# Echoes in genus three of Teichmüller curves in genus two

Thomas Le Fils

###### Abstract.

We classify the Teichmüller curves in the moduli space of genus three Riemann
surfaces $\mathcal{M}_{3}$ that are obtained by a covering construction from a
primitive Teichmüller curve in $\mathcal{M}_{2}$. We describe the action on
homology modulo two of the affine groups of translation surfaces generating
these primitive curves. We also classify the
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits of square-tiled surfaces in
$\mathcal{H}(2,2)$ that are a cover of a genus two one.

##  1\. Introduction

###  1.1. Teichmüller curves

The moduli space of algebraic curves $\mathcal{M}_{g}$ is a central object in
the geometry of surfaces that parametrises closed Riemann surfaces of genus
$g$. It is endowed with a complex structure and a complete metric, the
_Teichmüller metric_ , that corresponds to its Kobayashi metric. The geodesics
for this metric naturally come in families called _complex geodesics_ : for
each geodesic $\gamma\colon\mathbb{R}\to\mathcal{M}_{g}$, there exists a
totally geodesic immersion $f\colon\mathbb{H}^{2}\to\mathcal{M}_{g}$ such that
$\gamma=f_{|i\mathbb{R}_{>0}}$. Some exceptional complex geodesics have image
of finite area: the _Teichmüller curves_ 111We only consider Teichmüller
curves generated by squares of abelian differentials, see Section 2.. A
central problem in the theory of Teichmüller and flat surfaces is to classify
these rare objects, see for example the surveys [Mol18, McM23a].

Given a Teichmüller curve $V\subset\mathcal{M}_{g}$, one can obtain infinitely
many others in moduli spaces of curves of higher genera by covering
constructions: its _echoes_. A Teichmüller curve that is not an echo is called
_primitive_. In a series of groundbreaking articles [McM03, McM05, McM06b,
McM07], McMullen gave a complete classification of the primitive Teichmüller
curves in $\mathcal{M}_{2}$. There exists an infinite series of Teichmüller
curves $W_{D}\subset\mathcal{M}_{2}$ called the _Weierstrass curves_. They are
indexed by their discriminant, an integer $D\geqslant 5$ such that $D\equiv
0,1\mod 4$, and their spin $\epsilon\in\\{0,1\\}$ when $D>9$ satisfies
$D\equiv 1\mod 8$. We study the echoes of each Weierstrass curve in
$\mathcal{M}_{3}$.

######  Theorem 1.1.

Let $V\subset\mathcal{M}_{2}$ be a Weierstrass curve of discriminant $D$. If
$D\equiv 5\mod 8$ then $V$ has three echoes in $\mathcal{M}_{3}$, and five
otherwise.

McMullen showed in [McM06b] that the Weierstrass curves with non-square
discriminant account for all primitive Teichmüller curves in
$\mathcal{M}_{2}$, except for one curve, the decagon curve $\mathcal{D}$.

######  Theorem 1.2.

The curve $\mathcal{D}$ has three echoes in $\mathcal{M}_{3}$.

Teichmüller curves in genus three can be divided into three categories. The
first consists of the primitive ones, for which there is a conjectural
classification, see [McM23a, Problem 5.4]. The second category is formed by
the echoes of Teichmüller curves in genus two, and the third consists of the
echoes of the only Teichmüller curve in genus one. The present article is a
contribution to the classification of Teichmüller curves in genus three,
studying this second category.

###  1.2. The flat viewpoint

A translation surface is a surface obtained by gluing sides of polygons
together with translations. Equivalently, it is the datum of a pair
$(X,\omega)$ where $X\in\mathcal{M}_{g}$ and $\omega$ is a holomorphic one-
form on $X$, see Section 2. The group $\mathrm{SL}_{2}(\mathbb{R})$ acts on
the moduli space $\Omega\mathcal{M}_{g}$ of translation surfaces of genus $g$,
by acting on polygons. The Veech group of $(X,\omega)$ is its stabiliser for
this action, and is denoted $\mathrm{SL}(X,\omega)$. When
$\mathrm{SL}(X,\omega)$ is a lattice, $(X,\omega)$ is said to be a Veech
surface. The Teichmüller curves are obtained by projecting the orbits of Veech
surfaces to $\mathcal{M}_{g}$. This produces Teichmüller curves, since
$\mathbb{H}^{2}=\mathrm{SO}_{2}(\mathbb{R})\backslash\mathrm{SL}_{2}(\mathbb{R})$:

${\mathrm{SL}_{2}(\mathbb{R})}$${\Omega\mathcal{M}_{g}}$${\mathbb{H}^{2}/\mathrm{SL}(X,\omega)}$${\mathcal{M}_{g}.}$

Therefore our results can be expressed as a classification of the
$\mathrm{SL}_{2}(\mathbb{R})$-orbits of genus three Veech surfaces covering a
genus two translation surface. For example we can rephrase Theorem 1.2 as
follows.

######  Corollary 1.3.

The genus three Veech surfaces covering a genus two surface with two
singularities, and not covering a genus one surface, form three
$\mathrm{GL}^{+}_{2}(\mathbb{R})$-orbits.

###  1.3. Square-tiled surfaces

Our analysis includes the non-primitive Teichmüller curves in
$\mathcal{M}_{2}$ that are generated by a form in $\mathcal{H}(2)$: the
Weierstrass curves with square discriminant. They are generated by square-
tiled surfaces, that is translation surfaces obtained by gluing unit squares
along their sides. Hubert-Lelièvre [HL06] and McMullen [McM05] gave a complete
classification of their $\mathrm{SL}_{2}(\mathbb{R})$-orbits. More precisely,
they gave a classification of the square-tiled surfaces belonging to this
orbit, or in other words, of their $\mathrm{SL}_{2}(\mathbb{Z})$-orbit. We
analyse the lifts of these translation surfaces in genus three. We say that a
square-tiled surface is reduced if the set of its relative periods is
$\mathbb{Z}+i\mathbb{Z}$.

######  Theorem 1.4.

There are ten $\mathrm{SL}_{2}(\mathbb{Z})$-orbits of reduced square-tiled
surfaces with $2n$ tiles in $\mathcal{H}(2,2)$, that cover a genus two one for
all odd $n\geqslant 5$. There are five orbits if $n\geqslant 4$ is even or
$n=3$.

Similarly to [McM05, Corollary 1.5], we can rephrase this statement purely
topologically. Two branched covers $p_{1},p_{2}\colon S\to T$ are of the same
type if there exist orientation-preserving homeomorphisms $s,t$ such that the
following diagram commutes:

${S}$${S}$${T}$${T.}$$\scriptstyle{s}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{t}$

A square-tiled surface gives rise to a branched cover of the torus $T$. Only
some orbits of Theorem 1.4 give a surjection $\pi_{1}(S)\to\pi_{1}(T)$. We
characterise the ones that do and classify the types of the corresponding
branched covers.

######  Theorem 1.5.

Let $S$ be a genus three closed surface and $T$ a torus. For every $d\geqslant
4$ even, or $d=3$, there are exactly three types of degree $2d$ branched
covers $f\colon S\to\mathrm{T}$ branched over just two points and surjective
on homology, that are invariant by a fixed-point free involution. For
$d\geqslant 5$ odd, there are exactly seven types of these.

###  1.4. Monodromy representation

Let us consider the finite extension of the Veech group consisting of elements
of the mapping class group appearing in the identifications
$A\cdot\omega\simeq\omega$: the affine group $\mathrm{Aff}^{+}(X,\omega)$.
Since the mapping class group acts on homology, we get a _monodromy
representation_

$\rho\colon\mathrm{Aff}^{+}(X,\omega)\to\mathrm{Sp}\left(H_{1}(X,\mathbb{Z})\right).$

We will observe that classifying the echoes of Teichmüller curves amounts to
gaining understanding of the _monodromy groups_ of their generators: the
images of $\rho$. An important tool to analyse these groups comes from the
fact that they must commute with real multiplication. Let
$\mathcal{O}_{D}=\mathbb{Z}[x]/(x^{2}+ax+b)$, where $D=a^{2}-4b$. The Jacobian
$\mathrm{Jac}(X)$ of $X\in\mathcal{M}_{2}$ admits real multiplication by
$\mathcal{O}_{D}$ if there is an embedding
$\mathcal{O}_{D}\subset\mathrm{End}(\mathrm{Jac}(X))$ in the self-adjoint
endomorphisms, generating a proper subring of $\mathrm{End}(\mathrm{Jac}(X))$.
If $(X,\omega)$ generates a Weierstrass curve of discriminant $D$, then
$\mathrm{Jac}(X)$ admits real multiplication by $\mathcal{O}_{D}$, such that
$\mathcal{O}_{D}\cdot\omega\subset\mathbb{C}\omega$. See the reminders of
Section 2 for more details and references.

######  Proposition 1.6.

Every element in the monodromy group of $(X,\omega)$ generating a Weierstrass
curve commutes with real multiplication.

This observation is proven in [FL23, Lemma 6.2] when $D$ is not a square.
Commutation with real multiplication also appears in the work of Mukamel
[Muk14]. It will allow us to explicitly describe the _monodromy groups modulo
two_ : the images of the monodromy groups in
$\mathrm{Sp}\left(H_{1}(X,\mathbb{Z}/2\mathbb{Z})\right)$, recovering a
theorem of [GRP23].

######  Theorem 1.7.

The monodromy groups modulo two of Veech surfaces in $\mathcal{H}(2)$, and of
the regular decagon, are explicit finite dihedral groups.

The action of affine groups on homology has received a lot of attention
recently. McMullen studied them modulo two for regular polygons in [McM24,
McM23b]. Guitérrez-Romo and Pardo described the action of affine groups of
Veech surfaces in $\mathcal{H}(2)$ on Weierstrass points in [GRP23]. We
recover their result with new methods, and provide a uniform treatment for
both the arithmetic and non-arithmetic case. Freedman and Lucas in [FL23]
described in particular the monodromy groups modulo $p$ of primitive
Teichmüller curves in genus two, for an infinite set of primes $p\geqslant 3$,
that can be computed but depend on $D$.

###  1.5. Pure echoes

More generally, one can study Teichmüller curves that are generated by a
translation surface covering another via an unramified cover. We call these
Teichmüller cuves the _pure echoes_ of the original Teichmüller curve $V$. We
show as a consequence of our study that their numbers can be made arbitrarily
large. We also observe that it follows from a theorem of Funar and Lochak
[FL18] that for any primitive Teichmüller curve $V\to\mathcal{M}_{g}$, there
exist infinitely many $g^{\prime}$ and Teichmüller curves
$V\to\mathcal{M}_{g^{\prime}}$.

######  Theorem 1.8.

There can be arbitrarily large numbers of pure echoes of a Teichmüller curve
in $\mathcal{M}_{g}$. Any primitive Teichmüller curve has infinitely many
copies in higher genera moduli spaces.

###  1.6. Organisation of the article

In Section 2, we recall some known facts about translation surfaces, abelian
differentials, and the classification of Teichmüller curves in genus two.
Section 3 defines echoes, and relates them with actions of affine groups on
sets of covers. We give explicit generators for the monodromy groups of
abelian differentials generating the Weierstrass curves in Section 4. In
Section 5, we compute the monodromy group of the regular decagon, prove
Theorem 1.2 and deduce 1.3. Then in Section 6, we establish 1.6. In Section 7
we explicitly give the monodromy groups modulo two of generators of the
Weierstrass curves and prove Theorem 1.7. We deduce Theorem 1.1 and Theorem
1.4 from this result. We then determine the covers of reduced square-tiled
surfaces in $\mathcal{H}(2)$ that give rise to a surjection
$\pi_{1}(S)\to\pi_{1}(T)$ and show Theorem 1.5. Finally in Section 8 we
consider echoes in moduli spaces of higher genera and prove Theorem 1.8.

####  1.6.1. Acknowledgements

I wish to thank Erwan Lanneau for his encouragement and interest.

##  2\. Abelian differentials

In this section we recall some well-known facts about abelian differentials,
translation surfaces, Teichmüller curves, and their classification in genus
two. This presentation is classical and inspired by the survey [McM23a].

###  2.1. Teichmüller spaces

####  2.1.1. The classical Teichmüller space.

Let us fix $S$ a closed topological surface of genus $g\geqslant 2$. Let
$\mathrm{Homeo}^{+}(S)$ be the group of orientation-preserving homeomorphisms
of $S$, and $\mathrm{Homeo}_{0}(S)$ be its subgroup of elements isotopic to
the identity. Recall that the Teichmüller space $\mathcal{T}(S)$ of $S$ is the
space of complex structures on $S$, modulo the natural action of
$\mathrm{Homeo}_{0}(S)$, see _e.g._ [FM12, Section 11.1.1]. The Teichmüller
space $\mathcal{T}(S)$ carries a natural topology for which it is a manifold
homeomorphic to $\mathbb{R}^{6g-6}$, see [FM12, Theorem 10.6].

####  2.1.2. The Teichmüller space of abelian differentials

We now define the space parametrising the marked translation surfaces of genus
$g$, that will be naturally a vector bundle over $\mathcal{T}(S)$, with the
zero section removed. A translation surface gives a complex structure on the
underlying topological surface since the polygons are embedded in the plane.
It also gives a non-zero holomorphic one-form on it: the form $dz$ is
invariant by translations. The datum of a pair of a closed Riemann surface and
a holomorphic one-form on it is actually equivalent to the datum of a
translation surface, see e.g. [Zor06, Section 3.3].

######  Definition 2.1.

The Teichmüller space of abelian differentials on $S$, denoted by
$\Omega\mathcal{T}(S)$, is the space of $(X,\omega)$ where $X$ is a complex
structure on $S$ and $\omega$ is a non-zero holomorphic one-form on $X$,
modulo the action of $\mathrm{Homeo}_{0}(S)$.

We will sometimes denote by $\Omega\mathcal{T}_{g}$ the space
$\Omega\mathcal{T}(S)$ where $S$ is an implicit fixed topological surface of
genus $g$.

####  2.1.3. Linear charts

Integrating the form $\omega$ gives local _linear charts_ in $\mathbb{R}^{2}$
whose transition functions are translations. One can also recover $\omega$
from the charts. From this point of view, a translation surface can be seen as
a $(G,X)$-structure in the sense of Thurston, see [Thu97, Chapter 3].

###  2.2. Mapping class group and moduli space

We denote by $\mathrm{Mod}(S)$ the mapping class group of $S$:
$\mathrm{Mod}(S)=\mathrm{Homeo}^{+}(S)/\mathrm{Homeo}_{0}(S)$. Recall that the
moduli space of curves $\mathcal{M}_{g}$ is defined to be
$\mathcal{T}_{g}/\mathrm{Mod}(S)$. Similarly we define the moduli space of
abelian differentials $\Omega\mathcal{M}(S)$ to be the set of couples
$(X,\omega)$ modulo the action of the full group of homeomorphisms
$\mathrm{Homeo}(S)$, or in other words,

$\Omega\mathcal{M}(S)=\Omega\mathcal{T}(S)/\mathrm{Mod}(S).$

As before, we denote by $\Omega\mathcal{M}_{g}$ the space
$\Omega\mathcal{M}(S)$ where $S$ is a surface of genus $g$. When
$(X_{0},\omega_{0})$ and $(X_{1},\omega_{1})$ are two elements of
$\Omega\mathcal{T}(S)$, we write $\omega_{0}\simeq\omega_{1}$ to indicate that
they define the same element of the moduli space.

