# Geometry of the Wiman–Edge monodromy

Matthew Stover111This material is based upon work supported by Grant Number
DMS-1906088 from the National Science Foundation.
Temple University
<EMAIL_ADDRESS>
###### Abstract

The Wiman–Edge pencil is a pencil of genus $6$ curves for which the generic
member has automorphism group the alternating group $\mathfrak{A}_{5}$. There
is a unique smooth member, the _Wiman sextic_ , with automorphism group the
symmetric group $\mathfrak{S}_{5}$. Farb and Looijenga proved that the
monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular
group $\operatorname{SL}_{2}(\mathbb{Z}[\sqrt{5}])$. In this note, we give a
complete description of the monodromy by congruence conditions modulo $4$ and
$5$. The congruence condition modulo $4$ is new, and this answers a question
of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel
compactification of the locally symmetric manifold associated with the
monodromy is a projective surface of general type. Lastly, we give new
information about the image of the period map for the pencil.

##  1 Introduction

The Wiman–Edge pencil is a pencil of genus $6$ curves for which the generic
member has automorphism group the alternating group $\mathfrak{A}_{5}$. There
is a unique smooth member, the _Wiman sextic_ , whose automorphism group is
the symmetric group $\mathfrak{S}_{5}$. In a recent series of papers,
Dolgachev, Farb, and Looijenga [5] then Farb and Looijenga [6, 7] studied this
pencil from a modern perspective. See these papers for basic facts about the
pencil.

In [6], Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil
is commensurable with the Hilbert modular group
$\operatorname{SL}_{2}(\mathbb{Z}[\sqrt{5}])$. In this paper, we give a
complete description of the monodromy by congruence conditions modulo $4$ and
$5$. The congruence condition modulo $4$ is new, and this answers a question
of Farb–Looijenga [6, Qn. 4.4].

Before stating our main result, we need some notation. Let $\mathcal{O}$ be
the ring of integers in $\mathbb{Q}(\sqrt{5})$ and $\mathcal{O}_{o}$ denote
the index two subring $\mathbb{Z}[\sqrt{5}]$. There is a unique prime ideal
$\mathfrak{p}_{5}$ of $\mathcal{O}$ with residue field $\mathbb{F}_{5}$, the
field with five elements. The following is a precise description of the
Wiman–Edge monodromy.

######  Theorem A.

The monodromy group $\Gamma$ of the Wiman–Edge pencil can described
equivalently in $\operatorname{SL}_{2}(\mathcal{O})$ and
$\operatorname{SL}_{2}(\mathcal{O}_{o})$ by the following two properties:

  1. 1.

The reduction of $\Gamma$ modulo $\mathfrak{p}_{5}$ is unipotent.

  2. 2.

The reduction $\overline{\Gamma}$ of $\Gamma$ modulo $4\mathcal{O}$ fits into
an exact sequence

$1\to C_{4}\to\overline{\Gamma}\to\operatorname{SL}_{2}(\mathbb{F}_{2})\to 1$

where $C_{4}<\mathfrak{sl}_{2}(\mathbb{F}_{4})$ denotes the subgroup of
matrices $\left(\begin{smallmatrix}x_{1}&x_{2}\\\
x_{3}&x_{1}\end{smallmatrix}\right)\in\mathfrak{sl}_{2}(\mathbb{F}_{4})$ so
that $\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}(x_{1}+x_{2}+x_{3})=0$,
where $\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}$ denotes the trace.

The proof of Theorem A is in §3. Other than the explicit matrix generators for
$\Gamma$ given in [6], our proof that the monodromy has finite index in
$\operatorname{SL}_{2}(\mathcal{O})$ is completely independent of Farb and
Looijenga’s argument. We use a presentation for
$\operatorname{SL}_{2}(\mathcal{O})$ due to Yoshida [10] along with the
computer algebra program Magma [4] to describe $\Gamma$. It would be
interesting to understand the relationship between the condition on $\Gamma$
modulo $4$ and the geometry of the pencil.

######  Question 1.

Can one interpret the mod $4$ condition on $\Gamma$ in terms of properties of
the Wiman–Edge pencil analogous to the description of the mod
$\mathfrak{p}_{5}$ condition described in [6, Cor. 4.3]?

Our methods allow us to study other properties of the monodromy. In §4-5, we
study the smooth resolution $Y_{\Gamma}$ of the Baily–Borel compactification
of the locally symmetric manifold $X_{\Gamma}$ associated with the monodromy.
We prove:

######  Theorem B.

The smooth compactification $Y_{\Gamma}$ of $X_{\Gamma}$ is a smooth
projective surface of general type with Chern numbers:

$\displaystyle c_{1}^{2}(Y_{\Gamma})$ $\displaystyle=16$ $\displaystyle
c_{2}(\Gamma)$ $\displaystyle=56$

It has holomorphic Euler characteristic $\chi(\mathcal{O}_{Y_{\Gamma}})=6$,
irregularity $q(Y_{\Gamma})=0$, and geometric genus $p_{g}(Y_{\Gamma})=5$.

Finally, in §6, we study the period map of the monodromy, following the
suggestion of Farb–Looijenga that one should understand the image of the
period map in terms of the _Klein plane_ [6, §5]. Briefly, the period map can
be considered as a closed embedding of $\mathbb{P}^{1}$ into the Baily–Borel
compactification $Y^{\circ}$ of
$(\mathfrak{h}\times\mathfrak{h})/\operatorname{SL}_{2}(\mathcal{O}_{o})$. If
$X[2]$ denotes the quotient of $\mathfrak{h}\times\mathfrak{h}$ by the level
two congruence subgroup of $\operatorname{SL}_{2}(\mathcal{O})$ and $Y[2]$ its
smooth compactification, then there is an action of
$S_{3}=\operatorname{SL}_{2}(\mathbb{F}_{2})$ on $Y[2]$ inducing a rational
map to $Y^{\circ}$. We will show:

######  Theorem C.

The image of the period map of the Wiman–Edge pencil is determined by a unique
$S_{3}$-invariant curve of genus one on $Y[2]$.

See Proposition 6.1 for the precise statement. Hirzebruch famously identified
$Y[2]$ with the blowup of the Clebsch cubic surface at its ten Eckardt points
(e.g., see [9, §VIII.2]). The Klein plane is $\mathbb{P}^{2}$ realizing $Y[2]$
as the blowup of the plane at the sixteen points associated with an
icosahedron (where the blowup at these points is the Clebsch cubic) and a
dually inscribed dodecahedron (i.e., the points that transform to the Eckardt
points on the Clebsch cubic). In particular, we see that the image of the
period map is associated with a unique $S_{3}$-invariant plane curve of genus
one, which we call the _Wiman–Edge monodromy plane curve_.

We end the paper by proving a number of other general properties of the
monodromy plane curve, in particular regarding the combinatorics of how it
intersects the classical points and lines on $\mathbb{P}^{2}$ associated with
the icosahedron and its dual dodecahedron. We were not able to completely
describe the curve, most critically because we do not know if the monodromy
plane curve is smooth. If it is smooth, Proposition 6.6 proves that it must be
one of two smooth plane cubic curves passing through the unique $S_{3}$-orbit
of vertices of the dodecahedron with cardinality six. It would of course be
interesting to find further conditions that would precisely describe the
monodromy plane curve, and hence completely describe the image of the period
map.

We close the introduction with a question related to a curious discovery made
while writing this paper. The Wiman curve is naturally uniformized as a
quotient of the hyperbolic plane by a subgroup of the $(2,4,6)$ triangle group
(e.g., see [7, §4.3]). This implies that the Wiman curve is uniformized by a
cocompact arithmetic Fuchsian group with associated quaternion algebra the
unique quaternion algebra $A$ over $\mathbb{Q}$ ramified at the primes
$\\{2,3\\}$ (see [8, §13.3]). Classical embedding results for quaternion
algebras imply that $A$ embeds as a subalgebra of $\operatorname{M}_{2}(k)$,
since $2$ and $3$ are inert in $k$.

Specifically, there is a maximal order $\mathcal{O}$ of $A$ and a subgroup
$\Lambda$ of the group $N(\mathcal{O}^{1})$, the normalizer in
$\operatorname{SL}_{2}(\mathbb{R})$ of the group $\mathcal{O}^{1}$ of units of
norm one in $\mathcal{O}$, so that $C_{0}=\mathfrak{h}/\Lambda$ is the Wiman
curve. Let $\Lambda^{1}$ be the index two subgroup of $\Lambda$ given by
intersecting $\Lambda$ with $\mathcal{O}^{1}$. We found an embedding
$A\hookrightarrow\operatorname{M}_{2}(k)$ so that $\Lambda^{1}$ maps into
$\operatorname{SL}_{2}(\mathcal{O})$. However, the image is not contained in
$\operatorname{SL}_{2}(\mathcal{O}_{o})$, so this does not realize $\Lambda$
as a subgroup of the monodromy group $\Gamma$ of the Wiman–Edge pencil, only
some finite index subgroup of $\Lambda$. It follows that there is a finite
étale cover $C^{\prime}\to C_{0}$ and a closed immersion

$C^{\prime}\looparrowright X_{\Gamma}=(\mathfrak{h}\times\mathfrak{h})/\Gamma$

with image a totally geodesic submanifold of $X_{\Gamma}$. It would be
interesting to know if this curve $C^{\prime}$ has any significance for the
pencil or the image of the period map.

######  Question 2.

Let $D^{\prime}\subset X_{\Gamma}$ be an immersed totally geodesic curve
commensurable with the Wiman curve. Does the arithmetic Fuchsian subgroup of
the monodromy $\Gamma$ of the Wiman–Edge pencil associated with $D^{\prime}$
have special significance? What about the points of intersection of
$D^{\prime}$ with the image of the period map for the pencil?

#### Acknowledgments

I thank Benson Farb and Eduard Looijenga for a number of interesting
conversations about the Wiman–Edge pencil, and for their encouragement to
write up the results in this paper.

##  2 Basic facts and notation

We follow the basic notation in [6], which we now recall and fix for the
remainder of the paper. Set $k=\mathbb{Q}(X)$, where $X^{2}-X-1=0$. Then the
ring of integers $\mathcal{O}$ of $k$ is $\mathbb{Z}[X]$, and the unit group
$\mathcal{O}^{*}$ is isomorphic to $\mathbb{Z}\times\mathbb{Z}/2$ generated by
$X$ and $-1$, where $X^{-1}=X-1$.

In what follows $\mathbb{F}_{q}$ will always denote the field with $q$
elements. Two prime ideals of $\mathcal{O}$ that will appear throughout this
paper are $2\mathcal{O}$, which has residue field $\mathbb{F}_{4}$, and
$\mathfrak{p}_{5}=(1-2X)\mathcal{O}$, which has residue field
$\mathbb{F}_{5}$. Note that $(1-2X)^{2}=5$ which explicitly realizes the
isomorphism $k\cong\mathbb{Q}(\sqrt{5})$.

Let $\mathcal{O}_{o}$ denote the additive subgroup
$\mathbb{Z}[1,2X]=\mathbb{Z}[1,\sqrt{5}]$ and observe that
$2\mathcal{O}\subset\mathcal{O}_{o}$. The isomorphism
$\mathbb{F}_{4}\cong\mathbb{F}_{2}[X]$ characterizes $\mathcal{O}_{o}$ as the
pullback of $\mathbb{F}_{2}\subset\mathbb{F}_{4}$ under reduction modulo
$2\mathcal{O}$. This interpretation will be used often in what follows. Notice
moreover that $\mathcal{O}_{o}$ maps onto $\mathcal{O}/\mathfrak{p}_{5}$,
since it maps to a nontrivial cyclic additive subgroup of $\mathbb{F}_{5}$.

We now consider
$\operatorname{SL}_{2}(\mathcal{O}_{o})<\operatorname{SL}_{2}(\mathcal{O})$,
and begin by recording the following basic lemma that follows from the above
characterization of $\mathcal{O}_{o}$ as a subgroup of $\mathcal{O}$.

######  Lemma 2.1.

The reduction homomorphism

$h_{2}:\operatorname{SL}_{2}(\mathcal{O})\to\operatorname{SL}_{2}(\mathcal{O}/2\mathcal{O})\cong\operatorname{SL}_{2}(\mathbb{F}_{4})$

is surjective and $\operatorname{SL}_{2}(\mathcal{O}_{o})$ is the pullback of
the subgroup
$\operatorname{SL}_{2}(\mathbb{F}_{2})<\operatorname{SL}_{2}(\mathbb{F}_{4})$
induced by the field inclusion $\mathbb{F}_{2}\hookrightarrow\mathbb{F}_{4}$.

Following Yoshida [10], we now give a presentation for
$\operatorname{PSL}_{2}(\mathcal{O})$ and use it to derive one for
$\operatorname{SL}_{2}(\mathcal{O})$. First, consider the following matrices
in $\operatorname{SL}_{2}(\mathcal{O})$:

$\displaystyle\sigma$ $\displaystyle=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$
$\displaystyle\tau$ $\displaystyle=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$
$\displaystyle\mu$ $\displaystyle=\begin{pmatrix}X&0\\\ 0&X-1\end{pmatrix}$
$\displaystyle\eta$ $\displaystyle=\begin{pmatrix}1&X\\\ 0&1\end{pmatrix}$

Notice that $\langle\sigma,\tau\rangle=\operatorname{SL}_{2}(\mathbb{Z})$. Let
$\widehat{g}$ denote the image in $\operatorname{PSL}_{2}$ of a given
$g\in\operatorname{SL}_{2}$ and $[\,,\,]$ denote the commutator. Yoshida
showed:

######  Theorem 2.2 (Thm. 5.1 [10]).

The group $\operatorname{PSL}_{2}(\mathcal{O})$ is generated by
$\widehat{\sigma}$, $\widehat{\mu}$, $\widehat{\tau}$, and $\widehat{\eta}$
subject to the relations:

$\displaystyle\widehat{R}_{1}$ $\displaystyle=\widehat{\sigma}^{2}$
$\displaystyle\widehat{R}_{2}$
$\displaystyle=(\widehat{\sigma}\widehat{\tau})^{3}$
$\displaystyle\widehat{R}_{3}$
$\displaystyle=(\widehat{\sigma}\widehat{\mu})^{2}$
$\displaystyle\widehat{R}_{4}$ $\displaystyle=[\widehat{\tau},\widehat{\eta}]$
$\displaystyle\widehat{R}_{5}$
$\displaystyle=\widehat{\mu}\widehat{\tau}\widehat{\mu}^{-1}(\widehat{\tau}\widehat{\eta})^{-1}$
$\displaystyle\widehat{R}_{6}$
$\displaystyle=\widehat{\mu}\widehat{\eta}\widehat{\mu}^{-1}(\widehat{\tau}\widehat{\eta}^{2})^{-1}$
$\displaystyle\widehat{R}_{7}$
$\displaystyle=\widehat{\sigma}\widehat{\eta}\widehat{\sigma}(\widehat{\tau}\widehat{\eta}^{-1}\widehat{\sigma}\widehat{\eta}^{-1}\widehat{\mu})^{-1}$

We now use Theorem 2.2 to give a presentation for
$\operatorname{SL}_{2}(\mathcal{O})$ using the central element

$z_{0}=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}$

of order two. We then have the following.

