{pmatrix}1+c_{2}D_{2}&-c_{2}^{2}D_{2}\\\
D_{2}&1-c_{2}D_{2}\end{pmatrix},$ (25)

for some $D_{0},D_{1},D_{2}\in\mathbb{R}_{>0}.$

######  Lemma 5.

The constants $D_{0},D_{1},D_{2}$ can be determined in terms of the cusps
representatives $c_{1},c_{2}$ as follows

$D_{0}=\frac{1}{c_{1}(1-c_{2})},\quad D_{1}=\frac{1}{c_{1}(c_{2}-c_{1})},\quad
D_{2}=\frac{1}{(c_{2}-c_{1})(1-c_{2})}.$

###### Proof.

The above choice of cusp representatives fixes a fundamental domain
$\mathcal{F}$ for the action of $\Gamma$. It is well known that a free
generating set for $\Gamma$ is given by the Möbius transformations which pairs
the boundary geodesics of $\mathcal{F}.$ Note that among these transformations
there is always one which fixes one of the cusp representatives (see for
instance Fig 1, where the cusp $2/5$ is fixed by the transformation coloured
in green). In our case, the fixed cusp representative is $c_{2},$ since we set
$0<c_{1}<c_{2};$ call $S_{c_{2}}$ the transformation that fixes $c_{2}$ and
pairs the relative boundary geodesics.

The transformation $S_{2}$ also exchanges $c_{1}$ with its equivalent
$c_{1}^{\prime}>c_{2}.$ There is another transformation which exchanges
$c_{1}$ with $c_{1}^{\prime},$ namely the one that also sends $0$ to $1;$ call
it $P_{0,c_{1}}.$ The product

$S_{c_{1}}:=S_{c_{2}}^{-1}P_{0,c_{1}}$

gives a transformation that fixes $c_{1}.$ In the same way, the product
$S_{0}=P_{0,c_{1}}^{-1}T$ fixes $0.$ These matrices by construction satisfy
the parabolic relation (23)

$S_{c_{2}}S_{c_{1}}S_{0}T^{-1}=\mathrm{Id}.$ (26)

Moreover, since we constucted them from a generating set of $\Gamma,$ they
also give a generating set of $\Gamma.$ The matrices $S_{*},*=0,c_{1},c_{2}$
are of the form (25). We can compute the real numbers $D_{i}\,,i=0,1,2,$
solving the system given by the relation (26):

$\begin{pmatrix}1+c_{2}D_{2}&-c_{2}^{2}D_{2}\\\
D_{2}&1-c_{2}D_{2}\end{pmatrix}\begin{pmatrix}1+c_{1}D_{1}&-c_{1}^{2}D_{1}\\\
D_{1}&1-c_{1}D_{1}\end{pmatrix}\begin{pmatrix}1&0\\\
D_{0}&1\end{pmatrix}=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}.$

The statement follows after a straightforward computation. ∎

We remark that using a similar argument (but not the final computation) one
can construct a set of parabolic generators for a genus zero group with any
number of cusps. The idea is, as above, to start with a fixed fundamental
$\mathcal{F}$ domain and the unique matrix that fixes one of the cusp
representatives. Using the transformations that pairs the sides of
$\mathcal{F}$ we can compute step by step each matrix that fixes a certain
cusp representative; using the description in (25) at each step one can
express the new matrix in terms of the cusp representatives. Extra care must
be considered when the group $G$ contains irregular cusps (for instance if $n$
is odd), where the representation in $\eqref{eqn:D0}$ slightly changes.

###  4.4 Geometry of four-punctured spheres

For every choice of $\alpha\neq 0,1,$ the surface $X_{\alpha}$ admits a
nontrivial group $\mathrm{Aut}(X_{\alpha})$ of holomorphic automorphisms. In
general the group $\mathrm{Aut}(X_{\alpha})$ is a Klein four-group and is
generated by any two of the involutions

$\phi_{0}\colon\;t\mapsto\frac{1-\alpha
t}{\alpha(1-t)},\quad\phi_{1}\colon\;t\mapsto\frac{t-1}{\alpha{t}-1},\quad\phi_{2}\colon\;t\mapsto\frac{1}{\alpha
t},$ (27)

where $\phi_{0}=\phi_{1}\circ\phi_{2}.$ For exceptional choices of $\alpha$,
the automorphism group of $X_{\alpha}$ is larger: if $\alpha=-1,1/2,2$ then
$\mathrm{Aut}(X_{\alpha})$ has order $8;$ if $\alpha=1/2\pm i\sqrt{3}/2$ then
$\mathrm{Aut}(X_{\alpha})$ has order $12.$

