# On smooth plane models for modular curves of Shimura type

Samuele Anni  Samuele Anni, Aix-Marseille Université, CNRS, Centrale
Marseille, Institut de Mathématiques de Marseille case 907, 163 avenue de
Luminy, F13288 Marseille cedex 9, France<EMAIL_ADDRESS>,  Eran
Assaf  Eran Assaf, Department of Mathematics, Dartmouth College, 6188 Kemeny
Hall, Hanover, NH 03755, USA<EMAIL_ADDRESS>and  Elisa Lorenzo
García  Elisa Lorenzo García, Université de Neuchâtel, Institut de
Mathématiques, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland. – Univ
Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper we prove that there are finitely many modular curves that admit
a smooth plane model. Moreover, if the degree of the model is greater than or
equal to $19$, no such curve exists. For modular curves of Shimura type we
show that none can admit a smooth plane model of degree $5,6$ or $7$. Further,
if a modular curve of Shimura type admits a smooth plane model of degree $8$
we show that it must be a twist of one of four curves.

###### Key words and phrases:

Modular curves, congruence subgroups, cusp forms, canonical model, smooth
plane model, gonality.

###### 1991 Mathematics Subject Classification:

Primary 11G18, 14G35; Secondary 11F11, 14H45.

##  1\. Introduction

The compactification, by normalisation, of the quotient space of the complex
upper half plane by the action of a subgroup $\Gamma$ of
$\operatorname{SL}_{2}(\mathbb{Z})=\Gamma(1)$, a modular group, is called a
modular curve, $X_{\Gamma}$ and it admits the structure of a compact Riemann
surface. Serre’s GAGA theorem tells us that $X_{\Gamma}(\mathbb{C})$ is a
projective complex algebraic curve. Furthermore, Shimura, in
[SHI71]*Proposition 6.9, proved that modular curves admit the structure of
projective algebraic curves, see also [DISH]*§7.7. The problem of computing
equations for such curves and their projective embeddings has been a central
topic in numerous papers, motivated by a plethora of applications. We will
describe some of those while making a summary of the state of the art.

Modular curves are moduli spaces for elliptic curves with a given level
structure. Given a positive integer $N$ and a subgroup
$G\subseteq\mathrm{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$, the modular curve $X_{G}$
parametrises pairs $(E,\phi)$ up to isomorphism, where $E$ is an elliptic
curve and $\phi$ is a $G$-level structure on the $N$-torsion of $E$.
Therefore, the explicit knowledge of models of modular curves becomes key for
understanding properties of elliptic curves. This aspect leads to several
different applications, for example towards coding theory or for solving
Diophantine applications, starting with the proof of Fermat’s last theorem
where they appear in several steps. In this article we study whether it is
possible to have “nice” models, that is smooth plane models, for modular
curves defined over the rationals.

In general, finding equations for modular curves of large level is
computationally difficult, as it involves computing group actions on large
spaces and linear algebra over large cyclotomic fields. However, for some
groups it is easier to compute such curves, e.g. in [G1996] Galbraith computes
modular curves for the groups $\Gamma_{0}(N)$. One can look for a slightly
larger class of groups: given a positive integer $N$, a _group of Shimura
type_ of level $N$, as introduced originally by Shimura in [SHI71], is a
subgroup $\Gamma(H,t)\subseteq\operatorname{PSL}_{2}(\mathbb{Z})$ projection
of a subgroup $G(H,t)\subseteq\operatorname{SL}_{2}(\mathbb{Z})$ of the form

$G(H,t)=\left\\{\left(\begin{array}[]{cc}a&b\\\ 0&d\end{array}\right)\in
GL_{2}(\mathbb{Z}/N\mathbb{Z}):a\in H,t\mid b\right\\},$

where $H\subseteq(\mathbb{Z}/N\mathbb{Z})^{\times}$ is a subgroup and $t\mid
N$. We will call a modular curve of Shimura type any modular curve
corresponding to the choice of a group of Shimura type.

The main result of this article is the following:

######  Theorem 1.1.

There are finitely many modular curves which admit a smooth plane model over
the rationals. There is no modular curve which admits a smooth plane model of
degree greater or equal to $19$. Moreover, there is no modular curve of
Shimura type which admits a smooth plane model of degree $5,6$ or $7$. A
modular curve of Shimura type which admits a smooth plane model of degree $8$
must be a twist of one of four curves.

The four curves mentioned in Theorem 1.1 are listed in §5, 5.6.

Surprisingly, during the computations we found the example of a Galois
trigonal, i.e. superelliptic of degree $3$, model of a modular curve of genus
$6$.

###  1.1. State of the art

Galbraith in [G1996] presented several techniques to obtain explicit models of
modular curves by computing projective embeddings, relying on the computation
of spaces of modular forms. Let us recall Galbraith’s approach for the modular
curves $X_{0}(N)$ for a positive integer $N$. There is a well-known canonical
affine equation for $X_{0}(N)$ using $N$-modular polynomials that are
symmetric polynomials $\phi(x,y)\in\mathbb{Z}[x,y]$, of degree $N+1$ in each
variable, such that $\phi(j(\tau),j(N\tau))=0$, where $j(\tau)$ be the
classical modular $j$-function. These equations have very large degree, the
model is highly singular, and the coefficients involved are enormous.
Galbraith’s approach consists in obtaining equations via the canonical
embedding, which is suitable for practical computation since the differentials
on the curve correspond to the weight $2$ cusp forms for $\Gamma_{0}(N)$.
Chosen a basis $\\{f_{1},\dots,f_{g}\\}$ for the weight $2$ cusp forms, the
canonical map is translated into $\tau\mapsto[f_{1}(\tau):\dots:f_{g}(\tau)]$
and gives a map from $X_{0}(N)$ to $\mathbb{P}^{g-1}(\mathbb{C})$ from the
modularity of the forms $f_{i}(\tau)$. Galbraith’s strategy is a key element
in our approach towards the main theorem of this article.

Kohel in [K1997] presented a different method which involves quaternions and a
different approach towards the computation of the differentials. These
approaches have been used, together with others, to collect the database of
small modular curve models available in `Magma` [Magma].

Despite the lack of a general algorithm, models for several modular curves
have been found in the literature with a wide range of applications in mind.
We mention some of these, as well as their relevance towards various research
directions.

Baran found models for the isomorphic curves $X_{\mathrm{ns}^{+}}(13)$ and
$X_{s^{+}}(13)$ in [B2010]. The study of integral points on these curves
relates to the Serre’s uniformity question over $\mathbb{Q}$, as in [S1972].
More recently, Dose, Mercuri and Stirpe [DMS2017] proposed a new approach for
computing (singular) models in order to study Serre’s question.

Derickx, Najman and Siksek [DNS2020] proved that elliptic curves over totally
real cubic fields are “modular" meaning that their $L$-functions match the
$L$-function of the associated Hilbert modular forms. A key step to obtain
this result is the study of points on a plane (singular) model of
$X(\mathrm{b}5,\mathrm{ns}7)$.

Banwait and Cremona [BC2014] examined the failure of the local-to-global
principle for the existence of $\ell$-isogenies between elliptic curves over
number fields by, among others elements, determining a model for the
exceptional modular curve $X_{S_{4}}(13)$. Zywina, in [Z2020], generalised the
work of Banwait and Cremona, by relying on numerical approximation of
pseudo–eigenvalues of Atkin–Lehner operators. Through his approach it is
possible to determine $q$-expansions and models for modular curves.

Box in [B2021] described an algorithm [B2021]*Algorithm 4.13, that has been
implemented by the second named author, see [ASSAFgit], for computing the
canonical model for $X_{G}/\mathbb{Q}$ in the case where $G$ has surjective
determinant, $-I\in G$ and $G$ is normalised by
$J:=\left(\begin{smallmatrix}-1&0\\\ 0&1\end{smallmatrix}\right)$. In this
algorithm, one first determines the $q$-expansions of a basis for the
corresponding space of cusp forms and then a model, using the techniques
developed by Galbraith [G1996] when the genus is at least $2$. Box’s algorithm
presents the advantage that, for a finite groups $\mathcal{A}$ of the
automorphism group of $X_{G}$, it is possible to determine a model for the
quotient curve $X_{G}/\mathcal{A}$ directly, without computing $X_{G}$ first.
Box’s algorithm is another key ingredient to reach the conclusion of the main
result of this article.

Notice that for degree larger than $3$ all smooth plane curves are non-
hyperelliptic, see for example [Hart]*Ex. IV.5.1. In [BKX2013] the authors
prove that for $N\geq 8$ all geometrically connected curve modular curves
$X(N)$ defined over $\mathbb{Q}$ are neither hyperelliptic nor bielliptic.