######  Corollary 2.3.

The group $\operatorname{SL}_{2}(\mathcal{O})$ is generated by $z_{0}$,
$\sigma$, $\mu$, $\tau$, and $\eta$ subject to the relations:

$\displaystyle C_{0}$ $\displaystyle=z_{0}^{2}$ $\displaystyle C_{1}$
$\displaystyle=[z_{0},\sigma]$ $\displaystyle C_{2}$
$\displaystyle=[z_{0},\mu]$ $\displaystyle C_{3}$ $\displaystyle=[z_{0},\tau]$
$\displaystyle C_{4}$ $\displaystyle=[z_{0},\eta]$ $\displaystyle R_{1}$
$\displaystyle=\sigma^{2}z_{0}$ $\displaystyle R_{2}$
$\displaystyle=(\sigma\tau)^{3}$ $\displaystyle R_{3}$
$\displaystyle=(\sigma\mu)^{2}z_{0}$ $\displaystyle R_{4}$
$\displaystyle=[\tau,\eta]$ $\displaystyle R_{5}$
$\displaystyle=\mu\tau\mu^{-1}(\tau\eta)^{-1}$ $\displaystyle R_{6}$
$\displaystyle=\mu\eta\mu^{-1}(\tau\eta^{2})^{-1}$ $\displaystyle R_{7}$
$\displaystyle=\sigma\eta\sigma(\tau\eta^{-1}\sigma\eta^{-1}\mu)^{-1}z_{0}$

###### Proof.

The relations indicating that $z_{0}$ defines a central $\mathbb{Z}/2$
subgroup are clear. Since we have a central extension

$1\to\langle
z_{0}\rangle\cong\mathbb{Z}/2\to\operatorname{SL}_{2}(\mathcal{O})\to\operatorname{PSL}_{2}(\mathcal{O})\to
1,$

lifting each relation $\widehat{R}_{j}$ from Theorem 2.2 to a word in
$\sigma$, $\mu$, $\tau$, and $\eta$ must give either the identity or $z_{0}$.
This determines the given relation $R_{j}$.

We claim that $\operatorname{SL}_{2}(\mathcal{O})$ has no other relations.
Indeed, the abstract group $\Lambda$ with presentation in the statement of the
corollary defines a central extension of $\operatorname{PSL}_{2}(\mathcal{O})$
by $\mathbb{Z}/2$. Since these relations also hold in
$\operatorname{SL}_{2}(\mathcal{O})$, the projection
$\Lambda\to\operatorname{PSL}_{2}(\mathcal{O})$ defined by killing $z_{0}$
factors through a homomorphism onto $\operatorname{SL}_{2}(\mathcal{O})$
defined by setting the abstract generators equal to the given matrix. Since
this kernel has order two, $\operatorname{SL}_{2}(\mathcal{O})$ is isomorphic
to one of $\Lambda$ or $\operatorname{PSL}_{2}(\mathcal{O})$, and it is
definitely not isomorphic to the latter. ∎

######  Remark 2.4.

Observe that
$\operatorname{SL}_{2}(\mathcal{O})\to\operatorname{PSL}_{2}(\mathcal{O})$
does not split. One way to see this is using the well-known fact that
$\operatorname{SL}_{2}(\mathbb{Z})\to\operatorname{PSL}_{2}(\mathbb{Z})$ does
not split.

Let $\Gamma$ be the monodromy group of the Wiman–Edge pencil. We identify
$\Gamma$ with its image under the monodromy representation $\rho$ in [6],
which is a subgroup of $\operatorname{SL}_{2}(\mathcal{O}_{o})$. Farb and
Looijenga gave the following matrix generators.

######  Lemma 2.5.

The Wiman–Edge monodromy $\Gamma$ is generated by:

$\displaystyle\gamma_{\alpha}$ $\displaystyle=\begin{pmatrix}1&-1+2X\\\
0&1\end{pmatrix}$ $\displaystyle\gamma_{\alpha^{\prime}}$
$\displaystyle=\begin{pmatrix}1&0\\\ 1-2X&1\end{pmatrix}$
$\displaystyle=\tau^{-1}\eta^{2}$
$\displaystyle=\sigma\tau^{-1}\eta^{2}\sigma^{-1}$
$\displaystyle\gamma_{\beta}$ $\displaystyle=\begin{pmatrix}1+X^{3}&X^{3}\\\
-X^{3}&1-X^{3}\end{pmatrix}$ $\displaystyle\gamma_{\beta^{\prime}}$
$\displaystyle=\begin{pmatrix}1+X^{-3}&X^{-3}\\\
-X^{-3}&1-X^{-3}\end{pmatrix}$
$\displaystyle=\tau^{2}\eta^{-2}\sigma\mu^{3}\eta^{-2}\tau^{4}$
$\displaystyle=\eta^{-2}\tau^{-2}\sigma\mu^{-3}\eta^{-2}$

###### Proof.

See [6, Cor. 4.3] for the matrix representatives for the generators. One then
verifies by hand that the given words in our generators for
$\operatorname{SL}_{2}(\mathcal{O})$ multiply out to the appropriate matrix. ∎

######  Remark 2.6.

One can obtain relations for $\Gamma$ using a computer algebra program like
Magma [4]. We found the relations to be sufficiently complicated that their
inclusion wouldn’t add any value to one’s understanding of either $\Gamma$ or
the geometry of any of the objects associated with it.

##  3 Precise determination of the monodromy

The goal of this section is to give a more refined version of the following
theorem of Farb–Looijenga.

######  Theorem 3.1 (Thm. 1.1 [6]).

The monodromy group of the Wiman–Edge pencil is isomorphic to a finite index
subgroup of $\operatorname{SL}_{2}(\mathcal{O}_{o})$; in particular it is
arithmetic.

Our proof will be independent of [6] beyond our used of their matrix
generators in Lemma 2.5. Farb and Looijenga also proved that the reduction of
$\Gamma$ modulo $\mathfrak{p}_{5}$ maps onto a unipotent subgroup of
$\operatorname{SL}_{2}(\mathbb{F}_{5})$ [6, Cor. 4.3]. They asked whether this
congruence identity completely determines $\Gamma$ [6, Qn. 4.4], and our
results answer this question in the negative. In particular, we will see that
there is an additional congruence condition modulo $4\mathcal{O}$.

Before stating our precise determination of the monodromy, we need a
preliminary lemma that describes
$\operatorname{SL}_{2}(\mathcal{O}/4\mathcal{O})$ as a short exact sequence of
algebraic groups over $\mathbb{F}_{4}$. Observe first that
$\mathcal{R}_{4}=\mathcal{O}/4\mathcal{O}$ can be described as
$(\mathbb{Z}/4)[X]$ and reduction modulo $2\mathcal{R}_{4}$ is
$\mathbb{F}_{2}[X]\cong\mathbb{F}_{4}$.

######  Lemma 3.2.

There is a natural short exact sequence

$1\to\mathfrak{sl}_{2}(\mathbb{F}_{4})\to\operatorname{SL}_{2}(\mathcal{R}_{4})\to\operatorname{SL}_{2}(\mathbb{F}_{4})\to
1$

where $\mathfrak{sl}_{2}(\mathbb{F}_{4})$ denotes the Lie algebra of
$\operatorname{SL}_{2}(\mathbb{F}_{4})$, considered as an additive group. The
image $\operatorname{SL}_{2}(\mathcal{O}_{o}/4\mathcal{O})$ of
$\operatorname{SL}_{2}(\mathcal{O}_{o})$ in
$\operatorname{SL}_{2}(\mathcal{R}_{4})$ is the pullback of
$\operatorname{SL}_{2}(\mathbb{F}_{2})<\operatorname{SL}_{2}(\mathbb{F}_{4})$
under this sequence.

###### Proof.

The first statement is very standard and follows from studying matrices in
$\operatorname{SL}_{2}(\mathcal{R}_{4})$ of the form $\operatorname{Id}+2M$
and noticing that these are naturally represented by
$M\in\mathfrak{sl}_{2}(\mathbb{F}_{4})$, where we consider $\mathbb{F}_{4}$ as
$(\mathbb{Z}/2)[X]$ inside $\mathcal{R}_{4}\cong(\mathbb{Z}/4)[X]$. The second
statement follows from the fact that $\mathcal{O}_{o}$ is the pullback of
$\mathbb{F}_{2}\subset\mathbb{F}_{4}$. ∎

######  Remark 3.3.

Since $\mathbb{F}_{4}$ has characteristic $2$, it can be identified with the
additive group of matrices in $\operatorname{M}_{2}(\mathbb{F}_{4})$ of the
form $\left(\begin{smallmatrix}x_{1}&x_{2}\\\
x_{3}&x_{1}\end{smallmatrix}\right)$, i.e., matrices with trace zero. As a
group, $\mathfrak{sl}_{2}(\mathbb{F}_{4})\cong(\mathbb{Z}/2)^{6}$.

We can now restate our main result.

######  Theorem A.

The monodromy group $\Gamma$ of the Wiman–Edge pencil can described
equivalently in $\operatorname{SL}_{2}(\mathcal{O})$ and
$\operatorname{SL}_{2}(\mathcal{O}_{o})$ by the following two properties:

  1. 1.

The reduction of $\Gamma$ modulo $\mathfrak{p}_{5}$ is unipotent.

  2. 2.

The reduction $\overline{\Gamma}$ of $\Gamma$ modulo $4\mathcal{O}$ fits into
an exact sequence

$1\to C_{4}\to\overline{\Gamma}\to\operatorname{SL}_{2}(\mathbb{F}_{2})\to 1$

where $C_{4}<\mathfrak{sl}_{2}(\mathbb{F}_{4})$ denotes the subgroup of
matrices $\left(\begin{smallmatrix}x_{1}&x_{2}\\\
x_{3}&x_{1}\end{smallmatrix}\right)\in\mathfrak{sl}_{2}(\mathbb{F}_{4})$ so
that $\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}(x_{1}+x_{2}+x_{3})=0$,
where $\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}$ denotes the trace.

###### Proof.

The proof is carried out by interpreting computations done in the computer
algebra system Magma [4]. In particular, we describe how one gives arithmetic
significance to standard Magma calculations and refer the reader to [1] for
code that can be used to execute these calculations. Using the presentation
for $\operatorname{SL}_{2}(\mathcal{O})$ in Corollary 2.3 and the generators
for $\Gamma$ from Lemma 2.5, one obtains a presentation of $\Gamma$ as a
finite index subgroup of $\operatorname{SL}_{2}(\mathcal{O})$ with index
$480$.

The fact that $\Gamma$ has unipotent image modulo $\mathfrak{p}_{5}$ is
contained in [6, Cor. 4.3], but we give an alternate argument as a warm-up for
taking reductions modulo $2$ and $4$. One can check with Magma that there is a
unique normal subgroup of $\operatorname{SL}_{2}(\mathcal{O})$ with quotient
isomorphic to $\operatorname{SL}_{2}(\mathbb{F}_{5})$. Therefore this subgroup
must be the congruence subgroup
$\operatorname{SL}_{2}(\mathcal{O})[\mathfrak{p}_{5}]$ consisting of those
matrices congruent to the identity modulo $\mathfrak{p}_{5}$. The image of
$\Gamma$ in the quotient group has order $5$, which confirms our assertion,
since all subgroups of $\operatorname{SL}_{2}(\mathbb{F}_{5})$ isomorphic to
$\mathbb{Z}/5$ are unipotent.

Similarly, one sees that there are two normal subgroups of $\Gamma$ with
quotient isomorphic to $\operatorname{SL}_{2}(\mathbb{F}_{4})$, where the
second arises from its exceptional isomorphism with
$\operatorname{PSL}_{2}(\mathbb{F}_{5})$. The subgroup that does not contain
$\operatorname{SL}_{2}(\mathcal{O})[\mathfrak{p}_{5}]$ must be the congruence
subgroup $\operatorname{SL}_{2}(\mathcal{O})[2]$ of level $2\mathcal{O}$. One
checks that the image of $\Gamma$ in the quotient is isomorphic to
$\operatorname{PSL}_{2}(\mathbb{F}_{2})$. Since
$\Gamma\leq\operatorname{SL}_{2}(\mathcal{O}_{o})$, this image must be the
standard $\operatorname{SL}_{2}(\mathbb{F}_{2})$ in
$\operatorname{SL}_{2}(\mathbb{F}_{4})$.

We now study the reduction modulo $4\mathcal{O}$. From Lemma 3.2, one sees
that $\operatorname{SL}_{2}(\mathcal{R}_{4})$ has order $3840$. Magma confirms
that there is a unique normal subgroup of $\operatorname{SL}_{2}(\mathcal{O})$
with that index, hence it must be the level $4$ congruence subgroup
$\operatorname{SL}_{2}(\mathcal{O})[4]$. Note that
$\mathfrak{sl}_{2}(\mathbb{F}_{4})$ from Lemma 3.2 is the image of
$\operatorname{SL}_{2}(\mathcal{O})[2]$ under reduction modulo $4\mathcal{O}$.

