# Triangular modular curves of small genus

Juanita Duque-Rosero  Department of Mathematics, Dartmouth College, 6188
Kemeny Hall, Hanover, NH 03755, USA<EMAIL_ADDRESS>and
John Voight  Department of Mathematics, Dartmouth College, 6188 Kemeny Hall,
Hanover, NH 03755, USA<EMAIL_ADDRESS>
###### Abstract.

Triangular modular curves are a generalization of modular curves that arise
from quotients of the upper half-plane by congruence subgroups of hyperbolic
triangle groups. These curves also arise naturally as a source of Belyi maps
with monodromy $\text{PGL}_{2}(\mathbb{F}_{q})$ or
$\text{PSL}_{2}(\mathbb{F}_{q})$. We present a computational approach to
enumerate Borel-type triangular modular curves of low genus, and we carry out
this enumeration for prime level and small genus.

##  1\. Introduction

### Motivation

The study of modular curves has rewarded mathematicians for perhaps a century.
For an integer $N\geq 1$, let
$\Gamma_{0}(N),\Gamma_{1}(N)\leq\operatorname{SL}_{2}({\mathbb{Z}})$ be the
usual congruence subgroups and let $X_{0}(N),X_{1}(N)$ be the corresponding
quotients of the completed upper half-plane. The genera of $X_{0}(N)$ and
$X_{1}(N)$ as compact Riemann surfaces can be computed using the
Riemann–Hurwitz formula, and it can readily be seen that there are only
finitely many of any given genus $g\geq 0$.

The study of modular curves of small genus goes back at least to Fricke [9, p.
357]. At the end of the twentieth century, Ogg enumerated and studied elliptic
[17] and hyperelliptic [18] modular curves; the resulting Diophantine study
[19] informed Mazur’s classification of rational isogenies of elliptic curves
[15], where the curves of genus $0$ are precisely the ones with infinitely
many rational points. This explicit study continues today, extended to include
all quotients of the upper half-plane by congruence subgroups of
$\operatorname{SL}_{2}({\mathbb{Z}})$; the list up to genus $24$ was computed
by Cummins–Pauli [7]. Recent papers have studied curves with infinitely many
rational points in the context of Mazur’s _Program B_ —see
Rouse–Sutherland–Zureick-Brown [20] for further references and recent results
in this direction.

Given this rich backdrop, it is worthwhile to pursue generalizations. For
example, replacing $\operatorname{SL}_{2}({\mathbb{Z}})$ with its quaternionic
cousins, Voight [25] enumerated all Shimura curves of the form
$X_{0}^{1}(\mathfrak{D},\mathfrak{M})$ of genus at most $2$. In a similar
direction, Long–Maclachlan–Reid [12] enumerated all maximal arithmetic
Fuchsian groups of genus $0$ over ${\mathbb{Q}}$, corresponding to quotients
of Shimura curves by the full group of Atkin–Lehner involutions.

### Setup and main result

In this paper, we consider a different type of generalization: namely, from
the point of congruence subgroups of triangle groups as introduced by
Clark–Voight [3]. We briefly introduce this construction; for more detail, see
section 2.

Let $a,b,c\in{\mathbb{Z}}_{\geq 2}\cup\\{\infty\\}$, and suppose that
$1/a+1/b+1/c<1$ (where $1/\infty=0$). Then there is a triangle in the upper
half-plane $\mathcal{H}$ (completed if $\infty\in\\{a,b,c\\}$) with angles
$\pi/a$, $\pi/b$, and $\pi/c$, unique up to isometry. The reflections in the
sides of this triangle generate a discrete subgroup of
$\operatorname{PGL}_{2}({\mathbb{R}})$, and the orientation-preserving
subgroup (of index $2$) defines the triangle group
$\Delta=\Delta(a,b,c)\leq\operatorname{PSL}_{2}({\mathbb{R}})$, with
presentation

$\Delta(a,b,c)=\langle\delta_{a},\delta_{b},\delta_{c}\,|\,\delta_{a}^{a}=\delta_{b}^{b}=\delta_{c}^{c}=\delta_{a}\delta_{b}\delta_{c}=1\rangle$

(omitting the relation $\delta_{s}^{s}$ when $s=\infty$). The triangle group
acts properly by isometries on $\mathcal{H}$ and the quotient
$X(a,b,c)\colonequals\Delta(a,b,c)\backslash\mathcal{H}$ can be given the
structure of a compact Riemann surface of genus $0$, isomorphic to
$\mathbb{P}^{1}$ with a unique coordinate $t$ taking values $0,1,\infty$ at
the vertices labelled $a,b,c$, respectively. For example, we recover the
classical modular group as
$\Delta(2,3,\infty)\simeq\operatorname{PSL}_{2}({\mathbb{Z}})$, with
coordinate $t=j/1728$.

Let $m\colonequals\gcd(\\{a,b,c\\}\smallsetminus\\{\infty\\})$, with $m=1$ for
$a=b=c=\infty$. Attached to $(a,b,c)$ is an extension

(1.1) $E=E(a,b,c)\subseteq F=F(a,b,c)\subseteq{\mathbb{Q}}(\zeta_{2m})^{+}$

of totally real, abelian number fields. The field $F$ is the subfield of
${\mathbb{R}}$ generated by $\operatorname{Tr}\Delta$, and similarly $E$,
called the invariant trace field, is the subfield generated by
$\operatorname{Tr}\Delta^{(2)}$ where $\Delta^{(2)}\leq\Delta$ is the subgroup
generated by squares. Let ${\mathbb{Z}}_{E}\subset E$ be the ring of integers
and similarly ${\mathbb{Z}}_{F}\subset F$.

Let $\mathfrak{N}\subseteq{\mathbb{Z}}_{E}$ be a nonzero ideal. Then there is
a natural reduction homomorphism $\varpi_{\mathfrak{N}}$ with domain $\Delta$,
intuitively thought of as reducing matrix entries modulo $\mathfrak{N}$ but
with a rigorous quaternionic interpretation (see below). The kernel
$\Gamma(a,b,c;\mathfrak{N})\colonequals\ker\varpi_{\mathfrak{N}}$ is called
the principal congruence subgroup of level $\mathfrak{N}$. A subgroup
$\Gamma\leq\Delta(a,b,c)$ is said to be congruence if
$\Gamma\geq\Gamma(a,b,c;\mathfrak{N})$ for some $\mathfrak{N}$; the level of a
congruence subgroup is the minimal such $\mathfrak{N}$. Given a congruence
subgroup $\Gamma\leq\Delta(a,b,c)$, we call the quotient
$X(a,b,c;\Gamma)\colonequals\Gamma\backslash\mathcal{H}$ a triangular modular
curve, since they generalize the classical modular curves. The quotient map

(1.2) $\varphi_{\mathfrak{N}}\colon X(a,b,c;\Gamma)\to
X(a,b,c)\simeq\mathbb{P}_{\mathbb{C}}^{1}$

(generalizing the $j$-invariant) is a Belyi map, unramified away from
$\\{0,1,\infty\\}$ (by our normalization). Accordingly, the curve
$X(a,b,c;\mathfrak{N})$ descends to a number field [3, Theorem B].

In light of the motivation above, we focus now on a nice class of congruence
subgroups. The $E$-subalgebra $A\colonequals
E\langle\Delta^{(2)}\rangle\leq\operatorname{M}_{2}({\mathbb{R}})$ generated
by (any lift of) the image of
$\Delta^{(2)}\hookrightarrow\operatorname{PSL}_{2}({\mathbb{R}})$ is a
quaternion algebra, and
$\Lambda\colonequals{\mathbb{Z}}_{E}\langle\Delta\rangle$ is a
${\mathbb{Z}}_{E}$-order in $A$. Then there is a commutative square

(1.3)
$\textstyle{\Delta^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda^{1}/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{A^{\times}}(\Lambda)/E^{\times}}$

where
$\Lambda^{1}\colonequals\\{\gamma\in\Lambda:\operatorname{nrd}(\gamma)=1\\}$
are the elements of reduced norm $1$.

Suppose $\mathfrak{N}\subseteq{\mathbb{Z}}_{E}$ is coprime to
$\operatorname{discrd}(\Lambda)$ and $\mathfrak{d}_{F|E}$, the relative
discriminant of $F$ over $E$; for example, this holds if $\mathfrak{N}$ is
coprime to $2abc$. Then the reduction
$\Lambda\to\Lambda/\mathfrak{N}\Lambda\simeq\operatorname{M}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})$
gives a well-defined group homomorphism

(1.4)
$\pi_{\mathfrak{N}}\colon\Delta\to\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N}).$

Combined with (1.3), we obtain a commutative diagram (Proposition 3.6)

(1.5)
$\textstyle{\Delta^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{SL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathfrak{N}}}$$\textstyle{\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})}$

Let $G_{\mathfrak{N}}\colonequals\pi_{\mathfrak{N}}(\Delta)$ be the image. Let
$s^{\sharp}$ be the order of $\pi_{\mathfrak{N}}(\delta_{s})$ for $s=a,b,c$;
then $a^{\sharp},b^{\sharp},c^{\sharp}$ are the ramification degrees in
$\varphi_{\mathfrak{N}}$ above $0,1,\infty$, and the homomorphism
$\pi_{\mathfrak{N}}$ factors through
$\Delta(a^{\sharp},b^{\sharp},c^{\sharp})$. To avoid redundancy, we say that
$\mathfrak{N}$ is admissible for $(a,b,c)$ if $s^{\sharp}=s$ for all
$s=a,b,c$—so in particular, $s\neq\infty$.

Without loss of generality (but see Proposition 3.13), we now suppose that
$\mathfrak{N}$ is admissible for $(a,b,c)$. Then the main result of
Clark–Voight [3, Theorem A] (see Theorem 3.12) describes the group
$G_{\mathfrak{N}}$. For example, when $\mathfrak{N}=\mathfrak{p}$ is _prime_ ,
then

(1.6)
$G_{\mathfrak{N}}\simeq\operatorname{PXL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$

where $\operatorname{PXL}_{2}=\operatorname{PSL}_{2}$ if $\mathfrak{p}$
(necessarily unramified in $F$) splits completely in $F$, and otherwise
$\operatorname{PXL}_{2}=\operatorname{PGL}_{2}$.

With this in mind, in this paper we focus on the case where
$\mathfrak{N}=\mathfrak{p}$ is prime. This case is already a quite interesting
first step, and still relevant for our motivation (see the next section).
Moreover, the case of composite level $\mathfrak{N}$ builds on the prime level
case and at the same time introduces several new challenges that are not
present in prime level. We plan to pursue the general case in future work.

Returning now to our original motivation, the usual upper-triangular (Borel-
type) subgroups

(1.7) $H_{1,\mathfrak{p}}\leq
H_{0,\mathfrak{p}}\colonequals\begin{pmatrix}*&*\\\
0&*\end{pmatrix}\leq\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$

naturally include into $G_{\mathfrak{p}}$ via (1.6). We define the Borel-type
congruence subgroups of $\Delta$

(1.8) $\displaystyle\Gamma_{0}(a,b,c;\mathfrak{p})$
$\displaystyle\colonequals\pi_{\mathfrak{p}}^{-1}(H_{0,\mathfrak{p}})$
$\displaystyle\Gamma_{1}(a,b,c;\mathfrak{p})$
$\displaystyle\colonequals\pi_{\mathfrak{p}}^{-1}(H_{1,\mathfrak{p}})$

We write $X_{0}(a,b,c;\mathfrak{p})$ and $X_{1}(a,b,c;\mathfrak{p})$ for the
corresponding quotients. For $(a,b,c)=(2,3,\infty)$, we recover the classical
modular curves $X_{0}(p)$ and $X_{1}(p)$.

Our main result is as follows.

######  Theorem 1.9.

For any $g\in{\mathbb{Z}}_{\geq 0}$, there are only finitely many Borel-type
triangular modular curves of genus $g$ with admissible prime level
$\mathfrak{N}=\mathfrak{p}$. The number of curves of genus at most $2$ are as
follows:

${\begin{array}[]{c||c|c|c}\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\textup{genus}\hfil\lx@intercol\\\
&0&1&2\\\ \hline\cr\hline\cr X_{0}(a,b,c;\mathfrak{p})&69&248&453\\\ \hline\cr
X_{1}(a,b,c;\mathfrak{p})&6&9&11\end{array}}$

Our proof of Theorem 1.9 includes a complete enumeration, computed using an
implementation in Magma [2] available online [26] (including the list in
computer readable format with additional data). For the list with $g=0,1$ and
prime level, see Appendix A.

### Discussion

As for classical modular curves, Theorem 1.9 uses the Riemann–Hurwitz theorem.
We observe that the ramification at prime level takes a tidy form. We carry
out the explicit enumeration using the existence and classification results of
Clark–Voight [3], which themselves ultimately rest on work of Macbeath [13]
classifying two-generated subgroups of $\operatorname{SL}_{2}(\mathbb{F}_{q})$
in terms of trace triples. The case $a=2$ causes particular difficulties (see
Example 4.2).