###  2.3. Strata of abelian differentials

The spaces $\Omega\mathcal{T}_{g}$ and $\Omega\mathcal{M}_{g}$ are both
naturally stratified. Indeed an abelian differential $\omega$ on a Riemann
surface of genus $g$ has $2g-2$ zeroes counted with multiplicity. Therefore
these spaces are stratified by the subspaces where the number of zeroes and
their multiplicity are prescribed. The strata are denoted by
$\Omega\mathcal{T}(n_{1},\ldots,n_{k})$ and, respectively,
$\Omega\mathcal{M}(n_{1},\ldots,n_{k})$, where $\sum_{i}n_{i}=2g-2$. Another
common notation for the strata of the moduli space of abelian differentials is
the following:
$\mathcal{H}(n_{1},\ldots,n_{k})=\Omega\mathcal{M}(n_{1},\ldots,n_{k})$.

###  2.4. The $\mathrm{SL}_{2}(\mathbb{R})$-action

The moduli space of abelian differentials admits an action of
$\mathrm{SL}_{2}(\mathbb{R})$. Indeed, matrices act on polygons and this
action preserves parallelity. The projection of an orbit under the action of
$\mathrm{SL}_{2}(\mathbb{R})$ to the moduli space of curves defines a _complex
geodesic_ for the Teichmüller metric: a totally geodesic immersion of
$\mathbb{H}^{2}$, see for example [FM12, Proposition 11.18].

${\mathrm{SL}_{2}(\mathbb{R})}$${\Omega\mathcal{M}_{g}}$${\mathbb{H}^{2}=\mathrm{SO}_{2}\backslash\mathrm{SL}_{2}(\mathbb{R})}$${\mathcal{M}_{g}}$

The moduli space of quadratic differentials parametrises half-translation
surfaces: surfaces obtained by gluing sides of polygons with translations, and
$z\mapsto-z$. This space carries a similar
$\mathrm{SL}_{2}(\mathbb{R})$-action, and the complex geodesics in
$\mathcal{M}_{g}$ are all obtained by the discussion above, see [FM12,
Proposition 11.18]. In the present article, we only consider complex geodesics
generated by abelian differentials, or their squares in the space of quadratic
differentials.

###  2.5. Affine and Veech groups

The stabiliser of $(X,\omega)\in\Omega\mathcal{M}_{g}$ under the action of
$\mathrm{SL}_{2}(\mathbb{R})$ is called its _Veech group_ and is denoted by
$\mathrm{SL}(X,\omega)$:

$\mathrm{SL}(X,\omega)=\\{A\in\mathrm{SL}_{2}(\mathbb{R})\mid
A\cdot\omega\simeq\omega\\}.$

We define $\mathrm{Aff}^{+}(X,\omega)$ the _affine group_ of $(X,\omega)$ to
be the group of $f\in\mathrm{Homeo}^{+}(S)$ such that $f$ preserves the zeroes
of $\omega$ and is affine in linear charts. We have a short exact sequence,
where $\mathrm{Tr}(X,\omega)$ is the group of homeomorphisms of $S$ that are
translations in linear charts.

$1\to\mathrm{Tr}(X,\omega)\to\mathrm{Aff}^{+}(X,\omega)\to\mathrm{SL}(X,\omega)\to
1.$

We say that $(X,\omega)$ is a Veech surface when its Veech group is a lattice
in $\mathrm{SL}_{2}(\mathbb{R})$. We denote by $d$ the map taking
$f\in\mathrm{Aff}^{+}(X,\omega)$ to its derivative,
$d(f)\in\mathrm{SL}(X,\omega)$.

###  2.6. Periods

Given $(X,\omega)\in\Omega\mathcal{T}(S)$, we define its period map
$\mathrm{Per}\colon H_{1}(S,\mathbb{Z})\to\mathbb{C}$ to be the homomorphism

$\gamma\mapsto\int_{\gamma}\omega.$

We will sometimes refer to $\mathrm{Per}(H_{1}(S,\mathbb{Z}))$ as the set of
absolute periods, by opposition to the relative periods defined similarly
$\mathrm{RelPer}\colon H_{1}(S,Z,\mathbb{Z})\to\mathbb{C}$, where $Z\subset S$
is the set of zeroes of $\omega$. The relative periods give local
homeomorphisms
$U\subset\Omega\mathcal{T}(n_{1},\ldots,n_{k})\to\mathbb{C}^{2g+k-1}$, and
endow the strata with a manifold structure.

###  2.7. Dehn twist and cylinder decompositions

A saddle connection on a translation surface is a straight line joining two
conical points (i.e. two zeroes of the one-form), that has no conical point in
its interior. A Veech surface admits a cylinder decomposition in the direction
of each of its saddle connections. In other words, for any
$s\in\mathbb{S}^{1}$ such that there is a saddle connection with slope $s$,
the straight lines in the direction of $s$ decompose the surface into
cylinders $C_{i}$. The modulus of $C_{i}$ is $m_{i}=h_{i}/l_{i}$, where
$l_{i}$ is its length and $h_{i}$ its height. If the cylinder $C_{i}$ is
horizontal, then it is preserved by the right-handed Dehn twist:

$\begin{pmatrix}1&1/m_{i}\\\ 0&1\\\ \end{pmatrix}.$

The moduli $m_{i}$ are commensurable. Let $m$ be the lowest common multiple of
the $1/m_{i}$. The affine group of the surface contains the element whose
restriction to each $C_{i}$ is the right-handed Dehn twist to the power
$m\cdot m_{i}$.

###  2.8. Monodromy groups

There is a natural map $\mathrm{Aff}^{+}(X,\omega)\to\mathrm{Mod}(S)$. The
action on homology of $\mathrm{Mod}(S)$ respects the symplectic form
$(\gamma,\delta)\mapsto\gamma\cdot\delta$ given by the algebraic intersection.
This gives a surjection
$\mathrm{Mod}(S)\to\mathrm{Sp}\left(H_{1}(S,\mathbb{Z})\right)$, see for
example [FM12, Theorem 6.4]. By composing these maps one obtains an action on
homology, and the monodromy representation:

$\mathrm{Aff}^{+}(X,\omega)\to\mathrm{Sp}\left(H_{1}(S,\mathbb{Z})\right).$

We are mostly interested in the action on homology modulo two, hence we will
consider the projection
$\rho\colon\mathrm{Aff}^{+}(X,\omega)\to\mathrm{Sp}\left(H_{1}(S,\mathbb{Z}/2\mathbb{Z})\right)$.
Let us call the _monodromy group modulo two_ of $(X,\omega)$ the image of
$\rho$.

###  2.9. Teichmüller discs and Teichmüller curve

Given $(X,\omega)\in\Omega\mathcal{M}_{g}$, the _Teichmüller disc_ generated
by $\omega$ is $\mathrm{SL}_{2}(\mathbb{R})$-orbit of $\omega$ in
$\Omega\mathcal{M}_{g}$. A _Teichmüller curve_ is the projection of a
Teichmüller disc in $\mathcal{M}_{g}$ that has finite area. This happens
exactly when the forms $(X,\omega)$ generating the Teichmüller disc are Veech
surfaces.

###  2.10. The classification of McMullen in genus two

In a series of groundbreaking articles [McM03, McM05, McM06b, McM07], McMullen
gave a complete classification of the primitive Teichmüller curves in genus
two. Let us recall the definition of the Weierstrass curves. For any
$D\geqslant 5$, with $D\equiv 0,1\mod 4$, the Weierstrass curve $W_{D}$ is the
locus of Riemann surfaces $X\in\mathcal{M}_{2}$ such that

  1. (1)

The Jacobian of $X$ admits real multiplication by $\mathcal{O}_{D}$,

  2. (2)

There exists an abelian differential $\omega$ on $X$ with a single double
zero, such that $\mathcal{O}_{D}\cdot\omega\subset\mathbb{C}\omega$.

Recall that the Jacobian of $X$ is the complex torus
$\mathrm{Jac}(X)=\Omega^{\vee}(X)/H_{1}(S,\mathbb{Z})$. Here
$\Omega^{\vee}(X)$ is the dual of $\Omega(X)$, the vector space of holomorphic
one-forms on $X$, and $H_{1}(S,\mathbb{Z})$ is a lattice in $\Omega^{\vee}(X)$
by $\gamma\mapsto\int_{\gamma}\cdot$. An endomorphism of $\mathrm{Jac}(X)$
comes from a unique $\mathbb{C}$-linear map
$T\colon\Omega^{\vee}(X)\to\Omega^{\vee}(X)$ preserving the lattice
$H_{1}(S,\mathbb{Z})$. We say that $T$ is self-adjoint if for any
$\gamma,\delta\in H_{1}(S,\mathbb{Z})$, we have
$T(\gamma)\cdot\delta=\gamma\cdot T(\delta)$. The dual of $T$ acts on
$\Omega(X)$, giving a meaning to
$\mathcal{O}_{D}\cdot\omega\subset\mathbb{C}\omega$. The subring
$\mathcal{O}_{D}\subset\mathrm{End}(\mathrm{Jac}(X))$ is said to be proper
when for $T\in\mathrm{End}(\mathrm{Jac}(X))$, if $T$ is not in
$\mathcal{O}_{D}$, then neither is $nT$ for all
$n\in\mathbb{Z}\setminus\\{0\\}$. For more information on real multiplication
in this context, we refer to [McM07, McM23a]. McMullen showed that the
components of $W_{D}$ are Teichmüller curves, that are primitive if and only
if $D$ is not a square. He also proved [McM05] that they are connected unless
$D\equiv 1\mod 8$, $D>9$, in which case $W_{D}$ has two connected components,
distinguished by a spin inviariant $\epsilon\in\\{0,1\\}$. Veech proved in the
seminal article [Vee89] that the regular polygons with even number of sides
generate Teichmüller curves. In particular, the regular decagon defines a
Teichmüller disc in $\mathcal{H}(1,1)$ and gives a Teichmüller curve in
$\mathcal{M}_{2}$.

######  Theorem 2.2 (McMullen [McM06b]).

The curve generated by the decagon and the Weierstrass curves with non-square
discriminant account for all the primitive Teichmüller curves in
$\mathcal{M}_{2}$.

###  2.11. Square-tiled surfaces

Square-tiled surfaces, or origamis, form an important class of Veech surfaces
of combinatorial nature. Intuitively, a square-tiled surface is a translation
surface obtained by gluing sides of unit squares. A surface tiled by $n$
squares is entirely characterised by the following two permutations: one
taking a tile to its adjacent tile on its right, and the other taking it to
its adjacent tile on top of it. Square-tiled surfaces can thus be thought of
as pairs of permutations $(\alpha,\beta)\in\mathfrak{S}_{n}$, such that
$\langle\alpha,\beta\rangle$ acts transitively on $\\{1,\ldots,n\\}$, up to
simultaneous conjugacy, see for example [Mat22, Definition 6 and Remark 9].
Another way of defining them is to consider covers of tori. Let us denote by
$T$ the standard torus: the translation surface
$\mathbb{C}/{(\mathbb{Z}+i\mathbb{Z})}$ equipped with the form $\omega=dz$.

######  Definition 2.3.

A square-tiled surface is a translation surface obtained by pulling back the
standard torus via a cover $S\to T$ that sends its branch points to a single
point.

We say that a square-tiled surface is _reduced_ when the set of its relative
periods is $\mathbb{Z}+i\mathbb{Z}$. In [HL06], Hubert and Lelièvre classified
the $\mathrm{SL}_{2}(\mathbb{Z})$-orbits of reduced origamis in
$\mathcal{H}(2)$ tiled with a prime number of tiles. McMullen then classified
them for all numbers of tiles in [McM05].

######  Theorem 2.4.

(Hubert-Lelièvre, McMullen) There are two $\mathrm{SL}_{2}(\mathbb{Z})$-orbits
of reduced square-tiled surfaces in $\mathcal{H}(2)$ with $n$ squares if
$n\geqslant 5$ is odd. There is only one orbit for $n\geqslant 4$ even or
$n=3$.

There is still no similar classification for square-tiled surfaces in
$\mathcal{H}(1,1)$, but there are results in this direction in [Dur19].

##  3\. Lifts of abelian differentials

###  3.1. Covering maps

Throughout this article, unless otherwise stated, we will consider covering
maps up to upper equivalence. That is, we consider that the covers
$p_{i}\colon S\to\Sigma$ are equal for $0\leqslant i\leqslant 1$ when there
exists a homeomorphism $F\colon S\to S$ such that the following diagram
commutes

${S}$${S}$${\
\Sigma.}$$\scriptstyle{F}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{1}}$

The mapping class group $\mathrm{Mod}(\Sigma)$ acts on the set of unramified
covers $S\to\Sigma$ by pushforward. Namely if $p\colon S\to\Sigma$ and
$f\in\mathrm{Mod}(\Sigma)$, then $f\cdot p=f\circ p$.

###  3.2. Definitions

Let $f\colon S\to\Sigma$ be a possibly branched covering map. The lift, or
pullback, of $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ by $f$ is defined to be
the abelian differential in $\Omega\mathcal{M}(S)$ whose linear charts are
given by the composition of $f$ and charts of $(X,\omega)$. We denote by
$f^{*}(X,\omega)$, or $f^{*}(\omega)$, the lift of $(X,\omega)$ by $f$. We can
now define the echoes.

######  Definition 3.1.

An _echo_ of a Teichmüller curve generated by $(X,\omega)$ is a Teichmüller
curve generated by a lift of $(X,\omega)$.

It is easy to verify that this definition is consistent: being an echo of $V$
does not depend on the choice of the generator of $V$. In the present article,
we are mostly interested in unramified covers, and from now on we suppose that
$\Phi$ is the set of these covers $S\to\Sigma$. We define the lifting map:

$\mathcal{L}\colon\Omega\mathcal{T}(\Sigma)\times\Phi\to\Omega\mathcal{M}(S).$

The lift of a set $X\subset\Omega\mathcal{T}(\Sigma)$ is defined to be
$\mathcal{L}(X\times\Phi)$, and the lift of
$Y\subset\Omega\mathcal{M}(\Sigma)$ is the lift of the preimage of $Y$ by
$\Omega\mathcal{T}(\Sigma)\to\Omega\mathcal{M}(\Sigma)$. By definition, we
have $\mathcal{L}(f\cdot\omega,f\cdot p)=\mathcal{L}(\omega,p)$.

######  Lemma 3.2.

The map $\mathcal{L}$ is invariant by the diagonal action of
$\mathrm{Mod}(\Sigma)$.

Therefore in order to understand all the lifts of a specific
$(X,\omega)\in\Omega\mathcal{M}_{g}$, it suffices to consider one of its
representatives in $\Omega\mathcal{T}_{g}$ and all the possible different
covers.

###  3.3. Genus three surfaces covering a genus two one

We specialise to the case where $\Sigma$ is a closed surface of genus two,
whilst $S$ is of genus three. We fix these topological surfaces for the rest
of the article.

####  3.3.1. The set of covers.

Let us first observe that the covers $S\to\Sigma$ have no ramification point,
and that there are fifteen of them. This is also observed for example in
[LN18, Proposition 2.3], where Lanneau and Nguyen use this fact to show that
there is only a finite number of Teichmüller curves in the Prym locus in
$\mathcal{H}(2,2)^{\mathrm{hyp}}$ of fixed discriminant.

