# A note on the inverse problem for finite differential Galois groups

Camilo Sanabria Malagón<EMAIL_ADDRESS>Departamento de
Matemáticas
Universidad de los Andes
Bogotá, Colombia

###### Abstract

In this paper we revisit the following inverse problem: given a curve
invariant under an irreducible finite linear algebraic group, can we construct
an ordinary linear differential equation whose Schwarz map parametrizes it? We
present an algorithmic solution to this problem under the assumption that we
are given the function field of the quotient curve. The result provides a
generalization and an efficient implementation of the solution to the inverse
problem exposed by M. van der Put, C. Sanabria and J.Top [19]. As an
application, we show that there is no hypergeometric equation with
differential Galois group isomorphic to $H_{72}^{SL_{3}}$, thus completing
Beuker and Heckman’s answer [4] to the question of which irreducible finite
subgroup of $SL_{3}(\mathbb{C})$ are the monodromy of a hypergeometric
equation.

## Introduction

We consider a linear ordinary differential equation (LODE) $L(y)=0$, where

$L(y)=y^{(n)}+a_{n-1}y^{(n-1)}+\ldots+a_{1}y^{\prime}+a_{0}y$

and $a_{0},a_{1},\ldots,a_{n-1}\in\mathbb{C}(z)$. Let $U\subseteq\mathbb{C}$
be an open disk avoiding the singularities of the equation and let

$y_{j}:U\longrightarrow\mathbb{C},\quad j\in\\{1,\ldots,n\\}$

be a basis of solutions. The multivalued analytic extension of

$\displaystyle U$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{P}^{n-1}(\mathbb{C})$ $\displaystyle z$
$\displaystyle\longmapsto$
$\displaystyle\Big{[}y_{1}(z):\ldots:y_{n}(z)\Big{]}$

is the Schwarz map associated to our basis of solutions and its image is an
analytic curve in $\mathbb{P}^{n-1}(\mathbb{C})$. The monodromy group of
$L(y)=0$ defines a group of automorphisms of this curve.

If all the solutions are algebraic over $\mathbb{C}(z)$, then the monodromy
group is finite and the corresponding analytic curve is algebraic. The inverse
problem can be stated in the following terms. Let $G\subset
GL_{n}(\mathbb{C})$ be a finite linear algebraic group and $PG$ its projection
in $PGL_{n}(\mathbb{C})\simeq GL_{n}(\mathbb{C})/Z$. Given an algebraic curve
$C\subseteq\mathbb{P}^{n-1}(\mathbb{C})$, invariant under $PG$, can we
construct a linear ordinary differential equation such that, for some basis of
solutions, the image of its Schwarz map is the given curve and its monodromy
group contains $G$? In the case where we have an affirmative answer, we say
that the Schwarz map parametrizes $C$.

One of the first results in this direction was given by G.H. Halphen [8] and
A. Hurwitz [13] who independently solved the inverse problem in the case of
the Klein quartic and its automorphism group $G_{168}$. Subsequent results
arose from considering the same group but other invariant curves. For example,
R. Fricke [7] described a pencil of projective curves of degree $12$ invariant
under $G_{168}$ and for each curve in this pencil M. Kato [14] constructed an
ordinary linear differential equation with its Schwarz map parametrizing it.
Furthermore, M. Kato[15] showed that among the hypergeometric equations with
finite monodromy described by F. Beukers and G. Heckmann in [4], there is one
that solves the inverse problem for the Hessian of the Klein quartic.

More generally, curves invariant under other finite linear algebraic groups
have been studied. The Hesse pencil [10, 11], a family of elliptic curves
invariant under the normal abelian subgroup of order $9$ of the Hessian group
is one example. This pencil has regained much attention recently in different
areas [1, 2, 24, 28]. For each curve in it C. Sanabria [22] constructed a
linear ordinary differential equation with Schwarz map parametrizing it. Also,
L. Lachtin [18] considered curves invariant under the Valentiner group, the
perfect triple cover of $A_{6}$, and obtained a hypergeometric equation that
parametrizes one of these curves.

An important result closely related to this inverse problem is a classical
theorem of Klein [16, 17] stating that, given a finite primitive group
$G\subseteq SL_{2}(\mathbb{C})$, there exists a hypergeometric equation such
that every second order LODE whose projective monodromy group is isomorphic to
$PG$ is projectively equivalent to the pullback by a rational map of this
hypergeometric equation. Indeed, for each finite primitive group $G\subseteq
SL_{2}(\mathbb{C})$, there is a Gauss hypergeometric equation that
parametrizes $\mathbb{P}^{1}(\mathbb{C})$ and its monodromy contains $G$.
Naturally, these hypergeometric equations are in correspondence with Schwarz
spherical triangles. This classical result has been generalized to LODEs of
order $3$ by M. Berkenbosch [3], and to arbitrary order by C. Sanabria [21].

A generalization of this inverse problem is treated by Y. Haraoka and M. Kato
in [9], where they consider projective spaces $\mathbb{P}^{n}(\mathbb{C})$ and
reflection groups instead of curves and finite linear algebraic groups. Their
solution provides another generalization of Klein’s, one that applies to
Fuchsian systems of rank $n$ over $\mathbb{P}^{n}(\mathbb{C})$ with monodromy
isomorphic to a reflection group.