Enge and Schertz [ES2005] presented (singular) models for the modular curves
$X_{0}(N)$ for $N$ the product of two arbitrary primes using Dedekind’s $\eta$
functions. Kodrnja in [K2018], relying on the embeddings in projective space
through modular forms and modular functions presented by Muić in [M2014] for
computing models of modular curves, was able to find an explicit recipe to
obtain plane (singular) models for all modular curves $X_{0}(N)$ for $N\geq
2$. The equation of the model is the minimal polynomial of the modular
function $\Delta(Nz)/\Delta$ over $\mathbb{C}(j)$, where $\Delta$ is the
Ramanujan $\Delta$ function and $j$ is the modular $j$ function. Some plane
(singular) models for modular curves $X_{0}(N)$ were already found by Hasegawa
and Shimura in [HS1999] using different ideas, in particular studying the
gonality of modular curves.

Borisov, Gunnells and Popescu [BGP2001] showed that it is possible to
determine explicitly an embedding of the modular curve $X_{1}(p)$ into
$\mathbb{P}^{\frac{(p-3)}{2}}$, where $p\geq 5$ is a prime, using weight one
Eisenstein series. The equation obtained is a (singular) quadratic equation.
More recently, Baziz [BA2010] proposed different (singular) models for
$X_{1}(N)$ using $N$-division polynomials, and so with the advantage of
keeping track explicitly of the corresponding pairs $(E,P)$ parametrised by
the curve.

In this article we are interested only in modular curves as classically
presented: projective complex algebraic curves corresponding to the
compactification of the quotient space of the complex upper half plane by the
action of a modular subgroup. Nevertheless, it is possible to define curves
that are modular: a curve $C$ over $\mathbb{Q}$ is modular if it is dominated
by $X_{1}(N)$ for some $N$. Moreover, if in addition the image of the jacobian
of the curve in $J_{1}(N)$ is contained in the new subvariety of $J_{1}(N)$,
then $C$ is new-modular. Under this definition, the modular curves associated
to the classical modular groups $\Gamma_{0}(N)$ and $\Gamma_{1}(N)$, for some
positive integer $N$, are curves that are modular. In particular there are
infinitely many curves over $\mathbb{Q}$ that are modular and of genus $1$:
elliptic curves over $\mathbb{Q}$ are modular. Baker, González-Jiménez,
González and Poonen in [BGGP2005] showed that for each genus $g\geq 2$, the
set of curves over $\mathbb{Q}$ of genus $g$ that are new-modular curves is
finite and computable. In particular, by analysing the automorphism group of
the curve and the dominant map, they describe explicitly all curves that are
new-modular of genus $2$, and construct a list of new-modular hyperelliptic
curves of all genera (this list might be complete, but there are pathologies
presented in the last sections of the aforementioned paper). In [GO2010]
González-Jiménez and Oyono gave an algorithm to compute explicit equations for
non-hyperelliptic curves that are modular of genus $3$ over $\mathbb{Q}$.
Moreover they conjectured that the list of non-hyperelliptic curves that are
new-modular and of genus $3$ consists of $44$ curves, and provided equations
for all of them. The issue, as in [BGGP2005], is giving a bound for the
coefficients of the modular forms involved.

###  1.2. Structure of the paper

In §2 we prove that there are a finite number of modular curves admitting a
smooth plane model. To achieve this result, we bound the genus of such curves
and notice that there is a finite number of congruence subgroups of any given
genus. Moreover, we explicitly bound the level and the index of such groups.
These results give us a finite list of groups corresponding to modular curves
that may admit a smooth plane model. In §3 we discuss how to perform the
computation of the canonical model of the relevant modular curves. In
particular, we present the analysis regarding the runtime of the algorithm for
computing $q$-expansions, with the precision required to prove the correctness
of the resulting equations. Later, in §4 we present an algorithm that, given a
canonical model of a non-hyperelliptic curve, checks whether the curve admits
a smooth plane model and, if it is the case, computes it. Finally, in §5 we
present our computations regarding Shimura type modular forms and modular
curves.

###  1.3. Acknowledgements

We thank the organisers of the conference "Arithmetic Aspects of Explicit
Moduli Problems" held in Banff in June 2017 where this project was first
mentioned. We thank Peter Bruin, Bas Edixhoven and Noam Elkies for insightful
discussions during the aforementioned conference. We also thank John Voight
for putting us in contact with each other. Last but not least, we thank Wouter
Castryck and all the referees that read our paper and helped us to improve its
results and exposition; in particular, to the one detecting a crucial error in
the first version of Section 4.

The research of the first and third author is partially funded by the Melodia
ANR-20-CE40-0013 project. The second author was supported by a Simons
Collaboration grant (550029, to John Voight).

##  2\. A bound for the genus

In this section we prove the first two parts of the main theorem, Theorem 1.1.

######  Theorem 2.1.

There are a finite number of modular curves admitting a smooth plane model.
Moreover, the degree of such model is less or equal to $18$.

###### Proof.

The genus–degree formula tells us that a smooth plane curve of degree $d$ has
genus $g=\frac{(d-1)(d-2)}{2}$. The gonality (over the algebraic closure) of a
smooth plane curve of degree $d$ is $d-1$, see [COKA90]*Theorem A. The
gonality of a modular curve of genus $g$ is greater or equal to
$\frac{1}{2}\cdot 975(g-1)/4096$, see [POONEN07]*Remark 1.2 and
[BGGP2005]*Remark 4.5. Therefore, for a modular curve admitting a smooth plane
model we have that $975d^{2}-19309d+16384\leq 0$ and so

$1\leq d\leq 18\quad\mbox{ and }\quad
g\in\\{0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136\\}.$

There are a finite number of modular curves of a given genus, see [COPA84], so
there are a finite number of modular curves admitting a smooth plane model. ∎

For degree $1$ and $2$, i.e. genus $0$, the list of levels is given in
[COPA84]*Table 4.24.

For degree $3$, i.e. genus $1$, the complete list of the relevant congruence
subgroups is given in [CUPA03].

For degree $4$ we need to consider curves of genus $3$. The non-hyperelliptic
ones are given by smooth plane quartics. Indeed, we find modular curves of
genus 3 admitting a smooth plane model of degree $4$, see Table 6 for the
complete list we have computed.

Nevertheless, the following question arises naturally:

######  Question 2.2.

Is there any modular curve of genus greater than $3$ admitting a smooth plane
model?

For degrees $5$ and $6$ we did not find any example of a modular curve
admitting a smooth plane model, restricting to Shimura type modular curves,
see §5.

For each genus up to $24$ the complete explicit list of congruence subgroups
of $\operatorname{PSL}_{2}(\mathbb{Z})$ is known: see [CUPA03]*Theorem 2.8 and
the associated website111https://mathstats.uncg.edu/sites/pauli/congruence/.

One way to count how many modular curves may admit a smooth plane model is to
count congruence subgroups of $\operatorname{PSL}_{2}(\mathbb{Z})$ whose index
is bounded in terms of the degree of the model, as follows.

######  Proposition 2.3.

The index $\iota$ of a congruence subgroup in
$\operatorname{PSL}_{2}(\mathbb{Z})$ whose associated modular curve admits a
smooth plane model of degree $d\geq 3$ satisfies

(2.1) $6(d-1)(d-2)-12\leq\iota\leq 101(d-1)$

###### Proof.

On the one hand, combining [Z84]*Theorem 3 (see also [HS1999]*Theorem 4.3) and
an improvement presented in [CUPA03] due to Kim and Sarnak [Kim]*Appendix 2,
the index of a congruence subgroup in $\operatorname{PSL}_{2}(\mathbb{Z})$ is
bounded by $101$ times the gonality of the corresponding modular curve. By
assumption the modular curves admits a smooth plane model, so its gonality is
$d-1$. The index is therefore bounded by $101(d-1)$.

On the other hand, the genus $g$ of a modular curve admitting a smooth plane
model of degree $d\geq 3$ satisfies $g=\frac{(d-1)(d-2)}{2}>0$ and $g\leq
1+\frac{\iota}{12}$, where $\iota$ is the index of the corresponding
congruence subgroup, see [DISH]*Theorem 3.1.1.

Therefore the index $\iota$ is bounded above and below as in Equation 2.1. ∎

######  Remark 2.4.

The coefficient $101$ used in Proposition 2.3 is obtained by taking the floor
of a rational number $\alpha=2^{15}/325$. A sharper upper bound can be
obtained by rounding only after multiplication.