######  Lemma 3.3.

Every covering map $S\to\Sigma$ is unramified. The space of these covers is in
bijection with ${H}^{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})\setminus\\{0\\}$.

###### Proof.

If $f\colon S\to\Sigma$ is a covering map of degree $d$, with total branching
order $B$, then the Riemann-Hurwitz formula gives

$\chi(S)=d\chi(\Sigma)-B.$

Since $\chi(S)=-4$ and $\chi(\Sigma)=-2$, we must have $d\leqslant 2$.
Moreover if $d=2$ then $B=0$ and $f$ is unramified. Finally, it is impossible
to have $d=1$ since otherwise $f$ would be a homeomorphism. Every degree two
cover is normal thus the space of these covers is in bijection with the set of
epimorphisms $\pi_{1}(\Sigma)\to\mathbb{Z}/2\mathbb{Z}$, in other words with
the non-zero elements of ${H}^{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$. ∎

By duality, we may identify the non-zero elements of
${H}^{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ with the ones of
${H}_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$. We will in the rest of the text
identify the latter with the set of degree two covers $S\to\Sigma$. The cover
corresponding to $\gamma\in{H_{1}}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ is then
obtained by the following construction. Consider two copies of $\Sigma$ and a
simple closed curve $c\subset\Sigma$ on each copy representing $\gamma$.
Cutting the two copies along $c$ and gluing them back together produces the
cover associated with $\gamma$. The mapping class group acts transitively on
these covers.

######  Lemma 3.4.

The mapping class group $\mathrm{Mod}(\Sigma)$ acts transitively on the set of
covers $S\to\Sigma$.

###### Proof.

The action of the mapping class group on covers is conjugated to its natural
action on ${H}_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})\setminus\\{0\\}$. This
action is easily seen to be transitive, see for example [LF23, Proposition
3.2]. ∎

####  3.3.2. Lifts of strata of $\Omega\mathcal{M}_{2}$ in
$\Omega\mathcal{M}_{3}$

Kontsevich and Zorich gave a complete classification of the connected
components of the strata of $\Omega\mathcal{M}_{g}$ in [KZ03]. Calderon and
Calderon-Salter [Cal20, CS21] extended these results and studied the connected
components of the strata in $\Omega\mathcal{T}_{g}$. We will now identify the
connected components of the locus in the strata of $\Omega\mathcal{M}_{3}$ of
surfaces covering a genus two one. This was already done in [AN16, AN20], but
we provide a different and more direct proof. For this purpose, we will need
to know the connected components of $\Omega\mathcal{T}(1,1)$ and
$\Omega\mathcal{T}(2)$. In [Cal20, Remark 2.7], Calderon classifies these
connected components, and attributes this result to folklore knowledge.

######  Lemma 3.5 (Folklore).

The space $\Omega\mathcal{T}(1,1)$ is connected, while the space
$\Omega\mathcal{T}(2)$ has six connected components, corresponding to the six
Weierstrass points.

The space $\mathcal{H}(1,1)$ lifts to a connected set
$\widetilde{\mathcal{H}}(1,1)$ in $\mathcal{H}(1,1,1,1)$, as shown in [AN20,
Proposition 2.17]. Here we give a similar argument, that illustrates how we
can use the action of the mapping class group to study lifts.

######  Proposition 3.6.

The set $\widetilde{\mathcal{H}}(1,1)$ in $\mathcal{H}(1,1,1,1)$ is connected.

###### Proof.

Let us first restrict the lifting map to $\Omega\mathcal{T}(1,1)$:

$\mathcal{L}\colon\Omega\mathcal{T}(1,1)\times
H_{1}(\Sigma,\mathbb{Z}/2)\setminus\\{0\\}\to\mathcal{H}(1,1,1,1).$

Let us show that its image is connected. By Lemma 3.2,
$\mathcal{L}(\omega,\gamma)=\mathcal{L}(f\cdot\omega,f\cdot\gamma)$ for any
$(\omega,\gamma)$ and any $f\in\mathrm{Mod}(\Sigma)$. Recall that
$\mathrm{Mod}(\Sigma)$ acts transitively on
${H_{1}}(\Sigma,\mathbb{Z}/2)\setminus\\{0\\}$. Thus for $\gamma\in
H_{1}(\Sigma,\mathbb{Z}/2)\setminus\\{0\\}$, the image of $\mathcal{L}$ does
not change when restricted to $\Omega\mathcal{T}(1,1)\times\\{\gamma\\}$. The
space $\Omega\mathcal{T}(1,1)$ is connected hence so is its image
$\widetilde{\mathcal{H}}(1,1)$. ∎

It is proven in [KZ03] that $\mathcal{H}(2,2)$ has two connected components:
the hyperelliptic component $\mathcal{H}(2)^{\mathrm{hyp}}$, and the odd
component $\mathcal{H}(2)^{\mathrm{odd}}$. To tell which connected component
$(X,\omega)$ belongs to, one can proceed as follows. For each $\gamma\in
H_{1}(\Sigma,\mathbb{Z})$ primitive, consider $c\colon\mathbb{S}^{1}\to X$
representing $\gamma$, avoiding the zeroes of $(X,\omega)$ with non-zero
derivative everywhere. Taking local linear charts, we can compute
$c^{\prime}\colon\mathbb{S}^{1}\to\mathbb{C}\setminus\\{0\\}$. We then define
$\mathrm{Ind}(\gamma)$ to be the degree of the map
$\mathbb{S}^{1}\to\mathbb{S}^{1}$ given by
$s\to\frac{c^{\prime}(s)}{|c^{\prime}(s)|}$. Given a symplectic basis
$(a_{1},b_{1},a_{2},b_{2})$ of $X$, we define

$\mathrm{Arf}(X,\omega)=\sum_{i}\left(\mathrm{Ind}(a_{i})+1\right)\left(\mathrm{Ind}(b_{i})+1\right)\mod
2.$

This does not depend on the choices of the $c$ nor on the choice of the
symplectic basis and $(X,\omega)$ is in $\mathcal{H}(2)^{\mathrm{odd}}$ if and
only if $\mathrm{Arf}(X,\omega)=1$, see also [Zor06, Section 9.4]. We apply
this criterion to exhibit a surface in different connected components of
$\widetilde{\mathcal{H}}(2)$. Indeed, we easily verify by considering a basis
of homology on each of the surfaces of Figure 1 that they belong to different
components of $\mathcal{H}(2,2)$. They are both covering an $L$-shaped surface
in $\mathcal{H}(2)$, that we will study later, see Figure 2 in Section 4.

Figure 1. Translation surfaces, in $\widetilde{\mathcal{H}}(2)^{\mathrm{odd}}$
on the left, and in $\widetilde{\mathcal{H}}(2)^{\mathrm{hyp}}$ on the right.

We now show that the lift $\widetilde{\mathcal{H}}(2)\subset\mathcal{H}(2,2)$
has two connected components. This result is obtained in [AN16] as a
consequence of the classification of affine invariant submanifolds in
$\mathcal{H}(2,2)$.

######  Proposition 3.7.

The stratum $\mathcal{H}(2)$ lifts to two connected components
$\widetilde{\mathcal{H}}^{\mathrm{odd}}(2)$ and
$\widetilde{\mathcal{H}}^{\mathrm{hyp}}(2)$ in $\mathcal{H}(2,2)$, one in each
connected component of $\mathcal{H}(2,2)$.

Choose a complex structure $X$ on $\Sigma$ and let
$\iota\colon\Sigma\to\Sigma$ be a holomorphic involution representing the
hyperelliptic involution, the only non-trivial element in the centre of
$\mathrm{Mod}(\Sigma)$. The Weierstrass points of $X$ are the fixed points of
$\iota$. It follows from a remark of Arnold [Arn68] that a non-zero element in
$H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ can be identified with the choice of
two Weierstrass points on $X$.

######  Lemma 3.8.

The non-zero elements of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ can be
naturally identified with the unordered couples of Weierstrass points of $X$.

###### Proof.

By [Arn68, Lemma 1], the quotient by the hyperelliptic involution gives a well
defined injection $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})\to
H_{1}(S_{0}^{6},\mathbb{Z}/2\mathbb{Z})$ to the homology of the sphere with
six points removed. The latter homology group is identified with the
partitions of the Weierstrass points in two sets by [Arn68, Lemma 2]. We can
take a non-separating simple closed curve whose image singles out two
Weierstrass points. Since $\mathrm{Mod}(\Sigma)$ acts transitively on the non-
separating simple closed curves, it is the case of every such curve. The image
of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ is thus identified with the
$15={6\choose 2}$ unordered pairs of Weierstrass points. ∎

We can now prove 3.7.

###### Proof.

There are at least two of these connected components. Indeed we have seen that
the surfaces of Figure 1 do not belong to the same component of
$\mathcal{H}(2,2)$. The space $\Omega\mathcal{T}(2)$ has six connected
components, that correspond to the Weierstrass point on which the zero of the
form lies. Since $\mathrm{Mod}(\Sigma)$ acts transitively on
$H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})\setminus\\{0\\}$, the lift of
$\Omega\mathcal{T}(2)$ is covered by the sets
$\mathcal{L}(X_{i}\times\\{\gamma\\})$ where $X_{i}$ are the connected
components of $\Omega\mathcal{T}(2)$ for $1\leqslant i\leqslant 6$, and
$\gamma\in H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ is a fixed non-zero element.
The Dehn twists along a simple closed curve that single out two Weierstrass
points interchange them, and leave the others unchanged, see for example
[BB01, Proposition 3.3] or [MW19, Section 2.1]. A Dehn twist $T$ along a
simple closed curve $c$ representing $\gamma$ will interchange the two
Weierstrass points singled out by $\gamma$, and preserves $\gamma$. Therefore
$T$ interchanges the two connected components, say $X_{1}$ and $X_{2}$, of
$\Omega\mathcal{T}(2)$ corresponding to the Weierstrass points singled out by
$\gamma$. Thus they lift to the same sets in $\mathcal{H}(2,2)$:

$\mathcal{L}(X_{1}\times\\{\gamma\\})=\mathcal{L}(T(X_{1})\times\\{T(\gamma)\\})=\mathcal{L}\left(X_{2}\times\\{\gamma\\}\right).$

Similarly, Dehn twists along simple closed curves not intersecting $c$ will
permute the other four Weierstrass points, leaving $\gamma$ invariant.
Therefore the lift of $\mathcal{H}(2)$ has at most two connected components. ∎

Observe that our proof of 3.7 gives a criterion to determine the connected
component where $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ is sent to by
$\gamma\in H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$. Indeed it suffices to check
whether the zero of $(X,\omega)$ is one of the two Weierstrass points singled
out by $\gamma$. It is the case exactly when the lift is in
$\widetilde{\mathcal{H}}(2)^{\mathrm{hyp}}$. Therefore we have the following.

######  Corollary 3.9.

Among the fifteen lifts of $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$, exactly
five belong to the hyperelliptic component.

###  3.4. Telling the lifts appart

We now wish to determine when two lifts define the same abelian differential
in $\Omega\mathcal{M}_{3}$. Let us first show that, under an extra assumption,
an isomorphism of lifts comes from an isomorphism at the base level.

######  Lemma 3.10.

Let $\alpha_{i}\in\Omega\mathcal{T}(\Sigma)$, and $p_{i}\colon S\to\Sigma$ be
covers, for $0\leqslant i\leqslant 1$. Consider
$\omega_{0}=p_{0}^{*}(\alpha_{0})$ and $\omega_{1}=p_{1}^{*}(\alpha_{1})$.
Suppose that the forms $\omega_{i}$ have only one translation with quotient of
genus two. Every homeomorphism $F\colon S\to S$ such that
$F\cdot\omega_{0}=\omega_{1}$ is induced by a homeomorphism
$f\colon\Sigma\to\Sigma$ such that $f\circ p_{0}=p_{1}\circ F$ and
$f\cdot\alpha_{0}=\alpha_{1}$.

###### Proof.

We wish to check that $F$ passes through the quotient by the deck
transformations to a map $f\colon\Sigma\to\Sigma$. The deck transformations of
$p_{i}$ are translations with quotient of genus two. Since $\omega_{i}$ have
only one such translation, the deck transformations are conjugated by $F$.
Therefore $F$ passes to a map $f$. ∎

######  Remark 3.11.

Lemma 3.10 does not hold without this extra assumption. Indeed, there exist
non-isomorphic translation surfaces in genus two with isomorphic double
covers. For example, the escalator square-tiled surface defined by
$\alpha=(1,2)(3,4)(5,6)(7,8)$ and $\beta=(2,3)(4,5)(6,7)(8,1)$, see [SPWS17,
Figure 4]. It is a cover of both the origamis defined by $\alpha=(1,2)(3,4)$
and $\beta=(1,3)$ and $\alpha=(2,3)$, $\beta=(1,2)(3,4)$ that are not
isomorphic.

We can now describe, under our assumption, when two lifts define the same
element in $\Omega\mathcal{M}_{3}$. This happens exactly when the pairs
$(\omega,\gamma)$ differ by the diagonal action of the mapping class group on
$\Omega\mathcal{T}(\Sigma)\times H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$, by
Lemma 3.10 and the invariance of $\mathcal{L}$ by the diagonal action of
$\mathrm{Mod}(\Sigma)$.

######  Proposition 3.12.

Let $(X_{i},\omega_{i})\in\Omega\mathcal{T}(\Sigma)$ and $p_{i}\colon
S\to\Sigma$ be covering maps, for $0\leqslant i\leqslant 1$. Suppose that
$p^{*}(\omega_{i})$ have only one non-trivial translation. We have
$p_{0}^{*}(\omega_{0})\simeq p_{1}^{*}(\omega_{1})$ if and only if there
exists $f\in\mathrm{Mod}(\Sigma)$ such that both $f\cdot\omega_{0}=\omega_{1}$
and $f\cdot p_{0}=p_{1}$.

###  3.5. Lifting Teichmüller discs and Veech groups

We now derive from 3.12 a description of the Veech groups of forms covering
genus two ones. We will assume that the lifts only have one non-trivial
translation. Let $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$. Let us denote by
$[p]$ the orbit of the cover $p$ under the action of $\mathrm{Tr}(X,\omega)$.
Since $\mathrm{Tr}(X,\omega)$ is normal, the group
$\mathrm{Aff}^{+}(X,\omega)$ acts on these orbits. We thus obtain an action of
$\mathrm{SL}(X,\omega)$ on the set of orbits $[p]$.