A related inverse problem can be stated as follows: given a finite linear
algebraic group $G\subseteq GL_{n}(\mathbb{C})$, can we construct a linear
ordinary differential equation such that the monodromy group is isomorphic to
$G$? This inverse problem was studied by F. Beukers and G. Heckman in [4] for
reflection groups and by M. van der Put and F. Ulmer in [26] for equations of
order $3$. Their approaches obtain only equations over
$\mathbb{P}^{1}(\mathbb{C})$ with three singularities, and couldn’t determine
whether there is a hypergeometric equation with differential Galois group
$H_{72}^{SL_{3}}$.

In [19], M. van der Put, C. Sanabria and J. Top presented a solution to the
inverse problem in the case when the normalization of the quotient of the
curve by the group of $PG$ has genus $0$. More precisely, given any algebraic
curve $C\subseteq\mathbb{P}^{n-1}(\mathbb{C})$ invariant under a finite linear
algebraic group $G\subseteq GL_{n}(\mathbb{C})$ together with a generator of
$\mathbb{C}(C/G)$, they presented a procedure that constructs a linear
ordinary differential equation whose Schwarz map parametrizes $C$.
Furthermore, they showed that any irreducible linear ordinary differential
equation with rational coefficients and only algebraic solutions can be
obtained as a result of this procedure.

In this paper, we revisit the solution in [19] to the inverse problem but from
a geometric perspective. In this way, we obtain a generalization where we drop
the condition on the genus of the normalization of the quotient curve.
Moreover, we turn the solution into an efficient algorithm. Finally, as an
application of this alternative viewpoint, we prove that there is no
hypergeometric equation with differential Galois group isomorphic to the
elusive group $H_{72}^{SL_{3}}$, thus completing Beuker and G. Heckman answer
[4] to the question of which irreducible finite subgroup of
$SL_{3}(\mathbb{C})$ are the monodromy of a hypergeometric equation.

##  1 Preliminary

###  1.1 Space of orbits

We recall some elements of the space of orbits of actions of finite linear
groups on vector spaces. A more detailed exposition is found in [5].

Let $G\subseteq GL_{n}(\mathbb{C})$ be a finite linear algebraic group. Let
$R=\mathbb{C}[X_{1},\ldots,X_{n}]$ be the coordinate ring of $\mathbb{C}^{n}$.
We consider the natural left action of $G$ on $R$ by
$\mathbb{C}$-automorphisms

$g:X_{j}\mapsto\sum_{i=1}^{n}X_{i}g_{ij},$

for $g=(g_{ij})_{i,j=1}^{n}\in G$. This $G$-action on $R$ induces a right
$G$-action on $\mathbb{C}^{n}$ defined by

$g:(x_{1},\ldots,x_{n})\mapsto\Big{(}\sum_{i=1}^{n}x_{i}g_{i1},\ldots,\sum_{i=1}^{n}x_{i}g_{in}\Big{)}.$

Since $G$ is finite, it is reductive, thus the $G$-invariant polynomials in
$R$ separate the $G$-orbits in $\mathbb{C}^{n}$. Therefore, the coordinate
ring of the orbit space $\mathbb{C}^{n}/G$ is the finitely generated
$G$-invariant subring $R^{G}\subseteq R$.

Let $F_{1},\ldots,F_{N}$ be homogeneous generators of $R^{G}$. We can embed
$\mathbb{C}^{n}/G$ into $\mathbb{C}^{N}$ through the algebraic map

$\mathbf{x}\cdot
G\mapsto\Big{(}F_{1}(\mathbf{x}),\ldots,F_{N}(\mathbf{x})\Big{)},$

where $\mathbf{x}=(x_{1},\ldots,x_{n})$. We identify the orbit space
$\mathbb{C}^{n}/G$ with its embedding in $\mathbb{C}^{N}$. Since $G$ is
finite, the orbit space has dimension $n$ and therefore there exists an open
dense subset of $\mathbb{C}^{n}$ where the derivative of the quotient map

$\mathbf{x}\mapsto\Big{(}F_{1}(\mathbf{x}),\ldots,F_{N}(\mathbf{x})\Big{)}$

is non-singular.

Similarly, we get natural $G$-actions on $\mathbb{P}^{n-1}(\mathbb{C})$ and on
its homogeneous coordinate ring $R$, and we denote
$\mathbb{P}^{n-1}(\mathbb{C})/G$ the orbit space. We denote by
$\Pi:\mathbb{P}^{n-1}(\mathbb{C})\rightarrow\mathbb{P}^{n-1}(\mathbb{C})/G$
the quotient map. Let $\Lambda\in\mathbb{Z}_{>0}$ be such that
$R^{G}_{\Lambda}$, the homogeneous elements of $R^{G}$ of degree $\Lambda$,
form a homogeneous coordinate system for $\mathbb{P}^{n-1}(\mathbb{C})/G$,
i.e.