One can then compute the intersection of $\mathfrak{sl}_{2}(\mathbb{F}_{4})$
with the image $\overline{\Gamma}$ of $\Gamma$, and one obtains an index two
subgroup of $\mathfrak{sl}_{2}(\mathbb{F}_{4})$. Note that the group $C_{4}$
from the statement of the theorem is also an index two subgroup, since it is
defined by a linear equation over $\mathbb{Z}/2$, again considering
$\mathbb{F}_{4}$ as $(\mathbb{Z}/2)[X]$. One does not need Magma to see that
$\mu^{3}$ reduces modulo $4\mathcal{O}$ to an element of
$\mathfrak{sl}_{2}(\mathbb{F}_{4})$ not in $C_{4}$, hence it represents the
nontrivial coset representative for $C_{4}$ in
$\mathfrak{sl}_{2}(\mathbb{F}_{4})$. One then checks in Magma that the image
of $\mu^{3}$ in $\mathfrak{sl}_{2}(\mathbb{F}_{4})$ is also not in
$\mathfrak{sl}_{2}(\mathbb{F}_{4})\cap\overline{\Gamma}$, and it follows that
$\mathfrak{sl}_{2}(\mathbb{F}_{4})\cap\overline{\Gamma}=C_{4}$, as desired.

The above verifies that $\Gamma$ satisfies the congruence conditions given in
the statement of the theorem. We must check that the above conditions
completely describe $\Gamma$ as a subgroup of
$\operatorname{SL}_{2}(\mathcal{O})$. This is equivalent to showing that
$\Gamma$ contains the congruence subgroup
$\operatorname{SL}_{2}(\mathcal{O})[4\mathfrak{p}_{5}]$. Magma can easily
check this by showing that $\Gamma$ contains the intersection of
$\operatorname{SL}_{2}(\mathcal{O})[\mathfrak{p}_{5}]$ and
$\operatorname{SL}_{2}(\mathcal{O})[4]$. This completes our description of how
one verifies the theorem in Magma. ∎

##  4 The action on $\mathfrak{h}\times\mathfrak{h}$

Let $\mathfrak{h}$ be the upper-half plane. We now study the action of
$\Gamma$ on $\mathfrak{h}\times\mathfrak{h}$ and the geometry of the quotient
space $X_{\Gamma}=(\mathfrak{h}\times\mathfrak{h})/\Gamma$. Note that it is
$\operatorname{PSL}_{2}(\mathcal{O})$ that acts faithfully on
$\mathfrak{h}\times\mathfrak{h}$, so it will be convenient to replace $\Gamma$
with its image in $\operatorname{PSL}_{2}(\mathcal{O})$.

We first notice that $X_{\Gamma}$ is a manifold, by which we mean as a
Riemannian orbifold modeled on $\mathfrak{h}\times\mathfrak{h}$, not in the
weaker sense of the analytic quasi-projective variety underlying $X_{\Gamma}$
being nonsingular. This is well-known to follow immediately from the
following.

######  Lemma 4.1.

The monodromy group $\Gamma$ is torsion-free.

###### Proof.

Suppose $\gamma\in\operatorname{SL}_{2}(\mathcal{O})$ has finite order. Then
$\gamma$ has characteristic polynomial $t^{2}-\operatorname{tr}(\gamma)t+1$
and some root of unity $\zeta$ is a root of this polynomial. This implies that
$k(\zeta)$ is at most a quadratic extension of $k$. This reduces us to

$n\in\\{2,3,4,5,6,10\\}.$

An element of order $4$ must have square $z_{0}$, and $z_{0}\notin\Gamma$, so
we can eliminate that case.

Now note that the characteristic polynomial $p_{n}(t)$ over $k$ for an element
of order $n$ is

$p_{n}(t)=\begin{cases}(t+1)^{2}&n=2\\\ t^{2}+t+1&n=3\\\ t^{2}+Xt+1&n=5\\\
t^{2}-t+1&n=6\\\ t^{2}-Xt+1&n=10\end{cases}$

(For $n=5$, and $n=10$ there is technically also the Galois conjugate
polynomial $t^{2}\pm(1-X)t+1$.) Since the reduction of $\Gamma$ modulo $1-2X$
is unipotent, we must have that the reduction of $p_{n}(t)$ modulo $1-2X$
factors as $(t-1)^{2}$. This reduces us immediately to the case $n=5$.

Then, $p_{5}(t)$ does not factor over $\mathbb{F}_{4}=\mathbb{F}_{2}(X)$. This
implies that the congruence subgroup of level $2$ in
$\operatorname{SL}_{2}(\mathcal{O})$ contains no $5$-torsion. Thus if $\Gamma$
were to contain an element of order five, then its reduction modulo $2$ would
be a nontrivial element of $\operatorname{SL}_{2}(\mathbb{F}_{4})$ of order
five. However, the reduction of $\Gamma$ is
$\operatorname{SL}_{2}(\mathbb{F}_{2})$ has order six, so it contains no
element of order five. This proves that $\Gamma$ is torsion-free. ∎

######  Remark 4.2.

Eduard Looijenga also described a nice independent proof of Lemma 4.1 using
the geometry of the Wiman–Edge pencil. Briefly, the Torelli theorem implies
that torsion in $\Gamma$ would define a curve in the pencil with an
exceptional automorphism centralizing the action of $A_{5}$, but there is no
such curve.

We also have the following standard calculation.

######  Lemma 4.3.

The space $X_{\Gamma}$ has topological Euler number $e(X_{\Gamma})=16$.

###### Proof.

Let $\overline{\Gamma}$ be the image of $\Gamma$ in
$\operatorname{PSL}_{2}(\mathcal{O})$. Since
$\operatorname{PSL}_{2}(\mathcal{O})$ acts faithfully on
$\mathfrak{h}\times\mathfrak{h}$, we have

$e(X_{\Gamma})=[\operatorname{PSL}_{2}(\mathcal{O}):\overline{\Gamma}]\,e((\mathfrak{h}\times\mathfrak{h})/\operatorname{PSL}_{2}(\mathcal{O})),$

where
$e((\mathfrak{h}\times\mathfrak{h})/\operatorname{PSL}_{2}(\mathcal{O}))$
denotes the Euler–Poincaré characteristic. However, it is a classical fact
that

$e((\mathfrak{h}\times\mathfrak{h})/\operatorname{PSL}_{2}(\mathcal{O}))=2\zeta_{k}(-1)=\frac{1}{15},$

where $\zeta_{k}$ denotes the Dedekind zeta function; e.g., see [9, Thm.
IV.1.1]. Then $\Gamma$ has index $480$ in $\operatorname{SL}_{2}(\mathcal{O})$
and intersects the center of $\operatorname{SL}_{2}(\mathcal{O})$ trivially
(e.g., by Lemma 4.1), hence
$[\operatorname{PSL}_{2}(\mathcal{O}):\overline{\Gamma}]=240$ and the lemma
follows. ∎

Let $\Delta<\operatorname{SL}_{2}(\mathcal{O})$ be the subgroup of upper-
triangular matrices and $\overline{\Delta}$ denote the image of $\Delta$ in
$\operatorname{PSL}_{2}(\mathcal{O})$. Since $k$ has class number one, it
follows that
$(\mathfrak{h}\times\mathfrak{h})/\operatorname{SL}_{2}(\mathcal{O})$ has one
cusp, i.e., $\Delta$ determines the unique conjugacy class of parabolic
subgroups of $\operatorname{SL}_{2}(\mathcal{O})$ [9, Prop. I.1.1].

Before counting the cusps of $X_{\Gamma}$, we prove an easy lemma on the
structure of $\Delta$. The result is very well-known in much greater
generality (e.g., see [9, §II.1]), but we will need some calculations related
to the proof later so we include it for the reader’s convenience.

######  Lemma 4.4.

There is a natural isomorphism
$\Delta\cong\overline{\Delta}\times\mathbb{Z}/2$, where the $\mathbb{Z}/2$
factor is generated by the center $\langle z_{0}\rangle$ of
$\operatorname{SL}_{2}(\mathcal{O})$. The group $\overline{\Delta}$ is
generated by
$\widehat{\mu},\widehat{\tau},\widehat{\eta}\in\operatorname{PSL}_{2}(\mathcal{O})$
and has presentation on these generators with relations:

$\displaystyle\widehat{S}_{0}$ $\displaystyle=[\widehat{\tau},\widehat{\eta}]$
$\displaystyle\widehat{S}_{1}$
$\displaystyle=\widehat{\mu}\widehat{\tau}\widehat{\mu}^{-1}(\widehat{\tau}\widehat{\eta})^{-1}$
$\displaystyle\widehat{S}_{2}$
$\displaystyle=\widehat{\mu}\widehat{\eta}\widehat{\mu}^{-1}(\widehat{\tau}\widehat{\eta}^{2})^{-1}$

###### Proof.

We start by presenting $\overline{\Delta}$. Notice that
$\widehat{S}_{0},\widehat{S}_{1},\widehat{S}_{2}$ are just the relations
$\widehat{R}_{4},\widehat{R}_{5},\widehat{R}_{6}$ from Theorem 2.2. It is also
clear from the semi-direct product structure on the upper-triangular subgroup
of $\operatorname{PSL}_{2}(\mathbb{R})$ that we obtain a split short exact
sequence:

$1\to\langle\widehat{\tau},\widehat{\eta}\rangle\cong\mathbb{Z}^{2}\to\overline{\Delta}\cong\langle\widehat{\tau},\widehat{\eta}\rangle\rtimes\langle\mu\rangle\to\mathbb{Z}\to
1$ (1)

In particular, one must only understand the conjugation action of
$\widehat{\mu}$ on $\widehat{\tau}$ and $\widehat{\eta}$ to obtain a
presentation for $\widehat{\Delta}$. This is precisely what is described by
the relations $\widehat{S}_{1}$ and $\widehat{S}_{2}$. Finally, one checks
that lifting to $\mu,\tau,\sigma\in\Delta$ splits the central exact sequence

$1\to\langle z_{0}\rangle\cong\mathbb{Z}/2\to\Delta\to\overline{\Delta}\to 1$

which proves that $\Delta\cong\overline{\Delta}\times\mathbb{Z}/2$. ∎

We now determine the cusps of $X_{\Gamma}$ and identify the associated cusp
subgroups of $\Gamma$ as subgroups of $\Delta$.

######  Proposition 4.5.

The space $X_{\Gamma}$ has $6$ cusps. The associated conjugacy classes
$\Delta_{1},\dots,\Delta_{6}$ of parabolic subgroups of $\Gamma$ are conjugate
in $\operatorname{SL}_{2}(\mathcal{O})$ to the following subgroups of
$\Delta$:

$\displaystyle\Delta_{1},\Delta_{2}\sim\Lambda_{8}$
$\displaystyle=\langle\mu^{-2}\eta,\eta^{2},\tau^{2}\rangle$
$\displaystyle\Delta_{3}\sim\Lambda_{24}$
$\displaystyle=\langle\mu^{6},\tau^{2},\tau\eta^{2}\rangle$
$\displaystyle\Delta_{4},\Delta_{5}\sim\Lambda_{40}$
$\displaystyle=\langle\mu^{2}\tau^{-2}\eta^{-1},\tau^{2}\eta^{6},\eta^{10}\rangle$
$\displaystyle\Delta_{6}\sim\Lambda_{120}$
$\displaystyle=\langle\mu^{6},\tau\eta^{18},\eta^{20}\rangle$

Here $\sim$ denotes conjugacy in $\operatorname{SL}_{2}(\mathcal{O})$ and
$[\Delta:\Lambda_{j}]=j$. Specifically, we have:

$\displaystyle\Delta_{1}$
$\displaystyle=(\sigma\tau\mu^{2})^{-1}\Lambda_{8}(\sigma\tau\mu^{2})$
$\displaystyle\Delta_{2}$
$\displaystyle=(\tau\sigma\tau\sigma\mu^{-1})\Lambda_{8}(\tau\sigma\tau\sigma\mu^{-1})$
$\displaystyle\Delta_{3}$
$\displaystyle=(\sigma\tau)^{-1}\Lambda_{24}(\sigma\tau)$
$\displaystyle\Delta_{4}$
$\displaystyle=(\sigma\eta)^{-1}\Lambda_{40}(\sigma\eta)$
$\displaystyle\Delta_{5}$
$\displaystyle=(\mu\sigma\tau\mu)^{-1}\Lambda_{40}(\mu\sigma\tau\mu)$
$\displaystyle\Delta_{6}$ $\displaystyle=\Lambda_{120}$

###### Proof.

Again, we explain how one deduces the result using Magma. At the risk of some
confusion, we work in $\operatorname{PSL}_{2}(\mathcal{O})$ instead of
$\operatorname{SL}_{2}(\mathcal{O})$ but discard the $\widehat{g}$ notation in
favor of just $g$ for readability. For an ideal $\mathcal{J}$ of
$\mathcal{O}$, let

$X[\mathcal{J}]=(\mathfrak{h}\times\mathfrak{h})/\operatorname{PSL}_{2}(\mathcal{O})[\mathcal{J}]$

denote the quotient of $\mathfrak{h}\times\mathfrak{h}$ by the principal
congruence subgroup of $\operatorname{PSL}_{2}(\mathcal{O})$ of level
$\mathcal{J}$.

It is an easy consequence of the orbit-stabilizer theorem that
$X[\mathcal{J}]$ has
$[\operatorname{PSL}_{2}(\mathcal{O}/\mathcal{J}):r_{\mathcal{J}}(\Delta)]$
cusps, where

$r_{\mathcal{J}}:\operatorname{PSL}_{2}(\mathcal{O})\to\operatorname{PSL}_{2}(\mathcal{O}/\mathcal{J})$

is the reduction homomorphism. For $\mathcal{J}=4\mathfrak{p}_{5}$ this index
is $240$. One can then count cusps by identifying them with right coset
representatives of $r_{\mathcal{J}}(\Delta)$ in
$\operatorname{PSL}_{2}(\mathcal{O}/\mathcal{J})$. Then $\Gamma$ acts on these
cosets through $r_{\mathcal{J}}$, and we can identify cusps of $X_{\Gamma}$
with $\Gamma$-orbits under this action.