######  Corollary 3.13.

Let $(X_{0},\omega_{0})\in\Omega\mathcal{T}(\Sigma)$ and $p\colon S\to\Sigma$
be a covering map. Suppose that $(X,\omega)=p^{*}(X_{0},\omega_{0})$ has only
one translation with quotient of genus two. The Veech group of $(X,\omega)$ is

$\mathrm{SL}(X,\omega)=\\{A\in\mathrm{SL}(X_{0},\omega_{0})\mid
A\cdot[p]=[p]\\}.$

###### Proof.

Let $A\in\mathrm{SL}_{2}(\mathbb{R})$. We have $A\cdot
p^{*}(\omega_{0})=p^{*}(A\cdot\omega_{0})\simeq p^{*}(\omega_{0})$ if and only
if there exists $f\in\mathrm{Mod}(\Sigma)$ such that both $f\cdot p=p$ and
$f\cdot\omega_{0}=A\cdot\omega_{0}$ by 3.12. This happens exactly when
$A\in\mathrm{SL}(X_{0},\omega_{0})$ and $A\cdot[p]=[p]$. ∎

######  Remark 3.14.

If $p\colon S_{g}\to\Sigma_{h}$ is a covering map and
$\omega\in\Omega\mathcal{T}(\Sigma_{h})$, we always have the inclusion
$\\{A\in\mathrm{SL}(X,\omega)\mid
A\cdot[p]=[p]\\}\subset\mathrm{SL}(p^{*}(X,\omega))$. Indeed a map in
$\mathrm{Aff}^{+}(X,\omega)$ that stabilises $p$ lifts to a map in
$\mathrm{Aff}^{+}(p^{*}(X,\omega))$ with the same derivative.

We also derive from 3.12 a criterion to decide when two lifts lie in the same
Teichmüller disc.

######  Corollary 3.15.

Two lifts of $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ by the covers
$p_{i}\colon S\to\Sigma$, $0\leqslant i\leqslant 1$, with only one non-trivial
translation with quotient of genus two, belong to the same Teichmüller disc if
and only if $p_{1}$ is in the orbit of $p_{0}$ under the action of
$\mathrm{Aff}^{+}(X,\omega)$.

###### Proof.

The forms $p_{0}^{*}(\omega)$ and $p_{1}^{*}(\omega)$ belong to the same
Teichmüller disc if and only if there exists $A\in\mathrm{SL}_{2}(\mathbb{R})$
such that $A\cdot p_{1}^{*}(\omega)=p_{1}^{*}(A\cdot\omega)\simeq
p_{0}^{*}(\omega)$. It follows from 3.12 that this happens exactly when
$p_{1}$ is the orbit of $p_{0}$ under the action of
$\mathrm{Aff}^{+}(X,\omega)$. ∎

We will verify that the assumption of Lemma 3.10 is satisfied in our cases.

######  Lemma 3.16.

Let $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ be a generator of either a
Weierstrass curve, or the decagon curve. Any lift
$\tilde{\omega}\in\Omega\mathcal{M}(S)$ has only one translation with quotient
of genus two, unless $(X,\omega)$ generates $W_{9}$.

We postpone the proof of Lemma 3.16 to Section 4 for the Weierstrass curves,
and to Section 5 for the decagon one. Since a Teichmüller curve is generated
by a unique Teichmüller disc (up to scaling), the echoes of the curve
generated by $(X,\omega)$ are classified by the action of the
$\mathrm{Aff}^{+}(X,\omega)$ on homology in these cases.

######  Corollary 3.17.

Let $V\subset\mathcal{M}_{2}$ be either a Weierstrass curve or the decagon
curve. Fix $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ generating $V$. The echoes
of $V$ in $\mathcal{M}_{3}$ are in correspondence with the orbits of the
action of $\mathrm{Aff}^{+}(X,\omega)$ on
$H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})\setminus\\{0\\}$.

We prove 3.17 for $W_{9}$ in Section 4. The other cases follow from Lemma 3.16
and 3.15.

###  3.6. Echoes are covers

Under our assumption, each echo $\tilde{V}$ generated by $(X,\omega)$ of $V$
generated by $(X_{0},\omega_{0})$ is naturally a cover of $V$ since
$\tilde{V}=\mathbb{H}^{2}/\mathrm{SL}_{2}(X,\omega)$ and
$V=\mathbb{H}^{2}/\mathrm{SL}_{2}(X_{0},\omega_{0})$. The degree
$d=[\mathrm{SL}_{2}(X,\omega)\colon\mathrm{SL}_{2}(X_{0},\omega_{0})]$ of this
cover is the size of the orbit of the corresponding $[p]$.

##  4\. Generators of the monodromy groups of Weierstrass curves

In this section we exhibit elements of the monodromy groups of generators of
Weierstrass curves.

###  4.1. L-shaped billiards

In [McM05], McMullen gave explicit generators for the Teichmüller curves
$W_{D}\subset\mathcal{M}_{2}$. These generators are given as surfaces obtained
by unfolding a billiard table of the form $L(b,e)$: the L-shaped polygon of
Figure 2, having top square of side $\lambda=\frac{e+\sqrt{e^{2}+4b}}{2}$,
while the remaining rectangle has horizontal sides of length $b$ and vertical
side of unit length. It turns out that unfolding this billiard table leads,
after rescaling, to the translation surface that we still denote $L(b,e)$,
drawn in Figure 2. The integers $(e,b)$ are chosen so that $b\geqslant 1$,
$e\in\\{-1,0,1\\}$ and if $e=1$, then $b$ is even. We also assume that
$e+1<b$. We have $\lambda^{2}=e\lambda+b$.

$\lambda$$1$$\lambda$$b-\lambda$ Figure 2. Translation surface $L(b,e)$.

######  Theorem 4.1 (McMullen [McM05]).

The Teichmüller curve generated by $L(b,e)$ is a Weierstrass curve of
discriminant $D=e^{2}+4b$. When $D>9$ is such that $D\equiv 1\mod 8$, the
curves generated by $L(b,e)$ and $L(b,-e)$ do not have the same spin
invariant.

We use this explicit description to prove Lemma 3.16 for Weierstrass curves.
Recall that this proves 3.17 for $V$ a Weierstrass curve that is not $W_{9}$.

###### Proof.

Suppose that $\lambda\neq 1$. The saddle connections on $L$ lift on a double
cover $\tilde{L}$ to saddle connections with the same periods. There are two
saddle connections on $L$ with period $i$. Of those two, only one has the
property that the saddle connection defined by turning with an angle $\pi$
from its endpoint has period $i$. Indeed, turning by an angle $\pi$ at the
endpoint of the other will give a saddle connection with period $i\lambda$.
Namely it is the saddle connection going up from the only conical point in the
interior of the bottom edge of $L$. Therefore $\tilde{L}$ has exactly two
saddle connections satisfying this property. A translation of $\tilde{L}$ must
either fix them, or interchange them. Hence the group of translations of
$\tilde{L}$ is of order two. We have $\lambda=1$ if and only if
$2=e+\sqrt{e^{2}+4b}$, that is when $b=1-e$. We have assumed that $e+1<b$, so
the only possibility is $b=2$ and $e=-1$. The curve generated by $L(2,-1)$ is
$W_{9}$. ∎

We now prove 3.17 for $V=W_{9}$.

###### Proof.

The curve $W_{9}$ is generated by an origami $\mathcal{O}$ with three squares.
We check directly that among the fifteen lifts of $\mathcal{O}$, exactly one
has a group of translation of order larger than two. The action of
$\mathrm{Aff}^{+}(\mathcal{O})$ on covers $S\to\Sigma$ preserves the order of
the translation group of the associated lift, hence fixes this cover.
Moreover, for the other covers, there are only two translations and we can
apply 3.15. ∎

###  4.2. Marking and basis of homology

Let us first fix $(a_{1},b_{1},a_{2},b_{2})$, a basis for
$H_{1}(L,\mathbb{Z})$. We take for $a_{1}$ and $b_{1}$ respectively the
horizontal and vertical sides of the top square of side $\lambda$. For $a_{2}$
and $b_{2}$, we take the bottom horizontal side of $L$ and the vertical (blue)
side at the right of $L$, respectively. We orient $a_{1}$ and $a_{2}$ towards
the right, while $b_{1}$ and $b_{2}$ are oriented as going up. This basis is
symplectic. We fix an identification $\Sigma\simeq L$. We will use Dehn twists
in cylinders decompositions of $L$ as in Section 2.7 to study the action of
its affine group on homology.

###  4.3. Action on homology of Dehn twists

Suppose that $s$ is a direction that decomposes $L$ into two cylinders $C_{1}$
and $C_{2}$. Their moduli $m_{1}$ and $m_{2}$ are commensurable, since $L$ is
a Veech surface. Let $m$ be the lowest common multiple of $\frac{1}{m_{1}}$
and $\frac{1}{m_{2}}$ and write $m\cdot m_{1}={k_{1}}$ and $m\cdot
m_{2}={k_{2}}$. Since the action on homology modulo two of a Dehn twist in a
cylinder is of order two, the action of the Dehn twist of Section 2.7 only
depends on the parity of $k_{1}$ and $k_{2}$. Namely, the action is the same
as the one of the Dehn twist in $C_{1}$ (resp. $C_{2}$) if $k_{1}$ is odd and
$k_{2}$ is even (resp. $k_{1}$ is even and $k_{2}$ is odd). If both $k_{1}$
and $k_{2}$ are odd, then the action on homology is the same as the one of the
composition of a Dehn twist in $C_{1}$ and a Dehn twist in $C_{2}$. Observe
that we have $\frac{m_{1}}{m_{2}}=\frac{k_{1}}{k_{2}}$.

###  4.4. Horizontal cylinder decomposition

Let us compute the moduli of the cylinders that are in the horizontal
direction. The top square of side $\lambda$ has modulus $m_{1}=1$. The
rectangle below has modulus $m_{2}=\frac{1}{b}$. Thus the affine group of
$L(b,e)$ contains the map that is a right-Dehn twist in the bottom rectangle,
and the $b$-th power of a right-Dehn twist in the top square. Its action on
homology gives the following element of the monodromy group in our chosen
basis:

$H=\begin{pmatrix}1&b&0&0\\\ 0&1&0&0\\\ 0&0&1&1\\\ 0&0&0&1\end{pmatrix}.$

###  4.5. Vertical cylinder decomposition

The vertical direction gives a decomposition into two cylinders. The cylinder
on the left has modulus $m_{1}=\frac{\lambda}{\lambda+1}$, while the second
has modulus $m_{2}={b-\lambda}$. Since $\lambda^{2}=e\lambda+b$, their ratio
is

$\frac{m_{2}}{m_{1}}=\frac{\left(b-\lambda\right)\left(\lambda+1\right)}{\lambda}=\frac{b\lambda+b-\lambda^{2}-\lambda}{\lambda}=b-e-1.$

Similarly to the horizontal decomposition, we obtain an element of the
monodromy group:

$V=\begin{pmatrix}1&0&0&0\\\ 1&1&1&0\\\ 0&0&1&0\\\ 1&0&b-e&1\end{pmatrix}.$

We can obtain the third column of $V$ by remarking that the image by the Dehn
twist of $(-1,0,1,0)$ is $(-1,0,1,b-e-1)$.

###  4.6. Rotation

We want to prove that the monodromy group modulo two strictly contains the one
generated by $H$ and $V$, when $D\equiv 1\mod 8$.

######  Lemma 4.2.

If $D\equiv 1\mod 8$, then the monodromy group modulo two of $L$ is not
generated by the horizontal and vertical twists.

Recall that $D=e^{2}+4b$. We thus assume for the rest of the section that $b$
is even and $e\in\\{-1,1\\}$.

####  4.6.1. The case $e=-1$

There exists a simple argument to prove Lemma 4.2 when $e=-1$. In this case,
the following matrix belongs to the Veech group of $L$

$\begin{pmatrix}\frac{b}{\lambda+1}&0\\\
0&\frac{1}{\lambda}\end{pmatrix}\begin{pmatrix}0&&1\\\ -1&&0\end{pmatrix}.$

Indeed, the rotation of order four applied to $L$ gives Figure 3.

$\lambda$$1$$\lambda$$b-\lambda$ Figure 3. Translation surface $L(b,e)$
rotated.

Then the rescaling of the axes changes the bottom rectangle to a new one of
sides $\frac{b-\lambda}{\lambda}$ and $\frac{b}{\lambda+1}$. But since
$b=\lambda^{2}+\lambda$, it becomes a square of side $\lambda$. Similarly, the
top rectangle changes to a new one of width $b$ and height $1$. A
corresponding homeomorphism in the Veech group exchanges the two handles
$(a_{1},b_{1})$ and $(a_{2},b_{2})$. Its action on
$H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ will send $v=(0,1,0,0)$ to
$(-1,0,1,0)$, while $v$ is fixed by both the $H$ and $V$. This proves Lemma
4.2 when $e=-1$.

####  4.6.2. The case $e=1$

We suppose for the rest of the section that $e=1$. The
$\mathrm{GL}_{2}^{+}(\mathbb{R})$-orbit of $L(b,e)$ contains the $L$-shaped
translation surface $\mathcal{L}(b,e)$, that is the same as $L(b,e)$, but with
the top square of side $\lambda-2$ and the bottom rectangle of width $b-2$.
Indeed, apply the following matrix to $L(b,e)$:

$\begin{pmatrix}\frac{b-\lambda}{\lambda}&0\\\
0&\frac{1}{\lambda}\end{pmatrix}\begin{pmatrix}0&&1\\\ -1&&0\end{pmatrix}.$

After rotation and rescaling, the top rectangle has width

$\frac{(b-\lambda)(\lambda+1)}{\lambda}=b-e-1=b-2.$

The remaining rectangle is a square of side
$\frac{b-\lambda}{\lambda}=\lambda-e-1=\lambda-2$. We mark the surface
$\mathcal{L}(b,e)$ similarly to $L(b,e)$.

###  4.7. Diagonal cylinder decompositions

Our goal is to prove Lemma 4.2 in the remaining case $e=1$.

####  4.7.1. The case $b\equiv 2\mod 4$

Let us start by assuming that $b$ is not a multiple of four. Let us consider
the cylinder decomposition associated with the slope $s=\frac{2}{b}$. We
assume for now that $\frac{b}{2}>\lambda$. This is equivalent to $b>6$. This
requirement allows us to have a simple picture for the cylinder decomposition,
as shown in Figure 4.

$\lambda$$1$$\lambda$$b-\lambda$ Figure 4. Cylinder decomposition with slope
$s=2/b$.

The elements $\alpha=(0,0,1,2)$ and $\beta=(\frac{b}{2},1,0,0)$ of
$H_{1}(\Sigma,\mathbb{Z})$ represent saddle connections with slope $s$, see
the brown and violet lines in Figure 4. Let $\gamma=(1,0,0,1)$ and
$\delta=(1,0,0,0)$. The elements $(\alpha,\gamma)$ form a basis of the
homology of the handle formed by the smaller cylinder $C_{1}$, and
$(\alpha+\beta,\delta)$ form a basis of the homology of the handle formed by
the longer cylinder $C_{2}$. Let us compute the moduli of these cylinders. Let
us start with the cylinder $C_{1}$. The period of $\alpha$ is $b+2i$, while
the period of $\gamma$ is $\lambda+i$. We wish to compute the modulus of the
cylinder defined by these complex numbers. Multiplying both numbers by a
complex number does not change the modulus of the cylinder they form, hence we
can consider the parallelogram formed by $b^{2}+4$ and $(\lambda+i)(b-2i)$.
Thus the modulus $m_{1}$ of $C_{1}$ is:

$m_{1}=\frac{b-2\lambda}{b^{2}+4}.$

The period of $\alpha+\beta$ is $\frac{b}{2}\lambda+b+i(\lambda+2)$, and the
period of $\delta$ is $\lambda$. Thus the modulus $m_{2}$ of $C_{2}$ is:

$m_{2}=\frac{\lambda(\lambda+2)}{(\frac{b}{2}\lambda+b)^{2}+(\lambda+2)^{2}}=\frac{4\lambda}{(b^{2}+4)({\lambda}+2)}.$

Since $\lambda^{2}=e\lambda+b$, we have

$\frac{m_{1}}{m_{2}}=\frac{(b-2\lambda)(\lambda+2)}{4\lambda}=\frac{b\lambda+2b-2\lambda^{2}-4\lambda}{4\lambda}=\frac{\frac{b}{2}-e-2}{2}.$

Since $b$ is not a multiple of four, $\frac{b}{2}-e-2$ is even, and the twist
associated with the slope $s$ will send $\delta$ to $\delta+\alpha+\beta$ in
the monodromy modulo two group. Therefore, the action of this group on
$H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ must have $(1,0,0,0)$ in the orbit of
$(1,1,1,0)$. This is not done by any element of the group generated by the
horizontal and vertical twists. Indeed the vector $(1,1,1,0)$ is fixed by both
$V$ and $H$. We thus have proven Lemma 4.2 when $b>6$ is not a multiple of
four.