$\mathbb{P}(R^{G}_{\Lambda})\simeq\mathbb{P}^{n-1}(\mathbb{C})/G.$

Let $\Phi_{1},\ldots,\Phi_{M}$ be a basis of $R^{G}_{\Lambda}$. We can embed
$\mathbb{P}^{n-1}(\mathbb{C})/G$ into $\mathbb{P}^{M-1}(\mathbb{C})$ with the
map

$[\mathbf{x}]\cdot
G\mapsto\Big{[}\Phi_{1}(\mathbf{x}):\ldots:\Phi_{M}(\mathbf{x})\Big{]},$

where $[\mathbf{x}]=[x_{1}:\ldots:x_{n}]$. As before, we identify the orbit
space $\mathbb{P}^{n-1}(\mathbb{C})/G$ with its embedding in
$\mathbb{P}^{M-1}(\mathbb{C})$. Again, there exists an open dense subset of
$\mathbb{P}^{n-1}(\mathbb{C})$ where the derivative of the quotient map

$[\mathbf{x}]\mapsto\Big{[}\Phi_{1}(\mathbf{x}):\ldots:\Phi_{M}(\mathbf{x})\Big{]}$

is non-singular.

###  1.2 Schwarz maps

We now introduce Schwarz maps. A famous reference on the subject is [27]

Let $C_{0}$ be a compact Riemann surface and $K=\mathbb{C}(C_{0})$ the field
of meromorphic functions over $C_{0}$. Let $\delta:K\rightarrow K$ be any non-
trivial derivation. Note that $\delta$ can be uniquely extended to the sheaf
of meromorphic functions over open sets of $C_{0}$, and in particular to any
algebraic extension of $K$.

Let $L(y)=0$ be an linear ordinary differential equation of order $n$ where

$L(y)=\delta^{n}(y)+a_{n-1}\delta^{n-1}(y)+\ldots+a_{1}\delta(y)+a_{0}y,$

and $a_{0},a_{1},\ldots,a_{n-1}\in K$. Let $S_{0}\subset C_{0}$ be the set of
singular points of $L(y)$. Given a non-singular point $p\in C_{0}$ together
with a basis of solutions $\mathbf{y}=(y_{1},\ldots,y_{n})$ defined over a
neighborhood $U\subseteq C_{0}$ of $p$, we define the Schwarz map as the
analytic extension of

$\displaystyle[\mathbf{y}]:U$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{P}^{n-1}(\mathbb{C})$ $\displaystyle z$
$\displaystyle\longmapsto$
$\displaystyle\Big{[}y_{1}(z):\ldots:y_{n}(z)\Big{]}.$

The monodromy of the Schwarz map is the projection on $PGL_{n}(\mathbb{C})$ of
the monodromy group of the basis of solutions $(y_{1},\ldots,y_{n})$ . We will
denote it by $PG$ and call it the projective monodromy of $L(y)=0$. Let us
assume $PG$ is finite. Then, post-composing the Schwarz map with the quotient
by the action of $PG$ on $\mathbb{P}^{n-1}(\mathbb{C})$ we obtain a (single-
valued) rational map

$\psi:C_{0}\dashrightarrow\mathbb{P}^{n-1}(\mathbb{C})/PG,$

defined over $C_{0}\setminus S_{0}$, which we call the quotient Schwarz map.

###  1.3 Algebraic Picard-Vessiot theory

We briefly recall the results from Picard-Vessiot theory that we will use in
this paper. An extensive exposition of this subject is given in [25].

We assume that $L(y)=0$ is irreducible and that all its solutions are
algebraic over $K$. Under these assumptions, if $(y_{1},\ldots,y_{n})$ is a
basis of solutions, then a Picard-Vessiot extension of $K$ for $L(y)=0$ is the
field

$E=K[y_{1},\ldots,y_{n}].$

Let’s denote by $I$ the kernel of the evaluative $K$-morphism

$\displaystyle\Phi:K[Y_{1},\ldots,Y_{n}]$ $\displaystyle\longrightarrow$
$\displaystyle K$ $\displaystyle Y_{j}$ $\displaystyle\longmapsto$
$\displaystyle y_{j}.$

An isomorphic representation of the differential Galois group of $L(y)=0$ is
the finite group $G\subset GL_{n}(\mathbb{C})$ composed by the elements
$g=(g_{ij})_{i,j=1}^{n}$ that send the ideal $I$ into itself under the
$K$-automorphism

$\displaystyle g:K[Y_{1},\ldots,Y_{n}]$ $\displaystyle\longrightarrow$
$\displaystyle K[Y_{1},\ldots,Y_{n}]$ $\displaystyle Y_{j}$
$\displaystyle\longmapsto$ $\displaystyle\sum_{i=1}^{n}Y_{i}g_{ij}.$

By the Galois correspondence, if $P\in K[Y_{1},\ldots,Y_{n}]$ is $G$-invariant
then $\Phi(P)\in K$. Since $L(x)=0$ is irreducible, then $G$ is reductive and
we have the following result.

######  Theorem 1.

[Compoint’s theorem for the algebraic case [22, Theorem 2][19, Corollary 4.2]]
Let $L(y)=0$ be an irreducible linear ordinary differential equation. If the
differential Galois group of $L(y)=0$ is finite, then the ideal $I$ is
generated by the $G$-invariants contained in it. In particular, if
$P_{1},\ldots,P_{N}\in\mathbb{C}[Y_{1},\ldots,Y_{n}]$ is a set of generators
of the $G$-invariant subring $\mathbb{C}[Y_{1},\ldots,Y_{n}]^{G}$, then

$I=\langle P_{1}-f_{1},\ldots,P_{N}-f_{N}\rangle,$

where $f_{i}=\Phi(P_{i})$, $i=1,\ldots,N$.