Specifically, if we write

$\operatorname{PSL}_{2}(\mathcal{O}/\mathcal{J})=\coprod_{j=1}^{240}r_{\mathcal{J}}(\Delta)\epsilon_{j},$

let $[\epsilon_{j}]$ be the cusp associated with $\epsilon_{j}$. Then $h\in
r_{\mathcal{J}}(\Gamma)$ acts on the right by $[\epsilon_{j}]\cdot
h=[\epsilon_{h(j)}]$, where $\epsilon_{j}h=\delta(h,j)\epsilon_{h(j)}$ for
$\delta(h,j)\in r_{\mathcal{J}}(\Delta)$. The stabilizer in
$\operatorname{PSL}_{2}(\mathcal{O}/\mathcal{J})$ of $[\epsilon_{j}]$ is
$\epsilon_{j}^{-1}r_{\mathcal{J}}(\Delta)\epsilon_{j}$, and it follows that
$\epsilon_{j_{1}}$ is in the same $r_{\mathcal{J}}(\Gamma)$-orbit as
$\epsilon_{j_{2}}$ if and only if there is some $h\in r_{\mathcal{J}}(\Gamma)$
so that $\epsilon_{j_{1}}h\epsilon_{j_{2}}^{-1}\in r_{\mathcal{J}}(\Delta)$. A
simple for loop finds that there are six orbits, hence $X_{\Gamma}$ has six
cusps.

Using an abstract presentation for $\Delta$ derived from taking the quotient
of the presentation in Lemma 4.4 by the central $\mathbb{Z}/2$, one finds
subgroups $\Lambda_{k(j)}$ of $\Delta$ for which some conjugate of
$\Lambda_{k(j)}$ in $\Delta$ is the pullback to $\Delta$ of

$\epsilon_{j}\left(\epsilon_{j}^{-1}r_{\mathcal{J}}(\Delta)\epsilon_{j}\cap
r_{\mathcal{J}}(\Gamma)\right)\epsilon_{j}^{-1}.$ (2)

The groups in Equation (2) are the images under $r_{\mathcal{J}}$ of
$\Lambda_{8}$, $\tau^{-1}\Lambda_{8}\tau$, $\Lambda_{24}$, $\Lambda_{40}$,
$\mu^{-1}\Lambda_{40}\mu$, and $\Lambda_{120}$. Taking a representative
$e_{j}\in\operatorname{SL}_{2}(\mathcal{O})$ for $\epsilon_{j}$, we see that
$(e_{j}g)^{-1}\Lambda_{k(j)}(e_{j}g)$ is a subgroup of $\Gamma$ representing
the cusp associated with $\epsilon_{j}$. Finding generators for each
$\Lambda_{j}$ in a standard way completes the proof. ∎

##  5 The cusp resolutions

We retain the notation of §4, and now compute the standard smooth
compactification $Y_{\Gamma}$ of $X_{\Gamma}$. We refer the reader to [9, Ch.
II] for more details of the construction of the compactification. We briefly
sketch some general facts before moving on to describing each cusp resolution
for $X_{\Gamma}$ in detail. We work in $\operatorname{PSL}_{2}(\mathcal{O})$,
but discard the $\,\widehat{\,}\,$ notation for readability. At the conclusion
of this section, we will prove:

######  Theorem B.

The smooth compactification $Y_{\Gamma}$ of $X_{\Gamma}$ is a smooth
projective surface of general type with Chern numbers:

$\displaystyle c_{1}^{2}(Y_{\Gamma})$ $\displaystyle=16$ $\displaystyle
c_{2}(\Gamma)$ $\displaystyle=56$

It has holomorphic Euler characteristic $\chi(\mathcal{O}_{Y_{\Gamma}})=6$,
irregularity $q(Y_{\Gamma})=0$, and geometric genus $p_{g}(Y_{\Gamma})=5$.

Recall that we have the subgroup $\Delta=\langle\mu,\tau,\eta\rangle$ of
$\operatorname{PSL}_{2}(\mathcal{O})$. If $\Lambda\leq\Delta$ is a finite
index subgroup, then we can use the exact sequence in Equation (1) to see that

$\Lambda=\left\langle\mu^{a}y\,,\,\tau^{b_{1}}\eta^{b_{2}}\,,\,\tau^{c_{1}}\eta^{c_{2}}\right\rangle$

for $a,b_{1},b_{2},c_{1},c_{2}\in\mathbb{Z}$, some
$y\in\langle\tau,\eta\rangle$, and $a\neq 0$. Moreover, we can choose these
generators so that $\Lambda$ has index $|a(b_{1}c_{2}-b_{2}c_{1})|$ in
$\Delta$. This implies that

$T_{\Lambda}=\langle\tau^{b_{1}}\eta^{b_{2}}\,,\,\tau^{c_{1}}\eta^{c_{2}}\rangle=\Lambda\cap\langle\tau,\eta\rangle$

is the kernel of the map from $\Lambda$ to $\mathbb{Z}$ induced by Equation
(1). Note that the generators for each $\Lambda_{j}$ in the statement of
Proposition 4.5 are chosen to satisfy these properties.

We now describe how to resolve a cusp associated with $\Lambda$ as above.
Consider the standard embedding of $k$ into $\mathbb{R}^{2}$ using its two
archimedean places, which induces a discrete embedding of
$T_{\Lambda}\leq\mathbb{Z}[X]$ into $\mathbb{R}^{2}$. Let $T_{\Lambda}^{+}$ be
the elements of $T_{\Lambda}$ lying in the first quadrant of $\mathbb{R}^{2}$
and $\mathcal{C}_{\Lambda}$ be the convex hull of $T_{\Lambda}$.

Then $\mu^{a}y$ acts on $T_{\Lambda}$ by conjugation and preserves
$T_{\Lambda}^{+}$, hence it acts on $\mathcal{C}_{\Lambda}$. The action on the
boundary $\mathcal{B}_{\Lambda}$ of $\mathcal{C}_{\Lambda}$ determines a
combinatorial $n$-gon, where we associate edges of the $n$-gon with
$(\mu^{a}y)$-orbits of vertices of $\mathcal{B}_{\Lambda}$ and vertices of the
$n$-gon with edges of $\mathcal{B}_{\Lambda}$. See Figure 1 for a general
picture. Each edge of this $n$-gon represents a smooth rational curve in the
compactification.

$\mu^{a}z$ Figure 1: Compactification by a square of rational curves

It remains to determine the self-intersection of the rational curve $E_{v}$
associated with a vertex $v$ on $\mathcal{B}_{\Lambda}$. The two adjacent
vertices $v_{1},v_{-1}$ of $\mathcal{B}_{\Lambda}$ have the property that, as
elements of $\mathbb{Z}[X]$, $v_{1}+v_{-1}=bv$ for some integer $b>1$, and we
have $E_{v}^{2}=-b$. See [9, §II.2].

Finally, we note that how one resolves a cusp associated with some
$\Lambda\leq\Delta$ is independent of its conjugacy class in $\Delta$. In
particular, to resolve the cusps of $X_{\Gamma}$, it suffices to study the
groups $\Lambda_{j}$ in Proposition 4.5. The remainder of this section studies
these resolutions, and Mathematica code to assist the reader in verifying
assertions about various convex hulls is available on the author’s webpage
[2].

###  5.1 Degree $8$ cusps

Here we study the resolution of a cusp associated with the group

$\Lambda_{8}=\langle\mu^{-2}\eta,\eta^{2},\tau^{2}\rangle.$

In the above notation,
$T_{\Lambda}=\langle\eta^{2},\tau^{2}\rangle=2\mathcal{O}$ and $\mu^{-2}\eta$
has matrix

$\varphi=\begin{pmatrix}5&-3\\\ -3&2\end{pmatrix}$

for its action on $T_{\Lambda}$. The convex hull of $T_{\Lambda}^{+}$ is
pictured in Figure 2. Since $2$ is mapped to $4-6X$ under $\varphi$, the cusp
resolution is by a bigon of rational curves. Then

$\displaystyle(10-6X)+2$ $\displaystyle=3(4-2X)$ $\displaystyle(4-2X)+(2+2X)$
$\displaystyle=3(2)$

which implies that both rational curves in the compactification have self-
intersection $-3$.

$2$$4-2X$$2+2X$$10-6X$$4+6X$$26-16X$$10+16X$ Figure 2: Resolving the index $8$
cusps by two $-3$ curves

###  5.2 Degree $24$ cusp

We now study the resolution of a cusp associated with the group

$\Lambda_{24}=\langle\mu^{6},\tau^{2},\tau\eta^{2}\rangle.$

In the above notation, $T_{\Lambda}=\langle\tau^{2},\tau\eta^{2}\rangle$ and
$\mu^{6}$ has matrix

$\varphi=\begin{pmatrix}17&36\\\ 144&305\end{pmatrix}$

for its action on $T_{\Lambda}$. The convex hull of $T_{\Lambda}^{+}$ is
pictured in Figure 3. Then $\varphi$ sends $26-16X$ to $10+16X$, which implies
that the compactification is by a $16$-gon of rational curves. For any
adjacent vertices of the boundary of the convex hull of $T_{\Lambda}^{+}$ that
lie on a straight line, the associated curve has self-intersection $-2$,
since:

$\displaystyle(2+j-1+2(j-1)X)+(2+j+1+2(j+1)X)$ $\displaystyle=2(2+j+2jX)$
$\displaystyle(2+3(j-1)-2(j-1)X)+(2+3(j+1)-2(j+1)X)$
$\displaystyle=2(2+3j-2jX)$

There are fourteen such curves.

$(2+j)+2jX\quad j=0,\dots,8$$(2+3j)-2jX\quad j=0,\dots,8$ Figure 3: Resolving
the index $24$ cusp by sixteen curves

It remains to compute the self-intersections of the curves associated with $2$
and $10+16X$. In these cases,

$\displaystyle(3+2X)+(5-2X)$ $\displaystyle=4(2)$
$\displaystyle(9+14X)+(31+50X)$ $\displaystyle 4(10+16X)$

which implies that the remaining two curves have self-intersection $-4$. In
summary, the cusp associated with $\Lambda_{24}$ is compactified by a $16$-gon
of rational curves for which one pair of opposite sides determine curves with
self-intersection $-4$ and the remaining curves have self-intersection $-2$.

###  5.3 Degree $40$ cusps

We assume the reader is becoming familiar with the nature of these
calculations and start to skip some details. Considering the convex hull in
Figure 4, for $\Lambda_{40}$ we have that $\mu^{2}\tau^{-2}\eta^{-1}$ sends
$6-2X$ to $6+8X$. One then calculates that our cusp is compactified by a bigon
of $-3$ curves.

$14+22X$$6+8X$$4+2X$$6-2X$$14-8X$$36-22X$ Figure 4: Resolving the index $40$
cusps by two $-3$ curves

###  5.4 Degree $120$ cusp

This case is completely analogous to the $\Lambda_{24}$ case, and the cusp is
compactified by a $16$-gon consisting of two $-4$ curves and fourteen $-2$
curves. Again, the $-4$ curves are on opposite sides of the $16$-gon. The
boundary of the convex hull of $T_{\Lambda}^{+}$ is given by the points:

$\displaystyle(6+j)+(8-2j)X$ $\displaystyle 0\leq j\leq 8$
$\displaystyle(6+11j)+(8+18j)X$ $\displaystyle 0\leq j\leq 8$

The lists overlap at $j=0$, and the point $6+8X$ represents one of the curves
of self-intersection $-4$. At $j=8$, we have $X^{12}(14-8X)=94+152X$, which
implies that $\mu^{6}$ identifies these two points; this represents the other
curve of self-intersection $-4$. The picture of the convex hull is exactly as
in Figure 3, appropriately relabeled.

###  5.5 Proof of Theorem B

The computation of the Chern numbers of $Y_{\Gamma}$ follows from our above
calculations of the cusp resolutions, Lemma 4.3, and the formulas on [9,
p.63]. It is well known that $q(Y_{\Gamma})=0$ for any compactification of a
quotient of $\mathfrak{h}\times\mathfrak{h}$ by an irreducible lattice (e.g.,
see [9, Lem. I.6.3]). We then compute the remaining numerical invariants by:

$\displaystyle\chi(\mathcal{O}_{Y_{\Gamma}})$
$\displaystyle=\frac{1}{12}\left(c_{1}^{2}(Y_{\Gamma})+c_{2}(Y_{\Gamma})\right)$
$\displaystyle=1-q(Y_{\Gamma})+p_{g}(Y_{\Gamma})$

Since $Y_{\Gamma}$ is smooth and projective, it remains to show that
$Y_{\Gamma}$ is of general type. If it is minimal, this is clear from the
possibilities for $c_{1}^{2}$ and $c_{2}$ in the Enriques–Kodaira
classification [3, Ch. VI]. However, performing a sequence of blowdowns on
$Y_{\Gamma}$ only increases $c_{1}^{2}$, and a minimal surface with
$c_{1}^{2}\geq 16$ must be of general type. This completes the proof. ∎

##  6 Image of the period map

In this section, we study the image of the period map. Let
$\mathcal{B}^{\circ}$ be the base of the smooth locus of the pencil. Following
Farb and Looijenga [6, §5.1], we consider the period map $\Pi^{\circ}$ as a
map from $\mathcal{B}^{\circ}$ to

$X^{\circ}=(\mathfrak{h}\times\mathfrak{h})/\operatorname{SL}_{2}(\mathcal{O}_{o})$

through the orbifold covering $X_{\Gamma}\to X^{\circ}$. It is known that
$\Pi^{\circ}$ is a closed embedding by [6, Prop. 5.1].

As described in [6], it is natural to consider $X^{\circ}$ as the quotient of

$X[2]=(\mathfrak{h}\times\mathfrak{h})/\operatorname{SL}_{2}(\mathcal{O})[2]$

by an action of $\operatorname{SL}_{2}(\mathbb{F}_{2})$, where
$\operatorname{SL}_{2}(\mathcal{O})[2]$ denotes the level two congruence
subgroup of $\operatorname{SL}_{2}(\mathcal{O})$. We use the classical
connection between $X[2]$ and the Clebsch cubic surface to study the image of
the period map. In other words, we will study the diagram of maps in Figure 5,
where $X_{\Gamma}[2]$ denotes the minimal common covering of $X_{\Gamma}$ and
$X[2]$ and $\widetilde{\mathcal{B}}^{\circ}$ is the induced covering of
$\mathcal{B}^{\circ}$.