####  4.7.2. The case $b\equiv 0\mod 4$

We now turn to the case where $b$ is a multiple of four. To deal with this
case, we will for convenience use the model $\mathcal{L}(b,e)$ instead of
$L(b,e)$. Since they are in the same $\mathrm{GL}^{+}_{2}(\mathbb{R})$-orbit,
their monodromy groups modulo two are isomorphic. The horizontal and vertical
cylinder decomposition give the same matrices $H$ and $V$. Indeed when
computing the moduli of the associated cylinders, we obtain
$\frac{m_{1}}{m_{2}}=b-2$ for the horizontal one, and for the vertical one

$\frac{m_{2}}{m_{1}}=\frac{(b-\lambda)(\lambda-1)}{\lambda-2}=\frac{b\lambda-2b-e\lambda+\lambda}{\lambda-2}=b.$

We consider, similarly to the previous diagonal cylinder decomposition, the
slope $s=\frac{2}{b-2}$. We obtain a picture similar to Figure 4, provided
$\lambda-2<\frac{b-2}{2}$. This is actually always the case: the inequality is
satisfied as soon as $b>2$, and $b$ is a multiple of four. Let us compute the
moduli of the cylinders $C_{1}$ and $C_{2}$. We have as in the previous case,
setting $b^{\prime}=b-2$ and $\lambda^{\prime}=\lambda-2$:

$\frac{m_{1}}{m_{2}}=\frac{(b^{\prime}-2\lambda^{\prime})(\lambda^{\prime}+2)}{4\lambda^{\prime}}=\frac{(b-2\lambda+2)\lambda}{4(\lambda-2)}=\frac{b\lambda-2(e\lambda+b)+2\lambda}{4(\lambda-2)}=\frac{b}{4}.$

Since $b$ is a multiple of $4$, as before the twist in the cylinder $C_{2}$
will create an element of the monodromy group that is not in the group
generated by $H$ and $V$. We thus have proven Lemma 4.2 when $b$ is a multiple
of four.

####  4.7.3. The case $b=6$

The only cases remaining are when $b\leqslant 6$ is not a multiple of four.
Note that when $b=2$, since $e+1<b$, we have $e=-1$. This corresponds to the
fact that there is only one Weierstrass curve of discriminant $D=9$ by [McM05,
Theorem 1.1]. Hence we just need to consider the case $b=6$. This case
corresponds to a square-tiled surface, tiled by five squares. Let us give two
ways to prove Lemma 4.2 in this case. One way, that gives a uniform approach
to Lemma 4.2 but that is tedious, is to consider the cylinder decomposition
associated with the slope $s=\frac{1}{2}$, as pictured in Figure 5.

Figure 5. Cylinder decomposition of $L(6,1)$ with slope $s=1/2$.

We leave it to the reader to check that this cylinder decomposition has two
cylinders $C_{1}$ and $C_{2}$ and that both a twist in $C_{1}$ and $C_{2}$
send elements in $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ to others that are not
in their orbit under the action of $\langle H,V\rangle$.

An easier approach consists in remarking that, since $L(6,1)$ is square-tiled,
we may act by $\mathrm{SL}_{2}(\mathbb{Z})$ on it to find elements in its
Veech group directly. We will work with
$\mathcal{O}=\begin{pmatrix}\frac{1}{3}&0\\\ 0&1\end{pmatrix}\cdot L(6,1)$
instead of $L(6,1)$ since it is then tiled by five unit squares. We transport
our basis of homology to $\mathcal{O}$ with this rescaling of the $x$-axis.
The permutations in $\mathfrak{S}_{5}$ defining $\mathcal{O}$ can be taken to
be $\alpha=(1,2)$ and $\beta=(2,3,4,5)$. Let $L=\begin{pmatrix}1&0\\\
1&1\end{pmatrix}$ and $R=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$. The matrices
$L$ and $R$ transform $(\alpha,\beta)$ respectively to
$(\alpha\beta^{-1},\beta)$ and $(\alpha,\beta\alpha^{-1})$, see e.g. [Mat22,
Section 2.3]. This perspective allows us to find an element in the monodromy
group modulo two that is not in $\langle H,V\rangle$.

######  Lemma 4.3.

The matrix $M=LR^{3}L$ belongs to the Veech group of $\mathcal{O}$ and the
element in the monodromy group modulo two it induces does not leave $b_{1}$
invariant.

In Lemma 4.3, we can talk about _the_ element of the monodromy group induced
by $M$ since $\mathcal{O}$ does not have any translation, hence
$\mathrm{SL}(\mathcal{O})=\mathrm{Aff}^{+}(\mathcal{O})$. This is actually the
case for every square-tiled surface in $\mathcal{H}(2g-2)$, see for example
[Mat22, Proposition 39]. Concretely the matrix $M$ is

$M=\begin{pmatrix}4&3\\\ 5&4\end{pmatrix}.$

###### Proof.

Let us first show that $LR^{3}L$ is in the Veech group of $\mathcal{O}$. The
matrix $L$ changes $(\alpha,\beta)$ to $(\gamma,\beta)$ where
$\gamma=\alpha\beta^{-1}=(1\ 5\ 4\ 3\ 2)$. Then $R^{3}$ sends $(\gamma,\beta)$
to $(\gamma,\delta)$ where $\delta=\beta\gamma^{-3}=(1\ 4\ 3\ 2)$. Finally $L$
sends $(\gamma,\delta)$ to $(\epsilon,\delta)$ where
$\epsilon=\gamma\delta^{-1}=(1\ 5)$. The pair $(\epsilon,\delta)$ is
conjugated to $(\alpha,\beta)$ by $(1\ 2\ 5)(3\ 4)$. Therefore
$M\in\mathrm{SL}(\mathcal{O})$. In order to show that $b_{1}$ is not
stabilised by $M$, it suffices by 3.13 to show that the cover
$\tilde{\mathcal{O}}$ of $\mathcal{O}$ associated with $b_{1}$ does not
contain $M$ in its Veech group. The cover associated with $b_{1}$ has the form
of the right side of Figure 1. The permutations defining it can be taken to be
$\alpha^{\prime}=(1\ 2)(6\ 7)(3\ 8)(4\ 9)(5\ 10)$ and $\beta^{\prime}=(2\ 3\
4\ 5)(7\ 8\ 9\ 10)$. Applying the same transformations as before, one obtains

$(\epsilon^{\prime},\delta^{\prime})=\left((1\ 10)(5\ 6),(1\ 4\ 8\ 2)(3\ 7\ 6\
9)(5\ 10)\right).$

The pair $(\epsilon^{\prime},\delta^{\prime})$ is obviously not conjugated to
$(\alpha^{\prime},\beta^{\prime})$ thus
$M\notin\mathrm{SL}(\tilde{\mathcal{O}})$. ∎

This completes the proof of Lemma 4.2.

##  5\. Monodromy of the decagon

In [McM06b], McMullen established that the regular decagon $D$ is the only
abelian differential in $\mathcal{H}(1,1)$ that generates a primitive
Teichmüller curve, up to the action of $\mathrm{GL}_{2}^{+}(\mathbb{R})$. Let
us fix an identification $\Sigma\simeq D$. In this section we compute the
monodromy group of $D$, that is the image of the monodromy map

$\rho\colon\mathrm{Aff}^{+}(D)\to\mathrm{Sp}(H_{1}(\Sigma,\mathbb{Z})).$

Since the translation group of the decagon $\mathrm{Tr}(D)$ is trivial, we
identify the affine group with the Veech group $\mathrm{SL}(D)$ of $D$. The
seminal article of Veech [Vee89] describes the Veech groups of regular
polygons. In order to recall this description in our case, let us denote by
$R$ the matrix of the rotation of angle $\frac{\pi}{5}$, and by $T$ the
following matrix

$T=\begin{pmatrix}1&2\cot\frac{\pi}{10}\\\
0&1\end{pmatrix}=\begin{pmatrix}1&2\sqrt{5+2\sqrt{5}}\\\ 0&1\end{pmatrix}.$

It follows from [Vee89, Theorem 1.1] that the Veech group of the decagon is
generated by the matrices $R$ and $T$. Let us fix a basis
$(a_{1},b_{1},a_{2},b_{2})$ of $H_{1}(\Sigma,\mathbb{Z})$, as pictured in
Figure 6, where the curves are oriented from left to right.

$a_{1}$$b_{1}$$a_{2}$$b_{2}$ Figure 6. The decagon $D$.

Before computing the monodromy of $D$, let us prove Lemma 3.16.

###### Proof.

There is only one saddle connection on $D$ with period $2$. There are thus two
saddle connections on any cover of degree two $\tilde{D}$ with period $2$. Any
translation must either fix them, or interchange them. The translation group
of $\tilde{D}$ is thus of order two. ∎

This finishes the proof of 3.17. We now turn to the computation of the
monodromy. Let $\xi=e^{i\pi/5}$, and
$\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}$ be the periods of
$a_{1},b_{1},a_{2},b_{2}$, respectively. We can compute these numbers
directly:

$\displaystyle\alpha_{1}$
$\displaystyle=\xi+1=\frac{5+\sqrt{5}}{4}+i\sqrt{\frac{5-\sqrt{5}}{8}}$
$\displaystyle\beta_{1}$
$\displaystyle=\overline{\alpha_{1}}=\frac{5+\sqrt{5}}{4}-i\sqrt{\frac{5-\sqrt{5}}{8}}$
$\displaystyle\alpha_{2}$
$\displaystyle=\xi^{2}-\xi^{4}=\frac{\sqrt{5}}{2}+i\frac{\sqrt{5-2\sqrt{5}}}{2}$
$\displaystyle\beta_{2}$
$\displaystyle=\xi-\xi^{3}=\frac{\sqrt{5}}{2}-i\frac{\sqrt{5-2\sqrt{5}}}{2}.$

The numbers $\alpha_{1}$, $\beta_{1}$, $\alpha_{2}$, $\beta_{2}$ are linearly
independent.

######  Lemma 5.1.

The period map $\mathrm{Per}\colon H_{1}(\Sigma,\mathbb{Z})\to\mathbb{C}$ is
injective.

###### Proof.

Suppose $(x_{1},y_{1},x_{2},y_{2})\in\mathbb{Z}^{4}$ is such that

$x_{1}\alpha_{1}+y_{1}\beta_{1}+x_{2}\alpha_{2}+y_{2}\beta_{2}=0.$

The real part gives $x_{1}+y_{1}=0$ and $x_{2}+y_{2}=0$. The imaginary part
then gives

$x_{1}^{2}(5-\sqrt{5})=2x_{2}^{2}(5-2\sqrt{5}).$

Therefore $x_{1}=x_{2}=0$ and $y_{1}=y_{2}=0$. ∎

######  Remark 5.2.

Lemma 5.1 also follows from the fact that $\mathcal{D}$ is algebraically
primitive, see [FL23, Lemma 3.2].

This will allow us to derive the monodromy representation from the action of
the Veech group on periods. Indeed, the Veech group $\mathrm{SL}(D)$ acts on
$\mathbb{C}=\mathbb{R}^{2}$, and on periods since for $A\in\mathrm{SL}(D)$ and
$\gamma\in H_{1}(\Sigma,\mathbb{Z})$,

$A\cdot\mathrm{Per}(\gamma)=\mathrm{Per}(\rho(A)\cdot\gamma).$

Since $R$ is a rotation of angle $\frac{\pi}{5}$, we have
$R\beta_{1}=\xi\beta_{1}=\alpha_{1}$, and
$R\beta_{2}=\xi\beta_{2}=\alpha_{2}$. Hence $\rho(R)$ sends $b_{i}$ to $a_{i}$
for $1\leqslant i\leqslant 2$. From $\xi^{4}=-\xi^{-1}$, we get

$R\alpha_{1}=\xi(\xi+1)=\xi^{2}+\xi=(\xi+1)-(\xi^{-1}+1)+(\xi^{2}-\xi^{4}).$

Therefore $R\alpha_{1}=\alpha_{1}-\beta_{1}+\alpha_{2}$. Similarly,

$R\alpha_{2}=\xi(\xi^{2}-\xi^{4})=\xi^{3}+1=(\xi+1)-(\xi-\xi^{3}).$

Thus $R\alpha_{2}=\alpha_{1}-\beta_{2}$. We can now write the matrix of
$\rho(R)$ in the basis $(a_{1},b_{1},a_{2},b_{2})$

$\rho(R)=\begin{pmatrix}1&&1&&1&&0\\\ -1&&0&&0&&0\\\ 1&&0&&0&&1\\\
0&&0&&-1&&0\end{pmatrix}.$

Let us now turn to the computation of $\rho(T)$. Straightforward computations
give

$\displaystyle T\alpha_{1}$
$\displaystyle=3\cdot\frac{5+\sqrt{5}}{4}+i\sqrt{\frac{{5-\sqrt{5}}}{8}}$
$\displaystyle T\beta_{1}$
$\displaystyle=-\frac{5+\sqrt{5}}{4}-i\sqrt{\frac{5-\sqrt{5}}{8}}$
$\displaystyle T\alpha_{2}$
$\displaystyle=3\frac{\sqrt{5}}{2}+i\frac{\sqrt{5-2\sqrt{5}}}{2}$
$\displaystyle T\beta_{2}$
$\displaystyle=-\frac{\sqrt{5}}{2}-\frac{\sqrt{5-\sqrt{5}}}{2}.$

Therefore, we have $T\alpha_{1}=2\alpha_{1}+\beta_{1}$,
$T\beta_{1}=-\alpha_{1}$, $T\alpha_{2}=2\alpha_{2}+\beta_{2}$, and
$T\beta_{2}=-\alpha_{2}$. We now have the image of $T$ by the monodromy map:

$\rho(T)=\begin{pmatrix}2&&-1&&0&&0\\\ 1&&0&&0&&0\\\ 0&&0&&2&&-1\\\
0&&0&&1&&0\end{pmatrix}.$

Reducing modulo two, we easily see that the monodromy group modulo two of $D$
is dihedral of order ten. One could alternatively use results of [McM23b] to
describe this action. We are now set to prove Theorem 1.2: $\mathcal{D}$ has
three echoes in $\mathcal{M}_{3}$.