##  2 Schwarz maps for invariant curves

Let $C\subseteq\mathbb{P}^{n-1}(\mathbb{C})$ be an algebraic $G$-invariant
irreducible curve. Let $C_{0}\rightarrow C/G$ be a normalization of the
quotient curve. The following theorem shows that we can find a linear ordinary
differential equation, with coefficients in an abelian extension of
$\mathbb{C}(C_{0})$, whose Schwarz map parametrizes $C$. The result provides a
generalization of [22, Theorem 6] and [19, Procedure 4.3].

######  Theorem 2.

Let $G$ be a finite algebraic subgroup of $GL_{n}(\mathbb{C})$ and let
$C\subseteq\mathbb{P}^{n-1}(\mathbb{C})$ be an algebraic $G$-invariant
irreducible curve not contained in a projective hyperplane. Let
$\sigma:C_{0}\rightarrow C/G$ be a normalization. Let $K=\mathbb{C}(C_{0})$ be
the field of meromorphic functions of $C_{0}$. Then there exist (see Figure 1)

  * i)

a branched cover $\pi:C_{1}\rightarrow C_{0}$ where $C_{1}$ is a compact
Riemann surface and $K\subseteq E=\mathbb{C}(C_{1})$ is an abelian extension;

  * ii)

a linear differential equation $L(y)=0$ of order $n$ with coefficients in $E$
whose Schwarz map parametrizes $C$ and the quotient Schwarz map is
$\psi=\sigma\circ\pi$.

---
$\textstyle{C\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\imath}}$$\textstyle{\mathbb{P}^{n-1}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\mathbf{y}]}$$\scriptstyle{\pi}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\simeq}$$\textstyle{C/G\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\imath}$$\textstyle{\mathbb{P}^{n-1}(\mathbb{C})/G}$
Figure 1: Diagram for Theorem 2

###### Proof.

We recall our notation from section 1.1. The invariant ring
$R^{G}=\mathbb{C}[X_{1},\ldots,X_{n}]^{G}$, is generated by the homogeneous
polynomials $F_{1},\ldots,F_{N}$. The polynomials $\Phi_{1},\ldots,\Phi_{M}$
form a basis of the homogeneous coordinate system $R^{G}_{\Lambda}$ for
$\mathbb{P}^{n-1}(\mathbb{C})/G$.

For $j=1,\ldots,M$, let $r_{j,1},\ldots,r_{j,N}\in\mathbb{Z}_{\geq 0}$ be such
that
$r_{j,1}\deg\left(F_{1}\right)+\ldots+r_{j,N}\deg\left(F_{N}\right)=\Lambda$
and

$\Phi_{j}=\prod_{i=1}^{N}F_{i}^{r_{j,i}}.$

Let $\tilde{\imath}:C\rightarrow\mathbb{P}^{n-1}(\mathbb{C})$ denote the
inclusion map. Likewise, let
$\imath:C/G\rightarrow\mathbb{P}^{n-1}(\mathbb{C})/G$ be the inclusion and let
$d\in\\{1,\ldots,M\\}$ be such that $C/G\not\subseteq\\{\Phi_{d}=0\\}$. Then
there exist $\phi_{1},\ldots,\phi_{N}\in K$ such that for $i=1,\ldots,N$

$(\imath\circ\sigma)^{*}\left(\dfrac{\Phi_{i}}{\Phi_{d}}\right)=\dfrac{\phi_{i}}{\phi_{d}}.$

Let $f_{1},\ldots,f_{N}\in\overline{K}$ be such that

$\phi_{j}=\prod_{i=1}^{N}f_{i}^{r_{j,i}}.$

Then, the field $E=K(f_{1},\ldots,f_{n})$ is an abelian extension of $K$. Let
$C_{1}$ be a compact Riemann surface with field of meromorphic functions
$\mathbb{C}(C_{1})\simeq E$ and let $\pi:C_{1}\rightarrow C_{0}$ be the map
defined by the inclusion $K\subseteq E$.

Let $U\subseteq C/G$ be an open set not containing any ramification point of
$C\rightarrow C/G$, nor any singularity of $C/G$. Furthermore, we may take $U$
such that, if $U_{1}=(\sigma\circ\pi)^{-1}(U)$, then
$\sigma\circ\pi\upharpoonright_{U_{1}}$ is a bijection. Let $y_{1},\ldots
y_{n}$ be meromorphic functions on $U_{1}$ such that
$[\mathbf{y}]:z\mapsto[y_{1}(z):\ldots:y_{n}(z)]$ is a lifting of
$\sigma\circ\pi$, i.e.
$\Pi\upharpoonright_{C}\circ[\mathbf{y}]=\sigma\circ\pi$. So, for
$j=1,\ldots,M$, we have that