${X^{\circ}}$${X_{\Gamma}}$${\mathcal{B}^{\circ}}$${X[2]}$${X_{\Gamma}[2]}$${\widetilde{\mathcal{B}}^{\circ}}$$\scriptstyle{\Pi^{\circ}}$
Figure 5: Diagram of coverings for the period map

###  6.1 The curve on X[2]

Our first goal is to show that the image of the period map is associated with
a (possibly singular) curve on $X[2]$ with genus one and $18$ punctures. We
will then use the famous realization of a smooth compactification of $X[2]$ as
a blowup of $\mathbb{P}^{2}$ to show that the image of the period map is
determined by a plane curve of genus one that is invariant under an action of
the symmetric group $S_{3}$. The following result immediately implies Theorem
C.

######  Proposition 6.1.

With notation as in Figure 5, $X_{\Gamma}[2]$ is a Galois cover of
$X_{\Gamma}$ with group $\operatorname{SL}_{2}(\mathbb{F}_{2})$. The cover
$\widetilde{\mathcal{B}}^{\circ}\to\mathcal{B}^{\circ}$ is also Galois with
group $\operatorname{SL}_{2}(\mathbb{F}_{2})$, and
$\widetilde{\mathcal{B}}^{\circ}$ has genus one with $18$ punctures. The
diagram of maps in Figure 5 induces a closed embedding
$\widetilde{\Pi}^{\circ}:\widetilde{\mathcal{B}}^{\circ}\to X[2]$.

###### Proof.

The cover $X[2]\to X^{\circ}$ is Galois with group
$\operatorname{SL}_{2}(\mathbb{F}_{2})$ since
$\operatorname{SL}_{2}(\mathcal{O}_{o})$ is the pull-back of
$\operatorname{SL}_{2}(\mathbb{F}_{2})$ to
$\operatorname{SL}_{2}(\mathcal{O})$ under reduction modulo $2\mathcal{O}$. By
Theorem A, the monodromy $\Gamma$ also has image
$\operatorname{SL}_{2}(\mathbb{F}_{2})$ under reduction modulo $2\mathcal{O}$.
Elementary covering space theory then implies that the minimal common cover
$X_{\Gamma}[2]$ is a Galois cover of $X_{\Gamma}$ with group
$\operatorname{SL}_{2}(\mathbb{F}_{2})$ that makes the left-most square in
Figure 5 into a commutative square of covers.

The map $\pi_{1}(\mathcal{B}^{\circ})\to\Gamma$ is onto, hence the cover
$\widetilde{\mathcal{B}}^{\circ}$ of $\mathcal{B}^{\circ}$ induced by
$X_{\Gamma}[2]$ is connected with group
$\operatorname{SL}_{2}(\mathbb{F}_{2})$. Since $\Pi^{\circ}$ is a closed
embedding, we see that the induced map $\widetilde{\Pi}^{\circ}$ from
$\widetilde{\mathcal{B}}^{\circ}$ to $X[2]$ must also be a closed embedding.
It remains to show that $\widetilde{\mathcal{B}}^{\circ}$ is genus one with
$18$ punctures.

Using the generators for $\pi_{1}(\mathcal{B}^{\circ})$ in Lemma 2.5 and
defining reduction of $\Gamma$ modulo $2\mathcal{O}$ as a homomorphism to
$\operatorname{SL}_{2}(\mathbb{F}_{2})$, the number of lifts of each puncture
of $\mathcal{B}^{\circ}$ to $\widetilde{\mathcal{B}}^{\circ}$ is the index in
$\operatorname{SL}_{2}(\mathbb{F}_{2})$ of the subgroup generated by the image
of an element associated with a loop around that puncture. The images of
$\gamma_{\alpha}$, $\gamma_{\alpha^{\prime}}$, $\gamma_{\beta}$,
$\gamma_{\beta^{\prime}}$ all have order $2$, so each associated puncture has
three lifts. The element
$\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha^{\prime}}\gamma_{\beta^{\prime}}$
is the identity modulo $2\mathcal{O}$, so the associated puncture has six
lifts. We then have that $\widetilde{\mathcal{B}}^{\circ}$ has Euler
characteristic $-18$ and $18$ punctures, so it has genus one. This completes
the proof. ∎

###  6.2 The Clebsch cubic and the Klein plane

We now describe the smooth compactification $Y[2]$ of $X[2]$ and its
relationship to $\mathbb{P}^{2}$. See [9, §VIII.2] for details. The space
$Y[2]$ is the blowup of the famous Clebsch cubic surface at its ten Eckardt
points, and the map

$\sigma:Y[2]\to\mathbb{P}^{2}$

can be described as blowing up $\mathbb{P}^{2}$ at the vertices of an
icosahedron and its dually inscribed dodecahedron. Farb and Looijenga call the
image $\mathbb{P}^{2}$ the _Klein plane_. In the remainder of this subsection
we identify a number of the beautiful geometric features of this construction
that we will use in studying the image of the period map.

####  6.2.1 Icosahedron vertices and Eckardt points

Blowing up $\mathbb{P}^{2}$ at the six points associated with the vertices of
the icosahedron gives the Clebsch surface. The ten points associated with
vertices of the dodecahedron lift to the Eckardt points on the Clebsch
surface. On $\mathbb{P}^{2}$, we denote the six icosahedral vertex points as
$v_{1},\dots,v_{6}$ and the dodecahedral vertex points associated with Eckardt
points as $e_{1},\dots,e_{10}$. See Figures 6 and 7 for coordinates.

$\displaystyle v_{1}$ $\displaystyle=\begin{bmatrix}0\\\ 1\\\ X\end{bmatrix}$
$\displaystyle v_{3}$ $\displaystyle=\begin{bmatrix}X\\\ 0\\\ 1\end{bmatrix}$
$\displaystyle v_{5}$ $\displaystyle=\begin{bmatrix}1\\\ X\\\ 0\end{bmatrix}$
$\displaystyle v_{2}$ $\displaystyle=\begin{bmatrix}0\\\ 1\\\ -X\end{bmatrix}$
$\displaystyle v_{4}$ $\displaystyle=\begin{bmatrix}-X\\\ 0\\\ 1\end{bmatrix}$
$\displaystyle v_{6}$ $\displaystyle=\begin{bmatrix}1\\\ -X\\\ 0\end{bmatrix}$
Figure 6: Vertices of the icosahedron $\displaystyle e_{1}$
$\displaystyle=\begin{bmatrix}1\\\ 1\\\ 1\end{bmatrix}$ $\displaystyle e_{3}$
$\displaystyle=\begin{bmatrix}1\\\ -1\\\ 1\end{bmatrix}$ $\displaystyle e_{5}$
$\displaystyle=\begin{bmatrix}X-1\\\ 0\\\ X\end{bmatrix}$ $\displaystyle
e_{7}$ $\displaystyle=\begin{bmatrix}0\\\ X\\\ X-1\end{bmatrix}$
$\displaystyle e_{9}$ $\displaystyle=\begin{bmatrix}X\\\ X-1\\\
0\end{bmatrix}$ $\displaystyle e_{2}$ $\displaystyle=\begin{bmatrix}-1\\\ 1\\\
1\end{bmatrix}$ $\displaystyle e_{4}$ $\displaystyle=\begin{bmatrix}-1\\\
-1\\\ 1\end{bmatrix}$ $\displaystyle e_{6}$
$\displaystyle=\begin{bmatrix}1-X\\\ 0\\\ X\end{bmatrix}$ $\displaystyle
e_{8}$ $\displaystyle=\begin{bmatrix}0\\\ X\\\ 1-X\end{bmatrix}$
$\displaystyle e_{10}$ $\displaystyle=\begin{bmatrix}X\\\ 1-X\\\
0\end{bmatrix}$ Figure 7: Vertices of the dodecahedron

As a Hilbert modular surface, the exceptional curves on $Y[2]$ arising from
the Eckardt points are the compactifications of the ten lifts of
$\mathfrak{h}/\operatorname{PSL}_{2}(\mathbb{Z})$ to $X[2]$. Notice that each
lift is the thrice-punctured sphere arising from the quotient of
$\mathfrak{h}$ by the level $2$ congruence subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$.

####  6.2.2 The compactification curves

Recall that the cusps of $X[2]$ are in one-to-one correspondence with elements
of the projective like $\mathbb{P}^{1}(\mathbb{F}_{4})$ over $\mathbb{F}_{4}$,
or equivalently with the $\operatorname{SL}_{2}(\mathcal{O})[2]$-orbits of
lifts of these lines to $\mathbb{P}^{1}(k)$. Each of the five cusps of $X[2]$
is compactified by a triangle of lines, where each line has self-intersection
$-3$. Each line is the proper transform of the line of $\mathbb{P}^{2}$
associated with a pair of opposite edges of the icosahedron (or, dually, the
dodecahedron). In particular, the image on $\mathbb{P}^{2}$ of each cusp line
on $Y[2]$ meets exactly two of the six icosahedral vertices and exactly two of
the dodecahedral vertices. We denote the line through $e_{i}$ and $e_{j}$ by
$\ell_{i,j}$. See Figure 8 for an explicit parametrization of each line
$\ell_{i,j}$ and Figure 9 for which $v_{m}$ lies on each $\ell_{i,j}$.

$\displaystyle\ell_{i,j}=\Bigg{\\{}$ $\displaystyle re_{i}+se_{j}\ :\
\begin{bmatrix}r\\\ s\end{bmatrix}\in\mathbb{P}^{1}\Bigg{\\}}$
$\displaystyle\\{i,j\\}\in\big{\\{}$
$\displaystyle\\{1,5\\}\,,\,\\{1,7\\}\,,\,\\{1,9\\}\,,\,\\{2,6\\}\,,\,\\{2,7\\}\,,\,\\{2,10\\}\,,\,$
$\displaystyle\\{3,5\\}\,,\,\\{3,8\\}\,,\,\\{3,10\\}\,,\,\\{4,6\\}\,,\,\\{4,8\\}\,,\,\\{4,9\\}\,,\,$
$\displaystyle\\{5,6\\}\,,\,\\{7,8\\}\,,\,\\{9,10\\}\big{\\}}$ Figure 8: Edges
of the dodecahedron $\displaystyle v_{2},v_{5}$ $\displaystyle\in\ell_{1,5}$
$\displaystyle v_{3},v_{6}$ $\displaystyle\in\ell_{1,7}$ $\displaystyle
v_{1},v_{4}$ $\displaystyle\in\ell_{1,9}$ $\displaystyle v_{2},v_{6}$
$\displaystyle\in\ell_{2,6}$ $\displaystyle v_{4},v_{5}$
$\displaystyle\in\ell_{2,7}$ $\displaystyle v_{1},v_{3}$
$\displaystyle\in\ell_{2,10}$ $\displaystyle v_{1},v_{6}$
$\displaystyle\in\ell_{3,5}$ $\displaystyle v_{3},v_{5}$
$\displaystyle\in\ell_{3,8}$ $\displaystyle v_{2},v_{4}$
$\displaystyle\in\ell_{3,10}$ $\displaystyle v_{1},v_{5}$
$\displaystyle\in\ell_{4,6}$ $\displaystyle v_{4},v_{6}$
$\displaystyle\in\ell_{4,8}$ $\displaystyle v_{2},v_{3}$
$\displaystyle\in\ell_{4,9}$ $\displaystyle v_{3},v_{4}$
$\displaystyle\in\ell_{5,6}$ $\displaystyle v_{1},v_{2}$
$\displaystyle\in\ell_{7,8}$ $\displaystyle v_{5},v_{6}$
$\displaystyle\in\ell_{9,10}$ Figure 9: Vertices of the icosahedron on the
edge $\ell_{i,j}$

####  6.2.3 The $\operatorname{PSL}_{2}(\mathbb{F}_{2})$ action

The automorphism group of the Clebsch surface is the symmetric group $S_{5}$,
and the induced action on $Y[2]$ restricts to the automorphism group of
$X[2]$. This connects to the action of
$\operatorname{PSL}_{2}(\mathbb{F}_{4})$ on $X[2]$ through its isomorphism
with the alternating group $A_{5}$. (The remaining order $2$ automorphism of
$X[2]$ needed to generate $S_{5}$ is induced by the “swap map” exchanging the
two factors of $\mathfrak{h}\times\mathfrak{h}$.) The inclusion of
$\operatorname{PSL}_{2}(\mathbb{F}_{2})$ into
$\operatorname{PSL}_{2}(\mathbb{F}_{4})$ is then realized as an inclusion of
$S_{3}$ in $A_{5}$. See Figure 10 for matrices generating the $S_{3}$ action
and Figures 11 and 12 for vertex and edge orbits under this action.