Let $t\colon\mathbb{H}/\Gamma\to X_{\alpha}$ be a uniformization normalized as
in Proposition 4. Every automorphism $\phi\in\mathrm{Aut}(X_{\alpha})$ lifts
to an involution $\tilde{\phi}$ of the universal covering $\mathbb{H}$

$\begin{CD}\mathbb{H}@>{\tilde{\phi}}>{}>\mathbb{H}\\\ @V{t}V{}V@V{}V{t}V\\\
X_{\alpha}@>{\phi}>{}>X_{\alpha}\\\ \end{CD}$

Every such automorphism $\tilde{\phi}$ is given by a Möbius transformation,
and, being an involution, is represented by a traceless matrix
$W_{\phi}\in\mathrm{SL}_{2}(\mathbb{R}).$ The following fact is standard.

######  Lemma 6.

For every $\phi\in\mathrm{Aut}(X_{\alpha}),$ the matrix $W_{\phi}$ is in the
normalizer $N(\Gamma)$ of $\Gamma.$

It is easy to see that every $\phi\in\mathrm{Aut}(X_{\alpha})$ swaps pairs of
punctures of $X_{\alpha};$ it follows that the transformation $W_{\phi}$ does
the same with the corresponding cusps of $\Gamma.$ Consider for example the
automorphism $\phi_{0}$ in (27). The action of $\phi_{0}$ on the punctures
$\\{\infty,1,0,\alpha^{-1}\\}$ of $X_{\alpha}$ and of $W_{\phi_{0}}$ on the
cusps $\\{[\infty],[0],[c_{1}],[c_{2}]\\}$ of $\Gamma$ is displayed in the
following table (recall that $t(\infty)=0,\,t(0)=\alpha^{-1}$).

$\begin{array}[]{cc}\hline\cr\hline\cr\phi_{0}\text{ on
}\overline{X_{\alpha}}&W_{\phi_{0}}\text{ on }\overline{\mathbb{H}}\\\
\hline\cr 0\leftrightarrow\alpha^{-1}&[\infty]\leftrightarrow\ [0]\\\
\infty\leftrightarrow\ 1&[c_{2}]\leftrightarrow\ [c_{1}]\\\
\hline\cr\hline\cr\end{array}$

By our normalization, $T\in\Gamma.$ Since $W_{\phi}\leavevmode\nobreak\
\in\leavevmode\nobreak\ N(\Gamma)$ for every $\phi\leavevmode\nobreak\
\in\leavevmode\nobreak\ \mathrm{Aut}(X_{\alpha}),$ we have

$W_{\phi}TW_{\phi}^{-1}\in\Gamma.$

In particular, if $W_{\phi}$ sends the cusp $c$ to the cusp $\infty,$ the
element $S_{c}:\leavevmode\nobreak\ =\leavevmode\nobreak\
W_{\phi}TW_{\phi}^{-1}$ sends the cusp $c$ into itself, so it belongs to the
stabilizer $\Gamma_{c}.$ Actually, more is true.

######  Lemma 7.

The matrix $S_{c}=W_{\phi}TW_{\phi}^{-1}$ is the generator of $\Gamma_{c}.$

###### Proof.

This follows from the fact that $T$ generates $\Gamma_{\infty}$ and $W_{\phi}$
normalizes $\Gamma.$ If $S_{c}$ is not the generator of $\Gamma_{c},$ there
exists $n\in\mathbb{Z}$ such that $S_{c}=S^{n},$ where $S$ is the generator of
$\Gamma_{c}.$ But then, by construction, $W_{\phi}S^{n}W_{\phi}^{-1}=T$
implies

$T^{1/n}=W_{\phi}SW_{\phi}^{-1}\in\Gamma,$

since $S\in\Gamma$ and $W_{\phi}$ is in the normalizer $N(\Gamma).$ By
hypothesis the width of $\infty$ is one, then $T^{1/n}\in\Gamma$ only if
$n=\pm 1.$ ∎

Recall that the generator of $\Gamma_{c}$ is of the form

$S_{c}=\begin{pmatrix}1+cD_{c}&-c^{2}D_{c}\\\ D_{c}&1-cD_{c}\end{pmatrix}$
(28)

for some positive $D_{c}\in\mathbb{R}.$ From the lemma and the previous
discussion it follows that, if $\phi\in\mathrm{Aut}(X_{\alpha})$ sends $t(c)$
to $t(\infty)=0,$ then