######  Remark 2.5.

The result of Proposition 2.3 together with the bound for the degree presented
in Theorem 2.1, and the previous remark, implies in each case the following
lower and upper bounds for the index $\iota$:

degree | genus | index bound | degree | genus | index bound
---|---|---|---|---|---
3 | 1 | $0\leq\iota\leq 201$ | 11 | 45 | $528\leq\iota\leq 1008$
4 | 3 | $24\leq\iota\leq 302$ | 12 | 55 | $648\leq\iota\leq 1109$
5 | 6 | $60\leq\iota\leq 403$ | 13 | 66 | $780\leq\iota\leq 1209$
6 | 10 | $108\leq\iota\leq 504$ | 14 | 78 | $924\leq\iota\leq 1310$
7 | 15 | $168\leq\iota\leq 604$ | 15 | 91 | $1080\leq\iota\leq 1411$
8 | 21 | $240\leq\iota\leq 705$ | 16 | 105 | $1248\leq\iota\leq 1512$
9 | 28 | $324\leq\iota\leq 806$ | 17 | 120 | $1428\leq\iota\leq 1613$
10 | 36 | $420\leq\iota\leq 907$ | 18 | 136 | $1620\leq\iota\leq 1714$
Table 1. Index bounds

The logarithm of the number of congruence subgroups in
$\operatorname{PSL}_{2}(\mathbb{Z})$ of index bounded by $1714$ is
approximately $1897$, see [BNP21]*Proposition 8.1. Therefore naively listing
all subgroups would be not feasible, and the list of [CUPA03] contains only
groups of genus less than or equal to $24$.

Let us also remark that for any given genus we can bound the level $N$ of the
congruence subgroups occurring using the following formula, due to Cox and
Parry [COPA84]*Equation (4.22),

(2.2) $N\leq\begin{cases}168&\text{if }g=0\\\
12g+\frac{1}{2}(13\sqrt{48g+121}+145)&\text{if }g\geq 1\end{cases}$

Analysing the genera in Theorem 2.1 we produce the level bounds appearing in
Table 2.

genus | level bound | genus | level bound
---|---|---|---
1 | 169 | 45 | 922
3 | 214 | 55 | 1074
6 | 275 | 66 | 1237
10 | 351 | 78 | 1412
15 | 441 | 91 | 1600
21 | 542 | 105 | 1799
28 | 657 | 120 | 2010
36 | 784 | 136 | 2234
Table 2. Level bounds

It remains to check this finite number of possibilities, a task which we
proceed to describe in the rest of the paper.

##  3\. Computing Modular Curves

Let $\Gamma\subseteq\operatorname{PSL}_{2}(\mathbb{Z})$ be a congruence
subgroup of level $N$. Then the modular curve $X_{\Gamma}$ can be given the
structure of an algebraic curve over $\mathbb{Q}(\zeta_{N})$. This structure
depends on the choice of a group
$G\subseteq\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ such that the
projection of its pullback to $\operatorname{SL}_{2}(\mathbb{Z})$, denoted by
$PG$, coincides with $\Gamma$. We denote such a model by $X_{G}$. The Galois
action on the connected components of the curve $X_{G}$ is given by the
homomorphism $\sigma_{d}\mapsto\left(\begin{smallmatrix}d&0\\\
0&1\end{smallmatrix}\right)$, where $\sigma_{d}(\zeta_{N})=\zeta_{N}^{d}$.
Therefore, the field of definition of $X_{G}$ is the fixed field of
$\det(G)\subseteq(\mathbb{Z}/N\mathbb{Z})^{\times}=\operatorname{Gal}(\mathbb{Q}(\zeta_{N})|\mathbb{Q})$,
where $\det$ denotes the usual determinant map from
$\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ to
$(\mathbb{Z}/N\mathbb{Z})^{\times}$. The connected components of the curve
$X_{G}$ are indexed by $(\mathbb{Z}/N\mathbb{Z})^{\times}/\det(G)$, and each
component is defined over the field $\mathbb{Q}(\zeta_{N})^{\det(G)}$. In
particular, $X_{G}$ is geometrically connected and defined over $\mathbb{Q}$
if and only if $\det(G)=(\mathbb{Z}/N\mathbb{Z})^{\times}$. Therefore,
$X_{\Gamma}$, which is one of the components of $X_{G}$, admits a model over
$\mathbb{Q}$ only if there exists
$G\subseteq\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ such that
$PG=\Gamma$ and $\det(G)=(\mathbb{Z}/N\mathbb{Z})^{\times}$.

The methods of Galbraith and Box, described briefly in the introduction, for
computing modular curves use duality with modular symbols, and therefore
require $G$ also to be of real type, i.e. such that $JGJ=G$, where
$J=\left(\begin{smallmatrix}-1&0\\\ 0&1\end{smallmatrix}\right)$. Since $J$
acts via complex conjugation on the Fourier coefficients of modular forms, it
is equivalent to requiring the Fourier coefficients to be fixed by complex
conjugation.

We therefore restrict our attention to congruence subgroups $\Gamma$ such that
there exists $G$ of real type with surjective determinant and $PG=\Gamma$.
Note further that for these groups, when the degree is prime to $3$, it
suffices to check one such model $X_{G}$ by [BBLG2019]*Corollary 2.7. In the
range of degrees we are interested in, the only relevant case is that of
degree $6$, i.e. genus $10$. In this case, for groups of Shimura type, the
curve always admits a rational point, and so it is again enough to consider a
single model by [BBLG2019]*Corollary 2.2. For the other congruence subgroups
of genus $10$ for which we compute the curve, we verify that the resulting
curves indeed have rational points, hence in these cases it also suffices to
check a single model.

Our method of enumerating these subgroups of specific genus is to run over the
finite list of conjugacy classes of congruence subgroups of this genus in
$\operatorname{PSL}_{2}(\mathbb{Z})$, and for a representative
$\Gamma\subseteq\operatorname{PSL}_{2}(\mathbb{Z})$, we look at the projection
of its pullback $H\subseteq\operatorname{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$. As
for any compatible $G\subseteq\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$,
$H$ will be a normal subgroup, we start by looking for a conjugate
$H^{\prime}$ of $H$ in $\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ which
satisfies $JH^{\prime}J=H^{\prime}$, or equivalently $J\in N(H)$, where the
normalization takes place in $\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$.
Since $N(gHg^{-1})=gN(H)g^{-1}$, it suffices to consider conjugates of $N(H)$,
and look for one which contains $J$. We then note that if $G$ is such that
$G\cap SL_{2}(\mathbb{Z}/N\mathbb{Z})=H$, then $H\trianglelefteq G$, so that
$G\subseteq N(H)$. Thus, looking for $G$ with surjective determinant amounts
to enumerating the subgroups of $N(H)/H$ of order $\phi(N)$.

In Table 3 we list how many congruence subgroups $\Gamma$ exist, up to
conjugacy, for each degree $3\leq d\leq 8$, and how many of these admit a
model $G\subseteq\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ of real type
with surjective determinant. In Table 3 we also record the number of groups of
Shimura type of each degree $3\leq d\leq 8$.

degree | genus |  | congruence
---
subgroups
| real type &
---
surjective det
Shimura type
3 | 1 | 163 | 108 | 38
4 | 3 | 241 | 160 | 26
5 | 6 | 175 | 74 | 8
6 | 10 | 235 | 120 | 17
7 | 15 | 485 | 244 | 23
8 | 21 | 729 | 431 | 55
Table 3. Congruence subgroups of low genus

The methods we use for computing equations of modular curves make use of
explicit computation of the $q$-expansions and the canonical map. We briefly
recall the map and its properties.

###  3.1. The canonical map

Let $k$ be a perfect field. Let $C/k$ be a smooth projective curve of genus
$g\geq 2$ with canonical divisor $K$. Let $\\{z_{0},...,z_{g-1}\\}$ be a basis
defined over $k$ of the Riemann-Roch space $\mathcal{L}(K)$. The canonical map
of $C$ is given by

$\phi_{K}:\,C\rightarrow\mathbb{P}^{g-1},\quad\,P\mapsto(z_{0}(P):...:z_{g-1}(P)).$

The curve $C$ is non-hyperelliptic if and only if $\phi_{K}$ is an embedding.
In this case $\phi_{K}(C)$ is defined over $k$ and it is unique up to a linear
transformation of $\mathbb{P}^{g-1}$. Otherwise, when $\phi_{K}$ is not an
embedding, the curve $C$ is hyperelliptic and $\phi_{K}$ is the quotient by
the hyperelliptic involution: $\phi_{K}(C)\simeq\mathbb{P}^{1}$.