Our theorem has potential applications in arithmetic geometry analogous to
classical modular curves. Just as the quotient of the upper half-plane by
$\operatorname{PSL}_{2}({\mathbb{Z}})$ is the set of complex points of the
moduli space of elliptic curves (parametrized by the affine $j$-line),
Cohen–Wolfart [5, §3.3] and Archinard [1] showed that the curves $X(a,b,c)$
over ${\mathbb{C}}$ naturally parametrize _hypergeometric abelian varieties_ ,
certain Prym varieties of cyclic covers of $\mathbb{P}^{1}$ branched over
$\leq 4$ points. The name comes from the fact that their complex periods are
values of ${}_{2}F_{1}$-hypergeometric functions for the parameter
$t\in\mathbb{P}^{1}({\mathbb{C}})\smallsetminus\\{0,1,\infty\\}$. In
accordance with Manin’s “unity of mathematics” [4], their point counts are
defined by finite-field analogues of hypergeometric functions for
$t\in\mathbb{P}^{1}(\mathbb{F}_{q})\smallsetminus\\{0,1,\infty\\}$; these can
be packaged together (in an $\ell$-adic Galois representation) to define
hypergeometric $L$-functions attached to a motive for every
$t\in\mathbb{P}^{1}({\mathbb{Q}}^{\textup{al}})\smallsetminus\\{0,1,\infty\\}$.

More generally, just as classical modular curves parametrize elliptic curves
equipped with level structure, triangular modular curves parametrize
hypergeometric abelian varieties equipped with level structure: see upcoming
work of Kucharczyk–Voight [11] for the details, including a natural idelic
refinement and a notion of canonical model. In this light, our paper
classifies those situations where we might parametrize _infinitely many_ such
varieties with (nontrivial Borel-type) level structure for $t\in{\mathbb{Q}}$.

As shown by Takeuchi [22, 23], only finitely many triples $(a,b,c)$ give rise
to arithmetic Fuchsian groups; the remaining triples are _nonarithmetic_. Thus
almost all of the corresponding triangular modular curves are _thin_ subgroups
of the adelic points of a quaternionic group, so fall outside the usual scope
of the Langlands program.

As a final possible Diophantine application, we recall work of Darmon [8]: he
provides a dictionary between finite index subgroups of the triangle group
$\Delta(a,b,c)$ and approaches to solve the generalized Fermat equation
$x^{a}+y^{b}+z^{c}=0$. From this vantage point, the triangular modular curves
of low genus “explain” situations where the associated mod $\mathfrak{p}$
Galois representations are reducible.

In future work, we plan to compute equations for these curves (as Belyi maps)
using the methods of Klug–Musty–Schiavone–Voight [10] and then to study their
rational points. Even without these equations, we have verified that all but a
handful of the genus zero curves necessarily have a ramified rational point
(hence are isomorphic to $\mathbb{P}^{1}$ over any field of definition). It
would also be interesting to pursue cases when $\mathfrak{p}$ ramifies in $A$,
where the corresponding Galois covers will instead be solvable.

To conclude, we peek ahead to more general triangular modular curves, allowing
other subgroups $\Gamma\leq\Gamma(a,b,c;\mathfrak{N})$ (prescribing other
possible images of the corresponding Galois representations). For the case
$\Delta=\operatorname{PSL}_{2}({\mathbb{Z}})$, the story is a long and
beautiful one, originating with a conjecture of Rademacher that there are only
finitely many genus $0$ congruence subgroups of
$\operatorname{PSL}_{2}({\mathbb{Z}})$. Thompson [24] proved this for any
genus $g$, but the list of Cummins–Pauli relies upon difficult and delicate
$p$-adic methods of Cox–Parry [6] for an explicit bound on the level in terms
of the genus. We propose the following conjecture, which predicts a similar
result for triangular modular curves.

######  Conjecture 1.10.

For all $g\in{\mathbb{Z}}_{\geq 0}$, there are only finitely many admissible
triangular modular curves of genus $g$.

We consider our main result (Theorem 1.9) as partial progress towards this
conjecture—the Borel–type subgroups are the family with the smallest growing
index, thus likely to have the smallest genera. It would be interesting to see
if the rather delicate $p$-adic methods of Cox–Parry can be generalized from
$\operatorname{PSL}_{2}({\mathbb{Z}}/N{\mathbb{Z}})$ to groups of the form
$\operatorname{PXL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})$, as this would imply
1.10 in an effective way.

### Contents

In section 2, we set up triangular modular curves and as a warmup consider the
much easier Galois case $X(a,b,c;\mathfrak{p})$. In section 3, we extend the
work of Clark–Voight to understand the arithmetic requirements to construct
triangular modular curves. Then in section 4, for the case
$X_{0}(a,b,c;\mathfrak{p})$ with $a,b,c\in{\mathbb{Z}}$, we give an explicit
formula for the genus and we bound the norm of the level in terms of the
genus, proving finiteness; we then provide an algorithm to effectively
enumerate them in section 5. In section 6, we provide analogous results for
curves $X_{1}(a,b,c;\mathfrak{N})$, and finally we prove Theorem 1.9. We
conclude by providing the list in Appendix A.

### Acknowledgements

The authors would like to thank Asher Auel and Robert Kucharczyk for helpful
conversations and the anonymous referees for their constructive feedback. The
authors were supported by a Simons Collaboration grant (550029, to Voight).

##  2\. Setup and definitions

In this section, we give some basic setup and notation, define congruence
subgroups, and consider the enumeration problem in the Galois case; for
further reference, see Clark–Voight [3].

### Triangle groups

Beginning again, let $a,b,c\in{\mathbb{Z}}_{\geq 2}\cup\\{\infty\\}$. Let

(2.1) $\chi(a,b,c)\colonequals\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1$

so that $\chi(a,b,c)\pi$ measures difference from $\pi$ of the sum of the
angles of a triangle with angles $\pi/a,\pi/b,\pi/c$. If $\chi(a,b,c)\geq 0$,
then such a triangle is drawn on the sphere or Euclidean plane, and these are
very classical. Otherwise, we $\chi(a,b,c)<0$ and we say that the triple
$(a,b,c)$ is hyperbolic, as then the triangle lies in the (completed) upper
half-plane $\mathcal{H}$. For a hyperbolic triple $(a,b,c)$, we always have

(2.2) $\chi(a,b,c)\leq\chi(2,3,7)=-\frac{1}{42}$

bounded away from zero, by a simple maximization argument by cases.

As in the introduction, let $\Delta(a,b,c)$ be the subgroup of orientation-
preserving isometries of the group generated by reflections in the sides of
the triangle described above, drawn in the appropriate geometry. Then we have
a presentation

(2.3)
$\Delta(a,b,c)\colonequals\langle\delta_{a},\delta_{b},\delta_{c}\,|\,\delta_{a}^{a}=\delta_{b}^{b}=\delta_{c}^{c}=\delta_{a}\delta_{b}\delta_{c}=1\rangle.$

where $\delta_{s}$ corresponds to a counterclockwise rotation at the vertex
with angle $2\pi/s$. By cyclic permutation and inversion [3, Remark 2.2], we
can reorganize the generators and suppose without loss of generality that

(2.4) $a\leq b\leq c.$

From now on, we suppose that the triple $(a,b,c)$ is hyperbolic. Then there is
an associated map
$\Delta(a,b,c)\hookrightarrow\operatorname{PSL}_{2}({\mathbb{R}})$, unique up
to conjugation. We will often suppress the dependence on the triple from
notation, writing for example $\Delta=\Delta(a,b,c)$.

The group $\Delta$ is said to be cocompact if the quotient of the upper half-
plane by $\Delta$ is compact, else we say $\Delta$ is noncocompact. We have
$\Delta$ noncocompact if and only if at least one of $a,b,c$ is equal to
$\infty$.

Let $\Delta^{(2)}$ denote the subgroup of $\Delta$ generated by the set of
squares $\\{\delta^{2}:\delta\in\Delta\\}$. Then
$\Delta^{(2)}\trianglelefteq\Delta$ is a normal subgroup, in fact [3, (5.9)]
the quotient $\Delta/\Delta^{(2)}$ is represented by the elements $\delta_{s}$
with $s\in\\{a,b,c\\}$ such that either $s=\infty$ or $s\in{\mathbb{Z}}_{\geq
2}$ is even, hence

(2.5) $\Delta/\Delta^{(2)}\simeq\begin{cases}\\{0\\},&\text{if at least two of
$a,b,c$ are odd integers};\\\ {\mathbb{Z}}/2{\mathbb{Z}},&\text{if exactly one
of $a,b,c$ is an odd integer};\\\ ({\mathbb{Z}}/2{\mathbb{Z}})^{2},&\text{if
all of $a,b,c$ are even integers or $\infty$}.\end{cases}$

######  Lemma 2.6.

The group $\Delta^{(2)}$ is generated by the set

(2.7)
$\\{\delta_{s}^{-1}\delta_{t}^{2}\delta_{s}:s,t\in\\{a,b,c\\}\\}\cup\\{\delta_{s}\delta_{t}\delta_{s}^{-1}\delta_{t}^{-1}:s,t\in\\{a,b,c\\}\\}.$

###### Proof.

Follows from Takeuchi [22, Lemma 3, Proposition 5]: the generating set
presented there is smaller (depending on cases), whereas we collect these and
symmetrize to make a uniform statement. ∎

### Quaternions

For $s\in{\mathbb{Z}}_{\geq 2}\cup\\{\infty\\}$, let
$\zeta_{s}\colonequals\exp(2\pi i/s)$ and let
$\lambda_{s}\colonequals\zeta_{s}+1/\zeta_{s}=2\cos(2\pi/s)$, with
$\zeta_{\infty}=1$ and $\lambda_{\infty}=2$ by convention. Define the tower of
fields

(2.8)
${F=F(a,b,c)\colonequals{\mathbb{Q}}(\lambda_{2a},\lambda_{2b},\lambda_{2c})}$${E=E(a,b,c)\colonequals{\mathbb{Q}}(\lambda_{a},\lambda_{b},\lambda_{c},\lambda_{2a}\lambda_{2b}\lambda_{2c}).}$

The extension $F\supseteq E$ is abelian of exponent at most $2$ (since
$\lambda_{2s}^{2}=\lambda_{s}+2$) and has degree at most $4$. Let
${\mathbb{Z}}_{F}\supseteq{\mathbb{Z}}_{E}$ be the corresponding rings of
integers, and let $\mathfrak{d}_{F|E}$ be the relative discriminant of
$F\,|\,E$. The field $F$ is the trace field of the image of $\Delta$ in
$\operatorname{PSL}_{2}({\mathbb{R}})$, and $E$ the trace field for
$\Delta^{(2)}$, also called the invariant trace field (see Maclachlan–Reid
[14, Section 5.5]).

As above, we have a map
$\Delta\hookrightarrow\operatorname{PSL}_{2}({\mathbb{R}})$; the
$F$-subalgebra $B\colonequals
F\langle\Delta\rangle\leq\operatorname{M}_{2}({\mathbb{R}})$ generated by any
lift of the image (well-defined, since $-1\in F$) is a quaternion algebra,
similarly $\mathcal{O}\colonequals{\mathbb{Z}}_{F}\langle\Delta\rangle$ is a
${\mathbb{Z}}_{F}$-order in $B$ [21, Propositions 2–3]. The reduced
discriminant of $\mathcal{O}$ is a principal ideal of ${\mathbb{Z}}_{F}$
generated by [3, Lemma 5.4]

(2.9)
$\beta(a,b,c)\colonequals\lambda_{2a}^{2}+\lambda_{2b}^{2}+\lambda_{2c}^{2}+\lambda_{2a}\lambda_{2b}\lambda_{2c}-4=\lambda_{a}+\lambda_{b}+\lambda_{c}+\lambda_{2a}\lambda_{2b}\lambda_{2c}+2\in{\mathbb{Z}}_{E}.$

The same construction applies to $\Delta^{(2)}$, yielding a quaternion
$E$-algebra $A$ and a ${\mathbb{Z}}_{E}$-order $\Lambda$. Let
$\mathcal{O}^{1}\colonequals\\{\gamma\in\mathcal{O}:\operatorname{nrd}(\gamma)=1\\}$
be the elements of reduced norm $1$ in $\mathcal{O}$, and define $\Lambda^{1}$
similarly. Then we have a commutative square of group homomorphisms

(2.10)
$\textstyle{\Delta^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda^{1}/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}^{1}/\\{\pm
1\\}}$

In fact, the bottom map descends to the _normalizer_ $N_{A}(\Lambda)$ of
$\Lambda$ in $A$, as follows.