$W_{\phi}=\sqrt{D_{c}}\begin{pmatrix}c&\frac{-1-c^{2}D_{c}}{D_{c}}\\\
1&-c\end{pmatrix}.$ (29)

For example, for the automorphism $\phi_{0}$ in (27) we have $c=[0]$ and then

$W_{\phi_{0}}=\sqrt{D_{0}}\begin{pmatrix}0&-1/D_{0}\\\ 1&0\end{pmatrix}$

for some $D_{0}\in\mathbb{R}_{>0}.$ In other words, we know every lift of
$\phi\in\mathrm{Aut}(X_{\alpha})$ up to the positive real constants
$D_{0},D_{1},D_{2}$ in (25).

These lifts are the key to compute the constants $D_{0},D_{1},D_{2}$ and then
the uniformizing group. The reason is that the fixed points of
$W_{\phi},\,\phi\leavevmode\nobreak\ \in\leavevmode\nobreak\
\mathrm{Aut}(X_{\alpha})$ naturally gives cusp representatives of $\Gamma.$
Being an involution, each transformation $W_{\phi}$ has only one fixed point
on $\mathbb{H}$. If $W_{\phi}$ is related to the cusp $c$ in the sense of
(29), its fixed point on $\mathbb{H}$ is given by

$\tau_{\phi}=\hat{c}+i/\sqrt{D_{c}},$

where $\hat{c}\in\mathbb{R}$ is a well-defined cusp representatives of $c$.
For instance, in the case of the automorphism $\phi_{0}$ in (27) the cusp is
$c=[0].$ The action of $W_{\phi_{0}}$ is given by

$\tau\mapsto\frac{-1}{D_{0}\tau},$

and its fixed point on $\mathbb{H}$ is $\tau_{\phi_{0}}=i/\sqrt{D_{0}}.$ It
follows that the representative $\hat{c}_{0}$ of the cusp $[0]$ is
$\hat{c}_{0}=0,$ in accordance with our choice in Lemma 5.

###  4.5 Computation of the cusps representatives

Let $\tau_{\phi}$ be the fixed point in $\mathbb{H}$ of the lift $W_{\phi}.$
By construction there is an automorphism $\phi$ of $X_{\alpha}$ such that

$t(\tau_{\phi})=t(W_{\phi}(\tau_{\phi}))=\phi(t(\tau_{\phi})),$

which means $t(\tau_{\phi})$ is a fixed point of the automorphism
$\phi\in\mathrm{Aut}(X_{\alpha}).$

The fixed points of $\phi$ are in general determined by a quadratic equation:
for instance, the fixed points of $\phi_{0}$ are the solutions of

$\alpha{t}^{2}-2\alpha{t}+1=0.$ (30)

The image of $t(\tau_{\phi_{0}})$ is one of the two roots, and this it is
completely determined by the normalization $t.$ The correct root is chosed by
studying the mapping properties of $t.$ We show how it works in the case
$|\alpha^{-1}|<1$ and $\Re{(\alpha^{-1})},\Im{(\alpha^{-1})}>0.$ Using Möbius
transformations and complex conjugation, one can always reduce to this case.

Consider the involution $\phi_{0},$ whose fixed points are determined by the
equation above. We know that the fixed point of its involution on $\mathbb{H}$
is $\tau_{\phi_{0}}=i/\sqrt{D_{0}}.$ This means that it lies on the imaginary
axis, so its image on $X_{\alpha}$ via $t$ should belong to the curve,
determined by $t,$ which joines the punctures $0,\alpha^{-1}.$ Looking at the
two roots of (30) and considering the constraints on $\alpha^{-1}$ it follows
that

$t(\tau_{\phi_{0}})=z_{0}=1-\frac{\sqrt{\alpha(\alpha-1)}}{\alpha}.$

Now look at the involution $\phi_{1}\in\mathrm{Aut}(X_{\alpha})$ defined in
(27). The fixed points of this involutions are
$\alpha^{-1}\pm\bigl{(}\sqrt{1-\alpha}\bigr{)}/\alpha.$ Since $|\alpha|>1,$
none of these roots is real, and one lies above the real axis and the other
below. The fundamental domain for $\Gamma$ that we considered lies at the left
of the boundary geodesic between $\tau=i\infty$ and $\tau=0.$ This implies
that the image, via $t$, of the fundamental domain lies above the curve,
determined by $t,$ which joins the cusps $0,\alpha^{-1}.$ Then the root we
have to choose is the one with positive imaginary part. If
$\tau_{\phi_{1}}=\hat{c}_{\phi_{1}}+1/\sqrt{D_{1}}$ is the fixed point on
$\mathbb{H}$ of the lift of $\phi_{1},$ we have