######  Theorem 3.1.

(Noether-Enriques-Petri, [Noether]) Let $C$ be a smooth projective non-
hyperelliptic curve of genus $g$. The homogeneous ideal defining the canonical
curve $\phi_{K}(C)\subseteq\mathbb{P}^{g-1}$ is generated by its elements of
degree 2, except in the following cases:

  * •

$g=3$, so $C$ is a smooth plane quartic.

  * •

$g\geq 4$ and $C$ is a trigonal curve. In this case an element of degree 3 is
also needed to generate the ideal.

  * •

$g=6$ and $C$ is a smooth plane quintic. Again in this case an element of
degree 3 is also needed.

Therefore, to compute an equation for the modular curve, using the
identification
$S_{2}(\Gamma,\mathbb{Q}(\zeta_{N}))^{G}\simeq\Omega^{1}(X_{G})$, it suffices
to compute $q$-expansions up to sufficient precision and look for relations in
low degrees. We proceed by describing first the required precision.

###  3.2. Bounds

In order to distinguish modular forms we will use a finite number of
coefficients of the associated $q$-expansions thanks to the following result
due to Sturm [STU87]*Theorem 1, see also [STE07]*Section 9.4. Let us recall
that for a congruence subgroup
$\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Z})$ the width of the cusp
$\infty$ is the positive integer $h$ defined by
$\left(\begin{smallmatrix}1&h\mathbb{Z}\\\
0&1\end{smallmatrix}\right)=\Gamma\cap\left(\begin{smallmatrix}1&\mathbb{Z}\\\
0&1\end{smallmatrix}\right)$.

######  Theorem 3.2 ([STU87]*Theorem 1).

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$.
Let $h$ be the width of the cusp $\infty$ for $\Gamma$. Let $f$ be a modular
form on $\Gamma$ of weight $\kappa$, with coefficients in a discrete valuation
ring $R$ contained in $\mathbb{C}$. Let $\mathbb{F}$ be the residue field of
$R$. Suppose that the image $\sum a_{n}q^{n/h}$ in $\mathbb{F}[[q^{1/h}]]$ of
the $q$-expansion of $f$ has $a_{n}=0$ for all
$n\leq\kappa[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]/12$. Then $a_{n}=0$ for
all $n$, i.e. $f$ is congruent to $0$ modulo the maximal ideal of $R$.

Moreover, we can state the following corollary, derived from an observation at
the end of [STU87] and stated in this form in [RA09]:

######  Corollary 3.3 ([RA09]*Theorem 2.1).

Under the same hypotheses of the theorem above, let us assume furthermore that
$f$ is a cusp form. If the image $\sum a_{n}q^{n/h}$ in
$\mathbb{F}[[q^{1/h}]]$ of the $q$-expansion of $f$ has $a_{n}=0$ for all
$n\leq\kappa[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]/12-\\#(\text{cusps})$.
Then $a_{n}=0$ for all $n$, i.e. $f$ is congruent to $0$ modulo the maximal
ideal of $R$.

The integer $\kappa[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]/12$ (resp.
$\kappa[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]/12-\\#(\text{cusps})$) is
known as the Sturm bound (resp. Sturm bound for cusp forms) and we will use
the notation $B(\Gamma,\kappa)$ (resp. $B(\Gamma,\kappa)_{c}$) to refer to
such a bound.

###  3.3. Groups of Shimura type

For groups of Shimura type, the methods described in [A2022] can be used to
compute the $q$-expansions. Alternatively, conjugating by
$\alpha_{t}=\left(\begin{smallmatrix}1&0\\\ 0&t\end{smallmatrix}\right)$, we
see that

$\Gamma_{1}(Nt)\subseteq\alpha_{t}\Gamma(H,t)\alpha_{t}^{-1}\subseteq\Gamma_{0}(Nt).$

Moreover, if we decompose the space by Dirichlet characters as

$S_{2}(\Gamma_{1}(Nt))=\bigoplus_{\chi:(\mathbb{Z}/Nt\mathbb{Z})^{\times}\to\mathbb{C}^{\times}}S_{2}(\Gamma_{0}(Nt),\chi),$

then we obtain

$S_{2}(\alpha_{t}\Gamma(H,t)\alpha_{t}^{-1})=\bigoplus_{\chi:\chi(H)=1}S_{2}(\Gamma_{0}(Nt),\chi).$

The $q$-expansions for modular forms in the spaces in this decomposition are
then straightforward to compute.

In order to compute equations for all modular curves of Shimura type of genus
$1,3,6,10,15$, we will need to compute weight $2$ cusp forms and then check
quadratic and cubic relations, according to Theorem 3.1. The number of
coefficients of the $q$-expansions of the weight $2$ cusp forms needed to
certify the computation performed, is equal to $B(\Gamma,\kappa)_{c}$, where
$\kappa$ is either $4$ or $6$.

######  Proposition 3.4.

The level of a congruence subgroup in $\operatorname{PSL}_{2}(\mathbb{Z})$
associated to a Shimura type modular curve admitting a smooth plane model is
bounded by $1709$. More precisely, we can bound the level for each genus as
shown in Table 4.

###### Proof.

A congruence subgroup $\Gamma$ in $\operatorname{PSL}_{2}(\mathbb{Z})$
corresponding to a Shimura type modular curve is contained in $\Gamma_{0}(M)$
and contains $\Gamma_{1}(M)$ for an appropriate level $M$, after conjugation
in $\operatorname{GL}_{2}(\hat{\mathbb{Z}})$. Its index is bounded by $1714$
as in Table 1 and so by direct computation the level is bounded by $1709$. ∎

genus | level bound | genus | level bound
---|---|---|---
1 | 199 | 45 | 997
3 | 293 | 55 | 1103
6 | 401 | 66 | 1201
10 | 503 | 78 | 1307
15 | 601 | 91 | 1409
21 | 701 | 105 | 1511
28 | 797 | 120 | 1609
36 | 887 | 136 | 1709
Table 4. Level bounds for Shimura type modular curve admitting a smooth plane
model

###  3.4. Other congruence subgroups

For groups that are not of Shimura type, we apply the (generalization of the)
method of twists described by Box in [B2021]. We note that Box uses an
auxiliary divisor $M$ of the level $N$ such that $G_{M}=B_{0}(M)$, where
$B_{0}(M)$ is the Borel subgroup of
$\operatorname{GL}_{2}(\mathbb{Z}/M\mathbb{Z})$, but this constraint can be
relaxed to allow for $B_{1}(M)\subseteq G_{M}\subseteq B_{0}(M)$, where
$B_{1}(M)$ is the unipotent subgroup of $B_{0}(M)$, by decomposing according
to the action of Dirichlet characters. More precisely, if
$G^{\prime}\subseteq\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ is such
that $G\trianglelefteq G^{\prime}$ and $G^{\prime}/G$ is abelian, we can
decompose according to the characters of $G^{\prime}/G$, namely

$S_{2}(G)=\bigoplus_{\varepsilon:G^{\prime}/G\to\mathbb{C}^{\times}}S_{2}(G^{\prime},\varepsilon).$

In the cases where $G^{\prime}=B_{0}(M)$ and
$\Gamma_{1}(M)\cap\Gamma(K)\subseteq G$ for some relatively prime $K,M$, such
that $KM=N$, we may further identify

$S_{2}(G^{\prime},\varepsilon)=\bigoplus_{\begin{subarray}{c}\chi:\left(\mathbb{Z}/MK^{2}\mathbb{Z}\right)^{\times}\to\mathbb{C}^{\times}\\\
\chi|_{(K\mathbb{Z}+1)/MK^{2}\mathbb{Z}}=\varepsilon\end{subarray}}S_{2}(\Gamma_{0}(MK^{2}),\chi)^{G^{\prime}=\varepsilon}$

by conjugating with $\alpha_{K}=\left(\begin{smallmatrix}1&0\\\
0&K\end{smallmatrix}\right)$. We can then create the space of modular symbols
corresponding to $S_{2}(\Gamma_{0}(MK^{2}),\chi)$, and cut out the subspace on
which $G^{\prime}$ acts via $\varepsilon$ by a method similar to the algorithm
described in [B2021]*Algorithm 4.11. Putting together all these elements, we
obtain an algorithm that, given a group $G$ such that $B_{1}(M)\subseteq
G_{M}\subseteq B_{0}(M)$, returns the $q$-expansions of a basis for the space
of cusp forms $S_{2}(G)$. Denote by
$\pi_{M}:\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})\to\operatorname{GL}_{2}(\mathbb{Z}/M\mathbb{Z})$
the natural projection map.