######  Lemma 2.11.

The composition of the maps

$\Delta\hookrightarrow\frac{\mathcal{O}^{1}}{\\{\pm
1\\}}\hookrightarrow\frac{N_{B^{\times}}(\mathcal{O})}{F^{\times}}$

factors via the map

(2.12) $\displaystyle\Delta$
$\displaystyle\hookrightarrow\frac{N_{A^{\times}}(\Lambda)}{E^{\times}}$
$\displaystyle\delta_{s}$
$\displaystyle\mapsto\begin{cases}\delta_{s}^{2}+1=\lambda_{2s}\delta_{s},&\textup{if
$s\neq 2$};\\\
(\delta_{c}^{2}+1)(\delta_{b}^{2}+1)=\lambda_{2b}\lambda_{2c}\delta_{a},&\textup{if
$s=a=2$};\end{cases}$

followed by the natural inclusion
$N_{A^{\times}}(\Lambda)/E^{\times}\hookrightarrow
N_{B^{\times}}(\mathcal{O})/F^{\times}$.

###### Proof.

See Clark–Voight [3, Proposition 5.13]. (The description fails to be uniform
when $a=2$ because $\lambda_{4}=0$; since $a\leq b\leq c$ we must have $b>2$,
else $(a,b,c)$ is not hyperbolic. The map is nevertheless uniquely determined,
since $\delta_{a}\delta_{b}\delta_{c}=1$.) ∎

### Congruence subgroups: general definition

We now define congruence subgroups. Let
$\mathfrak{N}\subseteq{\mathbb{Z}}_{E}$ be a nonzero ideal. Then reducing
elements modulo $\mathfrak{N}$, as in (2.10) we obtain a commutative diagram

(2.13)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{(2)}(\mathfrak{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Lambda/\mathfrak{N}\Lambda)^{1}/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(\mathfrak{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi_{\mathfrak{N}}}$$\textstyle{(\mathcal{O}/\mathfrak{N}\mathcal{O})^{1}/\\{\pm
1\\}}$

but now with kernels in the rows: in particular, we have a group homomorphism

(2.14)
$\varpi_{\mathfrak{N}}\colon\Delta\to(\mathcal{O}/\mathfrak{N}\mathcal{O})/\\{\pm
1\\}$

with kernel

(2.15)
$\Gamma(\mathfrak{N})\colonequals\ker\pi_{\mathfrak{N}}=\\{\delta\in\Delta:\delta\equiv\pm
1\leavevmode\nobreak\ (\textup{\text{mod}}\leavevmode\nobreak\
{\mathfrak{N}\mathcal{O}})\\}\trianglelefteq\Delta$

called the principal congruence subgroup of level $\mathfrak{N}$. As in the
introduction, we define congruence subgroups of $\Delta$ to be those that
contain a principal congruence subgroup, and a triangular modular curve to be
a quotient of the (completed) upper half-plane by a congruence subgroup of a
triangle group, for example

(2.16)
$X(\mathfrak{N})=X(a,b,c;\mathfrak{N})\colonequals\Gamma(\mathfrak{N})\backslash\mathcal{H}$

are called the principal triangular modular curves.

######  Remark 2.17.

One could work more generally with ideals of ${\mathbb{Z}}_{F}$ instead,
arriving at the same definition of congruence subgroups but with a different
notion of level. In light of what follows, especially the robust failure of
$\varpi_{\mathfrak{N}}$ to be surjective, we prefer to work with levels in
${\mathbb{Z}}_{E}$.

Since $\Delta$ normalizes $\Delta^{(2)}$ and therefore $\Lambda$ and
$\mathfrak{N}\Lambda$, there is descent to the normalizer as in Lemma 2.11.
However, the precise description of $\Gamma(\mathfrak{N})$ depends on the
ramification behavior of the primes dividing $\mathfrak{N}$ in the extension
$F\,|\,E$ and in the algebras $A$ and $B$ (and this already introduces some
subtleties when $\mathfrak{N}$ is composite). We pursue this in the next
section.

### Galois case

Before proceeding, as a warmup we consider the curves $X(a,b,c;\mathfrak{p})$
corresponding to principal congruence subgroups, where
$X(a,b,c;\mathfrak{p})\to X(a,b,c)\simeq\mathbb{P}^{1}$ is a generically
Galois Belyi map.

Quite generally, for any generically Galois Belyi map with group $G$, the
ramification indices above each ramification point are equal. Without loss of
generality, we may suppose that $a,b,c$ are also the orders of the
ramification points. Thus the Riemann-Hurwitz formula gives

(2.18) $2g(X)-2=-2(\\#G)+\sum_{s=a,b,c}\frac{\\#G}{s}(s-1)$

which simplifies to

(2.19) $g(X)=1-\frac{\\#G}{2}\chi(a,b,c).$

From this genus formula and (2.2), we can conclude that, for any fixed genus
$g_{0}\geq 0$, there are finitely many hyperbolic $G$-Galois Belyi maps with
genus $g_{0}$.

We are of course interested in the special case where

$G=\Gamma(\mathfrak{p})\backslash\Delta\simeq\operatorname{PXL}_{2}(\mathbb{F}_{\mathfrak{p}})$

where $\mathbb{F}_{\mathfrak{p}}\colonequals{\mathbb{Z}}_{E}/\mathfrak{p}$ is
the residue field and $\operatorname{PXL}_{2}(\mathbb{F}_{\mathfrak{p}})$
denotes either $\operatorname{PSL}_{2}(\mathbb{F}_{\mathfrak{p}})$ or
$\operatorname{PGL}_{2}(\mathbb{F}_{\mathfrak{p}})$. (The major task in the
next section is to precisely investigate this arithmetically.) Plugging
$G=\operatorname{PXL}_{2}(\mathbb{F}_{q})$ into the above:

$84(g_{0}-1)\geq\\#G=q(q+1)(q-1)\cdot\begin{cases}1/2,&\text{ if
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ and $q$ is odd};\\\ 1,&\text{
otherwise}.\end{cases}$

Thus, there are no curves $X(a,b,c;\mathfrak{p})$ of genus at most $1$. For
genus 2, we can use the inequality to see that $q$ must be less than 6, so
$\\#G\leq 60$ and, if $g(X(a,b,c;\mathfrak{p}))=2$, then

$-\frac{1}{\chi(a,b,c)}\leq 30.$

This inequality implies that $a\leq b\leq c\leq 7$ and, by checking the genera
of these possibilities with (2.19), we conclude that there are no curves
$X(a,b,c;\mathfrak{p})$ of genus 2.

In fact, the smallest genus for a hyperbolic triple with
$a,b,c\in{\mathbb{Z}}_{\geq 2}$ is genus $3$ for $(a,b,c)=(2,3,7)$, yielding
the famed Klein quartic curve. More generally, see Clark–Voight [3, Table
10.5] for examples up to genus $24$.

##  3\. Triangular modular curves

In this section, we study triangular modular curves generalizing the classical
modular curves; the main results are Proposition 3.6, where we define the
relevant matrix representation of $\Delta$, and Theorem 3.12, describing its
image building on work of Clark–Voight [3]. Throughout, we retain our notation
from the previous section.

### Congruence subgroups: matrix case

We return to (2.13), and identify matrix groups. Recalling (2.9), we first
suppose that $\beta=\operatorname{discrd}\mathcal{O}$ is coprime to
$\mathfrak{N}$, so all primes $\mathfrak{p}\mid\mathfrak{N}$ are unramified in
$B$ but more strongly we have
$(\mathcal{O}/\mathfrak{N}\mathcal{O})^{1}/\\{\pm
1\\}\simeq\operatorname{SL}_{2}({\mathbb{Z}}_{F}/\mathfrak{N}{\mathbb{Z}}_{F})/\\{\pm
1\\}$.

For the order $\Lambda$, we recall Lemma 2.6: given $a,b,c$, we can compute
its ${\mathbb{Z}}_{E}$-module span in $A$ and therefore a
${\mathbb{Z}}_{E}$-pseudobasis for $\Lambda$, hence its reduced discriminant.
Since $\Lambda{\mathbb{Z}}_{F}\subseteq\mathcal{O}$, we have
$\beta\mid\operatorname{discrd}(\Lambda)$ [3, Corollary 5.17].

So we make the stronger assumption that $\mathfrak{N}$ is coprime to
$\operatorname{discrd}(\Lambda)$. Then from (2.10) we get

(3.1)
$\textstyle{\Delta^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Lambda/\mathfrak{N}\Lambda)^{1}/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{SL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})/\\{\pm
1\\}}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathfrak{N}}}$$\textstyle{(\mathcal{O}/\mathfrak{N}\mathcal{O})^{1}/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{SL}_{2}({\mathbb{Z}}_{F}/\mathfrak{N}{\mathbb{Z}}_{F})/\\{\pm
1\\}}$

To descend the bottom map to the normalizer as in Lemma 2.11, we restrict our
scope taking $\mathfrak{N}=\mathfrak{p}$ prime and work just a little bit
more.

Let

(3.2) ${\mathbb{Z}}_{E,(\mathfrak{p})}\colonequals\\{\alpha\in
E:\operatorname{ord}_{\mathfrak{p}}(\alpha)\geq 0\\}\subseteq E$

be the localization of ${\mathbb{Z}}_{E}$ at the ideal $\mathfrak{p}$ (all
elements coprime to $\mathfrak{p}$ become units).

######  Lemma 3.3.

Suppose that $\mathfrak{p}\nmid\mathfrak{d}_{F|E}$. Then for $s=a,b,c$, we can
write

(3.4) $\lambda_{s}+2=\upsilon_{s}\theta_{s}^{2}\in E^{\times}$

with:

  * •

$\upsilon_{s}\in{\mathbb{Z}}_{E,(\mathfrak{p})}^{\times}$, well-defined up to
multiplication by an element of ${\mathbb{Z}}_{E,(\mathfrak{p})}^{\times 2}$,
i.e., up to the square of an element of
${\mathbb{Z}}_{E,(\mathfrak{p})}^{\times}$, and

  * •

$\theta_{s}\in E^{\times}$, well-defined up to
${\mathbb{Z}}_{E,(\mathfrak{p})}^{\times}$.

If $\mathfrak{p}$ is coprime to $2abc$, then we may take $\theta_{s}=1$ and
$\upsilon_{s}=\lambda_{s}+2$.

Moreover, the prime $\mathfrak{p}$ (necessarily unramified in $F$) splits
completely in $F$ if and only if the Kronecker symbols
$(\upsilon_{s}\,|\,\mathfrak{p})=1$ are trivial for all $s=a,b,c$.

###### Proof.

First, a bit of generality: for $\alpha\in E^{\times}$ with even valuation at
all primes $\mathfrak{p}\mid\mathfrak{N}$, by weak approximation in $E$ we can
write

(3.5) $\alpha=\upsilon\theta^{2}\in E^{\times}$

with $\upsilon,\theta$ as in the statement of the lemma.

Now to apply this, we observe that
$F=E(\lambda_{2a},\lambda_{2b},\lambda_{2c})$ and recall that
$\lambda_{2s}^{2}=\lambda_{s}+2$. By hypothesis, we have
$\mathfrak{p}\nmid\mathfrak{d}_{F|E}$; in particular the elements
$\lambda_{s}+2$ must have even (nonnegative) valuation at $\mathfrak{p}$. Thus
(3.5) applies, giving (3.4). The final statement follows from the usual
splitting criterion in quadratic fields. ∎

We obtain the following result.

######  Proposition 3.6.

Suppose that
$\mathfrak{p}\nmid\operatorname{discrd}(\Lambda)\mathfrak{d}_{F|E}$. Then
there is a commutative diagram

(3.7)
$\textstyle{\Delta^{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{SL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})/\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathfrak{N}}}$$\textstyle{\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})}$

and the map
$\pi_{\mathfrak{N}}\colon\Delta\to\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})$
factors through $\varpi_{\mathfrak{N}}$.

We let
$G_{\mathfrak{p}}\colonequals\pi_{\mathfrak{p}}(\Delta)\leq\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$
be the image of $\pi_{\mathfrak{p}}$.

###### Proof.

Combine (3.1) with Lemma 3.3. ∎

######  Remark 3.8.

A similar argument works when $\mathfrak{N}$ is composite; however the right-
hand vertical map $\operatorname{SL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})/\\{\pm
1\\}\to\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{N})$ may no longer be
injective when $\mathfrak{N}$ is composite. This leads to certain ambiguities
about the definition which we will return to in future work.

### Admissibility, projectivity, and image

It can and does happen that two different triangular modular curves are
isomorphic (as curves and as covers of $\mathbb{P}^{1}$). The issue is simply
that in the homomorphism $\pi_{\mathfrak{N}}$ from $\Delta(a,b,c)$ to a matrix
group, the generators $\delta_{s}$ need not have order $s$ in the image (for
$s=a,b,c$). In other words, the reduction homomorphism factors through a
triangle group with a smaller triple. This happens for example when
$s=\infty$, as the order of $\pi_{\mathfrak{N}}(\delta_{s})$ is always finite!
To illustrate this phenomena, we present the following example.