$\displaystyle\dfrac{\Phi_{j}}{\Phi_{d}}\circ\tilde{\imath}\circ[\mathbf{y}]$
$\displaystyle=\dfrac{\Phi_{j}}{\Phi_{d}}\circ\Pi\circ\tilde{\imath}\circ[\mathbf{y}]=\dfrac{\Phi_{j}}{\Phi_{d}}\circ\imath\circ\Pi\upharpoonright_{C}\circ[\mathbf{y}]$
$\displaystyle=\dfrac{\Phi_{j}}{\Phi_{d}}\circ\imath\circ\sigma\circ\pi=\pi^{*}\circ(\imath\circ\sigma)^{*}\left(\dfrac{\Phi_{j}}{\Phi_{d}}\right)=\pi^{*}(\dfrac{\phi_{j}}{\phi_{d}})=\dfrac{\phi_{j}}{\phi_{d}},$

and therefore

$\dfrac{\Phi_{i}}{\Phi_{d}}(y_{1},\ldots,y_{n})=\dfrac{\phi_{i}}{\phi_{d}}.$

Replacing $(y_{1},\ldots,y_{n})\leftarrow\lambda(y_{1},\ldots,y_{n})$ if
necessary, where $\lambda$ is a radical of an element in $K$, we may assume
$\Phi_{j}(y_{1},\ldots,y_{n})=\phi_{j}$ for $j=1,\ldots,M$, and furthermore,
for $i=1,\ldots,N$

$F_{i}(y_{1},\ldots,y_{n})=f_{i}.$ (1)

Let $\delta:K\rightarrow K$ be a non-trivial derivation. Differentiating
$\eqref{Fifi}$, we obtain

$\frac{\partial F_{i}}{\partial
X_{1}}(\mathbf{y})\delta(y_{1})+\ldots+\frac{\partial F_{i}}{\partial
X_{n}}(\mathbf{y})\delta(y_{n})=\delta(f_{i}),$

where $\mathbf{y}=(y_{1},\ldots,y_{n})$. Since the derivative of
$\mathbf{x}\mapsto\Big{(}F_{1}(\mathbf{x}),\ldots,F_{N}(\mathbf{x})\Big{)}$ is
non-singular in a dense open set of $\mathbb{C}^{n}$, after rearranging
indices if necessary, we have that

$\frac{\partial(F_{1},\ldots,F_{n})}{\partial(X_{1},\ldots,X_{n})}(\mathbf{x})=\left[\begin{array}[]{ccc}\partial
F_{1}/\partial X_{1}&\ldots&\partial F_{1}/\partial X_{n}\\\
\vdots&\ddots&\vdots\\\ \partial F_{n}/\partial X_{1}&\ldots&\partial
F_{n}/\partial X_{n}\end{array}\right](\mathbf{x})$

is invertible in that set. Whenever
$\det\Big{(}\frac{\partial(F_{1},\ldots,F_{n})}{\partial(X_{1},\ldots,X_{n})}\Big{)}(\mathbf{y}(z))\neq
0$ for $z\in U_{1}$, we obtain a system of first order ordinary differential
equations characterizing the functions $y_{1},\ldots,y_{n}$

$\delta\left[\begin{array}[]{c}y_{1}\\\ \vdots\\\
y_{n}\end{array}\right]=\left[\frac{\partial(F_{1},\ldots,F_{n})}{\partial(X_{1},\ldots,X_{n})}(\mathbf{y})\right]^{-1}\
\delta\left[\begin{array}[]{c}f_{1}\\\ \vdots\\\ f_{n}\end{array}\right].$ (2)

Differentiating this system $n-1$ times, we obtain expressions for
$\delta^{k}\mathbf{y}$, $k=2,\ldots,n$, in terms of the derivatives of
$f_{1},\ldots,f_{n}$ and the partial derivatives of $F_{1},\ldots,F_{n}$. The
system (2) is $G$-invariant and therefore so are the linear dependence
relations

$c_{0}\mathbf{y}+c_{1}\delta\mathbf{y}+\quad\ldots\quad+c_{n-1}\delta^{n-1}\mathbf{y}+c_{n}\delta^{n}\mathbf{y}=0,$

hence the $n+1$ coefficients $c_{0},\ldots,c_{n}$ are in the field

$\left(F_{i}(\mathbf{y}),\delta^{k}f_{j}\Big{|}\
i,j,k=1,\ldots,n\right)\subseteq\mathbb{C}\left(\delta^{k}f_{i}\Big{|}\
i=1,\ldots,N,\ k=0,1,\ldots,n\right)\subseteq E.$

We have that $c_{n}\neq 0$ because $C$ is not contained in a projective
hyperplane. So, the linear differential equation

$\delta^{n}(y)+\frac{c_{n-1}}{c_{n}}\delta^{n-1}(y)+\ldots+\frac{c_{1}}{c_{n}}\delta(y)+\frac{c_{0}}{c_{n}}y=0$

has coefficients in $E$ and $(y_{1},\ldots,y_{n})$ as a basis of solutions. By
construction, the analytic extension of
$z\mapsto[\mathbf{y}]=[y_{1}(z):\ldots:y_{n}(z)]$ is a Schwarz map that
parametrizes $C$ and the quotient Schwarz map is $\sigma\circ\pi$. ∎

### The algorithm

We present an algorithm that computes the equation $L(y)=0$ from Theorem 2. We
assume that $K\subseteq\overline{\mathbb{Q}}(z)$ is a field such that we can
compute Gröbner basis on $K[X_{1},\ldots,X_{n}]$. Furthermore, we also assume
that the homogeneous generators $F_{1},\ldots,F_{N}$ of
$\mathbb{C}[X_{1},\ldots,X_{n}]^{G}$ have coefficients in
$\mathbb{K}=K\cap\overline{\mathbb{Q}}$ and that they are indexed so that
$\det\Big{(}\frac{\partial(F_{1},\ldots,F_{n})}{\partial(X_{1},\ldots,X_{n})}\Big{)}\neq
0$, as in the case where $F_{1},\ldots,F_{n}$ form a homogeneous system of
parameters of $\mathbb{C}[X_{1},\ldots,X_{n}]^{G}$. Details on Gröbner bases
and homogeneous system of parameters can be found in [6].