###### Proof.

By 3.17, it suffices to describe the action of the monodromy group on
$H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$. We form the graph whose vertices are
non-zero elements of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$, with an edge
between each vertex $v$ and $Mv$ for $M\in\\{\rho(R),\rho(T)\\}$. We verify
that this graph has three connected components, each of size five. ∎

Note that each of the three echoes is a cover of $\mathcal{D}$ of degree five.
We deduce from this analysis 1.3.

###### Proof.

Let $(X,\omega)$ be a genus three Veech surface covering
$(X_{0},\omega_{0})\in\mathcal{H}(1,1)$. If $(X,\omega)$ is not covering a
surface of genus one, then $\mathrm{SL}(X,\omega)$ is a subgroup of
$\mathrm{SL}(X_{0},\omega_{0})$, by [McM06a, Corollary 2.2]. Thus
$(X_{0},\omega_{0})$ is a Veech surface in $\mathcal{H}(1,1)$, generating a
primitive Teichmüller curve. It follows from [McM06b] that
$(X_{0},\omega_{0})$ is, after rescaling, in the
$\mathrm{SL}_{2}(\mathbb{R})$-orbit of $D$. Therefore $(X,\omega)$ belongs to
one of the three Teichmüller discs generated by lifts of $D$. ∎

####  5.0.1. Higher genera

Our techniques can be applied to study echoes of $\mathcal{D}$ in higher
genera moduli spaces. One can for example replace degree two covers by cyclic
covers of a given degree $n$, that is covers associated with a subgroup of the
form $\ker(\pi_{1}(\Sigma)\twoheadrightarrow\mathbb{Z}/n\mathbb{Z})$. Let us
call the _cyclic_ echoes of $\mathcal{D}$ the Teichmüller curves obtained by
pulling back $D$ with these covers. The cyclic echoes of $\mathcal{D}$ lie in
$\mathcal{M}_{n+1}$. We can mimic the proof of Lemma 3.16, and get that the
echoes are still classified by the action of the affine group on the covers as
in 3.17, see also 8.3 in Section 8. We compute the graphs as before, where
this time the vertices are the primitive elements of
$H_{1}(\Sigma,\mathbb{Z}/n\mathbb{Z})$. We can write a program that counts the
connected components of these graphs, and get the number $N(n)$ of cyclic
echoes of $\mathcal{D}$ in $\mathcal{M}_{n+1}$ for small values of $n$, see
Table 1.

$n=g-1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$N(n)$ | $3$ | $1$ | $3$ | $8$ | $3$ | $1$ | $3$ | $1$ | $24$ | $3$ | $3$ | $1$ | $3$ | $8$
Table 1. The numbers of cyclic echoes of $\mathcal{D}$ in $\mathcal{M}_{g}$.

##  6\. Real multiplication and monodromy groups

In this section, we prove 1.6. We first give a general proof, and then for the
specific case of Weierstrass curves with non-square discriminant, we indicate
an alternative and more direct approach.

###  6.1. Theoretical approach

####  6.1.1. Commutation with real multiplication

Let us fix $X\in\mathcal{T}(\Sigma)$ a complex structure on $\Sigma$. We
identify $\Omega(X)$ with $H^{1}(\Sigma,\mathbb{R})$ by the map that sends
$\alpha$ to $\mathrm{Re}(\int\alpha)$. Recall that
$[\alpha,\beta]\mapsto\frac{i}{2}\int\alpha\wedge\bar{\beta}$ is an inner
product on $\Omega(X)$. An endomorphism $T\colon H_{1}(\Sigma,\mathbb{Z})\to
H_{1}(\Sigma,\mathbb{Z})$ extends to an $\mathbb{R}$-linear endomorphism of
$H_{1}(\Sigma,\mathbb{R})$. Its dual $T^{\vee}\colon\Omega(X)\to\Omega(X)$ is
a real linear endomorphism. When $T^{\vee}$ is complex linear and self-
adjoint, we can diagonalise it in an orthogonal basis.

######  Lemma 6.1.

Let $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$, and $T$ be a complex linear
self-adjoint endomorphism of $H_{1}(\Sigma,\mathbb{Z})$ that admits $\omega$
as eigenvector. Its dual $T^{\vee}\colon\Omega(X)\to\Omega(X)$ is diagonal in
any orthogonal basis of the form $(\omega,\omega^{\prime})$.

###### Proof.

Since $T$ is self-adjoint, $T^{\vee}$ also is, and preserves
$\omega^{\perp}=\mathbb{C}\omega^{\prime}$. ∎

Let us verify that an element $f\in\mathrm{Aff}^{+}(X,\omega)$ preserves
$\mathbb{C}\omega$ and its orthogonal.

######  Lemma 6.2.

Let $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ and
$f\in\mathrm{Aff}^{+}(X,\omega)$. The endomorphism
$\rho(f)^{\vee}\colon\Omega(X)\to\Omega(X)$ preserves both $\mathbb{C}\omega$
and its orthogonal.

###### Proof.

Under our identification,
$\mathbb{C}\omega=\mathbb{R}\mathrm{Re}(\int\omega)+\mathbb{R}\mathrm{Im}(\int\omega)$.
The endomorphism $\rho(f)^{\vee}\colon\Omega(X)\to\Omega(X)$ sends
$\mathrm{Re}\int_{\cdot}\alpha$ to $\mathrm{Re}\int_{\rho(f)\cdot}\alpha$.
Since $f\in\mathrm{Aff}^{+}(X,\omega)$, there exists
$A\in\mathrm{SL}_{2}(\mathbb{R})$ such that
$\int_{\rho(f)\cdot}\omega=A\int_{\cdot}\omega$. In particular for
$z\in\mathbb{C}$,

$\rho(f)^{\vee}(z\omega)=\mathrm{Re}\left(zA\int\omega\right)\in\mathbb{R}\mathrm{Re}(\int\omega)+\mathbb{R}\mathrm{Im}(\int\omega).$

Therefore $\rho(f)^{\vee}$ preserves $\mathbb{C}\omega$. Since $\rho(f)$ is
symplectic, the adjoint of $\rho(f)^{\vee}$ is its inverse, and
$\rho(f)^{\vee}$ preserves $\omega^{\perp}$. ∎

We are now able to prove 1.6. Let us give a slightly more general statement.

######  Proposition 6.3.

Let $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ and
$f\in\mathrm{Aff}^{+}(X,\omega)$. Any self-adjoint complex linear endomorphism
$T\colon H_{1}(\Sigma,\mathbb{Z})\to H_{1}(\Sigma,\mathbb{Z})$ with real
eigenvalues such that $\omega$ is an eigenvector of $T^{\vee}$ commutes with
$\rho(f)$.

###### Proof.

Let $S=\rho(f)^{-1}T\rho(f)$. Let us pick $(\omega,\omega^{\prime})$ an
orthogonal basis of $\Omega(X)$. The spaces $\mathbb{C}\omega$ and
$\mathbb{C}\omega^{\prime}$ are composed of eigenvectors of $T$ by Lemma 6.1.
It follows from Lemma 6.2 that $\rho(f)^{\vee}$ preserves both of them.
Therefore if $T^{\vee}(\omega)=\lambda\omega$ and
$\rho(f)^{-1}(\omega)=\mu\omega$, shortening $S^{\vee}$ and $T^{\vee}$ by $S$
and $T$, we have

$S(\omega)=\rho(f)T\rho(f)^{-1}(\omega)=\rho(f)T(\mu\omega)=\lambda\rho(f)(\mu\omega)=\lambda\omega.$

Note that we used that $\lambda$ is real. Similarly,
$S^{\vee}(\omega^{\prime})=T^{\vee}(\omega^{\prime})$. Therefore $S=T$. ∎

####  6.1.2. Real multiplication on $L$

Recall that $W_{D}\subset\mathcal{M}_{2}$ is the locus of Riemann surfaces
whose Jacobian admits real multiplication by an order $\mathcal{O}_{D}$, and
an eigenform $\omega$ with a single zero. Let us find $T$ a generator of
$\mathcal{O}_{D}$ for $L(b,e)$ the surface Figure 2 of Section 4. We fix again
$\lambda=\frac{e+\sqrt{e^{2}+4b}}{2}$, and considering the periods, we let

$T=\begin{pmatrix}e&&0&&b&&0\\\ 0&&e&&0&&1\\\ 1&&0&&0&&0\\\ 0&&b&&0&&0\\\
\end{pmatrix}.$

We have $T^{2}=eT+b$, and $T$ is self-adjoint. It is actually a generator of
multiplication by $\mathcal{O}_{D}$. Indeed, let us recall an argument from
[McM07, Theorem 8.3]. A self-adjoint endomorphism has the following form

$S=\begin{pmatrix}\epsilon&0&\delta&-\beta\\\ 0&\epsilon&-\gamma&\alpha\\\
\alpha&\beta&\eta&0\\\ \gamma&\delta&0&\eta\end{pmatrix}.$

If $S$ is a generator of $\mathcal{O}_{D}$, we can replace it with $S-\eta I$
and suppose that $\eta=0$. Since $S^{\vee}(\omega)=\lambda\omega$, considering
periods, we get $S=T$. Therefore the monodromy group of $L$ commutes with $T$.
The matrix $T$ is also explicited in [McM05, Section 5].

###  6.2. A hands-on approach

We now indicate a more direct approach to show that the monodromy group of $L$
must commute with $T$, when $D$ is not a square. Let
$f\in\mathrm{Aff}^{+}(X,\omega)$, $A=d(f)$, and write

$A=\begin{pmatrix}a_{11}&a_{12}\\\ a_{21}&a_{22}\end{pmatrix}.$

We also write $\rho(f)=(u_{ij})_{1\leqslant i,j\leqslant 4}$. Let
$\gamma=a_{1}$. We have

$A\cdot\mathrm{Per}(\gamma)=A\cdot\begin{pmatrix}\lambda\\\
0\end{pmatrix}=\mathrm{Per}(\rho(f)\cdot\gamma)=\begin{pmatrix}u_{11}\lambda+u_{31}b\\\
u_{21}\lambda+u_{41}\end{pmatrix}$

and it follows from the relation $\lambda^{2}=\lambda e+b$ that

$\begin{pmatrix}a_{11}\\\
a_{21}\end{pmatrix}=\begin{pmatrix}u_{11}-u_{31}e+u_{31}\lambda\\\
u_{21}-u_{41}\frac{e}{b}+u_{41}\frac{\lambda}{b}\end{pmatrix}.$

Similarly, setting $\gamma=b_{1}$, we get

$A\cdot\begin{pmatrix}0\\\
\lambda\end{pmatrix}=\begin{pmatrix}u_{32}b+u_{12}\lambda\\\
u_{42}+u_{22}\lambda\end{pmatrix}.$

Just as before, we obtain

$\begin{pmatrix}a_{12}\\\
a_{22}\end{pmatrix}=\begin{pmatrix}u_{12}-u_{32}e+u_{32}\lambda\\\
u_{22}-\frac{e}{b}u_{42}+\frac{\lambda}{b}u_{42}\end{pmatrix}.$

Applying the same reasoning with $\gamma=a_{2}$ and $\gamma=b_{2}$ will give
us relations between the numbers $u_{ij}$. Namely, we obtain with
$\gamma=a_{2}$

$\begin{pmatrix}a_{11}\\\
a_{21}\end{pmatrix}=\frac{1}{b}\begin{pmatrix}u_{33}b+u_{13}\lambda\\\
u_{43}+u_{23}\lambda\end{pmatrix}.$

Finally for $\gamma=e_{4}$, we get

$\begin{pmatrix}a_{12}\\\
a_{22}\end{pmatrix}=\begin{pmatrix}u_{34}b+u_{14}\lambda\\\
u_{44}+u_{24}\lambda\end{pmatrix}.$

We now have four explicit equations relating the coefficients of $\rho(f)$.
Since $\lambda$ is not rational, these equations split and give eight linear
equations in the coefficients $u_{ij}$. We can verify directly that they are
another instance of the action on homology commuting with the matrix $T$.

######  Proposition 6.4.

These equations are satisfied if and only if $\rho(f)$ and $T$ commute.

This approach is more elementary in the sense that one does not need to know
about real multiplication to show that the matrix $T$ must commute with
$\rho(f)$.

##  7\. Echoes of the Weierstrass curves

In this section we compute the monodromy groups modulo two of the generators
of Weierstrass curves, and prove Theorem 1.7. Then we classify the echoes of
the components of $W_{D}$ and prove Theorem 1.1 and Theorem 1.4. After that,
we determine which covers of reduced square-tiled surfaces in $\mathcal{H}(2)$
have lattice of absolute periods $\mathbb{Z}+i\mathbb{Z}$, and prove Theorem
1.5. Finally, we remark that one can count the sizes of the orbits of reduced
square-tiled in $\widetilde{\mathcal{H}}(2)$ surfaces, applying a result of
Lelièvre and Royer.

###  7.1. Monodromy groups

We work again with the generators of $W_{D}$ given by the surfaces of Figure
2, and the marking of Section 4. Recall from 3.9 that there are five elements
of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ that lift $L$ to the hyperelliptic
component, while the other ten lift it to the odd component. Using the
criterion given in 3.9, we verify that in our basis
$(a_{1},b_{1},a_{2},b_{2})$ of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$, those
five elements are $(1,0,1,0)$, $(1,0,1,1)$, $(1,0,0,1)$, $(0,1,0,0)$ and
$(1,1,0,0)$. Note that it follows from 3.15 that the monodromy group of $L$
preserves these covers.

As a consequence of commutation with real multiplication, we show that the
monodromy groups modulo two of $L$ are generated by the horizontal and
vertical twists when $D\not\equiv 1\mod 8$.

######  Proposition 7.1.

Let $D\geqslant 5$ be such that $D\equiv 0,4,5\mod 8$ and
$(X,\omega)\in\Omega\mathcal{M}_{2}$ generating $W_{D}$. The monodromy group
modulo two of $(X,\omega)$ is dihedral, generated by two Dehn twists.

###### Proof.

Let us suppose that $D\equiv 0\mod 8$. Then $W_{D}$ is generated by the
$L$-shaped translation surface of Figure 2, where $e=0$ and $4b=D$. Since
$D\equiv 0\mod 8$, we have $b\equiv 0\mod 2$. The elements in the monodromy
group modulo two induced by the horizontal and vertical Dehn twists have been
computed in Section 4. They generate a group of order eight in
$\mathrm{Sp}(H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z}))$. The group
$\mathrm{Sp}(H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z}))$ is of order 720. The
monodromy group modulo two $G$ of $L$ is included in its subgroup of elements
that commute with $T$, and preserve the set of covers lifting to the
hyperelliptic component. This latter group can be checked to be of order
eight. Therefore $G=\langle V,H\rangle$. We proceed similarly for $D\equiv
4,5\mod 2$ with $(e,b)$ being respectively $(0,1),(1,1)$ modulo two. ∎

When $D\equiv 1\mod 8$, we have seen in Section 4 that the horizontal and
vertical Dehn twists do not generate the monodromy group modulo two of $L$.
However, we can still characterise it.

######  Proposition 7.2.

Let $(X,\omega)\in\Omega\mathcal{M}_{2}$ generating a Weierstrass curve of
discriminant $D$, with $D\equiv 1\mod 8$. The monodromy group modulo two of
$(X,\omega)$ is dihedral of order twelve.