######  Example 3.9.

Consider the triples $(2,3,c)$ with $c=p^{k}$, where $k\geq 1$ and $p\geq 5$
is prime. Then

(3.10) $E_{k}\colonequals
E(2,3,c)=F(2,3,c)={\mathbb{Q}}(\lambda_{2c})={\mathbb{Q}}(\zeta_{2c})^{+}$

and $\beta(2,3,c)=\lambda_{c}-1\in{\mathbb{Z}}_{E_{k}}^{\times}$. The prime
$p$ is totally ramified in $F$, so
$\mathbb{F}_{\mathfrak{p}_{k}}\simeq\mathbb{F}_{p}$ for $\mathfrak{p}_{k}\mid
p$. Thus $X(2,3,p^{k};\mathfrak{p}_{k})\simeq X(2,3,p;\mathfrak{p}_{1})$.

To avoid this redundancy, we make the following definition.

######  Definition 3.11.

Given a triple $(a,b,c)$, an ideal
$\mathfrak{p}\subseteq{\mathbb{Z}}_{E(a,b,c)}$ is admissible for $(a,b,c)$ if

  * •

$\mathfrak{p}\nmid\operatorname{discrd}(\Lambda)\mathfrak{d}_{F|E}$, and

  * •

the order of $\pi_{\mathfrak{p}}(\delta_{s})$ is equal to $s$ for all
$s=a,b,c$.

######  Theorem 3.12 (Clark–Voight).

We have
$\pi_{\mathfrak{p}}(G_{\mathfrak{p}}^{(2)})=\operatorname{PSL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$
and

$\pi_{\mathfrak{p}}(G_{\mathfrak{p}})=\operatorname{PXL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$

where $\operatorname{PXL}_{2}$ denotes $\operatorname{PSL}_{2}$ or
$\operatorname{PGL}_{2}$ according as $\mathfrak{p}$ splits in $F\supseteq E$
or not.

###### Proof.

We refer to Clark–Voight [3, Theorem A] for the case where $\mathfrak{p}\nmid
2abc$; but examining the argument given [3, Remark 5.24, proof of Theorem 9.1]
in light of the above, we see that it extends when
$\mathfrak{p}\nmid\operatorname{discrd}(\Lambda)\beta\mathfrak{d}_{F|E}$. ∎

### Hyperbolic triples reducing to non-hyperbolic triples

In considering admissible triples, we may lose the hypothesis that $(a,b,c)$
is hyperbolic; however, this situation is easy to characterize. We note that
in most cases, these groups do not contain
$\operatorname{PSL}_{2}(\mathbb{F}_{q})$, so they are not considered in this
paper.

######  Proposition 3.13.

Let $(a,b,c)\in({\mathbb{Z}}_{\geq 0}\cup\\{\infty\\})^{3}$ be a triple and
let $\mathfrak{p}\subset{\mathbb{Z}}_{E}$ be a nonzero prime ideal. Suppose
further that $(a^{\sharp},b^{\sharp},c^{\sharp})$ is a projective triple, but
not hyperbolic. Then $(a,b,c;p,q)$ is one of the elements listed in the
following table. In the table, $p$ lies below $\mathfrak{p}$ and $q$ is the
residue field degree of $\mathfrak{p}$.

(3.14)
${\begin{array}[]{c|c||c|c|c||c}\bm{(a,b,c)}&\text{{conditions}}&\;\;\bm{p}&\;\;\bm{q}&\bm{{\rm
PXL}}&\bm{E(a^{\sharp},b^{\sharp},c^{\sharp})}\\\
\hline\cr\hline\cr\begin{array}[]{c}(2^{k_{a}},2^{k_{b}},3\cdot 2^{k_{c}}),\\\
(3\cdot 2^{k_{c}},\infty,\infty),\\\ (2^{k_{a}},3\cdot
2^{k_{c}},\infty)\end{array}&1\leq k_{a}<k_{b}&2&2&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1\\\
\hline\cr\begin{array}[]{c}(3^{k_{a}},3^{k_{b}},3^{k_{c}}),\\\
(3^{k_{a}},\infty,\infty),\\\ (3^{k_{a}},3^{k_{b}},\infty),\\\
(\infty,\infty,\infty)\end{array}&1\leq k_{a}\leq k_{b}<k_{c}&3&3&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1\\\
\hline\cr\begin{array}[]{c}(2\cdot 3^{k_{a}},3^{k_{b}},3^{k_{c}}),\\\ (2\cdot
3^{k_{a}},3^{k_{b}},\infty),\\\ (2\cdot
3^{k_{a}},\infty,\infty)\end{array}&1\leq k_{b}\leq
k_{c},\,k_{a}k_{b}k_{c}\neq 1&3&3&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1\\\
\hline\cr\begin{array}[]{c}(2\cdot 3^{k_{a}},3^{k_{b}},4\cdot 3^{k_{c}}),\\\
(2\cdot 3^{k_{a}},4\cdot 3^{k_{b}},\infty)\end{array}&1\leq
k_{b},\,k_{a}k_{b}k_{c}\neq 1&3&3&-1&\href
https://www.lmfdb.org/NumberField/1.1.1.1\\\
\hline\cr\begin{array}[]{c}(2^{k_{a}},3\cdot 2^{k_{b}},5\cdot 2^{k_{c}}),\\\
(3\cdot 2^{k_{b}},5\cdot 2^{k_{c}},\infty)\end{array}&1\leq
k_{a},\,k_{a}k_{b}k_{c}\neq 1&2&4&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1\\\
\hline\cr\begin{array}[]{c}(2\cdot 5^{k_{a}},3\cdot 5^{k_{b}},5^{k_{c}}),\\\
(2\cdot 5^{k_{a}},3\cdot 5^{k_{b}},\infty)\end{array}&1\leq
k_{c},\,k_{a}k_{b}k_{c}\neq 1&5&5&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1\end{array}}$

Furthermore, the curves $X(a,b,c;\mathfrak{p})$ with $(a,b,c;\mathfrak{p})$ as
above all have genus 0.

###### Proof.

We first focus on the prime ideal case and make a case by case study. The only
triples $(a,b,c)\in({\mathbb{Z}}_{\geq 0}\cup\\{\infty\\})^{3}$ that are not
hyperbolic are

$(2,2,n)\text{ for
}n>1,\,(2,3,3),\,(2,3,4),\,(2,3,5),\,(2,3,6),\,(2,4,4),\text{ or }(3,3,3).$

Assume first that $(a^{\sharp},b^{\sharp},c^{\sharp})=(2,2,c)$ for $c>1$. The
image of
$\pi_{\mathfrak{p}}:\Delta(2,2,c)\to\operatorname{PGL}_{2}(\mathbb{F}_{q})$
must be dihedral. The only dihedral group that is isomorphic to
$\operatorname{PXL}_{2}(\mathbb{F}_{q})$ for any $q$ is
$D_{6}\simeq\operatorname{PSL}_{2}(\mathbb{F}_{2})$. Thus, we only have the
triple $(a^{\sharp},b^{\sharp},c^{\sharp})=(2,2,3)$ and prime
$\mathfrak{p}_{2}$.

The group $\Delta(2,3,6)$ is solvable since it fits in the exact sequence:

$1\to{\mathbb{Z}}^{2}\to\Delta(2,3,6)\to{\mathbb{Z}}/6{\mathbb{Z}}\to 1.$

The only solvable groups of the form $\operatorname{PXL}_{2}(\mathbb{F}_{q})$
are $S_{4}\simeq\operatorname{PGL}_{2}(\mathbb{F}_{3})$ and
$A_{4}\simeq\operatorname{PSL}_{2}(\mathbb{F}_{3})$. The triple $(2,3,6)$ is
not admissible for $q=2$ or $q=3$, so $(2,3,6)$ is not projective and
admissible for any prime ideal $\mathfrak{p}$. With the same analysis, we can
rule out $(2,4,4)$. We also have that the group $\Delta(3,3,3)$ is solvable.
Hence, the image of
$\pi_{\mathfrak{p}}:\Delta(3,3,3)\to\operatorname{PXL}_{2}(\mathbb{F}_{q})$
must be solvable. The only solvable groups of this form are
$A_{4}\simeq\operatorname{PSL}_{2}(\mathbb{F}_{3})$ and
$S_{4}\simeq\operatorname{PGL}_{2}(\mathbb{F}_{3})$. Thus, the only option is
that $\mathfrak{p}$ is a prime above $3$ with residue field $\mathbb{F}_{3}$.

The last triples to consider are $(2,3,3),\,(2,3,4)$ and $(2,3,5)$. These
triples are all _exceptional_. Tthe only projective linear groups that can
arise from exceptional triples [3, Remark 8.4] are the following:

$\operatorname{PSL}_{2}(\mathbb{F}_{3}),\operatorname{PGL}_{2}(\mathbb{F}_{3}),\operatorname{PGL}_{2}(\mathbb{F}_{4}),\operatorname{PSL}_{2}(\mathbb{F}_{5}).$

We now use this fact to finish the analysis. When
$(a^{\sharp},b^{\sharp},c^{\sharp})=(2,3,3)$, the admissible prime ideals
$\mathfrak{p}$ have residue field degree $3,4$ and $5$. The field $E(2,3,3)$
is the rational field, so
${\mathbb{Z}}_{E}/\mathfrak{p}_{2}\simeq\mathbb{F}_{2}$. In addition, the
ideal $2{\mathbb{Z}}_{E}$ is totally ramified in any field $E(2\cdot
2^{k_{a}},3\cdot 2^{k_{b}},3\cdot 2^{k_{c}})$, so $q\neq 4$. The only options
then are $q=3$ and $q=5$. A quick Magma [2] calculation shows that elements
with these orders cannot generate $\operatorname{PSL}_{2}(\mathbb{F}_{5})$.

Similarly, when $(a^{\sharp},b^{\sharp},c^{\sharp})=(2,3,4)$, the only
possibilities for $q$ which make the triple projective and admissible for
$\mathfrak{p}$ are $q=3$ or $q=5$. However, the field $E(2,3,4)$ is the
rational field and 5 is inert in $F$, so we would have
$G_{5{\mathbb{Z}}_{E}}\simeq\operatorname{PGL}_{2}(\mathbb{F}_{5})$, which is
not on the list of possible groups. The same happens for
$(a^{\sharp},b^{\sharp},c^{\sharp})=(2,3,5)$; the options of $q$ for an
admissible prime $\mathfrak{p}$ are $q=2,3,4,5$. The ideal $2{\mathbb{Z}}_{E}$
is inert in $E(2,3,5)$, an extension of ${\mathbb{Q}}$ of degree 2, thus $q=2$
is not possible. The ideal $3{\mathbb{Z}}_{E}$ is also inert in $E(2,3,5)$, so
an isomorphism with $\operatorname{PXL}_{2}(\mathbb{F}_{3})$ is not possible.
The only options for $q$ are $q=4$ and $q=5$.

For all of the possible triples $(a^{\sharp},b^{\sharp},c^{\sharp})$ and
primes $\mathfrak{p}$ described above, we certify that such map is possible by
exhibiting passports for each curve. Finally, we use Equation 2.19 to compute
the genus of each of these curves, finding that they all have genus 0. ∎

### Borel-type subgroups

As in the introduction, let

(3.15) $\left\\{\begin{pmatrix}a&b\\\
0&d\end{pmatrix}:a,b,d\in{\mathbb{Z}}_{E}/\mathfrak{p}\text{ and
}ad\in({\mathbb{Z}}_{E}/\mathfrak{p})^{\times}\right\\}\leq\operatorname{GL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$

be the upper-triangular matrices in
$\operatorname{GL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$, and let
$H_{0,\mathfrak{p}}$ be its image in the projection to
$\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$. Similarly, let

(3.16) $\left\\{\begin{pmatrix}1&b\\\
0&1\end{pmatrix}:b\in{\mathbb{Z}}_{E}/\mathfrak{p}\right\\}\leq\operatorname{GL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$

be the upper unipotent subgroup and $H_{1,\mathfrak{p}}$ again its image in
$\operatorname{PGL}_{2}({\mathbb{Z}}_{E}/\mathfrak{p})$.

We then define the subgroups

(3.17) $\displaystyle\Gamma_{0}(a,b,c;\mathfrak{p})$
$\displaystyle\colonequals\varphi^{-1}_{\mathfrak{p}}(H_{0,\mathfrak{p}}),$
$\displaystyle\Gamma_{1}(a,b,c;\mathfrak{p})$
$\displaystyle\colonequals\varphi^{-1}_{\mathfrak{p}}(H_{1,\mathfrak{p}}).$

and the corresponding quotients

(3.18) $\displaystyle X_{0}(a,b,c;\mathfrak{p})$
$\displaystyle\colonequals\Gamma_{0}(a,b,c;\mathfrak{p})\backslash\mathcal{H}=H_{0,\mathfrak{p}}\backslash
X^{\prime}(a,b,c;\mathfrak{p})$ $\displaystyle X_{1}(a,b,c;\mathfrak{p})$
$\displaystyle\colonequals\Gamma_{1}(a,b,c;\mathfrak{p})\backslash\mathcal{H}=H_{1,\mathfrak{p}}\backslash
X(a,b,c;\mathfrak{p}).$

Then we have natural quotient maps

(3.19) $X(a,b,c;\mathfrak{N})\to X_{1}(a,b,c;\mathfrak{N})\to
X_{0}(a,b,c;\mathfrak{N})\to X(a,b,c;1)\simeq\mathbb{P}^{1}.$

##  4\. Triangular modular curves $X_{0}(a,b,c;\mathfrak{p})$ of prime level

In this section, we exhibit a formula for the genus of the triangular modular
curves $X_{0}(a,b,c;\mathfrak{p})$ for $\mathfrak{p}$ prime. Using this
formula we show that there are only finitely many such curves with bounded
genus.