We take $\delta=d/dz$.

Algorithm
Input: $f_{1},\ldots,f_{N}\in K$ such that

  * i)

$I=\langle F_{1}-f_{1},\ldots,F_{N}-f_{N}\rangle$ is a maximal ideal of
$K[X_{1},\ldots,X_{n}]$, and

  * ii)

the projective curve $C\subseteq\mathbb{P}^{n-1}(\mathbb{C})$ defined by the
homogeneous elements in $I\cap\mathbb{K}[X_{1},\ldots,X_{n}]$ is not contained
in a projective hyperplane.

Output: $(a_{n-1},\ldots,a_{1},a_{0})\in K^{n}$ such that the linear ordinary
differential equation
$\delta^{n}(y)+a_{n-1}\delta^{n-1}(y)+\ldots+a_{1}\delta(y)+a_{0}y=0$ has a
fundamental system of solutions whose associated Schwarz map parametrizes $C$.

  1. 1.

Fix an admissible ordering on the monomials on $X_{1},\ldots,X_{n}$ and
construct a Gröbner basis $B$ of $\langle
F_{1}-f_{1},\ldots,F_{N}-f_{N}\rangle$.

  2. 2.

For $i=1,\ldots,n$, set $Y_{0,i}:=X_{i}$. Denote
$\mathbf{Y}_{0}=(Y_{0,1},\ldots,Y_{0,n})^{\intercal}$.

  3. 3.

Let $\mathbf{J}^{-1}$ be the remainder by $B$ of an inverse $\bmod I$ of the
jacobian matrix $\partial(F_{1},\ldots,F_{n})/\partial(X_{1},\ldots,X_{n})$.

  4. 4.

Let $Y_{1,1},\ldots,Y_{1,n}$ be such that $\mathbf{Y}_{1}$ is the remainder by
$B$ of $\mathbf{J}^{-1}\mathbf{f}$ where
$\mathbf{Y}_{1}=[Y_{1,1},\ldots,Y_{1,n}]^{\intercal}$ and
$\mathbf{f}=[f_{1},\ldots,f_{n}]^{\intercal}$.

  5. 5.

Extend $\delta$ to a derivative in $K(X_{1},\ldots,X_{n})$ by defining $\delta
X_{i}=0$ for $i=1,\ldots,n$.

  6. 6.

For $j=2,\ldots,n$ recursively let $Y_{j,1},\ldots,Y_{j,n}$ be such that, for
$i=1,\ldots,n$, $Y_{j,i}$ is the remainder by $B$ of

$\delta Y_{j-1,i}+\sum_{k=1}^{n}\dfrac{\partial{Y_{j-1,i}}}{X_{k}}Y_{1,k}.$

Denote $\mathbf{Y}_{j}=(Y_{j,1},\ldots,Y_{j,n})^{\intercal}$.

  7. 7.

Let $\mathbf{Y}^{-1}$ be the remainder by $B$ of an inverse $\bmod I$ of
$[\mathbf{Y}_{n-1}|\ldots|\mathbf{Y}_{1}|\mathbf{Y}_{0}]$.

  8. 8.

Return the remainder $\mathbf{a}$ by $B$ of $-\mathbf{Y}^{-1}\mathbf{Y}_{n}$.

We note that, as a corollary of Compoint’s theorem, we have that every
irreducible linear ordinary differential equation with coefficients in
$\mathbb{Q}(z)$ and finite differential Galois group can be obtained from the
algorithm.

##  3 The group $H_{72}^{SL_{3}}$

The group $H_{216}^{SL_{3}}$ is the preimage in $SL_{3}(\mathbb{C})$ of the
Hessian group $H_{216}\subseteq PSL_{3}(\mathbb{C})$.

As in [23], we defined the following matrices.

$S=\left[\begin{array}[]{ccc}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{array}\right],\quad T=\left[\begin{array}[]{ccc}0&1&0\\\
0&0&1\\\ 1&0&0\end{array}\right],\quad
U=\left[\begin{array}[]{ccc}\varepsilon&0&0\\\ 0&\varepsilon&0\\\
0&0&\varepsilon\omega\end{array}\right],$

and

$V=\rho\left[\begin{array}[]{ccc}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\end{array}\right].$

Here, $\varepsilon$ is a primitive $9$th root of unity. Thus,
$\varepsilon^{6}+\varepsilon^{3}+1=0$ and $\omega=-1-\varepsilon^{3}$ is a
primitive $3$rd root of unity. The scaling factor $\rho=1/(\omega-\omega^{2})$
is defined so that $\det(V)=1$. The generating matrices

The group $H_{216}^{SL_{3}}$ is generated by the matrices $S,T,U,V$. The
Hessian group has a normal subgroup of index $3$, $H_{72}$. The preimage of
this subgroup is $H_{72}^{SL_{3}}$ and it’s generated by $S$, $T$, $V$,
$UVU^{-1}$. The group $H_{72}$ has a normal subgroup of index $2$, $F_{36}$.
The preimage of this subgroup is $F_{36}^{SL_{3}}$, and it’s generated by $S$,
$T$ and $V$.