###### Proof.

As before, we consider the $L$-shaped surface of Figure 2, with $(e,b)$ such
that $e\in\\{-1,1\\}$ and $D=e^{2}+4b$. The group of elements in
$\mathrm{Sp}(H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z}))$ that commute with $T$ and
preserve the covers lifting to the hyperelliptic component is then checked to
be dihedral of order twelve. However the subgroup $\langle H,V\rangle$ is of
order six, and is strictly included in the monodromy group modulo two by Lemma
4.2. Therefore the monodromy group modulo two of $L$ is of order twelve. ∎

Concretely, this monodromy group is generated for example by the matrices $H$
and

$\begin{pmatrix}0&1&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\ 1&0&1&0\end{pmatrix}.$

###  7.2. Classification of the echoes

We now use the above generators of the monodromy group modulo two to classify
the echoes of Weierstrass curves. It suffices by 3.17 to classify the orbits
of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})\setminus\\{0\\}$ under the action of
these groups. In Table 2, we summarise results obtained by doing so, for the
different congruences of $D$ modulo eight. In the first line we recall the
order of $G$, the monodromy group modulo two of their generators. We then show
the sets of elements of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ that lift $L$
to generators of the same Teichmüller curve. For this purpose, we label the
elements of $H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ as follows. The element
$(x_{1},x_{2},x_{3},x_{4})\in H_{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$
corresponds to the number $\sum 2^{i}x_{i}$. For example, $(1,0,1,1)$ is
numbered $1+4+8=13$. We then write for example $\\{2,5\\}$ to indicate that
the Teichmüller curve generated by the lift of $L$ by the covers corresponding
to $(0,1,0,0)$ and $(1,0,1,0)$ are the same, and that these are the only
covers giving this curve.

$D\mod 8$ | $0$ | $1$ | $4$ | $5$
---|---|---|---|---
Order of $G$ | $8$ | $12$ | $8$ | $10$
Echoes in $\mathcal{H}(2,2)^{\mathrm{hyp}}$ |  $\\{2\\}$, $\\{3,5,9,13\\}$ |  $\\{2,5\\}$, $\\{3,9,13\\}$ |  $\\{2,3,9,13\\}$, $\\{5\\}$ | $\\{2,3,5,9,13\\}$
Echoes in $\mathcal{H}(2,2)^{\mathrm{odd}}$ |  $\\{1,7,11,15\\}$, $\\{4,6\\}$, $\\{8,10,12,14\\}$ |  $\\{1,6,8,11,12,15\\}$, $\\{4,10,14\\}$, $\\{7\\}$ |  $\\{1,4,11,14\\}$ $\\{6,7,8,12\\}$, $\\{10,15\\}$ |  $\\{1,8,11,12,14\\}$ $\\{4,6,7,10,15\\}$
Table 2. The different echoes of $W_{D}$.

The orbits in Table 2 correspond to the echoes of each Weierstrass curve,
which proves Theorem 1.1. It also allows us to prove Theorem 1.4.

###### Proof.

Reduced square-tiled surfaces belong to the same
$\mathrm{SL}_{2}(\mathbb{Z})$-orbit if and only if they generate the same
Teichmüller curve. They cover a square-tiled surface in $\mathcal{H}(2)$ with
$d$ squares if and only if the corresponding Teichmüller curve is an echo of a
component of $W_{D}$ with $D=d^{2}$. The $\mathrm{SL}_{2}(\mathbb{Z})$-orbits
of reduced square-tiled surfaces covering one in $\mathcal{H}(2)$ are thus
classified by the echoes of Weierstrass curves with discriminant $D$. Since
$D$ is a square, $D\equiv 0,1,4\mod 8$, and each component of $W_{D}$ has five
echoes. If $d$ is even or $d=3$, then $W_{D}$ has only one component. When
$d\geqslant 5$ is odd, then $W_{D}$ has two components, giving ten
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits of reduced square-tiled surfaces with
$2d$ tiles. ∎

When $D\not\equiv 5\mod 8$ and $(X,\omega)$ generates $W_{D}$, there is a
single lift of $(X,\omega)$ with Veech group $\mathrm{SL}(X,\omega)$. This
creates a copy of $W_{D}$ in $\mathcal{M}_{3}$. We will see in Section 8 that
primitive Teichmüller curves always have copies in higher genera moduli
spaces.

###  7.3. Primitivity

Let $(X,\omega)=L(b,e)$ be the $L$-shaped surface generating a Weierstrass
curve of discriminant $D$ from Section 4. Let us denote by
$\Lambda=\mathrm{Per}\left(H_{1}(\Sigma,\mathbb{Z})\right)$ the set of its
periods. If $p\colon S\to\Sigma$ is a covering, denote by $\Lambda_{p}$ the
set of periods of $p^{*}(\omega)$. In other words, if $p$ is the cover
associated with $f\in H^{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$, then

$\Lambda_{p}=\mathrm{Per}(\\{\gamma\in H_{1}(\Sigma,\mathbb{Z})\mid
f(\gamma)=0\\}).$

Note that if $p$ is the cover associated with $\gamma$, then
$f\colon\alpha\mapsto\alpha\cdot\gamma$. When $D$ is not a square, then
$\Lambda_{p}$ is always of index two in $\Lambda$.

######  Lemma 7.3.

The subgroup $\Lambda_{p}\subset\Lambda$ is of index at most two, and is of
index two when $D$ is not a square.

###### Proof.

Since $H_{1}(S,\mathbb{Z})$ is sent to an index two subgroup of
$H_{1}(\Sigma,\mathbb{Z})$ by $p$, the index of $\Lambda_{p}$ in $\Lambda$ is
at most two. If $D$ is not a square, then the period map $\mathrm{Per}\colon
H_{1}(\Sigma,\mathbb{Z})\to\Lambda$ is a group isomorphism, hence
$[\Lambda\colon\Lambda_{p}]=2$. ∎

We thus now assume that $D$ is a square and $D=d^{2}$. Since
$\lambda^{2}=e\lambda+b$, we have

$\Lambda=\lambda\mathbb{Z}+b\mathbb{Z}+i\left(\mathbb{Z}+\lambda\mathbb{Z}\right)=\lambda\mathbb{Z}+i\mathbb{Z}.$

Let us say that the cover $p$ is _primitive_ when $\Lambda_{p}=\Lambda$. In
other words, a cover is primitive if the lattice of absolute periods of the
cover $\tilde{L}=p^{*}(L)$ is the same as the one of $L$. We now identify the
covers $p$ that are primitive.

######  Lemma 7.4.

The covers associated with $\gamma$ of the form $(x,0,0,y)$ are primitive.

###### Proof.

Let us show in this case that $\Lambda_{p}=\Lambda$. The $f\in
H^{1}(\Sigma,\mathbb{Z}/2\mathbb{Z})$ corresponding to $\gamma$ is

$(x_{1},y_{1},x_{2},y_{2})\mapsto x_{2}y-y_{1}x.$

Therefore, $(1,0,0,0)$ is in $\ker f$. Its period is $\lambda$. Similarly,
$(0,0,0,1)$ is in $\ker f$. Its period is $i$. Thus
$\Lambda_{p}=\lambda\mathbb{Z}+i\mathbb{Z}$. ∎

They correspond to $(1,0,0,0)$, $(0,0,0,1)$ and $(1,0,0,1)$ which are labeled
in Table 2 by $1$, $8$, and $9$ respectively. Let us now consider the cover
labeled $5$.

######  Lemma 7.5.

The cover associated with $\gamma=(1,0,1,0)$ is primitive if and only if
$e+d\equiv 0\mod 4$.

###### Proof.

Here $f\colon(x_{1},y_{1},x_{2},y_{2})\mapsto-y_{1}-y_{2}$. We have
$\gamma=(1,0,0,0)\in\ker f$. Thus $\lambda\in\Lambda_{p}$. We have
$\Lambda_{p}=\Lambda$ if and only if $i\in\Lambda_{p}$. This is the case when
one can write $i=x_{1}\lambda+x_{2}b+i(y_{1}\lambda+y_{2})$ with
$y_{1}+y_{2}\in 2\mathbb{Z}$. That is, when $1=y_{1}\lambda+y_{2}$, with
$y_{1}+y_{2}\in 2\mathbb{Z}$. When $\lambda$ is odd, reducing modulo two gives
$y_{1}+y_{2}\equiv 1\mod 2$, and there is no solution. When $\lambda$ is even,
one can take $y_{1}=1$ and $y_{2}=1-\lambda$. The result follows since
$\lambda=\frac{e+d}{2}$ is even exactly when $e+d$ is a multiple of four. ∎

We now turn to the case of the cover labeled two.

######  Lemma 7.6.

The cover associated with $\gamma=(0,1,0,0)$ is primitive if and only if
$d-e\equiv 2\mod 4$.

###### Proof.

This cover is primitive if and only if one can write
$x_{1}\lambda+x_{2}b=\lambda$ with $x_{1}\in 2\mathbb{Z}$, that is,
$x_{1}+x_{2}(\lambda-e)=1$. There is no solution if $\lambda-e$ is even, and
$(x_{1},x_{2})=(1+\lambda-e,1)$ is a solution otherwise. Therefore the cover
is primitive if and only if $\lambda-e=\frac{d-e}{2}$ is odd, that is when
$d-e\equiv 2\mod 4$. ∎

Let us now turn to the cover number five.

######  Lemma 7.7.

The cover associated with $\gamma=(1,0,1,0)$ is primitive if and only if
$d+e\equiv 0\mod 4$.

###### Proof.

It is primitive if and only if $y_{1}\lambda+y_{2}=1$ has a solution with
$y_{1}+y_{2}\in 2\mathbb{Z}$. This happens exactly when $\lambda$ is even. ∎

We can now tell which of the covers lifting to the hyperelliptic component are
primitive. Indeed primitive covers are preserved by the action of
$\mathrm{Aff}^{+}(X,\omega)$. Moreover the orbits of covers lifting to the
hyperelliptic component are represented in every case by the covers numbered
$2,5$ and $9$, see Table 2. We now turn to the covers lifting to the odd
component, starting with the one numbered four.

######  Lemma 7.8.

The cover associated with $\gamma=(0,0,1,0)$ is primitive if and only if
$d+e\equiv 2\mod 4$.

###### Proof.

It is primitive if and only if $1=y_{1}\lambda+y_{2}$ with $y_{2}\in
2\mathbb{Z}$. This is possible exactly when $\lambda$ is odd, that is when
$d+e\equiv 2\mod 4$. ∎

We now consider the cover labeled seven.

######  Lemma 7.9.

The cover associated with $\gamma=(1,1,1,0)$ is primitive if and only if $e=0$
or $e+d\equiv 0\mod 4$.

###### Proof.

We have $f\colon(x_{1},y_{1},x_{2},y_{2})\mapsto y_{1}-x_{1}+y_{2}$. The cover
is primitive when the following two equations have a solution with
$(x_{1},y_{1},x_{2},y_{2})$ in $\ker f$

$\displaystyle i$ $\displaystyle=x_{1}\lambda+x_{2}b+i(y_{1}\lambda+y_{2}),$
$\displaystyle\lambda$
$\displaystyle=x_{1}\lambda+x_{2}b+i(y_{1}\lambda+y_{2}).$

There is a solution to the first equation when $\lambda$ is even:

$(x_{1},y_{1},x_{2},y_{2})=(0,1,0,1-\lambda).$

There is also a solution when $\lambda$ is odd and $e=0$:

$(x_{1},y_{1},x_{2},y_{2})=(1,\lambda,-\lambda,1-\lambda).$

However if $\lambda$ is odd and $e=\pm 1$, then from $x_{1}\lambda+x_{2}b=0$
we get $x_{1}+x_{2}(\lambda-e)=0$ and $x_{1}$ must be even. One must have
$y_{1}\lambda+y_{2}=1$ and $y_{1}+y_{2}\equiv 0\mod 2$, which has no solution.
For the second equation, there is a solution when $\lambda-e$ is odd:

$(x_{1},y_{1},x_{2},y_{2})=(1+\lambda-e,0,-1,0).$

When $e=0$ and $\lambda$ is even, we also have the solution

$(x_{1},y_{1},x_{2},y_{2})=(1+\lambda,1,-1,-\lambda).$

If $e=\pm 1$ and $\lambda$ is odd, a solution must have $x_{1}\equiv 1\mod 2$
and $y_{1}\lambda+y_{2}=0$ with $y_{1}+y_{2}\equiv 1\mod 2$. This is not
possible since $\lambda$ is odd. Summarising, we have a solution to both
equations if and only if $e=0$, or when $\lambda$ is even. ∎

Finally, we analyse the primitivity of the cover number ten.

######  Lemma 7.10.

The cover associated with $\gamma=(0,1,0,1)$ is primitive if and only if
$d-e\equiv 0\mod 4$.

###### Proof.

It is primitive when one can write $x_{1}\lambda+x_{2}b=\lambda$ with
$x_{1}+x_{2}\in 2\mathbb{Z}$. This happens when $1=x_{1}+x_{2}(\lambda-e)$,
which has a solution if and only if $\lambda-e$ is even. ∎

We can now tell which covers lifting to the odd component is primitive. Indeed
the orbits of these covers under the action of $\mathrm{Aff}^{+}(X,\omega)$
are represented by the covers $1,4,8,7,10$, see Table 2. We summarise in Table
3 the echoes of $L$ obtained by a primitive cover. Observe that if $e=\pm 1$,
then $d+e\equiv d-e+2\mod 4$.

$(d,d-e)\mod 4$ | $(0,0)$ | $(\pm 1,0)$ | $(2,2)$ | $(\pm 1,2)$
---|---|---|---|---
Echoes in $\mathcal{H}(2,2)^{\mathrm{hyp}}$ | $\\{3,5,9,13\\}$ | $\\{3,9,13\\}$ | $\\{2,3,9,13\\}$ |  $\\{2,5\\}$, $\\{3,9,13\\}$
Echoes in $\mathcal{H}(2,2)^{\mathrm{odd}}$ |  $\\{1,7,11,15\\}$, $\\{8,10,12,14\\}$ |  $\\{1,6,8,11,12,15\\}$, $\\{4,10,14\\}$ |  $\\{1,4,11,14\\}$ $\\{6,7,8,12\\}$ |  $\\{1,6,8,11,12,15\\}$, {7}
Table 3. The echoes of square-tiled surfaces by primitive covers.

Theorem 1.5 follows from this analysis.

###### Proof.

The types of branched covers of degree $d$ of Theorem 1.5 correspond to the
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits of square-tiled surfaces that are lifts
of a reduced square-tiled surface in $\mathcal{H}(2)$ by a primitive cover. ∎

It follows from [McM05, Theorem 5.3] that when $d$ is odd, we have $d-e\equiv
0\mod 4$ if and only if the spin invariant of the curve generated by
$L=L(\frac{d^{2}-1}{4},e)$ vanishes. This means that reduced square-tiled
surfaces in the $\mathrm{GL}_{2}^{+}(\mathbb{R})$-orbit of $L$ have exactly
one Weierstrass point on $\mathbb{Z}+i\mathbb{Z}$, see [McM05, Theorem 6.1],
while the ones in the other component of $W_{D}$ have three integral
Weierstrass points. This way of distinguishing the orbits of square-tiled
surfaces was introduced by Hubert and Lelièvre in [HL06].