$t(\tau_{\phi_{1}})=z_{1}=\frac{1}{\alpha}+\frac{\sqrt{1-\alpha}}{\alpha}.$

Similar considerations apply to the choice of the fixed points $z_{2}$ of the
third non-trivial involution of $X_{\alpha}.$

Recall that $Q$ denotes the local inverse function to $t.$ We know from
formula (22) in Section 4.2 that $Q=re^{2\pi i\tau}$ in the uniformizing case,
for some non-zero $r\in\mathbb{C}.$ Call $Q_{j}$ the image of $z_{j}$ via $Q.$
Then in the uniformizing case $Q_{j}$ will have the form

$Q_{j}=re^{2\pi i\tau_{\phi_{j}}},\quad j=0,1,2,$

where $\tau_{\phi_{j}}$ are the fixed points in $\mathbb{H}$ of the lifts of
the involutions $\phi_{j}\leavevmode\nobreak\ \in\leavevmode\nobreak\
\mathrm{Aut}(X_{\alpha}).$ Then we see that to compute the cusp
representatives $\hat{c}_{\phi_{1}},\hat{c}_{\phi_{2}}$ it is enough to
consider the real part of

$\log(Q_{j}/Q_{0})/(2\pi
i)=\tau_{j}-\tau_{0}=\hat{c}_{\phi_{j}}+i(1/\sqrt{D_{j}}-1/\sqrt{D_{0}}),\quad
j=1,2.$

Finally, using Lemma 5, from these representative of the cusps we can compute
$D_{0},D_{1},D_{2}$ and the generators of the group $\Gamma.$ Moreover, we can
compute the $r$ appearing in the relation $Q=re^{2\pi i\tau}$ simply by

$r=\frac{Q_{0}}{\exp(-2\pi/\sqrt{D_{0}})}.$ (31)

###  4.6 Determination of the Fuchsian value

In the last two sections, to set up the correspondence between cusps
representatives $\hat{c}_{i}$ and fixed points $z_{i}$ of
$\phi_{i}\in\mathrm{Aut}(X_{\alpha})$, we assumed that we already had an
Hauptmodul $t\colon\mathbb{H}\to X_{\alpha}.$ In general, if we start from the
differential equation (18), we do not have the Hauptmodul $t,$ and everything
depends on the accessory parameter $\rho.$

However, the constructions of the previous sections still make sense even if
$\rho$ is a formal parameter, in the sense that we can compute everything (the
numbers $\hat{c}_{i}$ and the matrices defined from these in Lemma 5) as
functions of $\rho.$ The matrices $S_{c_{i}}(\rho),\,i=1,2,3,$ (25) that we
obtain are functions of $\rho,$ as the group $\Gamma(\rho)$ they generate.

Classical uniformization theory affirms that there exists a unique value
$\rho_{F}$ of $\rho$ which makes the group $\Gamma(\rho)$ a Fuchsian group
such that the function $t=t(\rho,Q)|_{\rho=\rho_{F}}$ in (20), is an
Hauptmodul $t\colon\mathbb{H}/\Gamma\to X_{\alpha}.$ The theory in Section 1.3
gives an equivalent statement: the Fuchsian value $\rho_{F}$ is the unique
value that makes the function $f(\rho,Q)$ defined in (21) a weight one modular
form with respect to the group $\Gamma(\rho).$ This characterization of
$\rho_{F}$ is the one that we exploit now. If, in analogy with (31) we set

$r(\rho):=Q(\rho,z_{0})/\exp(-2\pi/\sqrt{D_{0}(\rho)}),$

we can define, for every fixed $\tau^{*}\in\mathbb{H}$, new functions of
$\rho$ by