#### Invariants of $H_{72}^{SL_{3}}$

As in [20], we define $P=X_{1}X_{2}X_{3}$, $S=X_{1}^{3}+X_{2}^{3}+X_{3}^{3}$,
$Q=X_{1}^{3}X_{2}^{3}+X_{1}^{3}X_{3}^{3}+X_{2}^{3}X_{3}^{3}$,
$R=(X_{1}^{3}-X_{2}^{3})(X_{1}^{3}-X_{3}^{3})(X_{2}^{3}-X_{3}^{3})$,
$F_{6}=S^{2}-12Q$, $\Phi_{6}=S^{2}-18P^{2}-6PS$, and $F_{12}=S^{4}+216P^{3}S$.

The ring of invariants of $H_{72}^{SL_{3}}$ is

$\mathbb{C}[X_{1},X_{2},X_{3}]^{H_{72}^{SL_{3}}}=\mathbb{C}[F_{6},R,F_{12},\Phi_{6}^{2}].$

As a ring, it’s isomorphic to

$\mathbb{C}[Z_{6},Z_{9},Z_{12},X_{12}]/(T_{36}),$

where

$\displaystyle T_{36}$
$\displaystyle=(432Z_{9}^{2}+3Z_{6}Z_{12}-Z_{6}^{3})^{2}-4(X_{12}^{3}-3Z_{12}X_{12}^{2}+3Z_{12}^{2}X_{12})$
$\displaystyle=(432Z_{9}^{2}+3Z_{6}Z_{12}-Z_{6}^{3})^{2}-4X_{12}\big{(}(\xi+1)Z_{12}-X_{12}\big{)}\big{(}(\overline{\xi}+1)Z_{12}-X_{12}\big{)}$

and $\xi$ is a $6$th root of unity.

The polynomials $F_{6},R,F_{12}$ form a homogeneous system of parameters of
$\mathbb{C}[X_{1},X_{2},X_{3}]^{H_{72}^{SL_{3}}}$ [20, Théorème 4]. The
jacobian of the map $(X_{1},X_{2},X_{3})\mapsto(F_{6},R,F_{12})$ is

$\displaystyle\det\left(\frac{\partial(F_{6},R,F_{12})}{\partial(X_{1},X_{2},X_{3})}\right)$
$\displaystyle=$ $\displaystyle\left|\begin{array}[]{ccc}\partial
F_{6}/\partial X_{1}&\partial F_{6}/\partial X_{2}&\partial F_{6}/\partial
X_{3}\\\ \partial R/\partial X_{1}&\partial R/\partial X_{2}&\partial
R/\partial X_{3}\\\ \partial F_{12}/\partial X_{1}&\partial F_{12}/\partial
X_{2}&\partial F_{12}/\partial X_{3}\\\ \end{array}\right|$ $\displaystyle=$
$\displaystyle 18(F_{12}-\Phi_{6}^{2})^{2}$

The ring of invariants of $F_{36}^{SL_{3}}$ is

$\mathbb{C}[X_{1},X_{2},X_{3}]^{F_{36}^{SL_{3}}}=\mathbb{C}[F_{6},\Phi_{6},R,F_{12},\Psi_{12}],$

where $\Psi_{12}=PS^{3}+3P^{2}S^{2}-18P^{3}S$. As a ring, it’s isomorphic to

$\mathbb{C}[Z_{6},X_{6},Z_{9},Z_{12},X_{12}]/(T_{18},T_{24}),$

where

$\displaystyle T_{18}$
$\displaystyle=432Z_{9}^{2}-Z_{6}^{3}+3Z_{6}Z_{1}2-2X_{6}^{3}-36X_{6}X_{12},$
$\displaystyle T_{24}$
$\displaystyle=Z_{12}^{2}-X_{6}^{2}(Z_{12}+12X_{12})+12X_{12}^{2}$

For future reference, we remark that $\Phi_{6}$ factors into semi-invariants
of $F_{36}^{SL_{3}}$ as $\Phi_{6}=(S+\xi^{2}P)(S+\xi^{-2}P)$. The factors are
semi-invariants of order $4$, and the $\mathbb{C}$-vector space generated by
them is irreducible under the action of $H_{72}^{SL_{3}}$.

### Differential equations for $H_{72}^{SL_{3}}$

F. Beuker and G. Heckman in [4] and by M. van der Put and F. Ulmer in [26]
obtained different families of hypergeometric equations of degree $3$ with
finite differential Galois group. They have hypergeometric equations for each
finite primitive subgroup of $SL_{3}(\mathbb{C})$ but for $H_{72}^{SL_{3}}$.
We now show that there isn’t any.