###  7.4. Counting square-tiled surfaces

Remark that our proof gives the indices of the Veech groups of the covers. In
particular, since Lelièvre and Royer in [LR06] counted the sizes of the
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits of reduced square-tiled surface in
$\mathcal{H}(2)$, we have formulas for the sizes of
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits of square-tiled surface in
$\widetilde{\mathcal{H}}(2)$. As an example, let us count the number of
square-tiled surfaces in $\widetilde{\mathcal{H}}(2)$, with $d=22$ squares,
odd Arf invariant and lattice of absolute periods $\mathbb{Z}+i\mathbb{Z}$. By
the results gathered in Table 3, they form four
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits. From [LR06], the two
$\mathrm{SL}_{2}(\mathbb{Z})$-orbits of reduced square-tiled surfaces in
$\mathcal{H}(2)$ with an odd number $n$ of squares are of sizes
$a_{n}=\frac{3}{16}(n-1)n^{2}\prod_{p\mid n}(1-\frac{1}{p^{2}})$ and
$b_{n}=\frac{n-3}{n-1}a_{n}$. The numbers $a_{n}$ are the sizes of the orbits
with vanishing spin invariant, while $b_{n}$ are the sizes of the others. Thus
the $a_{n}$, resp. $b_{n}$, correspond to the orbits whose lifts are
classified on the second, resp. last, column of Table 2. For $n=11$, we have
$a_{n}=225$ and $b_{n}=180$. It follows from 3.13 that the sizes of our orbits
lifting origamis with vanishing spin invariant are $1350=6\cdot 225$ and
$900=4\cdot 225$. The remaining orbits are of sizes $1080=6\cdot 180$ and
$180$.

##  8\. Remarks in higher genus

In this section we make some remarks on Teichmüller curves in higher genus,
obtained by covering constructions. We first show that there are arbitrarily
large numbers of echoes of Teichmüller curves of genus two, in higher genus.
We then show that, as a corollary of a theorem of Funar and Lochak, one can
find copies of a given primitive Teichmüller curve in moduli spaces of higher
genera.

###  8.1. There are arbitrarily large numbers of pure echoes

As a consequence of 1.6, we show that a Teichmüller curve
$V\subset\mathcal{M}_{g}$ can have an arbitrarily large number of _pure_
echoes in higher genera moduli spaces. Since square-tiled surfaces are dense
in each moduli space $\Omega\mathcal{M}_{g}$, there can be infinite numbers of
echoes of a given Teichmüller curve in a given $\mathcal{M}_{g}$. However,
this is not possible with pure echoes.

######  Lemma 8.1.

Let $V\subset\mathcal{M}_{g}$ be a Teichmüller curve. There is only a finite
number of pure echoes of $V$ in $\mathcal{M}_{g^{\prime}}$ for each
$g^{\prime}$.

###### Proof.

There is only one Teichmüller curve in $\mathcal{M}_{1}$, thus we suppose that
$g\geqslant 2$. There are only finitely many covers of genus $g^{\prime}$ of a
given genus $g$ surface. ∎

Recall that we call cyclic echoes of a Teichmüller curve
$V\subset\mathcal{M}_{g}$ the Teichmüller curves generated by $p^{*}(\omega)$
where $\omega$ generates $V$ and $p$ is a normal cover with cyclic deck
transformation group. The number of cyclic echoes of Teichmüller curves
generated by a square-tiled surface in $\mathcal{H}(2)$ can be made
arbitrarily large.

######  Proposition 8.2.

Let $b>1$ be a square and $D=4b$. For any odd $n\geqslant 3$ coprime with $b$,
the number of cyclic echoes of $W_{D}$ in $\mathcal{M}_{n+1}$ is at least
$\varphi(n)$.

In 8.2, the function $\varphi$ is the Euler’s totient function. Let us first
observe that as in 3.17, the echoes are classified by the action of
$\mathrm{Aff}^{+}(X,\omega)$ on the set of primitive elements of
$H_{1}(\Sigma,\mathbb{Z}/n\mathbb{Z})$. Actually, the proof of Lemma 3.16
shows the following more general result.

######  Corollary 8.3.

Let $(X,\omega)\in\Omega\mathcal{T}(\Sigma)$ with a unique double zero, and
$p\colon S_{g}\to\Sigma$ be an unramified normal cover with deck
transformation group $G$. Suppose that $(X,\omega)$ generates a Teichmüller
curve $V$ that is not $W_{9}$.

  1. (1)

The translation group of $p^{*}(\omega)$ is isomorphic to $G$.

  2. (2)

The echoes of $V$ by such covers are in correspondence with the action of
$\mathrm{Aff}^{+}(X,\omega)$ on the set of these covers.

###### Proof.

The group of deck transformations $G$ is naturally embedded in the translation
group. On the translation surface $p^{*}(\omega)$, there are $|G|$ saddle
connections $\sigma_{1},\ldots,\sigma_{|G|}$ lifting the one of the proof of
Lemma 3.16. Any translation must permute the $\sigma_{i}$, and is determined
by the image of $\sigma_{1}$. The translation group has thus order at most
$|G|$. For the second affirmation, it suffices to notice that any isomorphism
$F\colon S_{g}\to S_{g}$ sending one lift to the other must then conjugate the
corresponding deck transformations as in Lemma 3.10. ∎

We can now prove 8.2.

###### Proof.

Let $b^{\prime}=\sqrt{b}$. The matrix $T$ admits $u_{1}=(b^{\prime},0,1,0)$
and $u_{2}=(0,b^{\prime},0,1)$ as eigenvectors associated with the eigenvalue
$b^{\prime}$. It also admits $u_{3}=(b^{\prime},0,-1,0)$ and
$(0,b^{\prime},0,-1)$ as eigenvectors with eigenvalue $-b^{\prime}$. Since the
monodromy group commutes with $T$, any matrix
$A\in\rho(\mathrm{Aff}^{+}(X,\omega))$ must preserve these sets.

The following matrix is invertible modulo $n$ since its determinant is $4b$:

$\begin{pmatrix}b^{\prime}&0&b^{\prime}&0\\\ 0&b^{\prime}&0&b^{\prime}\\\
1&0&-1&0\\\ 0&1&0&-1\end{pmatrix}.$

Thus $(u_{1},u_{2},u_{3},u_{4})$ forms a basis of
$H_{1}(\Sigma,\mathbb{Z}/n\mathbb{Z})$, and we now write $A$ in this basis.
The matrix $A$ is now block-diagonal. Thus the elements $(1,0,x,0)\in
H_{1}(\Sigma,\mathbb{Z}/n\mathbb{Z})$ with $x\in\mathbb{Z}/n\mathbb{Z}$ are
not in the same orbit for $x$ generating different subgroups of
$\mathbb{Z}/n\mathbb{Z}$. There are $\varphi(n)+1$ of these groups. ∎

Note that one could improve the bound to $2\varphi(n)+1$ in the preceding
argument by also considering the elements of the form $(x,0,1,0)$. One can
then count primitive elements of the form $(x,0,y,0)$, where both $x$ and $y$
do not generate $\mathbb{Z}/n\mathbb{Z}$. We can also use the fact that $A$ is
symplectic to restrict the possibilities of its action on homology even
further.

###  8.2. Any Teichmüller curve appears in infinitely many moduli spaces

Given $V$ a hyperbolic surface of finite type, it is known that there are only
finitely many Teichmüller curves $V\to\mathcal{M}_{g}$ for each genus $g$, see
[McM09]. A result of Funar and Lochak [FL18, Theorem 1.4] states that every
closed surface $\Sigma_{g}$ of genus $g\geqslant 2$ admits infinitely many
surjections $\rho\colon\pi_{1}(\Sigma_{g})\to G$ with $G$ simple and finite,
such that $\ker\rho$ is a characteristic subgroup of $\pi_{1}(\Sigma_{g})$. As
a consequence of this result and our observations, we show that for a
Teichmüller curve $V\to\mathcal{M}_{g}$, there exist infinitely many
$g^{\prime}>g$ and Teichmüller curves $V\to\mathcal{M}_{g^{\prime}}$. It is
proven in [Her06] that there are infinitely many copies of the Teichmüller
curve in $\mathcal{M}_{1}$ generated by any torus, namely
$\mathbb{H}^{2}/\mathrm{SL}_{2}(\mathbb{Z})$.

######  Proposition 8.4.

There are infinitely many copies of a given primitive Teichmüller curve in
higher genera moduli spaces.

###### Proof.

This is proven by Herrlich in [Her06] for the only Teichmüller curve in
$\mathcal{M}_{1}$, hence we fix $\Sigma_{g}$ a closed surface of genus
$g\geqslant 2$. Let $V\subset\mathcal{M}_{g}$ be a primitive Teichmüller curve
and $\rho\colon\pi_{1}(\Sigma_{g})\to G$ a surjection onto a finite group with
characteristic kernel. Fix $(X,\omega)\in\Omega\mathcal{T}(\Sigma_{g})$
generating $V$. Let $p\colon S^{\prime}\to\Sigma_{g}$ be the cover associated
with $\ker\rho$. The Teichmüller curve generated by $p^{*}(\omega)$ is
isomorphic to $V$. Indeed, since $\ker\rho$ is characteristic, $p$ is
stabilised by the whole group $\mathrm{Mod}(\Sigma_{g})$. Therefore by 3.14,
the Veech group of $p^{*}(\omega)$ contains $\mathrm{SL}(X,\omega)$. By
results of Möller [Mol06, Theorem 2.6], see also [McM06a, Corollary 2.2], the
Veech group of $p^{*}(\omega)$ is a subgroup of $\mathrm{SL}(X,\omega)$. We
thus get a copy of $V=\mathbb{H}^{2}/\mathrm{SL}(X,\omega)$ in
$\mathcal{M}_{g^{\prime}}$, where $g^{\prime}=(g-1)|G|+1$. There exist
arbitrarily large such $G$ by [FL18]. ∎

## References

  * [AN16] David Aulicino and Duc-Manh Nguyen.  Rank two affine submanifolds in $\mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$.  Geom. Topol., 20(5):2837–2904, 2016.
  * [AN20] David Aulicino and Duc-Manh Nguyen.  Rank 2 affine manifolds in genus 3.  J. Differential Geom., 116(2):205–280, 2020.
  * [Arn68] Vladimir I. Arnold.  A remark on the branching of hyperelliptic integrals as functions of the parameters.  Funkcional. Anal. i Priložen., 2(3):1–3, 1968.
  * [BB01] Stephen J. Bigelow and Ryan D. Budney.  The mapping class group of a genus two surface is linear.  Algebr. Geom. Topol., 1:699–708, 2001.
  * [Cal20] Aaron Calderon.  Connected components of strata of Abelian differentials over Teichmüller space.  Comment. Math. Helv., 95(2):361–420, 2020.
  * [CS21] Aaron Calderon and Nick Salter.  Higher spin mapping class groups and strata of abelian differentials over Teichmüller space.  Adv. Math., 389:Paper No. 107926, 56, 2021.
  * [Dur19] Eduard Duryev.  Teichmüller curves in genus two: Square-tiled surfaces and modular curves.  Preprint, 2019.
  * [FL18] Louis Funar and Pierre Lochak.  Profinite completions of Burnside-type quotients of surface groups.  Comm. Math. Phys., 360(3):1061–1082, 2018.
  * [FL23] Sam Freedman and Trent Lucas.  Veech fibrations.  Preprint, 2023.
  * [FM12] Benson Farb and Dan Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [GRP23] Rodolfo Gutiérrez-Romo and Angel Pardo.  Permutation of periodic points of Veech surfaces in $\mathcal{H}(2)$.  Preprint, 2023.
  * [Her06] Frank Herrlich.  Teichmüller curves defined by characteristic origamis.  In The geometry of Riemann surfaces and abelian varieties, volume 397 of Contemp. Math., pages 133–144. Amer. Math. Soc., Providence, RI, 2006.
  * [HL06] Pascal Hubert and Samuel Lelièvre.  Prime arithmetic Teichmüller discs in $\mathcal{H}(2)$.  Israel J. Math., 151:281–321, 2006.
  * [KZ03] Maxim Kontsevich and Anton Zorich.  Connected components of the moduli spaces of Abelian differentials with prescribed singularities.  Invent. Math., 153(3):631–678, 2003.
  * [LF23] Thomas Le Fils.  Holonomy of complex projective structures on surfaces with prescribed branch data.  J. Topol., 16(1):430–487, 2023.
  * [LN18] Erwan Lanneau and Duc-Manh Nguyen.  Connected components of Prym eigenform loci in genus three.  Math. Ann., 371(1-2):753–793, 2018.
  * [LR06] Samuel Lelièvre and Emmanuel Royer.  Orbitwise countings in $\mathcal{H}(2)$ and quasimodular forms.  Int. Math. Res. Not., pages Art. ID 42151, 30, 2006.
  * [Mat22] Carlos Matheus.  Three lectures on square-tiled surfaces.  In Teichmüller theory and dynamics, volume 58 of Panor. Synthèses, pages 77–99. Soc. Math. France, Paris, 2022.
  * [McM03] Curtis T. McMullen.  Billiards and Teichmüller curves on Hilbert modular surfaces.  J. Amer. Math. Soc., 16(4):857–885, 2003.
  * [McM05] Curtis T. McMullen.  Teichmüller curves in genus two: discriminant and spin.  Math. Ann., 333(1):87–130, 2005.
  * [McM06a] Curtis T. McMullen.  Prym varieties and Teichmüller curves.  Duke Math. J., 133(3):569–590, 2006.
  * [McM06b] Curtis T. McMullen.  Teichmüller curves in genus two: torsion divisors and ratios of sines.  Invent. Math., 165(3):651–672, 2006.
  * [McM07] Curtis T. McMullen.  Dynamics of ${\rm SL}_{2}(\mathbb{R})$ over moduli space in genus two.  Ann. of Math. (2), 165(2):397–456, 2007.
  * [McM09] Curtis T. McMullen.  Rigidity of Teichmüller curves.  Math. Res. Lett., 16(4):647–649, 2009.
  * [McM23a] Curtis T. McMullen.  Billiards and Teichmüller curves.  Bull. Amer. Math. Soc. (N.S.), 60(2):195–250, 2023.
  * [McM23b] Curtis T. McMullen.  Billiards in regular polygons.  Prepint, 2023.
  * [McM24] Curtis T. McMullen.  Triangle groups: Cusps, congruence and chaos.  Preprint, 2024.
  * [Mol06] Martin Moller.  Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve.  Invent. Math., 165(3):633–649, 2006.
  * [Mol18] Martin Moller.  Geometry of Teichmüller curves.  In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, pages 2017–2034. World Sci. Publ., Hackensack, NJ, 2018.
  * [Muk14] Ronen E. Mukamel.  Orbifold points on Teichmüller curves and Jacobians with complex multiplication.  Geom. Topol., 18(2):779–829, 2014.
  * [MW19] Julien Marché and Maxime Wolff.  Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group.  Groups, Geometry and Dynamics, pages 731–766, 2019.
  * [SPWS17] Jan-Christoph Schlage-Puchta and Gabriela Weitze-Schmithüsen.  Finite translation surfaces with maximal number of translations.  Israel J. Math., 217(1):1–15, 2017.
  * [Thu97] William P. Thurston.  Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1997.
  * [Vee89] W. A. Veech.  Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards.  Invent. Math., 97(3):553–583, 1989.
  * [Zor06] Anton Zorich.  Flat surfaces.  In Frontiers in number theory, physics, and geometry. I, pages 437–583. Springer, Berlin, 2006.