### Setup

Let $(a,b,c)$ be a hyperbolic triple and $\mathfrak{p}$ be an admissible prime
of $E=E(a,b,c)$ with residue field $\mathbb{F}_{\mathfrak{p}}$. Let
$q\colonequals\\#\mathbb{F}_{\mathfrak{p}}$, so
$\mathbb{F}_{\mathfrak{p}}\simeq\mathbb{F}_{q}$. Because $E$ is Galois over
${\mathbb{Q}}$, all primes $\mathfrak{p}$ have the same ramification and
splitting type; it follows that the genus of $X_{0}(a,b,c;\mathfrak{p})$ only
depends on the prime number $p\in{\mathbb{Z}}$ below $\mathfrak{p}$ (and the
inertial degree of $\mathfrak{p}$ over $p$).

Let $G\colonequals G_{\mathfrak{p}}$ be as in Theorem 3.12. Then the group
$H_{0}=H_{0,\mathfrak{p}}$ consists of the image in $G$ of the upper-
triangular matrices of $\operatorname{SL}_{2}(\mathbb{F}_{q})$ or
$\operatorname{GL}_{2}(\mathbb{F}_{q})$, depending on $G$. By construction,
the curves $X_{0}(a,b,c;\mathfrak{p})$ and $X(a,b,c;\mathfrak{p})$ fit in the
following diagram.

$\textstyle{X(a,b,c;\mathfrak{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H_{0}}$$\scriptstyle{G}$$\textstyle{X_{0}(a,b,c;\mathfrak{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}}$

We first compute the index $[G:H_{0}]$, which corresponds to the degree of the
cover $X_{0}(a,b,c;\mathfrak{p})\to\mathbb{P}^{1}$. If
$G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$, up to multiplication by a scalar
matrix, it is possible to choose representatives of elements of $H_{0}$ that
have 1 on the first entry of the matrix. Thus, $\\#H_{0}=q(q-1)$ and
$[G:H_{0}]=q+1.$ When $q$ is even, we have an isomorphism
$\operatorname{PSL}_{2}(\mathbb{F}_{q})\simeq\operatorname{PGL}_{2}(\mathbb{F}_{q})$,
so the index $[G:H_{0}]$ is the same as above. Finally, if
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ with $q$ odd, then representatives
can be chosen to have 1 on the first entry of the matrix as above. Also, the
upper triangular matrices are defined up to multiplication by $-1$. Hence
$\\#H_{0}=\frac{1}{2}q(q-1)$ and $[G:H_{0}]=q+1.$

Via the projection of the first column of the matrix to
$\mathbb{P}^{1}(\mathbb{F}_{q})$, the set of cosets $G/H_{0}$ is naturally in
bijection with $\mathbb{P}^{1}(\mathbb{F}_{q})$. With this bijection, the
action of $\pi_{\mathfrak{p}}(\Delta)$ on $G/H_{0}$ becomes simply matrix
multiplication. The ramification of the cover
$X_{0}(a,b,c;p)\to\mathbb{P}^{1}$ then depends on the cycle decomposition of
the corresponding elements (in $G$) as an element of
$\operatorname{Sym}(\mathbb{P}^{1})\simeq S_{q+1}$.

### Cycle structure and genus formula

The following lemma describes the cycle structure using only the order of the
elements. Recall we write $\operatorname{PXL}_{2}$ for either
$\operatorname{PSL}_{2}$ or $\operatorname{PGL}_{2}$.

######  Lemma 4.1.

Let $G=\operatorname{PXL}_{2}(\mathbb{F}_{q})$ with $q=p^{r}$ for a prime
number $p$. Let $\overline{\sigma}_{s}\in G$ have order $s\geq 2$, and if
$s=2$ suppose $p=2$. Then the action of $\overline{\sigma}_{s}$ on
$\mathbb{P}^{1}(\mathbb{F}_{q})$ has:

  1. (i)

two fixed points and $(q-1)/s$ orbits of length $s$ if $s\mid(q-1)$;

  2. (ii)

one fixed point and $q/p$ orbits of length $p$ if $s=p$ (this is the case when
$s\mid q$); and

  3. (iii)

(no fixed points and) $(q+1)/s$ orbits of length $s$ if $s\mid(q+1)$.

###### Proof.

We note that each class in $G$ is represented by matrices that are
diagonalizable over $\mathbb{F}_{q}$, diagonalizable only over
$\mathbb{F}_{q^{2}}$, or not diagonalizable. We prove the Lemma by studying in
detail each case. Let $\sigma_{s}$ be an element of
$\operatorname{GL}_{2}(\mathbb{F}_{q})$ whose projection to $G$ is
$\overline{\sigma}_{s}$. If $\sigma_{s}$ is diagonalizable, then we say that
$\overline{\sigma}_{s}$ is split semisimple, and $\sigma_{s}$ is conjugate to
say the diagonal matrix $\begin{pmatrix}u&0\\\ 0&v\end{pmatrix}$. We must have
$u\neq v$ because otherwise $\overline{\sigma}_{s}$ would be the identity in
$G$, contradicting that $s\geq 2$. The order of $\overline{\sigma}_{s}$ is
$s$, so $s$ is the order of $uv^{-1}$ in $\mathbb{F}_{q}^{\times}$. To find
the orbits of the action of $\overline{\sigma}_{s}$ on
$\mathbb{P}^{1}(\mathbb{F}_{q})$, we use that

$\begin{pmatrix}u&0\\\ 0&v\end{pmatrix}{\begin{pmatrix}{1}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {0}\end{pmatrix}}={\begin{pmatrix}{1}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {0}\end{pmatrix}},\hskip
19.91684pt\begin{pmatrix}u&0\\\ 0&v\end{pmatrix}{\begin{pmatrix}{x}\\\
\rotatebox[origin={c}]{90.0}{:}\\\
{1}\end{pmatrix}}={\begin{pmatrix}{uv^{-1}x}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {1}\end{pmatrix}},$

for any $x\in\mathbb{F}_{q}$. Hence, the action of $\overline{\sigma}_{s}$ has
two fixed points: ${\begin{pmatrix}{1}\\\ \rotatebox[origin={c}]{90.0}{:}\\\
{0}\end{pmatrix}}$ and ${\begin{pmatrix}{0}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {1}\end{pmatrix}}$, and $(q-1)/s$ orbits
with $s$ elements.

The element $\overline{\sigma}_{s}$ is unipotent if and only if it is
conjugate to $\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}$ in $G$ for some
$u\in\mathbb{F}_{q}^{\times}$. This is the case when the characteristic
polynomial of $\sigma_{s}$ has two equal roots and $\sigma_{s}$ is not
diagonalizable over $\mathbb{F}_{q}^{2}$. This happens if and only if $s=p$.
In this case, we have

$\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}{\begin{pmatrix}{1}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {0}\end{pmatrix}}={\begin{pmatrix}{1}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {0}\end{pmatrix}},\hskip
19.91684pt\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}{\begin{pmatrix}{x}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {1}\end{pmatrix}}={\begin{pmatrix}{x+u}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {1}\end{pmatrix}},$

where $x\in\mathbb{F}_{q}$. There is only one fixed point and there are $q/p$
orbits of size $p$.

If the characteristic polynomial of $\sigma_{s}$ does not split in
$\mathbb{F}_{q}$, we call $\overline{\sigma}_{s}$ non-split semisimple. The
action of $\overline{\sigma}_{s}$ has no fixed points because this would imply
that $\sigma_{s}$ has an eigenvector. The splitting field of the
characteristic polynomial of $\sigma_{s}$ is $\mathbb{F}_{q^{2}}$. Let
$\alpha_{1},\alpha_{2}\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ be the
roots of this polynomial. Then $\sigma_{s}$ is conjugate with the diagonal
matrix $[\alpha_{1},\alpha_{2}]$ with
$\sigma_{s}=T^{-1}[\alpha_{1},\alpha_{2}]T$ for some invertible matrix $T$.
For all $m\in{\mathbb{N}}$ such that $\overline{\sigma}_{s}^{m}$ fixes
$(x:y)^{t}\in\mathbb{P}^{1}(\mathbb{F}_{q})$, we have that

$\begin{pmatrix}\alpha_{1}^{m}&0\\\
0&\alpha_{2}^{m}\end{pmatrix}\left(T{\begin{pmatrix}{x}\\\
\rotatebox[origin={c}]{90.0}{:}\\\
{y}\end{pmatrix}}\right)=\left(T{\begin{pmatrix}{x}\\\
\rotatebox[origin={c}]{90.0}{:}\\\ {y}\end{pmatrix}}\right).$

From the analysis of the split semisimple case, we conclude that every orbit
has length $s$. Thus, the action of $\sigma_{s}$ on
$\mathbb{P}^{1}(\mathbb{F}_{q})$ has $(q+1)/s$ orbits of length $s$. ∎

The previous lemma does not consider the case when $s=2$ and $q$ is odd. The
ambiguity arises since if $s=2$ then $s\mid(q-1)$ and $s\mid(q+1)$, so
$\overline{\sigma}_{2}$ can be either split or non-split (semisimple).

######  Example 4.2.

For $(a,b,c)=(2,3,8)$ and $G=\operatorname{PGL}_{2}(\mathbb{F}_{7})$, we have
$\sigma_{2}$ split. On the other hand, for $(2,6,6)$ and
$G=\operatorname{PGL}_{2}(\mathbb{F}_{7})$, we have $\sigma_{2}$ non-split.

The following lemma partially solves this problem.

######  Lemma 4.3.

Let $G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ with $q$ odd, and let
$\overline{\sigma}_{2}\in G$ be an element of order $2$. Then the action of
$\overline{\sigma}_{2}$ on $\mathbb{P}^{1}(\mathbb{F}_{q})$ has:

  1. (i)

two fixed points and $(q-1)/2$ orbits of size $2$ if $-1$ is a square modulo
$q$; and

  2. (ii)

(no fixed points and) $(q+1)/2$ orbits of size $2$, otherwise.

###### Proof.

Let $\overline{\sigma}_{2}$ be a matrix of order 2 in
$\operatorname{PSL}_{2}(\mathbb{F}_{q})$. Pick a lift
$\sigma_{2}\in\operatorname{SL}_{2}(\mathbb{F}_{q})$ of
$\overline{\sigma}_{2}$. Because $\sigma_{2}^{4}$ is the identity, its
characteristic polynomial must be a quadratic polynomial dividing $x^{4}-1$.
In addition, the constant of this polynomial must be 1 since this is the
determinant of $\sigma_{2}$. The only possibility for such a polynomial is
$x^{2}+1$. If $-1\in\mathbb{F}_{q}^{\times 2}$, then this characteristic
polynomial splits with distinct roots, so we are in the split semisimple case
of Lemma 4.1. Otherwise, $-1$ is not a square and we are in the non-split
semisimple case. ∎

Now we are ready to give a formula for the genus $g$ of $X_{0}(a,b,c;p)$. For
$x\in{\mathbb{R}}$, we write $\lfloor x\rceil$ for the rounding down of $x$,
so $\lfloor 3/2\rceil=1$.

######  Theorem 4.4.

Let $(a,b,c)$ be a hyperbolic admissible triple and $\mathfrak{p}$ be a prime
of $E$ above a rational prime $p$. Then the genus of
$X_{0}(a,b,c;\mathfrak{p})$ is given by

(4.5)
$g(X_{0}(a,b,c;\mathfrak{p}))=-q+\frac{1}{2}\sum_{s\in\\{a,b,c\\}}\left\lfloor\frac{q}{s}\right\rceil(s-1)+\epsilon(a,b,c;\mathfrak{p})$

where $q\colonequals\operatorname{Nm}(\mathfrak{p})$ and
$\epsilon(a,b,c;\mathfrak{p})\in\\{0,1/2\\}$ is uniquely determined by
$g(X_{0}(a,b,c;\mathfrak{p}))\in{\mathbb{Z}}$. Moreover, we have
$\epsilon(a,b,c;\mathfrak{p})=0$ unless $a=2$ and $q$ is odd.

In the latter case ($a=2$ and $q$ odd), Lemma 4.3 implies that when
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$, we have
$\epsilon(a,b,c;\mathfrak{p})=0$ if and only if $q\equiv 1\pmod{4}$ (case
(i)).