$F_{j}(\rho):=f\bigl{(}\rho,r(\rho)e^{2\pi
iS_{c_{j}}(\rho)(\tau^{*})}\bigr{)}-f\bigl{(}\rho,r(\rho)e^{2\pi
i\tau^{*}}\bigr{)}J(S_{c_{j}}(\rho),\tau^{*})\quad j=1,2,3,$

where

$S_{c_{j}}(\rho)(\tau^{*}):=\frac{s_{j,1}(\rho)\tau^{*}+s_{j,2}(\rho)}{s_{j,3}\tau^{*}+s_{j,4}},\quad
S_{c_{j}}(\rho)=\begin{pmatrix}s_{j,1}(\rho)&s_{j,2}(\rho)\\\
s_{j,3}(\rho)&s_{j,4}(\rho)\end{pmatrix}$

and $J(S_{c_{j}}(\rho),\tau^{*}):=(s_{j,3}(\rho)\tau^{*}+s_{j,4}(\rho)).$

Then the accessory parameter problem reduces to the following one: the
Fuchsian value $\rho_{F}$ is the unique zero of the system of equations
$F_{j}(\rho)=0.$ We can find it numerically by computing the unique number
$\rho_{F}$ such that

$F_{j}(\rho_{F})=0,\quad\text{for all }j=1,2,3.$

For a given $X_{\alpha}$ we can write the corresponding differential equation
(8), compute the functions introduced above as functions of $\rho$ and use
Newton’s algorithm to find numerically the common zero of the functions
$F_{j},\,j=1,2,3.$ This algorithm can be implemented, for instance in PARI,
and used to compute to high precision the uniformization of any four-punctured
sphere, and of spheres with higher number of punctures, provided that they
have enough automorphisms to make the above algorithm work. The output gives:
the Fuchsian value to desired precision, the generators of the uniformizing
group $\Gamma$ and a set of generators of the ring of modular forms on
$\Gamma.$

An example of the use of this algorithm is given in the next section.

##  5 Example: local expansion of the Fuchsian value function

In this section we consider the function that associate to a $n$-punctured
sphere $X$ the modular Fuchsian value $\rho_{F}(X)$ of its uniformization. We
can see it as a function on the space
$W_{n}=\\{(w_{1},\dots,w_{n-3})\;|\;w_{i}\neq w_{j}\text{ if }i\neq
j,\,w_{i}\neq 0,1\\}$ as follows

$\rho_{F}\colon W_{n}\to\mathbb{C}^{n-3},\quad
w=(w_{1},\dots,w_{n-3})\mapsto\rho_{F}(w)=(\rho_{1},\dots,\rho_{n-3}),$

where $\rho_{F}=(\rho_{1},\dots,\rho_{n-3})$ is the modular Fuchsian value for
the surface

$X=\mathbb{P}^{1}\smallsetminus\\{w_{1},w_{2},\dots,w_{n-3},0,1,\infty\\}.$

Kra [15] proved that this function is real-analytic, but non complex-analytic;
this means that if $z$ is a local parameter on $W_{n},$ then the function
$\rho_{F}$ has a local expansion around every point $z_{0}\in W_{n}$ of the
form

$\rho_{F}(z_{0}+z)=\sum_{j,k\geq 0}{a_{j,k}z^{i}\bar{z}^{j}},\quad
a_{j,k}\in\mathbb{C}.$ (32)

In the following we consider the case $n=4$ and exploit the algorithm
presented in the previous section to compute the local expansion of the
(modular) Fuchsian value function. In this case it is simply a function

$\rho_{F}\colon\mathbb{C}\setminus\\{0,1\\}\to\mathbb{C},\quad
z\mapsto\rho_{F}(z).$

We expand the function $\rho_{F}$ around the point $1/2,$ corresponding to the
four-punctured sphere $\mathbb{P}^{1}\setminus\\{\infty,1,0,\/2\\}.$ We choose
this point since we know exactly that $\rho_{F}(1/2)=1$ (see for instance
[9]).

First a few words about how we obtained the expansion. We implemented in PARI
the algorithm presented before. We then computed many values of the function
$\rho_{F}$ near the point $z_{0}=1/2$ along the lines
$L_{n}:\,\Im(z)=\Re(z)/n-1/(2n),$ for different values of $n\in\mathbb{N}.$
The expansion in this case depends only on one real variable $x,$ since $z\in
L_{n}$ if $z=1/2+x(1+i/n)$ and

$\rho_{F}(1/2+z)=\sum_{j,k\geq 0}{a_{j,k}}(1+i/n)^{j}(1-i/n)^{k}x^{k+j},$ (33)

Here the $a_{jk}$ are real numbers, because of the easily proved relation
$\rho_{F}(\bar{z})=\overline{\rho_{F}(z)}$. Then we had to compute enough
expansions along different $L_{n}$ (the number of expansions depends on the
number of $a_{ij}$ one wants to compute) and solve some linear systems to
determine numerically the coefficients.

We present the coefficients of the expansion (32) we obtained with this
method. The constant term $a_{00}$ equals $1$ since it is simply the value
$\rho_{F}(1/2)$ The other coefficients are given in the following table.