From Theorem 2, to an LODE with differential Galois group isomorphic to
$H_{72}^{SL_{3}}$ corresponds a $\mathbb{C}$-algebra homomorphism
$\psi^{*}:\mathbb{C}[Z_{6},Z_{9},Z_{12},X_{12}]/(T_{36})\rightarrow\mathbb{C}[z]$
induced by the quotient Schwarz map. We denote by $f_{6}$, $r_{9}$, $f_{12}$
and $\phi_{6}^{2}$ the images of $Z_{6}$, $Z_{9}$, $Z_{12}$ and $X_{12}$
respectively.

Let’s first see that that $\phi_{6}^{2}\neq 0$. Indeed, if $\phi_{6}^{2}=0$,
we have $\Phi_{6}(\mathbf{y})=0$ for some basis of solutions
$\mathbf{y}=(y_{1},y_{2},y_{3})$ of the LODE. Whence
$(S+\xi^{2}P)(\mathbf{y})=0$ or $(S+\xi^{-2}P)(\mathbf{y})=0$, and so
$S+\xi^{2}P$ and $S+\xi^{-2}P$ are semi-invariants of the Galois group, and
therefore, the Galois group would be isomorphic to a subgroup of
$F_{36}^{SL_{3}}$ instead of $H_{72}^{SL_{3}}$.

Let’s now see that $\phi_{6}^{2}$ must have at least one zero of odd
multiplicity. For, if not, we would have $\phi_{6}\in\mathbb{C}[z]$ and, since
$\phi_{6}^{2}\neq 0$, from the syzygy $T_{18}$, we have
$\psi_{12}=(432r_{9}^{2}-f_{6}^{3}+3f_{6}f_{12}-2\phi_{6}^{3})/(36\phi_{6})\in\mathbb{C}(z)$,
therefore $\Phi_{6}$ and $\Psi_{12}$ would be invariants and, again, the
Galois group would be isomorphic to a subgroup of $F_{36}^{SL_{3}}$ instead of
$H_{72}^{SL_{3}}$.

Now, since $\psi^{*}(T_{36})=0$, we have

$\displaystyle(432r_{9}^{2}+3f_{6}f_{12}-f_{6}^{3})^{2}$
$\displaystyle=4(\phi_{6}^{6}-3f_{12}\phi_{6}^{4}+3f_{12}^{2}\phi_{6}^{2})$
$\displaystyle=4\phi_{6}^{2}\big{(}\phi_{6}^{2}-(\xi+1)f_{12}\big{)}\big{(}\phi_{6}^{2}-(\overline{\xi}+1)f_{12}\big{)}.$

The right-hand side is a square in $\mathbb{C}[z]$, and, in particular, each
of its zeroes has even multiplicity. So, if $z_{0}\in\mathbb{C}$ is a zero of
odd multiplicity of one of the three factors $\phi_{6}^{2}$,
$(\xi+1)f_{12}-\phi_{6}^{2}$ or $(\overline{\xi}+1)f_{12}-\phi_{6}^{2}$, then
it must be a zero of odd multiplicity of another of these three factor but not
of both. But since any of the three factors is a linear combination over
$\mathbb{C}$ of the two others, if $z_{0}$ is a zero of two factors, it’s a
zero of the third factor too. Thus, any zero of odd multiplicity of one of the
three factors, since it’s a zero of odd multiplicity of two factors, then it’s
a zero of even multiplicity of the third factor. Similarly, a zero of two
factors is a zero of $\phi_{6}^{2}$ and $f_{12}$ because these two values are
linear combinations over $\mathbb{C}$ of any of the two factors.

Let $z_{0}$ be a zero of odd multiplicity of $\phi_{6}^{2}$. Assume first that
$z_{0}$ is a zero of odd multiplicity of the factor
$\phi_{6}^{2}-(\xi+1)f_{12}$, and so a zero of even multiplicity of the factor
$\phi_{6}^{2}-(\overline{\xi}+1)f_{12}$. Since $\phi_{6}^{2}-(\xi+1)f_{12}$
and $\phi_{6}^{2}-(\overline{\xi}+1)f_{12}$ are conjugates, $\overline{z_{0}}$
is a zero of odd multiplicity of $\phi_{6}^{2}-(\overline{\xi}+1)f_{12}$ and a
zero of even multiplicity of $\phi_{6}^{2}-(\xi+1)f_{12}$. Therefore,
$z_{0}\neq\overline{z_{0}}$, and in particular $z_{0}\neq 0,1$. Since $z_{0}$
and $\overline{z_{0}}$ are zeroes of two of the factors, $z_{0}$ is a zero of
$\phi_{6}^{2}$ and of $f_{12}$, and so of the jacobian
$18(f_{12}-\phi_{6}^{2})^{2}$. Hence, $z_{0}$ is a singularity of the equation
different from $0$ and $1$, and so the equation is not hypergeometric. And
similarly, if we assume that $z_{0}$ is a zero of odd multiplicity of
$\phi_{6}^{2}-(\overline{\xi}+1)f_{12}$. As a corollary, we obtain that every
third order LODE with differential Galois group $H_{72}^{SL_{3}}$ and rational
coefficients has non-real singularities. Two such equations can be found in
[12] and [19].

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