######  Proposition 3.5.

The running time complexity of the algorithm described above for a group
$G=\pi_{M}^{-1}(G_{M})\cap\pi_{K}^{-1}(G_{K})$ with $B_{1}(M)\subseteq
G_{M}\subseteq B_{0}(M)$ of genus $g$ is given by

$\widetilde{O}\left([G_{M}:B_{1}(M)](M^{3}K^{6}+MK^{4}g^{2})\right).$

For a group of Shimura type $G=G(H,t)$, it is given by
$\widetilde{O}\left(\frac{\phi(N)}{|H|}Nt^{2}g^{2}\right)$.

###### Proof.

We note that the complexity of the algorithm is dominated by the linear
algebra operations performed in these spaces of modular forms. Specifically,
since the algorithm requires computing the Hecke decomposition, to obtain the
modular symbols corresponding to the eigenform, our complexity is dominated by
$\widetilde{O}(d^{3}+dL^{2})$, where $d=\dim S_{2}(\Gamma_{0}(MK^{2}),\chi)$
is the dimension of the space and $L$ is the precision required for the
$q$-expansions, see [BBCCCDLLRSV2020]*Table 5.2.3. By Corollary 3.3, as the
cusp width $h$ is bounded by $K$, we see that the required precision to
ascertain our linear relations indeed hold is bounded by $L\leq
KB(\Gamma,\kappa)_{c}$, where $\kappa$ is the maximal weight in which we look
for an equation and $\Gamma$ is the pullback of $G$ to
$\operatorname{SL}_{2}(\mathbb{Z})$. By Theorem 3.1, $\kappa\in\\{4,6\\}$ in
all considered cases, with $\kappa=6$ occurring only if $g\leq 6$. Finally, by
[Z1991]*Theorem 2.3, the index
$[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]=O(g)$, hence $L=O(Kg)$. As
$d=O(MK^{2})$ (see [M2005] for a more precise and detailed asymptotic
analysis), it follows that the running time complexity of the algorithm on the
space corresponding to each Dirichlet character $\chi$ is
$\widetilde{O}(M^{3}K^{6}+MK^{4}g^{2})$, and summing over all Dirichlet
characters we obtain the result. For a group of Shimura type, we can simply
compute for each of the direct summands the Hecke operators up to the required
precision, which now satisfies $L=O(tg)$, as the cusp width at $\infty$ is
precisely $h=t$. ∎

As a result, when looking for smooth plane models of general congruence
subgroups, we will have to restrict ourselves to reasonable ranges of the
parameter $MK^{2}$. We therefore treat in this paper only groups that are of
Shimura type or such that $MK^{2}\leq 500$. We further note that different
representatives in the conjugacy class of $\Gamma$, and different groups $G$
which pull back to $\Gamma$ give rise to different values of $M,K$. We find
for each conjugacy class of the congruence subgroup $\Gamma$, a corresponding
group $G$ with the maximal value of $M$ (and so the minimal for $MK^{2}$).
Moreover, by their definition, for groups of Shimura type we may choose
$M=N/t$ and $K=t$, making them the easiest to compute using this method as
well.

##  4\. From a canonical model to a smooth plane model

In this section we propose an algorithm to check whether a smooth irreducible
projective curve $C$ of genus $g$ and defined over a perfect field $k$ does
not admits a smooth plane model over $\bar{k}$. In the case where we cannot
rule out this possibility, we propose a strategy to compute such smooth plane
model. We do not focus on the minimal fields of definitions for these models,
but we point the interested reader to [BBLG2019].

###  4.1. The low genus cases

For $g=0,1$ there is always a smooth plane model. For genus $2$, or more
generally, for hyperelliptic curves, there is never one. For genus $3$ non-
hyperelliptic there is always a smooth plane model and it is given by the
canonical model. The next genus to check is $6$, for which we have another
necessary condition in order to have a smooth plane model of degree $5$: the
canonical ideal $I_{C}$ defining $\phi_{K}(C)$ is not generated only by degree
$2$ elements, see Theorem 3.1. Still in this situation we need to distinguish
between trigonal curves and smooth plane quintics, see Subsection 5.3 for a
detailed example. Let us recall the following classical result, coming from
the description of the regular differential forms of a smooth plane curve: The
canonical model of a degree $d$ smooth plane curve is given by the composition
of $C\hookrightarrow\mathbb{P}^{2}$ with the $(d-3)$-Veronese embedding
$\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{g-1}$.

######  Lemma 4.1.

Let $C$ be a smooth plane quintic curve. Then the degree $2$ elements of the
canonical ideal $I_{C}$ defining $\phi_{K}(C)$ define a $\mathbb{P}^{2}$. A
bijective parametrization of it, evaluated at a degree $3$ non-trivial
generator of $I_{C}$, gives the smooth plane quintic model.

###### Proof.

In the quintic case $C:\,F(x,y,z)=0\subseteq\mathbb{P}^{2}$ with
$\operatorname{deg}(F)=5$ and the Canonical model is given by the composition
with the $2$-Veronese embedding $\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{5}$.
The canonical image $\phi_{K}(C)$ is generated by the equations defining
$\phi_{K}(\mathbb{P}^{2})$ that can all be taken of degree $2$ and the $3$
degree $3$ equations, corresponding to $xF(x,y,z)=0$, $yF(x,y,z)=0$ and
$zF(x,y,z)=0$. ∎

We deal next with the higher genus situations.

###  4.2. The minimal free resolution

A smooth curve $C$ of genus $g$ admits a smooth plane model if and only if it
has a (unique up to linear equivalence) very ample complete $g^{2}_{d}$-linear
series, i.e. a very ample divisor $D$ such that $\text{deg}(D)=d$ and
$\ell(D)=3$. Given a basis $\\{x,y,z\\}$ of $\mathcal{L}(D)$, the plane model
is given by the image of
$C\rightarrow\mathbb{P}^{2}:\,P\mapsto(x(P):y(P):z(P))$.

######  Theorem 4.2 ([Green]*Appendix).

If a smooth curve $C$ of genus $g=\frac{(d-1)(d-2)}{2}$ with $d\geq 5$ and
canonical divisor $K$, has a $g^{2}_{d}$-linear series then the Koszul
cohomology group $\mathcal{K}_{\frac{(d-3)(d-2)}{2},1}(C,K)\neq 0$.

This theorem proves a special case of one of the directions of Conjecture 5.1
in [Green]. In terms of graded Betti numbers [schreyer]*p. 84 we have:

$\operatorname{dim}(\mathcal{K}_{\frac{(d-3)(d-2)}{2},1}(C,K_{C}))=\beta_{d-4,d-2}.$

Let $C/\mathbb{C}$ be a smooth curve of genus $g$ and $\phi_{K}(C)$ its image
by the canonical map given by the ideal $I_{C}$ in
$S=\mathbb{C}[z_{0},z_{1},...,z_{g-1}]$. Let $S_{C}=S/I_{C}$ be the
homogeneous coordinate ring of $\phi_{K}(C)$. We consider the minimal free
resolution:

$0\leftarrow S_{C}\leftarrow S\leftarrow F_{1}\leftarrow
F_{2}\leftarrow...\leftarrow F_{g-2}\leftarrow 0.$

Noether proved that $F_{i}=S(-i-1)^{\beta_{i,i+1}}\oplus
S(-i-2)^{\beta_{i,i+2}}$ for $i=1,...,g-3$, i.e. that $F_{i}$ is a module
generated by elements of degree $i+1$ and $i+2$. These Betti numbers can be
computed with `Magma` [Magma]. In order to speed up these calculations, we
compute the Betti numbers for the reduction of the curve modulo a prime of
good reduction: in this case the Betti numbers are the same for both curves,
see [milneLEC]*Thm. 20.5.

When $\beta_{d-4,d-2}\neq 0$, we still need to check whether a smooth plane
model exists. As in the proof of Lemma 4.1, when $g\geq 6$ the ideal $I_{C}$
is generated by the degree 2 equations defining
$\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{g-1}$ by the $(d-3)$-Veronese
embedding plus the (this time) degree 2 equations $x^{a}y^{b}z^{c}F(x,y,z)=0$
with $a+b+c=d-6$. In order to recover the putative smooth model we aim to
determine the degree 2 equations defining the $\mathbb{P}^{2}$. Then we
compute a bijective parametrization and plug it into any other equation of
$I_{C}$, not defining the $\mathbb{P}^{2}$, and, therefore, we should obtain
the smooth plane model we are looking for. If not such a model is found, it
means that it does not exists. This strategy to recover the $\mathbb{P}^{2}$
is the one in the proof of Theorem 4.1 in [schreyer] that gives a proof of the
reverse implication of Conjecture 5.1 in [Green] for $d=6$. The idea is to
recover the exceptional surface, so the $\mathbb{P}^{2}$, by finding relations
with a certain shape and the standard basis techniques presented in the
Appendix of [schreyer]. We present an implementation of this algorithm in
[ASSAFgit].