###### Proof.

Consider elements
$\overline{\sigma}_{a},\,\overline{\sigma}_{b},\,\overline{\sigma}_{c}\in\operatorname{PXL}_{2}(\mathbb{F}_{q})$
of orders $a^{,}\,b,$ and $c$, respectively, such that
$\sigma_{a}\sigma_{b}\sigma_{c}=1$. We recall that the map
$X_{0}(a,b,c;\mathfrak{p})\to X(1)$ has degree $q+1$ since $[G:H_{0}]=q+1$.
The Riemann–Hurwitz formula implies

(4.6) $2g-2=-2(q+1)+\epsilon_{a}+\epsilon_{b}+\epsilon_{c},$

where $\epsilon_{s}$ is the ramification index at the points that ramify. We
can compute $\epsilon_{s}$ from Lemma 4.1 and Lemma 4.3, with
$\epsilon_{s}=k_{s}(s-1)$, where

(4.7) $k_{s}=\begin{cases}(q-1)/s,&\text{ if $s\mid(q-1)$};\\\ q/s,&\text{ if
$s\mid q$};\\\ (q+1)/s&\text{ if $s\mid(q+1)$};\end{cases}$

if $s\neq 2$ or ($s=a=2$ and $q$ is even); whereas if $s=a=2$ and $q$ is odd,
then either $k_{2}=(q+1)/2$ or $k_{2}=(q-1)/2$ is determined by the fact that
$g\in{\mathbb{Z}}$, since they differ by $1$. ∎

######  Remark 4.8.

Instead of using parity, in the $\operatorname{PGL}_{2}(\mathbb{F}_{q})$ and
$q$ odd case, we can always explicitly compute elements
$\overline{\sigma}_{2}$, $\overline{\sigma}_{b}$, $\overline{\sigma}_{c}\in
G$, of orders $2$, $b$, and $c$ respectively, such that
$\overline{\sigma}_{2}\overline{\sigma}_{b}\overline{\sigma}_{c}=1$. We can
then decide if $\overline{\sigma}_{2}$ is split or non-split and use Lemma 4.1
to compute the ramification.

### Algorithm

We present an implementation of Theorem 4.4.

######  Algorithm 4.9.

Let $(a,b,c)$ be a hyperbolic triple and let
$\mathfrak{p}\subseteq{\mathbb{Z}}_{E(a,b,c)}$ be a nonzero prime ideal. This
algorithm computes the genus of $X_{0}(a,b,c;\mathfrak{p})$ and the Galois
group $G_{\mathfrak{p}}$ of the cover
$X(a,b,c;\mathfrak{p})\to\mathbb{P}^{1}$.

  1. 1.

Compute the residue field of $\mathfrak{p}$ and set
$q\colonequals\\#\mathbb{F}_{\mathfrak{p}}$.

  2. 2.

Compute the residue field ${\mathbb{Z}}_{F}/\mathfrak{p}_{F}$, where
$\mathfrak{p}_{F}$ is a prime of $F(a,b,c)$ above $\mathfrak{p}$. If
$\mathbb{F}_{q}\simeq{\mathbb{Z}}_{F}/\mathfrak{p}_{F}$, then
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$. Otherwise set
$G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$.

  3. 3.

Compute $g$ using Theorem 4.4.

###### Proof of correctness.

Correctness follows from the formula in Theorem 4.4. Steps 1 and 2 can be
performed by constructing the algebraic number field; it can also be done
purely in terms of the prime number $p$ below $\mathfrak{p}$ as in Algorithm
5.1. ∎

### Bounding the genus

Our goal remains to show that, for fixed genus $g_{0}$, there are finitely
many admissible curves $X_{0}(a,b,c;\mathfrak{p})$ of genus $g\leq g_{0}$. We
first characterize the hyperbolic triples $(a,b,c)$ such that the curve
$X(a,b,c)$ has Galois group $\operatorname{PXL}_{2}(\mathbb{F}_{q})$, for a
given $q$.

In the prime case, the notion of admissible ideal can be turned around, as
follows.

######  Definition 4.10.

Let $q\colonequals p^{r}$ be a power of a prime number $p$. A hyperbolic
triple $(a,b,c)$ is $q$-admissible if $s$ divides at least one integer in the
set $\\{q-1,\,p,\,q+1\\}$ for all $s\in\\{a,b,c\\}$, not including $\infty$.

######  Lemma 4.11.

For any triangular modular curve $X_{0}(a,b,c;\mathfrak{p})$ with
$q\colonequals\operatorname{Nm}\mathfrak{p}$ and $\mathfrak{p}$ admissible for
$(a,b,c)$, the triple $(a,b,c)$ is $q$-admissible.

###### Proof.

As shown in the proof of Lemma 4.1, the order of every element in
$\operatorname{PXL}_{2}(\mathbb{F}_{q})$ needs to divide one of
$\\{q-1,p,q+1\\}$. ∎

######  Proposition 4.12.

Let $g$ be the genus of the triangular modular curve
$X_{0}(a,b,c;\mathfrak{p})$ and set
$q\colonequals{\mathbb{Z}}_{E}/\mathfrak{p}$. Then,

$q\leq 84(g+1)+1.$

###### Proof.

Let $s^{\sharp}$ be the order of $\pi_{\mathfrak{p}}(\delta_{s})$. The cases
where $(a^{\sharp},b^{\sharp},c^{\sharp})$ is not hyperbolic are handled in
Proposition 3.13: we get $g=0$, and the inequality holds. So we may suppose
without loss of generality that $s^{\sharp}=s$ for $s=\\{a,b,c\\}$, and still
that $(a,b,c)$ is hyperbolic.

We study the Belyi map $X_{0}(a,b,c;\mathfrak{p})\to\mathbb{P}^{1}$. Let
$\epsilon_{a},\epsilon_{b},\epsilon_{c}$ be the ramification degrees of this
map. Using Lemma 4.1, we have that for $s\in\\{a,b,c\\}$,

(4.13)
$(q-1)-\frac{q-1}{s}=\frac{(s-1)(q-1)}{s}\leq\epsilon_{s}\leq\frac{(s-1)(q+1)}{s}=(q+1)-\frac{q+1}{s}.$

Because of these bounds and (4.6),

(4.14) $\displaystyle g(X_{0}(a,b,c;\mathfrak{p}))$
$\displaystyle\geq-(q+1)+\frac{(a-1)(q-1)}{2a}+\frac{(b-1)(q-1)}{2b}+\frac{(c-1)(q-1)}{2c}+1$
$\displaystyle=(q-1)\left(-1+\frac{3}{2}-\frac{1}{2a}-\frac{1}{2b}-\frac{1}{2c}\right)-1$
$\displaystyle=\frac{q-1}{2}\left|\chi(a,b,c)\right|-1,$

where $\chi(a,b,c)$ is as in (2.1). The result then follows from the previous
inequality and (2.2). ∎

######  Corollary 4.15.

For a fixed genus $g_{0}\in{\mathbb{Z}}_{\geq 0}$, there are only finitely
many hyperbolic triples $(a,b,c)$ and admissible primes $\mathfrak{p}$ such
that the curves $X_{0}(a,b,c;\mathfrak{p})$ have genus $g\leq g_{0}$.

###### Proof.

By Proposition 4.12, we obtain an upper bound on the rational prime $p$ given
by $q\leq 84(g_{0}+1)+1$. Also, for $(a,b,c)$ to be $q$-admissible,
necessarily $s\leq q+1$ for all $s\in\\{a,b,c\\}$. This leaves only finitely
many possibilities. ∎

######  Remark 4.16.

To make computations more efficient, we can consider a bound on $q$ that
depends on $\chi(a,b,c)$. For the genus of $X_{0}(a,b,c;\mathfrak{p})$ to be
less than or equal to $g_{0}$, it is necessary that

(4.17) $q\leq\frac{2(g_{0}+1)}{\left|\chi(a,b,c)\right|}+1.$

This inequality also shows that

(4.18) $0<\left|\chi(a,b,c)\right|\leq\frac{2(g_{0}+1)}{q-1}.$

Therefore, we can bound $a$, $b$, and $c$ whenever $q$ is fixed.

##  5\. Enumerating curves of low genus

We present the main algorithms that use the theory developed in section 4. The
goal of this section is to effectively enumerate the curves
$X_{0}(a,b,c;\mathfrak{p})$ of bounded genus. The number of curves is finite
from Corollary 4.15. As explained in section 2, if $\mathfrak{p}$ is
admissible, then $G$ is given by $\operatorname{PXL}_{2}(\mathbb{F}_{q})$. The
first condition (coprimality) in admissibility can be expensive to check, so
we first check the easier necessary (but not sufficient) condition that
$\mathfrak{p}\nmid\beta(a,b,c)$.

######  Algorithm 5.1.

Let $(a,b,c)$ be a hyperbolic triple and $p$ be a prime number. This algorithm
returns `true` if there exists a prime
$\mathfrak{p}\subseteq{\mathbb{Z}}_{E(a,b,c)}$ above $p$ such that
$\mathfrak{p}\nmid\beta(a,b,c)$.

  1. 1.

If $p\nmid 2abc$, then return `true`.

  2. 2.

Find $\mathbb{F}_{\mathfrak{p}}=\mathbb{F}_{q}$, where $\mathfrak{p}$ is any
prime of $E$ above $p$.

  3. 3.

Set $m\colonequals\operatorname{lcm}(a,b,c)$. Construct
$\mathbb{F}_{q}(\zeta_{2m})$. Set $z\colonequals\zeta_{2m}$.

  4. 4.

For every $i\in({\mathbb{Z}}/2m{\mathbb{Z}})^{\times}$, and set
$l_{2s}\colonequals z^{im/s}+1/z^{im/s}$ for $s\in\\{a,b,c\\}$. Compute

$\beta_{i}\colonequals l_{2a}^{2}+l_{2b}^{2}+l_{2c}^{2}+l_{2a}l_{2b}l_{2c}-4.$

If $\beta_{i}\neq 0$ and whenever $p\mid s$ we have $s=p$, then return `true`.
Otherwise, return `false`.

###### Proof of correctness.

Let $\mathfrak{p}$ be a prime of ${\mathbb{Z}}_{E(a,b,c)}$ above $p$. If
$p\nmid 2abc$ then $\mathfrak{p}\nmid\beta(a,b,c)$ [3, Lemma 5.5]. When
$\mathfrak{p}\mid abc$, checking that $\mathfrak{p}$ does not divide
$\beta(a,b,c)$ is more involved. We do this in steps 2 to 4 by computing
$\beta$ in the residue field of $\mathfrak{p}$. This computation is
independent of the prime $\mathfrak{p}$ chosen above $p$ because $E$ is Galois
over ${\mathbb{Q}}$. ∎

Now we are ready to present the main algorithm that ties the results of
section 4 into an explicit enumeration.

######  Algorithm 5.2.

Returns a list `lowGenus` of all hyperbolic triples
$(a,b,c)\in{\mathbb{Z}}_{\geq 2}^{3}$ and norms of prime ideals $\mathfrak{p}$
of $E(a,b,c)$ that are admissible such that the genus of
$X_{0}(a,b,c;\mathfrak{p})$ is at most $g_{0}$.

  1. 1.

Loop over the list of possible powers $q=p^{r}$, where $p$ is a prime number
and $q\leq 84(g_{0}+1)+1$.

  2. 2.

For each $q$ from step 1, find all $q$-admissible hyperbolic triples $(a,b,c)$
(as in Definition 4.10).

  3. 3.

For each $q$-admissible triple $(a,b,c)$ from step 2, check if $\chi(a,b,c)$
satisfies (4.18) and if $\mathfrak{p}$ does not divide $\beta(a,b,c)$ using
Algorithm 5.1. If yes, compute the candidate genus $g$ of
$X_{0}(a,b,c;\mathfrak{p})$ using Algorithm 4.9.

  4. 4.

If $g\leq g_{0}$, check that
$\mathfrak{p}\nmid\operatorname{discrd}(\Lambda)\mathfrak{d}_{F|E}$. If yes,
add $(a,b,c;q)$ to the list `lowGenus`.

###### Proof of correctness.

For step 1, see Proposition 4.12. Every hyperbolic $q$-admissible triple gives
rise to one such curve. The correctness of the rest of the algorithm follows
from the work done in section 4. ∎

We list the CPU time (in seconds) for our implementation to compute the list
of curves $X_{0}(a,b,c;\mathfrak{p})$ of genus up to bounds 0, 1, and 2 on a
standard laptop:

${\begin{array}[]{c||c|c|c}\text{Genus bound}&0&1&2\\\
\hline\cr\hline\cr\text{Time (s)}&1.7&9.7&1110.3\end{array}}$

##  6\. Triangular modular curves $X_{1}(a,b,c;\mathfrak{p})$

In this section, we use section 4 to give analogous results for triangular
modular curves $X_{1}(a,b,c;\mathfrak{p})$, completing the proof of our main
result.