$\displaystyle\sigma_{1}$ $\displaystyle=\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\
0&0&1\end{bmatrix}$ $\displaystyle\sigma_{2}$
$\displaystyle=\begin{bmatrix}-\frac{X}{2}&-\frac{1}{2}&\frac{X-1}{2}\\\\[6.0pt]
-\frac{1}{2}&\frac{X-1}{2}&-\frac{X}{2}\\\\[6.0pt]
\frac{X-1}{2}&-\frac{X}{2}&-\frac{1}{2}\end{bmatrix}$ $\displaystyle\tau$
$\displaystyle=\begin{bmatrix}\frac{X}{2}&\frac{1}{2}&\frac{1-X}{2}\\\\[6.0pt]
\frac{1}{2}&\frac{1-X}{2}&\frac{X}{2}\\\\[6.0pt]
\frac{X-1}{2}&-\frac{X}{2}&-\frac{1}{2}\end{bmatrix}=\sigma_{1}\sigma_{2}$
$\displaystyle\sigma_{3}$
$\displaystyle=\tau\sigma_{1}=\sigma_{1}\sigma_{2}\sigma_{1}$

$S_{3}=\left\langle\sigma_{1},\sigma_{2},\tau\ |\
\sigma_{1}^{2},\sigma_{2}^{2},\tau^{3},\tau=\sigma_{1}\sigma_{2}\right\rangle$

Figure 10: The $S_{3}$ action $\displaystyle\sigma_{1}\quad\quad$
$\displaystyle\\{v_{1},v_{2}\\},\\{v_{3},v_{4}\\},\\{v_{5}\\},\\{v_{6}\\},$
$\displaystyle\\{e_{1},e_{4}\\},\\{e_{2},e_{3}\\},\\{e_{5},e_{6}\\},\\{e_{7},e_{8}\\},\\{e_{9}\\},\\{e_{10}\\}$
$\displaystyle\sigma_{2}\quad\quad$
$\displaystyle\\{v_{1}\\},\\{v_{2},v_{6}\\},\\{v_{3},v_{5}\\},\\{v_{4}\\},$
$\displaystyle\\{e_{1}\\},\\{e_{2},e_{6}\\},\\{e_{3},e_{8}\\},\\{e_{4},e_{10}\\},\\{e_{5},e_{7}\\},\\{e_{9}\\}$
$\displaystyle\sigma_{3}\quad\quad$
$\displaystyle\\{v_{1},v_{6}\\},\\{v_{2}\\},\\{v_{3}\\},\\{v_{4},v_{5}\\},$
$\displaystyle\\{e_{1},e_{10}\\},\\{e_{2},e_{7}\\},\\{e_{3},e_{5}\\},\\{e_{4}\\},\\{e_{6},e_{8}\\},\\{e_{9}\\}$
$\displaystyle\tau\quad\quad$
$\displaystyle\\{v_{1},v_{2}\\},\\{v_{2},v_{6}\\},\\{v_{3},v_{5}\\},$
$\displaystyle\\{e_{1},e_{4},e_{10}\\},\\{e_{2},e_{5},e_{8}\\},\\{e_{3},e_{7},e_{6}\\},\\{e_{9}\\}$
Figure 11: Vertex orbits under $S_{3}$ $\displaystyle\sigma_{1}\quad\quad$
$\displaystyle\\{\ell_{1,5},\ell_{4,6}\\},\\{\ell_{1,7},\ell_{4,8}\\},\\{\ell_{1,9},\ell_{4,9}\\},\\{\ell_{2,6},\ell_{3,5}\\},$
$\displaystyle\\{\ell_{2,7},\ell_{3,8}\\},\\{\ell_{2,10},\ell_{3,10}\\},\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{5,6}}\\},\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{7,8}}\\},\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ell_{9,10}}\\}$
$\displaystyle\sigma_{2}\quad\quad$
$\displaystyle\\{\ell_{1,5},\ell_{1,7}\\},\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ell_{1,9}}\\},\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{2,6}}\\},\\{\ell_{2,7},\ell_{5,6}\\},\\{\ell_{2,10},\ell_{4,6}\\},$
$\displaystyle\\{\ell_{3,5},\ell_{7,8}\\},\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{3,8}}\\},\\{\ell_{3,10},\ell_{4,8}\\},\\{\ell_{4,9},\ell_{9,10}\\}$
$\displaystyle\sigma_{3}\quad\quad$
$\displaystyle\\{\ell_{1,5},\ell_{3,10}\\},\\{\ell_{1,7},\ell_{2,10}\\},\\{\ell_{1,9},\ell_{9,10}\\},\\{\ell_{2,6},\ell_{7,8}\\},$
$\displaystyle\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{2,7}}\\},\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{3,5}}\\},\\{\ell_{3,8},\ell_{5,6}\\},\\{\ell_{4,6},\ell_{4,8}\\},\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ell_{4,9}}\\}$
$\displaystyle\tau\quad\quad$
$\displaystyle\\{\ell_{1,5},\ell_{4,8},\ell_{2,10}\\},\\{\ell_{1,7},\ell_{4,6},\ell_{3,10}\\},\\{\ell_{1,9},\ell_{4,9},\ell_{9,10}\\},$
$\displaystyle\\{\ell_{2,6},\ell_{3,5},\ell_{7,8}\\},\\{\ell_{2,7},\ell_{5,6},\ell_{3,8}\\}$
Figure 12: Edge orbits under $S_{3}$. Blue indicates nontrivial $\mathbb{Z}/2$
action and Red indicates trivial action.

The six icosahedron vertex points $\\{v_{i}\\}$ are partitioned into two
$S_{3}$ orbits with three elements:

$\displaystyle\mathcal{P}_{1}$ $\displaystyle=\\{v_{1},v_{2},v_{6}\\}$
$\displaystyle\mathcal{P}_{2}$ $\displaystyle=\\{v_{3},v_{4},v_{5}\\}$

The dodecahedron vertex points $\\{e_{i}\\}$ are partitioned as:

$\displaystyle\mathcal{Q}_{1}$ $\displaystyle=\\{e_{1},e_{4},e_{10}\\}$
$\displaystyle\mathcal{Q}_{2}$
$\displaystyle=\\{e_{2},e_{3},e_{5},e_{6},e_{7},e_{8}\\}$
$\displaystyle\mathcal{Q}_{3}$ $\displaystyle=\\{e_{9}\\}$

Recall that the lines on $\mathbb{P}^{2}$ associated with the edges of the
icosahedron and dodecahedron are the lines arising from the smooth
compactification of $X[2]$. These are naturally partitioned into two orbits of
triangles, one of size three and one of size two. If $\ell_{i,j}$ is the line
containing $e_{i}$ and $e_{j}$, we have orbits:

$\displaystyle\mathcal{L}_{1}$
$\displaystyle=\\{\ell_{1,5},\ell_{1,7},\ell_{2,10},\ell_{3,10},\ell_{4,6},\ell_{4,8}\\}$
$\displaystyle\mathcal{L}_{2}$
$\displaystyle=\\{\ell_{1,9},\ell_{4,9},\ell_{9,10}\\}$
$\displaystyle\mathcal{L}_{3}$
$\displaystyle=\\{\ell_{2,6},\ell_{3,5},\ell_{7,8}\\}$
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=\\{\ell_{2,7},\ell_{3,8},\ell_{5,6}\\}$

This allows us to visualize each $S_{3}$ orbit of triangles as in Figure 13
and Figure 14.

$\mathcal{L}_{2}$$\mathcal{L}_{3}$$\mathcal{L}_{4}$$\mathcal{Q}_{1}$$\mathcal{P}_{1}$$\mathcal{P}_{2}$$\mathcal{Q}_{3}$$\mathcal{Q}_{2}$$\mathcal{P}_{1}$$\mathcal{Q}_{2}$$\mathcal{P}_{1}$$\mathcal{Q}_{2}$$\mathcal{P}_{2}$$\mathcal{Q}_{2}$$\mathcal{P}_{2}$
Figure 13: The cusp triangle orbit of size three
$\mathcal{L}_{1}$$\mathcal{L}_{1}$$\mathcal{L}_{1}$$\mathcal{Q}_{1}$$\mathcal{P}_{1}$$\mathcal{Q}_{2}$$\mathcal{P}_{2}$$\mathcal{P}_{2}$$\mathcal{Q}_{1}$$\mathcal{P}_{1}$$\mathcal{Q}_{2}$$\mathcal{P}_{1}$$\mathcal{Q}_{2}$$\mathcal{P}_{2}$$\mathcal{Q}_{1}$
Figure 14: The cusp triangle orbit of size two

###  6.3 The monodromy plane curve

Since $\operatorname{PSL}_{2}(\mathbb{F}_{2})$ preserves the image of the
lifted period map $\widetilde{\Pi}^{\circ}$, we conclude that
$\widetilde{\Pi}^{\circ}(\widetilde{\mathcal{B}}^{\circ})$ is an
$S_{3}$-invariant curve of genus one on $X[2]$ with $18$ punctures. Consider
the completion

$\widetilde{\Pi}:\widetilde{\mathcal{B}}\to Y[2]$

and let $\mathcal{C}=(\sigma\circ\widetilde{\Pi})(\widetilde{\mathcal{B}})$ be
the image curve on $\mathbb{P}^{2}$. We call $\mathcal{C}$ the _(Wiman–Edge)
monodromy plane curve_. Given the maps and group actions described above, to
completely determine the image in $X^{\circ}$ of the period map for the
Wiman–Edge monodromy, it suffices to describe the monodromy plane curve.
Proposition 6.1 implies the following.

######  Corollary 6.2.

The monodromy plane curve $\mathcal{C}$ is a (possibly singular) plane curve
of genus one.

####  6.3.1 Intersection of the monodromy plane curve with $\ell_{i,j}$

We now study the intersections of $\mathcal{C}$ with each line $\ell_{i,j}$.
Since $\mathcal{C}$ is $S_{3}$-invariant, this only depends on how
$\mathcal{C}$ meets a representative for each orbit. Notice that points on
$\mathcal{C}\cap\ell_{i,j}$ associated with a vertex of the icosahedron or
dodecahedron lift to points on
$\widetilde{\Pi}^{\circ}(\widetilde{\mathcal{B}}^{\circ})$, not punctures.
Conversely, points on $\mathcal{C}\cap\ell_{i,j}$ that are not a vertex of
either the icosahedron or the dodecahedron lift to the cusp divisor for
$Y[2]$, hence they are associated with the punctures of
$\widetilde{\Pi}^{\circ}(\widetilde{\mathcal{B}}^{\circ})$; we call these
_puncture points_ on $\ell_{i,j}$. The following should be compared with the
description in [6, §5.2] of where the punctures of $\mathcal{B}^{\circ}$ go in
$X^{\circ}$.

######  Lemma 6.3.

  1. 1.

There are either one or two puncture points on each
$\ell_{i,j}\in\mathcal{L}_{1}$.

  2. 2.

Each line in $\mathcal{L}_{2}$ contains exactly three or four puncture points.

  3. 3.

Puncture points on lines in $\mathcal{L}_{3}$ or $\mathcal{L}_{4}$ are fixed
points for the $\mathbb{Z}/2$ action on the line, where $\mathbb{Z}/2$ is the
stabilizer in $S_{3}$ of the relevant triangle of lines as in Figure 13. In
particular, there are at most two puncture points on any such line.

The number of puncture points on any line $\ell_{i,j}$ is an invariant of its
$S_{3}$-orbit.

###### Proof.

The last statement in the lemma is obvious.

The orbit of $\mathcal{L}_{1}$ is associated with the cusp of
$(\mathfrak{h}\times\mathfrak{h})/\operatorname{SL}_{2}(\mathcal{O}_{o})$
denoted $\infty_{X}$ in [6, §3.3]. Indeed, this is the orbit consisting of
exactly two triangles of lines on $Y[2]$. In terms of the action of $\Gamma$
on $\mathbb{P}^{1}(k)$, $\infty_{X}$ is associated with the lines $[X:1]$ and
$[1:X]$ in $\mathbb{P}^{1}(k)$, which represent distinct cusps of
$\operatorname{SL}_{2}(\mathcal{O})[2]$ but the same cusp of
$\operatorname{SL}_{2}(\mathcal{O}_{o})$. Since
$\gamma_{\alpha}\gamma_{\beta}\gamma_{\alpha^{\prime}}\gamma_{\beta^{\prime}}$
stabilizes $[X:1]$, we see that the six relevant punctures of
$\widetilde{\Pi}^{\circ}(\widetilde{\mathcal{B}}^{\circ})$ are given by
intersections of $\mathcal{C}$ with the lines in $\mathcal{L}_{1}$.

Similarly, the orbits $\mathcal{L}_{2}$, $\mathcal{L}_{3}$, and
$\mathcal{L}_{4}$ are associated with the cusp called $\infty_{0}$ in [6,
§3.3], which is the orbit consisting of exactly three triangles of lines on
$Y[2]$. With respect to the action of $\Gamma$ on $\mathbb{P}^{1}(k)$, this
cusp is associated with the lines $[0:1]$, $[1:0]$, $[1:1]$, and $[-1:1]$ in
$\mathbb{P}^{1}(k)$, which all represent distinct cusps of
$\operatorname{SL}_{2}(\mathcal{O})[2]$ but the same cusp of
$\operatorname{SL}_{2}(\mathcal{O}_{o})$. Since $\gamma_{\alpha}$ fixes
$[1:0]$, $\gamma_{\alpha^{\prime}}$ fixes $[0:1]$, and $\gamma_{\beta}$,
$\gamma_{\beta^{\prime}}$ both fix $[-1:1]$, the twelve relevant punctures of
$\widetilde{\Pi}^{\circ}(\widetilde{\mathcal{B}}^{\circ})$ are given by
intersections of $\mathcal{C}$ with the lines in $\mathcal{L}_{2}$,
$\mathcal{L}_{3}$, and $\mathcal{L}_{4}$.

Returning to $\mathcal{L}_{1}$, the nature of the $S_{3}$ action implies that
each of the two triangles of lines represented by Figure 14 determines exactly
three punctures of $\widetilde{\Pi}^{\circ}(\widetilde{\mathcal{B}}^{\circ})$.
Invariance under the $\mathbb{Z}/3$ action on each individual triangle implies
that there are two possibilities: either each line contains a unique puncture
point, and that point lies on no other line in the triangle, or the puncture
points on the triangle are the three intersection points between the lines,
hence each line contains exactly two puncture points. This proves that
$\mathcal{C}$ meets each $\ell_{i,j}\in\mathcal{L}_{1}$ as claimed in the
statement of the lemma.

Now, consider the triangle associated with lines in
$\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4}$. There are exactly four
points in the intersection of $\mathcal{C}$ with the triangle that are
associated with punctures, and these four points are fixed points for the
action of $\mathbb{Z}/2$. Since $\mathbb{Z}/2$ acts trivially on the lines in
$\mathcal{L}_{2}$ and as $-1$ on the curves in $\mathcal{L}_{3}$ and
$\mathcal{L}_{4}$ (under the appropriate identification with
$\mathbb{P}^{1}$), puncture points must arise from points on the line in the
orbit $\mathcal{L}_{2}$ along with possibly the point at the intersection of
the lines in $\mathcal{L}_{3}$ and $\mathcal{L}_{4}$. This leaves only the
possibilities in the statement of the lemma. ∎

######  Lemma 6.4.

The monodromy plane curve $\mathcal{C}$ meets one of the $S_{3}$ orbits
$\mathcal{Q}_{j}$ associated with the Eckardt points of the Clebsch cubic.

###### Proof.

This follows from the description in [6, §5.2] of the involution $\iota$,
which implies that $\mathcal{C}$ must meet one of the lifts of
$\mathfrak{h}/\operatorname{SL}_{2}(\mathbb{Z})$ to $X[2]$. As mentioned
above, these lifts map precisely to the points on $\mathbb{P}^{2}$ in the
orbits $\mathcal{Q}_{i}$ for each $i=1,2,3$. ∎

####  6.3.2 Smooth monodromy plane curve

In this section, we study the possibility that the monodromy plane curve is
smooth. Mathematica code to assist the reader in verifying various assertions
that follow is available from the author’s webpage [2]. Since $\mathcal{C}$ is
smooth of genus one, it is a cubic plane curve by the genus-degree formula. We
first use $S_{3}$-invariance to restrict the possibilities for the equation of
the curve.