###  4.3. The algorithm

Following the discussion in the previous subsections, we present an algorithm,
Algorithm 1, which allows to determine whether a curve admits a smooth plane
model.

Data: $C/k$ a genus $g$ curve given with its canonical model

Result: determining whether $C$ admits a smooth plane model and, when
possible, returning such model

$M\leftarrow 0$;

if  _$g$ is $0$ or $1$_ then

$T\leftarrow true$;

else

if  _$g$ is $3$_ then

if  _$g(\phi_{K}(C))=3$_ then

$T\leftarrow true$;

$M\leftarrow\phi_{K}(C)$;

else

$T\leftarrow false$;

end if

else

if  _$g$ is $6$_ then

if  _$I_{C}$ generated by quadrics_ then

$T\leftarrow false$;

else

compute $M$ with Lemma 4.1;

if  _$M$ is a smooth plane quintic_ then

$T\leftarrow true$;

else

$T\leftarrow false$;

end if

end if

else

$T\leftarrow false$;

if  _$\exists d\in\mathbb{N}$ with $g=(d-1)(d-2)/2$ and $g(\phi_{K}(C))\neq 0$
_ then

Apply Theorem 4.2;

if  _$\beta_{d-4,d-2}\neq 0$ _ then

compute $M$ with Schreyer’s strategy [schreyer];

if  _$M$ is smooth_ then

$T\leftarrow true$;

end if

end if

end if

end if

end if

end if

return _$T,M$_

Algorithm 1 Existence of a smooth plane model

###  4.4. Other strategies

Sometimes, in order to prove that a certain curve does not admit a smooth
plane model, we can try some less computationally expensive techniques. For
instance, when the curves under considerations have some involutions:

######  Theorem 4.3.

[Remark 2.1 (i) & Theorem 2.2 with $n=2$ in [Harui]] Let $C$ be a smooth plane
curve of degree $d$ and $\sigma$ an involution of $C$. Then the involution
$\sigma$ has $f=d+\frac{1-(-1)^{d}}{2}$ fixed points and the quotient
$C/\langle\sigma\rangle$ has gonality $\lfloor\frac{d}{2}\rfloor$.

Other ways of using the knowledge of some quotients to prove the non-existence
of smooth plane models are the following results:

######  Lemma 4.4.

Let $C$ be a smooth curve admitting a degree $n$ morphism to a hyperelliptic
curve. Then $C$ does not admit a smooth plane model of degree greater than
$2n+1$.

###### Proof.

The gonality of a smooth plane curve of degree $d$ is $d-1$ and the gonality
of a hyperelliptic curve is $2$. ∎

######  Theorem 4.5.

[Theorem 3.1 with $r=1$ in [Greco]] A smooth plane curve of degree $d$ does
not admit any rational map to $\mathbb{P}^{1}$ of degree $n$ such that

$(a-1)d+1\leq n\leq ad-(a^{2}+1)$

for some $a\in\mathbb{N}$.

##  5\. Computations

We compute equations for all modular curves of Shimura type of genus
$1,3,6,10$ and $15$, i.e. possibly admitting a smooth plane model of degree
$3,4,5,6$ or $7$. We run the algorithm in previous section on all of them, in
order to check which ones do admit a smooth plane model.

We also compute equations for modular curves of these genera which are not of
Shimura type, when the congruence subgroup is $\Gamma=PG$, with
$G\subseteq\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ satisfies
$G=\pi_{M}^{-1}(G_{M})\cap\pi_{K}^{-1}(G_{K})$ with $B_{1}(M)\subseteq
G_{M}\subseteq B_{0}(M)$ and $G_{K}\subseteq GL_{2}(\mathbb{Z}/K\mathbb{Z})$,
and such that $MK^{2}\leq 500$. Note that $(MK^{2})^{3}$ is the dominant
factor in the running time complexity, and indeed for larger values of
$MK^{2}$, the linear algebra becomes the bottleneck.

We discuss the results in the following subsections. We use the congruence
subgroup labels introduced in [CUPA03]. All computations were done using
`Magma` [Magma] and the full results are available online at [ASSAFgit].

###  5.1. Genus 1

In the case of modular curves of genus $1$ they always admit a smooth plane
model of degree $3$. However, it is not given by the canonical model since
they are not non-hyperelliptic. In this case we note that all groups of
Shimura type give rise to elliptic curves, since the cusp at $\infty$ is
rational. We may further compute models for some of the other congruence
subgroups, using the methods in [RZB2015] to obtain the $j$-map and a model
for the curve. For example, we see that for the congruence subgroups of level
$6$, the groups labeled 6A1, 6C1, 6D1 all yield elliptic curves, with
equations $y^{2}=x^{3}-27$, $y^{2}=x^{3}+1$ and $y^{2}=x^{3}+1$. We also find
the $q$-expansion of the unique eigenform for all $98$ congruence subgroups of
genus $1$ for which $MK^{2}\leq 500$.

###  5.2. Genus 3

Among the $26$ groups of Shimura type of genus $3$, we find that there are
$11$ modular curves which are hyperelliptic, these are listed below. We note
that $7$ of these curves belong to the $X_{0}(N)$ family, and indeed we
recover the models of Galbraith [G1996] for all these curves except for
$X_{0}(35)$ and $X_{0}(41)$ that we find different ones. These, of course,
give isomorphic curves to the Galbraith ones.

label | name | Hyperelliptic model
---|---|---
12K3 |  | $y^{2}=x^{8}+14x^{4}+1$
20J3 |  | $y^{2}=x^{8}+8x^{6}-2x^{4}+8x^{2}+1$
21D3 |  | $y^{2}=x^{8}-6x^{6}+4x^{5}+11x^{4}-24x^{3}+22x^{2}-8x+1$
24V3 |  | $y^{2}=x^{8}+14x^{4}+1$
30K3 | $X_{0}(30)$ | $y^{2}=x^{8}+6x^{7}+9x^{6}+6x^{5}-4x^{4}-6x^{3}+9x^{2}-6x+1$
33C3 | $X_{0}(33)$ | $y^{2}=x^{8}+10x^{6}-8x^{5}+47x^{4}-40x^{3}+82x^{2}-44x+33$
35A3 | $X_{0}(35)$ | $y^{2}=x^{8}-12x^{7}+50x^{6}-108x^{5}+131x^{4}-76x^{3}-10x^{2}+44x-19$
39A3 | $X_{0}(39)$ | $y^{2}=x^{8}-6x^{7}+3x^{6}+12x^{5}-23x^{4}+12x^{3}+3x^{2}-6x+1$
40F3 | $X_{0}(40)$ | $y^{2}=x^{8}+8x^{6}-2x^{4}+8x^{2}+1$
41A3 | $X_{0}(41)$ | $y^{2}=x^{8}-12x^{7}+48x^{6}-82x^{5}+60x^{4}-8x^{3}-27x^{2}+16x-4$
48J3 | $X_{0}(48)$ | $y^{2}=x^{8}+14x^{4}+1$
Table 5. Hyperelliptic Shimura type modular curves of genus $3$

######  Remark 5.1.

The curves corresponding to the groups labeled 12K3, 24V3 and 48J3 are
isomorphic. The curves corresponding to the groups labeled 20J3 and 40F3 are
isomorphic. No other curves in Table 5 are isomorphic. This phenomenon is
explained by the fact that the corresponding groups are conjugate in
$\operatorname{GL}_{2}(\hat{\mathbb{Z}})$.