We recall that $X_{1}(a,b,c;\mathfrak{p})$ is defined in (3.18) as the
quotient of $\mathcal{H}$ by $\Gamma_{1}(a,b,c;\mathfrak{p})$.

######  Corollary 6.1.

For any integer $g_{0}\geq 0$, there are finitely many triangular modular
curves $X_{1}(a,b,c;\mathfrak{p})$ with $\mathfrak{p}$ admissible.

###### Proof.

For every triple $(a,b,c)\in({\mathbb{Z}}_{\geq 2}\cup\\{\infty\\})^{3}$ and
prime ideal $\mathfrak{p}$ of $E(a,b,c)$, there is a cover
$X_{1}(a,b,c;\mathfrak{p})\to X_{0}(a,b,c;\mathfrak{p})$. All curves
$X_{1}(a,b,c;\mathfrak{p})$ of genus bounded above by $g_{0}$ cover curves
$X_{0}(a,b,c;\mathfrak{p})$ of genus bounded above by $g_{0}$. Because of
Corollary 4.15, there are finitely many admissible triples $(a,b,c)$ and prime
ideals $\mathfrak{p}$ that give rise to curves $X_{0}(a,b,c;\mathfrak{p})$ of
genus bounded above by $g_{0}$. ∎

We now focus on explicitly enumerating all curves of bounded genus. The goal
first is to prove group-theoretic results that describe the degree and
ramification of the cover $X_{1}(\mathfrak{p})\to X(1)$. We describe the
structure of the quotient $\operatorname{PXL}_{2}(\mathbb{F}_{q})$ modulo
$H_{1,\mathfrak{p}}$ and then describe the action of
$\pi_{\mathfrak{p}}(\delta_{s})$ on this quotient. The main difference with
section 4 is that the quotient $G/H_{0,\mathfrak{p}}$ does not depend on $G$
being isomorphic to $\operatorname{PSL}_{2}(\mathbb{F}_{q})$ or
$\operatorname{PGL}_{2}(\mathbb{F}_{q})$, whereas the structure of
$G/H_{1,\mathfrak{p}}$ depends on the choice of $G$. Let $H_{1}\colonequals
H_{1,\mathfrak{p}}$.

######  Lemma 6.2.

Let $G=\operatorname{PXL}_{2}(\mathbb{F}_{q})$, where
$\mathbb{F}_{q}\colonequals{\mathbb{Z}}_{E}/\mathfrak{p}$. The quotient
$G/H_{1}$ can be described as follows.

  1. (i)

If $G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$, then
$G/H_{1}\simeq(\mathbb{F}_{q}\times\mathbb{F}_{q}\setminus\\{(0,0)\\})/\langle\pm
1\rangle$: explicitly, the class of
$(x,z)\in\mathbb{F}_{q}\times\mathbb{F}_{q}$ maps to the coset of
$\begin{pmatrix}x&y\\\ z&w\end{pmatrix}$, where $y,w\in\mathbb{F}_{q}$ satisfy
$xw-yz=1$.

  2. (ii)

If $G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$, then
$G/H_{1}\simeq(\mathbb{F}_{q}\times\mathbb{F}_{q}\setminus\\{(0,0)\\})/\langle\pm
1\rangle\times\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}$: explicitly,
for $\mu\in\mathbb{F}_{q}^{\times}\smallsetminus\mathbb{F}_{q}^{\times 2}$ the
class of
$((x,y),u)\in(\mathbb{F}_{q}\times\mathbb{F}_{q}\setminus\\{(0,0)\\})\times\mathbb{F}_{q}^{\times}$
maps to the coset of $\begin{pmatrix}x&y\\\ z&w\end{pmatrix}$, where
$z,w\in\mathbb{F}_{q}$ satisfy $xw-yz=1$ if $u$ is a square and $xw-yz=\mu$
otherwise.

###### Proof.

Let $G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ with $q$ odd. Because
$\\#H_{1}=\\#\mathbb{F}_{q}$, we have that $[G:H_{1}]=(q^{2}-1)/2.$ The coset
representatives of $G/H_{1}$ can be parameterized by
$(x,z)\in(\mathbb{F}_{q}\times\mathbb{F}_{q})/\langle\pm 1\rangle$. Indeed,
two elements in $\operatorname{PSL}_{2}(\mathbb{F}_{q})$ are in the same coset
of $G/H_{1}$ if and only if there is $\alpha\in\mathbb{F}_{q}$ such that

$\pm\begin{pmatrix}x&y\\\ z&w\end{pmatrix}\begin{pmatrix}1&\alpha\\\
0&1\end{pmatrix}=\pm\begin{pmatrix}x&x\alpha+y\\\
z&z\alpha+w\end{pmatrix}=\begin{pmatrix}x^{\prime}&y^{\prime}\\\
z^{\prime}&w^{\prime}\end{pmatrix},$

which is the case if and only if $(x,z)=\pm(x^{\prime},z^{\prime}).$ Thus, the
map $(\mathbb{F}_{q}\times\mathbb{F}_{q})/\langle\pm 1\rangle\to G/H_{1}$
defined by the parametrization is a well-defined, injective homomorphism. By a
cardinality comparison it follows that it is an isomorphism.

Now we let $G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$, so $[G:H_{1}]=q^{2}-1$.
We claim that the quotient $G/H_{1}$ is isomorphic to
$(\mathbb{F}_{q}\times\mathbb{F}_{q}\setminus\\{(0,0)\\})/\\{\pm
1\\}\times\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}$. To present this
isomorphism, we fix a non-square
$\mu\in\mathbb{F}_{q}^{\times}\smallsetminus\mathbb{F}_{q}^{\times 2}$. For
any
$\pm(x,z)\in(\mathbb{F}_{q}\times\mathbb{F}_{q}\setminus\\{(0,0)\\})/\langle\pm
1\rangle$, and any
$u\in\\{1,\mu\\}\simeq\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}$, we
choose values of $y,w\in\mathbb{F}_{q}$ such that $xw-yz=u$ and map $\pm(x,z)$
to the class of the matrix $\begin{pmatrix}x&y\\\ z&w\end{pmatrix}$ in
$\operatorname{PGL}_{2}(\mathbb{F}_{q})$. Given two different choices
$y,w\in\mathbb{F}_{q}$ and $y^{\prime},w^{\prime}\in\mathbb{F}_{q}$, if $x\neq
0$, then

$\pm\begin{pmatrix}x&y\\\ z&w\end{pmatrix}=\begin{pmatrix}x&y^{\prime}\\\
z&w^{\prime}\end{pmatrix}\begin{pmatrix}1&x^{-1}(y-y^{\prime})\\\
0&1\end{pmatrix}.$

If $x=0$, then $z\neq 0$ and $0\neq u=yz=y^{\prime}z$. Thus, $y=y^{\prime}$.
Also,

$\pm\begin{pmatrix}0&y\\\ z&w\end{pmatrix}=\begin{pmatrix}0&y\\\
z&w^{\prime}\end{pmatrix}\begin{pmatrix}1&z^{-1}(w-w^{\prime})\\\
0&1\end{pmatrix}.$

Thus, the map $(\mathbb{F}_{q}\times\mathbb{F}_{q}\setminus\\{(0,0)\\})/\\{\pm
1\\}\times\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}\to G/H_{1}$ is a
well defined homomorphism. In addition, multiplication by elements in $H_{1}$
does not change the square class of the determinant or the first column of the
matrix, so the homomorphism described above is injective. Since the
cardinalities of the domain and range are equal, we conclude that this is an
isomorphism. ∎

We proceed to describe the ramification of the cover
$X_{1}(a,b,c;\mathfrak{p})\to\mathbb{P}^{1}$. This lemma is similar to Lemma
4.1. The main difference is that in certain cases there are more fixed points
than strictly necessary.

######  Lemma 6.3.

Let $\overline{\sigma}_{s}\in G=\operatorname{PXL}_{2}(\mathbb{F}_{q})$ and
assume that the order of $\overline{\sigma}_{s}$ is $s$. The structure of the
action of $\overline{\sigma}_{s}$ on $G/H_{1}$ is as follows:

  1. (i)

if $\sigma_{s}$ is semisimple, then there are (no fixed points and)
$\frac{[G:H_{1}]}{s}$ orbits of length $s$,

  2. (ii)

if $\sigma_{s}$ is unipotent, then:

    1. (a)

if $G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ and $q$ is odd, there are
$(q-1)/2$ fixed points and $(q^{2}-q)/(2p)$ orbits of length $p$,

    2. (b)

otherwise, there are $q-1$ fixed points and $(q^{2}-q)/p$ orbits of length
$p$.

###### Proof.

We use the description of the quotient $G/H_{1}$ given in Lemma 6.2. Let
$\sigma_{s}$ be any element of $\operatorname{GL}_{2}(\mathbb{F}_{q})$ that
maps to $\overline{\sigma}_{s}$ in the quotient to $G$.

If $\sigma_{s}$ is split semisimple, then it is conjugate over
$\mathbb{F}_{q}$ to a diagonal matrix with entries $u,v$. Because the order of
$\overline{\sigma}_{s}$ is $s$, then $s$ is the order of $uv^{-1}$. We pick a
class in the quotient $G/H_{1}$ represented by a matrix $M$. If the class of
$M$ is fixed by the action of $\overline{\sigma}_{s}$, then the first column
of $M$ is, up to multiplication by $\pm 1$, fixed by multiplication by the
diagonal matrix. This implies that $(u,v)=\pm(1,1)$, contradicting that $s\geq
2$. Thus, there are no fixed points of the action of $\overline{\sigma}_{s}$
on $G/H_{1}$. A similar argument shows that orbits of elements that are not
fixed cannot have length less than $s$. Thus, every element belongs to an
orbit of length $s$.

If $\sigma_{s}$ is non-split semisimple, then $\sigma_{s}$ is split in a
quadratic extension of $\mathbb{F}_{q}$. We assume that
$\sigma_{s}=T^{-1}[\alpha_{1},\alpha_{2}]T$ in this extension. If
$\sigma_{s}^{r}$ fixes an element for $r\geq 1$, then we have

$\pm\begin{pmatrix}\alpha_{1}^{r}&0\\\
0&\alpha_{2}^{r}\end{pmatrix}T\begin{pmatrix}x&y\\\
z&w\end{pmatrix}=T\begin{pmatrix}x&y^{\prime}\\\ z&w^{\prime}\end{pmatrix}.$

Multiplication by $T$ does not change the equality in $G/H_{1}$. Thus, we are
back to the split semisimple case and the orbits of the action of
$\overline{\sigma}_{s}$ all have size $s$.

If $\sigma_{s}$ is unipotent, then $\sigma_{s}$ can be chosen (by multiplying
by scalar matrices) to be conjugate to an upper diagonal matrix with ones in
the diagonal. Then,

$\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}\begin{pmatrix}x&y\\\
z&w\end{pmatrix}=\begin{pmatrix}x+uz&y+uw\\\ z&w\end{pmatrix},$

so the class of this matrix in $G/H_{1}$ is fixed by multiplication by
$\sigma_{s}$ if and only if $uz=0$. Since $s\geq 2$, then $z$ must be $0$. We
note that if $z\neq 0$, then the orbit of the element has length $p$. In
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ there are $(q-1)/2$ representatives
for which $z=0$, i.e. fixed points. Similarly, if
$G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$, then there are $q-1$ fixed points.
∎

######  Corollary 6.4.

Let $(a,b,c)\in{\mathbb{Z}}_{\geq 2}^{3}$ be a $q$-admissible hyperbolic
triple. Let $\mathfrak{p}$ be a prime ideal of $E(a,b,c)$ above a rational
prime $p$. Then the genus of $X_{1}(a,b,c;\mathfrak{p})$ is given by

$g(X_{1}(a,b,c;\mathfrak{p}))=-[G:H_{1}]+1+\frac{1}{2}\sum_{s\in\\{a,b,c\\}}k_{s}(s-1),$

where

$k_{s}=\begin{cases}(q^{2}-q)/(2p),&\text{ if $s=p$ and
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$};\\\ (q^{2}-q)/p,&\text{ if $s=p$
and $G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$};\\\ (q^{2}-1)/s,&\text{ if
$s\neq p$ and $G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$};and\\\
(q^{2}-1)/(2s),&\text{ if $s\neq p$ and $q$ is odd and
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$}.\end{cases}$

###### Proof.

This formula is given by using the Riemann-Hurwitz formula on
$X_{1}(\mathfrak{p})\to\mathbb{P}^{1}$ and Lemma 6.3. ∎

Now we are ready to present an algorithm that enumerates all curves
$X_{1}(a,b,c;\mathfrak{p})$.

######  Algorithm 6.5.

Returns a list `lowGenusX1` of all hyperbolic triples $(a,b,c)$ and admissible
ideals $\mathfrak{p}$ such that the genus of $X_{1}(a,b,c;\mathfrak{p})$ is
$g\leq g_{0}$.

  1. 1.