######  Lemma 6.5.

Suppose that $\mathcal{C}$ is a smooth plane cubic curve that is invariant
under the action of $S_{3}$ described in §6.2.3 with coordinates
$[z_{1}:z_{2}:z_{3}]$ on $\mathbb{P}^{2}$. Then $\mathcal{C}$ is the vanishing
set of the homogeneous cubic

$\displaystyle F\left([z_{1}:z_{2}:z_{3}]\right)=a_{1}z_{1}^{3}$
$\displaystyle+a_{2}z_{2}^{3}+a_{3}z_{2}z_{3}^{2}$
$\displaystyle+\Big{(}(2X-2)a_{1}-Xa_{2}-(X-1)a_{3}\Big{)}z_{1}^{2}z_{2}$
$\displaystyle+\Big{(}(2-X)a_{1}+a_{2}+(X-1)a_{3}\Big{)}z_{1}z_{2}^{2}$
$\displaystyle+\Big{(}(3X-5)a_{1}+(X+1)a_{2}+(2-X)a_{3}\Big{)}z_{1}z_{3}^{2}$

for some $[a_{1}:a_{2}:a_{3}]\in\mathbb{P}^{2}$.

###### Proof.

Using invariance of $F$ under $\sigma_{1}$, we see that either
$F\circ\sigma_{1}=F$ and every term of $F$ of the form
$z_{1}^{j_{1}}z_{2}^{j_{2}}z_{3}^{j_{3}}$ has $j_{1}+j_{2}$ even, or
$F\circ\sigma_{1}=-F$ and every term of $F$ of the form
$z_{1}^{j_{1}}z_{2}^{j_{2}}z_{3}^{j_{3}}$ has $j_{1}+j_{2}$ odd. In the first
case, one can use a computer algebra program like Mathematica to see that
invariance of $F$ under $\sigma_{2}$ implies that $F$ is the singular
homogeneous cubic

$z_{3}\Big{(}(X-2)z_{1}^{2}+2z_{1}z_{2}-(X+1)z_{2}^{2}+z_{3}^{2}\Big{)}.$

Under similar analysis in the second case, one sees that $F\circ\sigma_{2}=-F$
(since $\sigma_{1}$ and $\sigma_{2}$ are conjugate in $S_{3}$) and furthermore
that any such $F$ invariant under $\sigma_{1}$ and $\sigma_{2}$ must then be
of the form given in the statement of the lemma. Since $\sigma_{1}$ and
$\sigma_{2}$ generate $S_{3}$, this proves the lemma. ∎

Note that some of the homogeneous cubics of the form given in the statement of
Lemma 6.5 are singular. We will not need the finer distinction of which
coefficients give a smooth cubic curve.

Bezout’s Theorem implies that $\mathcal{C}$ meets every $\ell_{i,j}$ in two or
three points, where the case of two intersection points occurs if and only if
$\ell_{i,j}$ is tangent to $\mathcal{C}$. We begin by completely determining
$\mathcal{C}$ in the case where no $\ell_{i,j}$ is tangent to $\mathcal{C}$.

######  Proposition 6.6.

Suppose that the monodromy plane curve $\mathcal{C}$ is smooth. Then
$\mathcal{C}$ is the plane cubic curve with one of the following equations:

$\displaystyle z_{1}^{3}+(13-8X)z_{2}^{3}$
$\displaystyle+(3-X)z_{1}^{2}z_{2}+(18-11X)z_{1}z_{2}^{2}$
$\displaystyle+(3X-5)z_{1}z_{3}^{2}+(X-2)z_{2}z_{3}^{2}$ $\displaystyle
z_{1}^{3}+(5-3X)z_{2}^{3}$
$\displaystyle+Xz_{1}^{2}z_{2}+(4-5X)z_{1}z_{2}^{2}$
$\displaystyle+(3X-5)z_{1}z_{3}^{2}-z_{2}z_{3}^{2}$

Equivalently, $\mathcal{C}$ is either

  1. 1.

the unique $S_{3}$-invariant smooth plane cubic tangent to the lines in
$\mathcal{L}_{1}$ at the points in $\mathcal{Q}_{2}$, or

  2. 2.

the unique $S_{3}$-invariant smooth plane cubic containing the orbit
$\mathcal{Q}_{2}$ (and no other $\mathcal{P}_{i}$ or $\mathcal{Q}_{i}$) and
the intersection points between pairs of lines in the orbit $\mathcal{L}_{1}$
represented by the vertices of the triangle in Figure 14.

###### Proof.

Invariance under $S_{3}$ implies that $\mathcal{C}$ meets one vertex of the
triangle in Figure 14 if and only if it meets all the vertices. We break the
proof into two cases, accordingly.

Case 1: No vertex in the triangle from Figure 14 is on $\mathcal{C}$.

Lemma 6.3 then implies that each line in $\mathcal{L}_{1}$ contains exactly
one puncture point on $\mathcal{C}$. We first assume that $\mathcal{C}$
intersects each line in $\mathcal{L}_{1}$ transversally, hence in exactly
three points by Bezout’s theorem. Then $\mathcal{C}$ meets exactly two of the
orbits $\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{Q}_{1}$,
$\mathcal{Q}_{2}$. However, $\mathcal{C}$ cannot simultaneously meet
$\mathcal{P}_{1}$ and $\mathcal{Q}_{2}$ or $\mathcal{P}_{2}$ and
$\mathcal{Q}_{2}$, since otherwise it would meet lines in $\mathcal{L}_{3}$ or
$\mathcal{L}_{4}$ in four points, respectively. This leaves four cases.

Case 1(a): $\mathcal{P}_{1}\cup\mathcal{P}_{2}\subset\mathcal{C}$

Since $\mathcal{C}$ does not contain $\mathcal{Q}_{1}$ or $\mathcal{Q}_{2}$,
Lemma 6.4 implies that $\mathcal{Q}_{3}\subset\mathcal{C}$. One checks with a
computer algebra program like Mathematica that there is no cubic curve with
the form given in Lemma 6.5 containing all of
$\mathcal{P}_{1}\cup\mathcal{P}_{2}\cup\mathcal{Q}_{3}$. Thus this case is
impossible.

Case 1(b): $\mathcal{P}_{1}\cup\mathcal{Q}_{1}\subset\mathcal{C}$

The only cubic as in Lemma 6.5 vanishing on
$\mathcal{P}_{1}\cup\mathcal{Q}_{1}$ is:

$z_{1}^{3}-z_{2}^{3}+(X+1)z_{1}^{2}z_{2}+(X-2)z_{1}z_{2}^{2}-(X+1)z_{1}z_{3}^{2}+(2-X)z_{2}z_{3}^{2}$

However, this cubic also vanishes at $\mathcal{P}_{2}$, contradicting our
observation that $\mathcal{C}$ meets exactly two of the orbits
$\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{Q}_{1}$, $\mathcal{Q}_{2}$.
This case is therefore eliminated.

Case 1(c): $\mathcal{P}_{2}\cup\mathcal{Q}_{1}\subset\mathcal{C}$

The only possibility in this case is the cubic considered in Case 1(b), which
eliminates this case as well.

Case 1(d): $\mathcal{Q}_{1}\cup\mathcal{Q}_{2}\subset\mathcal{C}$

Now, a computer algebra computation shows that the only possibility is the
cubic:

$(1+X)z_{1}^{3}+z_{2}^{2}+(1+3X)z_{1}^{2}z_{2}+(1-2X)z_{1}z_{2}^{2}+(X-2)z_{1}z_{3}^{2}-(2+3X)z_{2}z_{3}^{2}$

We rule this case out using the triangle of curves in Figure 13. The
associated puncture points on the cubic curve are at the intersection of the
lines in $\mathcal{L}_{3}$ and $\mathcal{L}_{4}$ and two points on the line in
$\mathcal{L}_{2}$ that do not lie on the other two lines. This implies that
the triangle only determines three puncture points, but it must determine
four. Indeed, the three triangles making up this $S_{3}$-orbit of must account
for the $12$ punctures associated with $\gamma_{\alpha}$, $\gamma_{\beta}$,
$\gamma_{\alpha^{\prime}}$, and $\gamma_{\beta^{\prime}}$ (cf. the proof of
Lemma 6.3). This contradiction eliminates this case.

We now consider the possibility that $\mathcal{C}$ is tangent to a line in
$\mathcal{L}_{1}$. Here $\mathcal{C}$ meets each line in $\mathcal{L}_{1}$ in
exactly two points, hence $\mathcal{C}$ meets exactly one of the orbits
$\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{Q}_{1}$, $\mathcal{Q}_{2}$.

Case 1(e): The lines in $\mathcal{L}_{1}$ are tangent to $\mathcal{C}$ at
either $\mathcal{P}_{1}$, $\mathcal{P}_{2}$, or $\mathcal{Q}_{1}$.

For each of these three possibilities, there is no $S_{3}$-invariant cubic
passing through the relevant orbit that is tangent to $\mathcal{L}_{1}$ at the
specified points. Therefore, this case is not possible.

Case 1(f): The lines in $\mathcal{L}_{1}$ are tangent to $\mathcal{C}$ at
$\mathcal{Q}_{2}$.

The only possibility is the first cubic in the statement of the proposition.
Then $\mathcal{C}$ meets the triangle in Figure 13 in four puncture points,
namely at the intersection of the lines in $\mathcal{L}_{3}$ and
$\mathcal{L}_{4}$ along with three points on the line in $\mathcal{L}_{2}$.
Therefore this case is a possibility.

Case 1(g): The lines in $\mathcal{L}_{1}$ are tangent to $\mathcal{C}$ at a
puncture point.

The cubic must also meet $\mathcal{L}_{1}$ in exactly one of $\mathcal{P}_{i}$
or $\mathcal{Q}_{i}$ ($i=1,2$). For $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$,
the cubic must also meet $\mathcal{Q}_{3}$ by Lemma 6.4. For $i=1,2$, the
unique cubics through $\mathcal{P}_{i}$ and $\mathcal{Q}_{3}$ are:

$\displaystyle
z_{1}\big{(}z_{1}^{2}+(2X-2)z_{1}z_{2}+(2-X)z_{2}^{2}+(3X-5)z_{3}^{2}\big{)}$
$\displaystyle
z_{2}\big{(}z_{2}^{2}+(X+1)z_{1}^{2}-2Xz_{1}z_{2}-(3X+2)z_{3}^{2}\big{)}$

These are singular, which rules these cases out.

For $\mathcal{Q}_{1}$, there are cubics that satisfy the required properties
with respect to the triangle of lines in $\mathcal{L}_{1}$. However, these
cubics all would give seven puncture points on the triangle of lines in Figure
13. This contradicts Lemma 6.3, and hence rules out this case. There is no
cubic with the required form that meets $\mathcal{Q}_{2}$ and is tangent to
the lines in $\mathcal{L}_{1}$ in a point not in $\mathcal{Q}_{2}$. This
completes the analysis of this case.

The final remaining option for Case 1 is:

Case 1(h): The lines in $\mathcal{L}_{1}$ meet $\mathcal{C}$ in a triple
point.

The triple point must be a puncture point of $\mathcal{C}$, so $\mathcal{C}$
does not meet $\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{Q}_{1}$, or
$\mathcal{Q}_{2}$. Lemma 6.4 implies that $\mathcal{C}$ meets
$\mathcal{Q}_{3}$. One checks that there is no cubic of the appropriate form
that meets each line in $\mathcal{L}_{1}$ in a triple point, and this
completes the analysis of Case 1.

Case 2: Each vertex in the triangle from Figure 14 is on $\mathcal{C}$.

Note that a vertex cannot be an intersection point with multiplicity three.
Indeed, then each line in $\mathcal{L}_{1}$ would meet $\mathcal{C}$ in two
points, one having multiplicity three, and this contradicts Bezout’s theorem.
We continue as in the first case by first assuming that no line in
$\mathcal{L}_{1}$ is tangent to $\mathcal{C}$. A line in $\mathcal{L}_{1}$
then contains two puncture points, hence $\mathcal{C}$ meets exactly one of
the orbits $\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{Q}_{1}$, or
$\mathcal{Q}_{2}$.

Case 2(a): $\mathcal{P}_{i}\subset\mathcal{C}$, $i=1$ or $2$

Lemma 6.4 implies that $\mathcal{C}$ also meets $\mathcal{Q}_{3}$. One checks
that there is no such cubic with the form in Lemma 6.5, hence this case is
impossible.

Case 2(b): $\mathcal{Q}_{1}\subset\mathcal{C}$

This case implies that the cubic is:

$z_{1}^{3}+(1+X)z_{2}^{3}+Xz_{1}^{2}z_{2}-Xz_{1}z_{2}^{2}+(3X-1)z_{1}z_{3}^{2}-(1+4X)z_{2}z_{3}^{2}$

This cubic curve meets the triangle in Figure 14 in what would be seven
puncture points. In particular, it meets the triangle at the intersection of
the lines in $\mathcal{L}_{3}$ and $\mathcal{L}_{4}$, two additional points on
each of those lines, and two points on the line in $\mathcal{L}_{2}$ (neither
of which is $e_{9}\in\mathcal{Q}_{3}$). This contradicts the fact that this
triangle must determine exactly four puncture points, as we saw in Case 1(d).
This case is therefore eliminated.

Case 2(c): $\mathcal{Q}_{2}\subset\mathcal{C}$

The only possibility here is the second cubic given in the statement of the
proposition. As in Case 1(h), $\mathcal{C}$ then meets the triangle in Figure
13 in four puncture points, namely at the intersection of the lines in
$\mathcal{L}_{3}$ and $\mathcal{L}_{4}$ along with three points on the line in
$\mathcal{L}_{2}$. Therefore this case is a possibility.

Case 2(d): The lines in $\mathcal{L}_{1}$ are tangent to $\mathcal{C}$.