$\begin{array}[]{lcr}\hline\cr\hline\cr
a_{1,0}=-1.231129697228372059&a_{0,1}=0.063875489913862273&\\\ \hline\cr
a_{2,0}=2.46225939445674411&a_{1,1}=-0.127750979827724546&a_{0,2}=0\\\
\hline\cr a_{3,0}=-4.823691858558769882&a_{2,1}=0.189062079397880349&\\\
a_{1,2}=0.011749087780820557&a_{0,3}=0.063020693132626783&\\\ \hline\cr
a_{4,0}=9.6473837171175397652587926&a_{3,1}=-0.3781241587957606984328002&\\\
a_{2,2}=-0.0234981755616411140683394&a_{1,3}=-0.1260413862652535661442666&a_{0,4}=0\\\
\hline\cr a_{5,0}=-19.09466584514482511&a_{4,1}=0.667376927634543472&\\\
a_{3,2}=0.046637902612272536&a_{2,3}=0.188862509928182591&\\\
a_{1,4}=0.023318951306136268&a_{0,5}=0.133475385526908694&\\\
\hline\cr\hline\cr\end{array}$

We immediatly notice the following relations among the coefficients:

$\displaystyle a_{2,0}=-2a_{1,0},\quad a_{1,1}=-2a_{0,1},$ (34) $\displaystyle
a_{0,2}=0=a_{0,4},$ (35) $\displaystyle a_{2,1}=3a_{0,3},$ (36) $\displaystyle
a_{3,2}=2a_{1,4},\quad a_{4,1}=5a_{0,5}.$ (37)

It turns out that these relations can be explained using the symmetry of
$\rho_{F}$ near $1/2,$ and a result of Takhtajan and Zograf [22].

The point $z_{0}=1/2$ is the fixed point of the involution $z\mapsto 1-z.$ It
is known [16] that the following identity holds

$\rho_{F}(1-z)=\frac{z\rho_{F}(z)-1}{z-1}.$

It follows that, near the point $z_{0}=1/2,$ one has

$(z-1/2)\rho_{F}(1/2-z)=(1/2+z)\rho_{F}(1/2+z)-1,$

which gives

$\sum_{i,j\geq 0}{a_{i,j}[1+(-1)^{i+j}]z^{i}\overline{z}^{j}}+2\sum_{i,j\geq
0}{a_{i,j}[1-(-1)^{i+j}]z^{i+1}\overline{z}^{j}}-2=0.$ (38)

The above relation implies that

$\displaystyle a_{0,2n}=0\quad\text{if }n\geq 1,$ $\displaystyle
a_{i+1,j}=-2a_{i,j}\quad\text{if }i+j\text{ is odd.}$

This explain why $a_{0,2}=a_{0,4}=0.$

The result of Takhtajan and Zograf (formula 4.1 in [22]) reduces to the
following identity in the four-punctured case

$\frac{\partial
m_{\alpha}}{\partial\bar{z}}=\overline{\biggl{(}{\frac{\partial
m_{\alpha}}{\partial\bar{z}}}\biggr{)}},$ (39)

where $m_{\alpha}=m_{\alpha}(X_{\alpha})$ is the Fuchsian parameter function,
defined from the differential equation (2), associated to the puncture
$\alpha^{-1}\neq 0,1,\infty.$ The Fuchsian parameter function we are
considering is not $m_{\alpha},$ but the modular one $\rho_{F}$. They are
related by the simple formula (14)

$m_{\alpha}=\frac{2\alpha\rho_{F}-1}{\alpha(1-\alpha)}.$

This, together with (39) and the fact that in our notation (33)
$\alpha=1/2+z,$ implies that $\rho_{F}(z,\overline{z})$ satisfies the
following differential equation

$(1-2z)\overline{\biggl{(}\frac{\partial\rho_{F}}{\partial\overline{z}}\biggr{)}}=(1-2\overline{z})\frac{\partial\rho_{F}}{\partial\overline{z}}.$

This implies the following relations between the coefficients of the local
expansion of $\rho_{F}:$

$(j+1)a_{i,j+1}-2ja_{i,j}=(i+1)a_{j,i+1}-2ia_{j,i},\quad i,j\geq 0.$

It is easy to check that the relations (34) come from this one and from (38).
For instance, from the one above for $(i,j)=(0,1)$ we get

$2a_{0,2}-2a_{0,1}=a_{1,1}.$

This, together with $a_{0,2}=0,$ gives the first identity in (34).

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