Loop over all hyperbolic triples $(a,b,c)$ and prime ideals $\mathfrak{p}$
such that $X_{0}(a,b,c;\mathfrak{p})$ has genus bounded above by $g_{0}$. This
list can be obtained from Algorithm 5.2.

  2. 2.

For each triple $(a,b,c)$ and ideal $\mathfrak{p}$ of the previous step,
compute the genus $g$ of $X_{1}(a,b,c;\mathfrak{p})$ with Corollary 6.4. If
$g\leq g_{0}$, then add $(a,b,c;\mathfrak{p})$ to the list `lowGenusX1`.

###### Proof of correctness.

For all triples $(a,b,c)$ and prime ideals $\mathfrak{p}$ there are maps
$X_{1}(a,b,c;\mathfrak{p})\to X_{0}(a,b,c;\mathfrak{p})$. Thus, the only
curves $X_{1}(a,b,c;\mathfrak{p})$ that can have genus $g\leq g_{0}$ must be
covering curves $X_{0}(a,b,c;\mathfrak{p})$ of genus bounded above by $g_{0}$.
∎

### Proof of theorem

We conclude the paper by proving our main result.

###### Proof of Theorem 1.9.

By Corollary 4.15, there are only finitely many curves
$X_{0}(a,b,c;\mathfrak{p})$ with nontrivial admissible prime level
$\mathfrak{p}$ and genus $g\leq g_{0}$. Since every curve
$X_{1}(a,b,c;\mathfrak{p})$ covers $X_{0}(a,b,c;\mathfrak{p})$, the same is
true for $X_{1}(a,b,c;\mathfrak{p})$ (see Corollary 6.1).

For the computation, we run Algorithm 5.2 with $g_{0}=2$, adding extra cases
according to Proposition 3.13. To finish, we run Algorithm 6.5. The
implementation of this computation can be found in our Magma code [26]. ∎

##  Appendix A Tables

We present tables of all hyperbolic triples $(a,b,c)$ and admissible primes
$\mathfrak{p}$ such that the curve $X_{0}(a,b,c;\mathfrak{p})$ has genus 0 or
1. The list with additional data is available online [26].

To record $\mathfrak{p}$, we list the prime number $p$ below $\mathfrak{p}$.
We describe the group $G=\operatorname{PXL}_{2}(\mathbb{F}_{q})$ by presenting
$q$ and writing $1$ in the $\operatorname{PXL}$ field if
$G=\operatorname{PSL}_{2}(\mathbb{F}_{q})$ and $-1$ if
$G=\operatorname{PGL}_{2}(\mathbb{F}_{q})$. We also record the information
about the field $E(a,b,c)$ and the number of different prime ideals of $E$
above $p$.

Nugent–Voight [16] define an invariant, the arithmetic dimension
$\operatorname{adim}(a,b,c)$, to be the dimension of a quaternionic Shimura
variety attached to $\Delta(a,b,c)$ given by the number of split real places
of $E(a,b,c)$ of the quaternion algebra $A=E\langle\Delta^{(2)}\rangle$. In
particular, the triangle group $\Delta(a,b,c)$ is arithmetic if and only if
$\operatorname{adim}(a,b,c)=1$.

One subtlety is that there can be an isomorphism between the cover coming from
a nonarithmetic group and the cover coming from an arithmetic group. This can
only happen when the arithmetic group is of noncompact type, with

$\displaystyle(a,b,c)$
$\displaystyle=(2,3,\infty),(2,4,\infty),(2,6,\infty),(2,\infty,\infty),(3,3,\infty),(3,\infty,\infty),$
$\displaystyle\qquad(4,4,\infty),(6,6,\infty),(\infty,\infty,\infty)$

by Takeuchi [23]. All of these arise from finite-index subgroups of
$\operatorname{PSL}_{2}({\mathbb{Z}})$, so they are related to classical
modular curves, and are defined over ${\mathbb{Q}}$. The ramification of the
curve $X_{0}(a,b,c;\mathfrak{p})$ for $a,b,c\in{\mathbb{Z}}_{\geq
2}\cup\\{\infty\\}$ replaces any occurrence of $\infty$ by $p$; this allows
one to readily identify when this extra isomorphism applies. We record this by
adding $(1)$ to the arithmetic dimension entry on the table.

For the arithmetic triangle groups $\Delta(a,b,c)$ such that
$\Delta\simeq\Lambda^{1}$, the corresponding list of curves is contained in
[25, Tables 4.1–4.7]. We confirmed that the intersection is in agreement.

Finally, for noncocompact triples see Proposition 3.13.

### Genus 0, $X_{0}(a,b,c;\mathfrak{p})$

${\begin{array}[]{c|c||c|c||c|c|c}\bm{(a,b,c)}&\bm{p}&\bm{q}&\bm{{\rm
PXL}}&\bm{{\rm adim}}&\bm{E(a,b,c)}&\\#\text{ of }\mathfrak{p}\\\
\hline\cr\hline\cr(2,3,7)&7&7&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&1\\\ \hline\cr(2,3,7)&2&8&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&1\\\
\hline\cr(2,3,7)&13&13&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,7)&29&29&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,7)&43&43&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,8)&7&7&-1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,3,8)&3&9&-1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(2,3,8)&17&17&1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,3,8)&5&25&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(2,3,9)&19&19&1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,3,9)&37&37&1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,3,10)&11&11&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,3,10)&31&31&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,3,12)&13&13&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&2\\\
\hline\cr(2,3,12)&5&25&1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&1\\\
\hline\cr(2,3,13)&13&13&1&2\,(1)&\href
https://www.lmfdb.org/NumberField/6.6.371293.1&1\\\
\hline\cr(2,3,15)&2&16&1&2\,(1)&\href
https://www.lmfdb.org/NumberField/4.4.1125.1&1\\\
\hline\cr(2,3,18)&19&19&-1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,4,5)&5&5&-1&1&\href https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,4,5)&3&9&1&1&\href https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
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\hline\cr(2,4,5)&41&41&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\ \hline\cr(2,4,6)&5&5&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\ \hline\cr(2,4,6)&7&7&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(2,4,6)&13&13&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\ \hline\cr(2,4,8)&3&9&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(2,4,8)&17&17&1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,4,12)&13&13&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&2\\\ \hline\cr(2,5,5)&5&5&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,5,5)&11&11&1&1&\href
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\hline\cr(2,5,10)&11&11&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\ \hline\cr(2,6,6)&7&7&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(2,6,6)&13&13&1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(2,6,7)&7&7&-1&2\,(1)&\href
https://www.lmfdb.org/NumberField/3.3.49.1&1\\\
\hline\cr(2,8,8)&3&9&-1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(3,3,4)&7&7&1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(3,3,4)&3&9&1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(3,3,4)&5&25&1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(3,3,5)&2&4&1&1&\href https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(3,3,6)&13&13&1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&2\\\
\hline\cr(3,4,4)&5&5&-1&1&\href https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(3,4,4)&13&13&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&1\\\ \hline\cr(3,6,6)&7&7&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\ \hline\cr(4,4,4)&3&9&1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&1\end{array}}$

### Genus 1, $X_{0}(a,b,c;\mathfrak{p})$

This long table is split into three tables (over the next three pages).

${\begin{array}[]{c|c||c|c||c|c|c}\bm{(a,b,c)}&\bm{p}&\bm{q}&\bm{{\rm
PXL}}&\bm{{\rm adim}}&\bm{E}&\\#\text{ of }\mathfrak{p}\\\
\hline\cr\hline\cr(2,3,7)&3&27&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&1\\\
\hline\cr(2,3,7)&41&41&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,7)&71&71&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,7)&97&97&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,7)&113&113&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,7)&127&127&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,8)&23&23&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,3,8)&31&31&1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,3,8)&41&41&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,3,8)&73&73&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,3,8)&97&97&1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&2\\\ \hline\cr(2,3,9)&2&8&1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&1\\\
\hline\cr(2,3,9)&17&17&1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,3,9)&73&73&1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,3,10)&3&9&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,3,10)&19&19&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,3,10)&41&41&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,3,10)&61&61&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,3,11)&11&11&1&1&\href
https://www.lmfdb.org/NumberField/5.5.14641.1&1\\\
\hline\cr(2,3,11)&23&23&1&1&\href
https://www.lmfdb.org/NumberField/5.5.14641.1&5\\\
\hline\cr(2,3,12)&11&11&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&2\\\
\hline\cr(2,3,12)&37&37&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&2\\\
\hline\cr(2,3,12)&7&49&1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&1\\\
\hline\cr(2,3,13)&5&25&1&2&\href
https://www.lmfdb.org/NumberField/6.6.371293.1&3\\\
\hline\cr(2,3,13)&3&27&1&2&\href
https://www.lmfdb.org/NumberField/6.6.371293.1&2\\\
\hline\cr(2,3,14)&13&13&-1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,14)&29&29&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,14)&43&43&-1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,3,15)&31&31&1&2&\href
https://www.lmfdb.org/NumberField/4.4.1125.1&4\\\
\hline\cr(2,3,16)&17&17&-1&1&\href
https://www.lmfdb.org/NumberField/4.4.2048.1&4\\\
\hline\cr(2,3,17)&2&16&1&2&\href
https://www.lmfdb.org/NumberField/8.8.410338673.1&2\\\
\hline\cr(2,3,17)&17&17&1&2&\href
https://www.lmfdb.org/NumberField/8.8.410338673.1&1\\\
\hline\cr(2,3,18)&37&37&1&1&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,3,19)&19&19&1&3\,(1)&\href
https://www.lmfdb.org/NumberField/9.9.16983563041.1&1\\\
\hline\cr(2,3,20)&19&19&-1&2&\href
https://www.lmfdb.org/NumberField/4.4.2000.1&4\\\
\hline\cr(2,3,22)&23&23&-1&2&\href
https://www.lmfdb.org/NumberField/5.5.14641.1&5\\\
\hline\cr(2,3,24)&5&25&-1&1&\href
https://www.lmfdb.org/NumberField/4.4.2304.1&2\\\
\hline\cr(2,3,26)&3&27&-1&2&\href
https://www.lmfdb.org/NumberField/6.6.371293.1&2\\\
\hline\cr(2,3,30)&31&31&-1&1&\href
https://www.lmfdb.org/NumberField/4.4.1125.1&4\\\
\lx@intercol\hfil\vdots\hfil\lx@intercol\end{array}}\\\ $

${\begin{array}[]{c|c||c|c||c|c|c}\bm{(a,b,c)}&\bm{p}&\bm{q}&\bm{{\rm
PXL}}&\bm{{\rm adim}}&\bm{E}&\bm{\\#\text{ of }\mathfrak{p}}\\\
\hline\cr\hline\cr(2,4,5)&19&19&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,4,5)&29&29&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,4,5)&31&31&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\ \hline\cr(2,4,5)&7&49&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,4,5)&61&61&-1&1&\href
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\hline\cr(2,4,6)&11&11&-1&1&\href
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\hline\cr(2,4,6)&17&17&-1&1&\href
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\hline\cr(2,4,6)&19&19&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(2,4,6)&29&29&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(2,4,6)&31&31&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\
\hline\cr(2,4,6)&37&37&-1&1&\href
https://www.lmfdb.org/NumberField/1.1.1.1&1\\\ \hline\cr(2,4,7)&7&7&1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&1\\\
\hline\cr(2,4,7)&13&13&-1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,4,7)&29&29&-1&1&\href
https://www.lmfdb.org/NumberField/3.3.49.1&3\\\
\hline\cr(2,4,8)&7&7&-1&1&\href https://www.lmfdb.org/NumberField/2.2.8.1&2\\\
\hline\cr(2,4,8)&5&25&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.8.1&1\\\
\hline\cr(2,4,9)&17&17&1&2&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,4,9)&19&19&-1&2&\href
https://www.lmfdb.org/NumberField/3.3.81.1&3\\\
\hline\cr(2,4,10)&3&9&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,4,10)&11&11&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,4,11)&11&11&-1&2\,(1)&\href
https://www.lmfdb.org/NumberField/5.5.14641.1&1\\\
\hline\cr(2,4,12)&5&25&1&1&\href
https://www.lmfdb.org/NumberField/2.2.12.1&1\\\
\hline\cr(2,4,13)&13&13&-1&3\,(1)&\href
https://www.lmfdb.org/NumberField/6.6.371293.1&1\\\
\hline\cr(2,4,14)&13&13&-1&2&\href
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\hline\cr(2,4,16)&17&17&-1&2&\href
https://www.lmfdb.org/NumberField/4.4.2048.1&4\\\
\hline\cr(2,4,17)&17&17&1&4\,(1)&\href
https://www.lmfdb.org/NumberField/8.8.410338673.1&1\\\
\hline\cr(2,5,5)&3&9&1&1&\href https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,5,5)&31&31&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
\hline\cr(2,5,5)&41&41&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\ \hline\cr(2,5,6)&5&5&-1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&1\\\
\hline\cr(2,5,6)&11&11&1&1&\href
https://www.lmfdb.org/NumberField/2.2.5.1&2\\\
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