The other groups of Shimura type of genus 3 give rise to smooth plane
quartics. In table 6 we present the plane quartics obtained.

label | name | Plane quartic model
---|---|---
7A3 | $X(7)$ | $-xy^{3}+x^{3}z+yz^{3}=0$
8A3 |  | $x^{3}z+4xz^{3}-y^{4}=0$
12O3 |  | $x^{3}z-3x^{2}z^{2}-xy^{3}+4xz^{3}+2y^{3}z-2z^{4}=0$
15E3 |  | $x^{3}z-x^{2}y^{2}+xyz^{2}-y^{3}z-5z^{4}=0$
16H3 |  | $-y^{4}+x^{3}z+4xz^{3}=0$
20S3 | $X_{1}(20)$ | $x^{3}z-x^{2}y^{2}-3x^{2}z^{2}+xy^{3}+4xz^{3}-2z^{4}=0$
24X3 |  | $\begin{array}[]{c}x^{3}z-2x^{2}yz-x^{2}z^{2}-xy^{3}+2xy^{2}z\\\ +6xyz^{2}+2y^{3}z-2y^{2}z^{2}-4xz^{3}=0\end{array}$
24Y3 |  | $\begin{array}[]{c}x^{3}z-x^{2}y^{2}-x^{2}z^{2}+xz^{3}-xy^{2}z\\\ -3xyz^{2}+y^{3}z+2y^{2}z^{2}+yz^{3}=0\end{array}$
32J3 |  | $-y^{4}+x^{3}z+4xz^{3}=0$
34C3 | $X_{0}(34)$ | $\begin{array}[]{c}-x^{2}y^{2}+2xy^{3}-y^{4}+x^{3}z+3xy^{2}z+4y^{3}z-\\\ 3x^{2}z^{2}-3xyz^{2}-6y^{2}z^{2}+4xz^{3}+4yz^{3}-2z^{4}=0\end{array}$
36K3 |  | $-xy^{3}+x^{3}z+2y^{3}z-3x^{2}z^{2}+4xz^{3}-2z^{4}=0$
43A3 | $X_{0}(43)$ | $\begin{array}[]{c}-2x^{2}y^{2}+xy^{3}-9y^{4}+x^{3}z+2x^{2}yz+3xy^{2}z+24y^{3}z-\\\ 2x^{2}z^{2}-5xyz^{2}-28y^{2}z^{2}+3xz^{3}+16yz^{3}-4z^{4}=0\end{array}$
45D3 | $X_{0}(45)$ | $-x^{2}y^{2}+x^{3}z-y^{3}z+xyz^{2}-5z^{4}=0$
49A3 |  | $-xy^{3}+x^{3}z+yz^{3}=0$
64B3 | $X_{0}(64)$ | $-y^{4}+x^{3}z+4xz^{3}=0$
Table 6. Plane quartic Shimura type modular curves of genus $3$

######  Remark 5.2.

The curves corresponding to the groups labeled by 7A3 and 49A3 are isomorphic.
The curves corresponding to the groups labeled by 8A3, 16H3, 32J3, 64B3 are
isomorphic. The curves corresponding to the groups labeled by 12O3 and 36K3
are isomorphic. The curves corresponding to the groups labeled by 15E3 and
45D3 are isomorphic. No other curves in Table 6 are isomorphic. Again, the
explanation for these isomorphisms is that the congruence subgroups are
conjugate in $\operatorname{GL}_{2}(\hat{\mathbb{Z}})$.

We have also computed models for $92$ out of the $105$ congruence subgroups
that are not of Shimura type and have $MK^{2}\leq 500$. An example of a plane
quartic occurs for the group labeled $9A3$, cut out by the quartic
$81x^{4}-54x^{3}y-27x^{2}y^{2}+3xy^{3}+y^{4}-729xz^{3}+486yz^{3}$.

###  5.3. Genus 6

Among the $8$ groups of Shimura type occurring, none admits a smooth plane
model. We have also computed models for $19$ out of the $29$ congruence
subgroups that are not of Shimura type and have $MK^{2}\leq 500$.

Except the curve corresponding to 18A6, all the other ones are of genus $6$
and have a canonical model generated by quadrics, which means that they are
non-hyperelliptic and that they do not admit a smooth plane model.

For the curve 18A6, we also need cubic equations to define its canonical
model. According to Theorem 3.1, this implies that it does admit a smooth
plane model or that it is a trigonal curve. We first found an explicit
birational equivalence between $\mathbb{P}^{2}$ and the locus of the quadrics
in the ideal generating the canonical model. This birational map gives a
parametrization of the locus of the quadrics. We plugged it into the degree 3
equations, and, after a suitable scaling of the variables, we found the
following equation:

$y^{3}=(x-3)(x+1)(x^{2}+3)(x+3)^{2}(x^{2}+6x+21)^{2}.$

Interestingly, the curve (that it is not a smooth plane quintic) is not only
trigonal, but also superelliptic. Notice that this is a quite remarkable
exception since the dimensions of the moduli space of curves of genus $6$, and
the locus of trigonal, plane and superelliptic ones of degree 3 inside it are:
$\dim(\mathcal{M}_{6})=15$, $\dim(\mathcal{M}^{trig}_{6})=13$,
$\dim(\mathcal{M}_{6}^{plane})=12$ and $\dim(\mathcal{M}_{6}^{superell,3})=5$.
Actually, we could have guessed the existence of an automorphism of order $3$
of the curve, since $\operatorname{Aut}(G)$ contains an element $\sigma$ of
order $3$, which induces an automorphism of the curve $X_{G}$. The induced
action of the automorphism on the $q$-expansions is via
$\sigma(f)(q)=f(\zeta_{3}q)$. Therefore, we can readily compute its action on
the curve, and observe that it corresponds to the action $x\mapsto x$ and
$y\mapsto\zeta_{3}y$ in the model we found.

###  5.4. Genus 10

We have checked all the $17$ groups of Shimura type of genus $10$. None of
them admits a smooth plane model. This was verified by computing the graded
Betti number $\beta_{2,4}$. In $15$ out of the $17$ cases we have
$\beta_{2,4}=0$. Therefore, by Theorem 4.2, these curves do not admit a
$g_{6}^{2}$, or equivalently a smooth plane model. The remaining cases, of the
groups 46A10 and 92A10, are both isomorphic to the curve $X_{0}(92)$. In this
case, we obtain $\beta_{2,4}=27$, which by [schreyer]*Corollary 4.2 implies
either that the curve is a smooth plane curve or that it is a double cover of
an elliptic curve. However, in this case one checks that the quotient of the
curve by the Atkin-Lehner involution $W_{23}$ yields an elliptic curve, hence
by Lemma 4.4 it does not admit a smooth plane model.

In most cases, projecting to $\mathbb{P}^{2}$ using the three divisors of
maximal valuation at the cusp at $\infty$, which is a flex, one obtains a
singular curve of degree $10$ or $11$ with coefficients of large height.
However, for the groups in Table 7 we obtain a smooth cubic (an elliptic
curve).

label | name | Image in $\mathbb{P}^{2}$
---|---|---
9A10 | $X(9)$ | $y^{2}=x^{3}+16$
18E10 |  | $y^{2}=x^{3}-12x^{2}+48x$
27B10 |  | $y^{2}=x^{3}+16$
36Q10 |  | $y^{2}=x^{3}-12x^{2}+48x$
54A10 |  | $y^{2}=x^{3}-12x^{2}+48x$
81A10 |  | $y^{2}=x^{3}+16$
108F10 | $X_{0}(108)$ | $y^{2}=x^{3}-12x^{2}+48x$
Table 7. Elliptic curves admitting a morphism from a Shimura type modular
curve of genus $10$

###  5.5. Genus 15

We have not been able to check the existence of a smooth plane model for any
of the $23$ groups of Shimura type of genus $15$ using Algorithm 1, as
computing the corresponding Betti numbers $\beta_{3,5}$ turned out to be
beyond our computational ability.

However, we can rule out the existence of smooth plane models for all these
curves by looking at their Atkin-Lehner quotients. Indeed, any congruence
subgroup $\Gamma(H,1)$ of Shimura type of level $N$ and parameter $t=1$
satisfies $\Gamma_{1}(N)\subseteq\Gamma\subseteq\Gamma_{0}(N)$, and a subset
of the Atkin-Lehner operators on $\Gamma_{0}(N)$ normalize $\Gamma$ as well.
These induce automorphisms of the curve $X_{G}$ (for $G=G(H,1)$), and we
denote the quotients by a subset $W$ of them by $X_{G}/W$. Whenever $|W|=4$
and $X_{G}/W$ is hyperelliptic, we have a morphism $X_{G}\to\mathbb{P}^{1}$ of
degree $8$, and we can deduce from Theorem 4.5 that $X_{G}$ does not admit a
smooth plane model (which must be of degree $d=7$).