In this case, Bezout’s theorem implies that $\mathcal{C}$ meets the triangle
of lines in $\mathcal{L}_{1}$ in only the three vertex points. Specifically,
it suffices to study plane cubics passing through $\ell_{1,5}\cap\ell_{4,8}$
that are tangent to either $\ell_{1,5}$ or $\ell_{4,8}$. These are:

$\displaystyle
z_{1}^{3}+(X-1)z_{2}^{3}+Xz_{1}^{2}z_{2}+(X-2)z_{1}z_{2}^{2}+(X-1)z_{1}z_{3}^{2}+(1-2X)z_{2}z_{3}^{2}$
$\displaystyle
z_{1}^{3}+(1-X)z_{2}^{3}+Xz_{1}^{2}z_{2}+(2-X)z_{1}z_{2}^{2}+(X-3)z_{1}z_{3}^{2}+z_{2}z_{3}^{2}$

Neither cubic can vanish at any other point on the triangle, hence it cannot
vanish on the orbits $\mathcal{Q}_{1}$ or $\mathcal{Q}_{2}$. Lemma 6.4 then
implies that it must vanish on $\mathcal{Q}_{3}=\\{e_{9}\\}$. Neither cubic is
zero at this point, hence we have eliminated the possibility that
$\mathcal{C}$ is tangent to a line in $\mathcal{L}_{1}$. This completes the
analysis of Case 2, and hence the proof of the proposition. ∎

## References

  * [1] Magma code associated with this paper.  Available at:
http://math.temple.edu/$\sim$mstover/WE.m.

  * [2] Mathematica supplement to this paper.  Available at:
http://math.temple.edu/$\sim$mstover/WE.mx.

  * [3] W. Barth, C. Peters, and A. Van de Ven.  Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3).  Springer-Verlag, 1984.
  * [4] W. Bosma, J. Cannon, and C. Playoust.  The Magma algebra system. I. The user language.  J. Symbolic Comput., 24(3-4):235–265, 1997.  Computational algebra and number theory (London, 1993).
  * [5] I. Dolgachev, B. Farb, and E. Looijenga.  Geometry of the Wiman-Edge pencil, I: algebro-geometric aspects.  Eur. J. Math., 4(3):879–930, 2018.
  * [6] B. Farb and E. Looijenga.  Arithmeticity of the monodromy of the Wiman–Edge pencil.  arXiv:1911.01210; to appear in Ann. Inst. Fourier (Grenoble).
  * [7] B. Farb and E. Looijenga.  Geometry of the Wiman-Edge pencil and the Wiman curve.  Geom. Dedicata, 208:197–220, 2020.
  * [8] C. Maclachlan and A. Reid.  The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics.  Springer-Verlag, 2003.
  * [9] G. van der Geer.  Hilbert modular surfaces, volume 16 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3).  Springer-Verlag, 1988.
  * [10] H. Yoshida.  On some problems concerning discrete subgroups.  Comment. Math. Univ. St. Pauli, 60(1-2):231–253, 2011.


10 | 1.54621 | $0.00000\%$ | $+1.44942\%$ | $+1.24680\%$ | $+1.29418\%$ | $+1.03556\%$ | $-4.35472\%$
11 | 1.55761 | $0.00000\%$ | $-1.03066\%$ | $-1.23426\%$ | $-1.12582\%$ | $-1.33133\%$ | $+10.88780\%$
12 | 1.60705 | $0.00000\%$ | $+3.14540\%$ | $+2.86421\%$ | $+3.10455\%$ | $+2.94187\%$ | $-4.74928\%$
13 | 1.63913 | $0.00000\%$ | $-1.90326\%$ | $-2.10939\%$ | $-2.04352\%$ | $-2.26721\%$ | $-0.21303\%$
14 | 1.64878 | $0.00000\%$ | $-1.36754\%$ | $-1.49240\%$ | $-1.55215\%$ | $-1.81234\%$ | $+2.85737\%$
15 | 1.66128 | $0.00000\%$ | $-1.97332\%$ | $-2.09528\%$ | $-1.45453\%$ | $-0.71927\%$ | $+3.14082\%$
16 | 1.66200 | $0.00000\%$ | $-2.97875\%$ | $-3.12205\%$ | $+0.74136\%$ | $+1.98271\%$ | $-2.06459\%$
17 | 1.66257 | $0.00000\%$ | $-0.78257\%$ | $-1.27512\%$ | $-1.38717\%$ | $-1.69549\%$ | $+2.29955\%$
18 | 1.71999 | $0.00000\%$ | $+0.24199\%$ | $+0.12119\%$ | $+0.46545\%$ | $+0.57221\%$ | $-0.77614\%$
19 | 1.72518 | $0.00000\%$ | $+0.08472\%$ | $-0.03743\%$ | $+1.32173\%$ | $+2.26348\%$ | $-0.64065\%$
20 | 1.73728 | $0.00000\%$ | $+2.00660\%$ | $+1.82355\%$ | $+1.82582\%$ | $+1.53470\%$ | $-3.40562\%$
21 | 1.75121 | $0.00000\%$ | $+1.02305\%$ | $+0.85984\%$ | $+0.98778\%$ | $+0.80087\%$ | $-1.83097\%$
22 | 1.76028 | $0.00000\%$ | $-0.30472\%$ | $+0.54420\%$ | $-0.60213\%$ | $-0.80462\%$ | $-1.47586\%$
23 | 1.78722 | $0.00000\%$ | $-1.09432\%$ | $-0.47427\%$ | $-0.60620\%$ | $-0.73652\%$ | $+2.72516\%$
24 | 2.04493 | $0.00000\%$ | $-1.47188\%$ | $-1.95233\%$ | $-2.31021\%$ | $-1.34374\%$ | $+3.07357\%$
25 | 2.10664 | $0.00000\%$ | $+0.34389\%$ | $-0.49598\%$ | $-0.99791\%$ | $-1.31535\%$ | $+1.44756\%$
26 | 2.22927 | $0.00000\%$ | $-4.59012\%$ | $-4.67559\%$ | $-3.56198\%$ | $-3.26678\%$ | $+1.83404\%$
27 | 2.30707 | $0.00000\%$ | $-2.46154\%$ | $-2.54728\%$ | $-2.64045\%$ | $-2.63787\%$ | $+1.71891\%$
28 | 2.70816 | $0.00000\%$ | $-1.59644\%$ | $-1.78781\%$ | $-1.65567\%$ | $-1.77671\%$ | $+2.98058\%$
29 | 3.15152 | $0.00000\%$ | $-1.30373\%$ | $-1.37948\%$ | $-1.54499\%$ | $-1.73567\%$ | $+1.03780\%$
30 | 3.32601 | $0.00000\%$ | $-0.74133\%$ | $-0.73786\%$ | $-1.51039\%$ | $-1.65202\%$ | $+1.16945\%$
Table 9: Relative changes in community metrics for different values of $\mathcal{K}$. Relative changes indicate difference between the final output of the solution algorithm and the initial metric for that community. Community | Accessibility Metric Rel. Diff.
---|---
ID | $\beta=0.5$ | $\beta=1.0$ | $\beta=1.5$ | $\beta=2.0$
1 | $+5.02349\%$ | $+10.23670\%$ | $+17.66870\%$ | $+27.54780\%$
2 | $+1.15748\%$ | $+2.30643\%$ | $+3.29771\%$ | $+3.98499\%$
3 | $+0.12525\%$ | $+0.06377\%$ | $-0.45141\%$ | $-1.40403\%$
4 | $-0.20341\%$ | $-0.04187\%$ | $+0.13229\%$ | $-0.06179\%$
5 | $+2.63748\%$ | $+4.18293\%$ | $+4.64409\%$ | $+4.32943\%$
6 | $+2.51144\%$ | $+3.40944\%$ | $+0.13601\%$ | $-2.39743\%$
7 | $+2.20261\%$ | $+5.73177\%$ | $+10.51130\%$ | $+17.15080\%$
8 | $+0.09534\%$ | $+0.13326\%$ | $-0.91723\%$ | $-3.30634\%$
9 | $+0.78419\%$ | $+0.92850\%$ | $-0.18574\%$ | $-0.68032\%$
10 | $+0.03285\%$ | $+1.44942\%$ | $+3.48831\%$ | $+6.81105\%$
11 | $-0.98510\%$ | $-1.03066\%$ | $-1.44343\%$ | $-2.22022\%$
12 | $+1.53500\%$ | $+3.14540\%$ | $+4.46802\%$ | $+5.46441\%$
13 | $-1.29035\%$ | $-1.90326\%$ | $-1.83503\%$ | $-1.42341\%$
14 | $-0.97666\%$ | $-1.36754\%$ | $-1.14218\%$ | $-0.34302\%$
15 | $-1.32526\%$ | $-1.97332\%$ | $-1.62264\%$ | $-1.07859\%$
16 | $-2.05815\%$ | $-2.97875\%$ | $-2.76270\%$ | $-1.87015\%$
17 | $-0.24560\%$ | $-0.78257\%$ | $-0.35359\%$ | $+0.59357\%$
18 | $-0.52186\%$ | $+0.24199\%$ | $+2.54721\%$ | $+6.52475\%$
19 | $-0.48299\%$ | $+0.08472\%$ | $+1.03859\%$ | $+2.05805\%$
20 | $+0.39778\%$ | $+2.00660\%$ | $+3.79349\%$ | $+6.22916\%$
21 | $+0.49524\%$ | $+1.02305\%$ | $+1.01055\%$ | $-0.19708\%$
22 | $-0.26372\%$ | $-0.30472\%$ | $+0.19110\%$ | $+0.98391\%$
23 | $+0.12488\%$ | $-1.09432\%$ | $-1.27936\%$ | $-1.31398\%$
24 | $-0.83268\%$ | $-1.47188\%$ | $-1.30115\%$ | $-0.89414\%$
25 | $+0.44535\%$ | $+0.34389\%$ | $+0.25383\%$ | $+0.10043\%$
26 | $-2.24379\%$ | $-4.59012\%$ | $-5.91311\%$ | $-7.05280\%$
27 | $-1.56020\%$ | $-2.46154\%$ | $-2.23398\%$ | $-1.84788\%$
28 | $-1.07935\%$ | $-1.59644\%$ | $-1.40627\%$ | $-0.78129\%$
29 | $-0.91016\%$ | $-1.30373\%$ | $-1.28732\%$ | $-1.00115\%$
30 | $-0.35668\%$ | $-0.74133\%$ | $-0.37780\%$ | $-0.11457\%$
Table 10: Relative changes in community metrics for different values of $\beta$. Relative changes indicate difference between the final output of the solution algorithm and the initial metric for that community, both measured using the corresponding value of $\beta$. Trial | Solution Vector
---|---
|  | E0 | E1 | E2 | E3 | E4 | E5 | E6 | E7 | N0 | N1 | N2 | N3 | N4 | N5 | N6 | N7 | N8 | N9
| Initial | 8 | 12 | 12 | 11 | 12 | 14 | 11 | 12 | 12 | 10 | 13 | 13 | 12 | 11 | 6 | 7 | 12 | 12
| $\epsilon=0.00$ | 8 | 12 | 12 | 11 | 12 | 14 | 11 | 12 | 12 | 10 | 13 | 13 | 12 | 11 | 6 | 7 | 12 | 12
$*$ | $\epsilon=0.01$ | 1 | 16 | 4 | 8 | 12 | 14 | 11 | 12 | 12 | 10 | 20 | 10 | 7 | 20 | 6 | 5 | 20 | 12
| $\epsilon=0.05$ | 1 | 16 | 2 | 11 | 12 | 14 | 11 | 12 | 12 | 10 | 20 | 10 | 7 | 20 | 6 | 4 | 20 | 12
| $\epsilon=0.10$ | 1 | 16 | 3 | 10 | 12 | 14 | 11 | 12 | 12 | 10 | 20 | 10 | 7 | 20 | 6 | 4 | 20 | 12
| $\epsilon=\infty$ | 1 | 16 | 2 | 11 | 12 | 14 | 11 | 12 | 12 | 10 | 20 | 10 | 7 | 20 | 6 | 4 | 20 | 12
| $\mathcal{K}=1$ | 8 | 12 | 12 | 11 | 12 | 14 | 11 | 12 | 12 | 10 | 13 | 13 | 12 | 11 | 6 | 7 | 12 | 12
$*$ | $\mathcal{K}=6$ | 1 | 16 | 4 | 8 | 12 | 14 | 11 | 12 | 12 | 10 | 20 | 10 | 7 | 20 | 6 | 5 | 20 | 12
| $\mathcal{K}=12$ | 19 | 11 | 8 | 2 | 12 | 14 | 10 | 1 | 2 | 1 | 20 | 6 | 6 | 20 | 20 | 5 | 20 | 12
| $\mathcal{K}=18$ | 1 | 11 | 7 | 8 | 12 | 14 | 11 | 1 | 13 | 4 | 20 | 8 | 16 | 20 | 10 | 5 | 20 | 12
| $\mathcal{K}=24$ | 13 | 11 | 8 | 1 | 12 | 14 | 1 | 11 | 3 | 1 | 20 | 17 | 20 | 20 | 6 | 4 | 20 | 11
| $\mathcal{K}=30$ | 12 | 11 | 14 | 7 | 12 | 14 | 11 | 5 | 11 | 9 | 8 | 20 | 7 | 5 | 8 | 20 | 6 | 12
| $\beta=0.5$ | 10 | 20 | 11 | 2 | 9 | 13 | 11 | 1 | 1 | 2 | 20 | 6 | 3 | 18 | 18 | 5 | 20 | 12
$*$ | $\beta=1.0$ | 1 | 16 | 4 | 8 | 12 | 14 | 11 | 12 | 12 | 10 | 20 | 10 | 7 | 20 | 6 | 5 | 20 | 12
| $\beta=1.5$ | 1 | 17 | 5 | 10 | 12 | 14 | 11 | 12 | 12 | 10 | 16 | 10 | 7 | 20 | 6 | 5 | 20 | 12
| $\beta=2.0$ | 1 | 20 | 5 | 10 | 12 | 14 | 11 | 12 | 12 | 10 | 12 | 10 | 7 | 20 | 6 | 6 | 20 | 12
Table 11: Solution vectors for all artificial trial sets as well as the
initial solution vector. Rows listing a parameter $\epsilon$, $\mathcal{K}$,
or $\beta$ correspond to the trial set in which the listed parameter was
varied. The three starred ($*$) rows represent a single trial with
$\epsilon=0.01$, $\mathcal{K}=6$, and $\beta=1.0$.