Moreover, each of the congruence subgroups of Shimura type of genus $15$ is
isomorphic to a group of type $\Gamma(H,1)$. Table 8 summarizes our findings,
where for groups which are conjugate in
$\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$ we have written down both
labels.

label | name | $W$ | $g(X_{G}/W)$
---|---|---|---
35C15, 175A15 | $X_{0}(175)$ | $\langle w_{25},w_{7}\rangle$ | 3
40W15, 80R15 |  | $\langle w_{16},w_{5}\rangle$ | 1
40X15, 80T15 |  | $\langle w_{16},w_{5}\rangle$ | 1
43A15 |  | $\langle w_{43}\rangle$ | 7
51A15, 153A15 | $X_{0}(153)$ | $\langle w_{9},w_{17}\rangle$ | 2
60AC15 |  | $\langle w_{3},w_{5}\rangle$ | 1
60AD15 |  | $\langle w_{3},w_{5}\rangle$ | 1
67A15 |  | $\langle w_{67}\rangle$ | 7
68D15, 136D15 | $X_{0}(136)$ | $\langle w_{8},w_{17}\rangle$ | 3
85A15 |  | $\langle w_{5},w_{17}\rangle$ | 2
85B15 |  | $\langle w_{5},w_{17}\rangle$ | 2
102C15 | $X_{0}(102)$ | $\langle w_{2},w_{3},w_{17}\rangle$ | 1
110A15 | $X_{0}(110)$ | $\langle w_{5},w_{11}\rangle$ | 1
141D15 | $X_{0}(141)$ | $\langle w_{3},w_{47}\rangle$ | 1
155A15 | $X_{0}(155)$ | $\langle w_{5},w_{31}\rangle$ | 1
161A15 | $X_{0}(161)$ | $\langle w_{7},w_{23}\rangle$ | 2
179A15 | $X_{0}(179)$ | $\langle w_{179}\rangle$ | 3
193A15 | $X_{0}(193)$ | $\langle w_{193}\rangle$ | 7
Table 8. Atkin-Lehner quotients of modular curves of Shimura type of genus
$15$

Note that, with the exception of
$X_{0}(102),X_{0}(136),X_{0}(175),X_{0}(179),X_{0}(193)$ and the two curves
corresponding to labels 43A15 and 67A15, the genus of all Atkin-Lehner
quotients is either $1$ or $2$, implying that they are hyperelliptic, and
$|W|=4$. By [H1997] we deduce that $X_{0}^{*}(136)=X_{0}(136)/W$ is also
hyperelliptic and thus we are left with six curves for which we were not able
to rule out the existence of a smooth plane model using Theorem 4.5, namely
$X_{0}(102),X_{0}(175),X_{0}(179),X_{0}(193)$ and the two curves corresponding
to labels 43A15 and 67A15. Note that by [FH1999] there is no quotient of
either of $X_{0}(175),X_{0}(179),X_{0}(193)$ which is hyperelliptic.

In order to rule out the six remaining curves, we use Theorem 4.3 with a
simple application of Riemann-Hurwitz as follows. If $X_{G}$ admits a smooth
plane model (of degree $d=7$), and $w\in W$ is any Atkin-Lehner involution, by
Theorem 4.3 $w$ has $8$ fixed points. Riemann-Hurwitz then implies that the
genus of the quotient is $g(X_{G}/\langle w\rangle)=6$. Therefore, if we find
some $w\in W$ where the genus of the quotient is not $6$, it does not admit a
smooth plane model. Looking again at Table 8 we see that this rules out
$X_{0}(179),X_{0}(193)$ and the two curves corresponding to labels 43A15 and
67A15. Finally, for $X_{0}(102)$ we see that $g(X_{0}(102)/\langle
w_{2}\rangle)=7$, and for $X_{0}(175)$ we have $g(X_{0}(175)/\langle
w_{7}\rangle)=8$, showing that they also do not admit a smooth plane model.

The map to $\mathbb{P}^{2}$ obtained by using the cusp at $\infty$ as the flex
point always yields a singular curve of degree $16$, except the following
cases. For the group 60AC15, we obtain the elliptic curve

$y^{2}=x^{3}+4x^{2}-16x.$

For the groups 40W15 and 80R15 (which induce isomorphic curves) we obtain a
septic with $3$ singular points, namely

$\displaystyle x^{5}y^{2}$
$\displaystyle-x^{6}z-5x^{4}y^{2}z+x^{2}y^{4}z+4x^{5}z^{2}+12x^{3}y^{2}z^{2}-4xy^{4}z^{2}-6x^{4}z^{3}$
$\displaystyle-16x^{2}y^{2}z^{3}+4y^{4}z^{3}+4x^{3}z^{4}+12xy^{2}z^{4}-x^{2}z^{5}-4y^{2}z^{5}=0.$

###  5.6. Genus 21

We have not been able to check the existence of a smooth plane model for any
of the $55$ groups of Shimura type of genus $21$ using Algorithm 1, as
computing the corresponding Betti numbers $\beta_{4,6}$ turned out to be
beyond our computational ability.

However, we can rule out the existence of smooth plane models for all but five
of these curves by looking at their Atkin-Lehner quotients, and using Theorem
4.3 and Riemann-Hurwitz as before. In this case, if $X_{G}$ admits a smooth
plane model (of degree $d=8$), and $w\in W$ is any Atkin-Lehner involution, by
Theorem 4.3 $w$ has $8$ fixed points. Riemann-Hurwitz then implies that the
genus of the quotient is $g(X_{G}/\langle w\rangle)=9$. Table 9 shows, for
each of these curves, the chosen Atkin-Lehner involution and the genus of the
corresponding quotient, where for groups which are conjugate in
$\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$ we have written down all labels
in a single line.

The curves left after this analysis are $X_{0}(256)$ and the curves
corresponding to labels 41A21, 91A21, 91B21, 137A21. The modular curve
$X_{0}(256)$ has a single Atkin-Lehner involution, and its quotient by that
involution has genus $9$. By Theorem 4.3, if this curve admitted a smooth
plane model, the corresponding quotient would have gonality $4$. However, by
computing the Betti table of the quotient, we see that the
$a(X_{0}(256)/\langle w\rangle)$ invariant of the table is $3$. Then,
Corollary 9.7 in [Eisenbud] implies that
$\operatorname{Cliff}(X_{0}(256)/\langle w\rangle)\geq 4$, so the gonality of
$X_{0}(256)/\langle w\rangle$ is at least $6$. On the other hand, it is also
bounded by $6$ because of the genus being $9$. This proves that $X_{0}(256)$
does not admit a smooth plane model. For the other four curves, at the present
time, no algorithm is implemented to compute their Atkin-Lehner quotients.

label | name | $W^{\prime}\subseteq W$ | $g(X_{G}/W^{\prime})$
---|---|---|---
21C21, 147B21 |  | $\langle w_{3}\rangle$ | 10
24A21, 48CE21, 96BA21, 192P21 | $X_{0}(192)$ | $\langle w_{64}\rangle$ | 11
28M21, 56L21, 112E21 |  | $\langle w_{112}\rangle$ | 10
30G21, 90O21 |  | $\langle w_{10}\rangle$ | 11
33C21 | $X_{1}(33)$ | $\langle w_{33}\rangle$ | 10
34A21 | $X_{1}(34)$ | $\langle w_{34}\rangle$ | 10
35B21, 245A21 | $X_{0}(245)$ | $\langle w_{245}\rangle$ | 8
36F21, 72AC21 |  | $\langle w_{72}\rangle$ | 10
39B21, 117A21 |  | $\langle w_{13}\rangle$ | 11
42H21, 84Q21 |  | $\langle w_{3}\rangle$ | 10
45F21 |  | $\langle w_{45}\rangle$ | 10
52C21, 104B21 |  | $\langle w_{13}\rangle$ | 11
55B21 |  | $\langle w_{5}\rangle$ | 3
56M21 |  | $\langle w_{8}\rangle$ | 6
65A21 |  | $\langle w_{13}\rangle$ | 6
69A21, 207A21 | $X_{0}(207)$ | $\langle w_{23}\rangle$ | 8
72AC21 |  | $\langle w_{72}\rangle$ | 10
78C21 |  | $\langle w_{26}\rangle$ | 8
92A21, 184A21 | $X_{0}(184)$ | $\langle w_{23}\rangle$ | 5
97A21 |  | $\langle w_{97}\rangle$ | 10
111A21 |  | $\langle w_{111}\rangle$ | 7
115A21 |  | $\langle w_{23}\rangle$ | 6
119A21 |  | $\langle w_{7}\rangle$ | 6
133B21 |  | $\langle w_{133}\rangle$ | 10
138A21 | $X_{0}(138)$ | $\langle w_{23}\rangle$ | 5
154B21 | $X_{0}(154)$ | $\langle w_{22}\rangle$ | 11
165B21 | $X_{0}(165)$ | $\langle w_{3}\rangle$ | 11
178A21 | $X_{0}(178)$ | $\langle w_{89}\rangle$ | 8
201A21 | $X_{0}(201)$ | $\langle w_{67}\rangle$ | 11
215A21 | $X_{0}(215)$ | $\langle w_{43}\rangle$ | 11
247A21 | $X_{0}(247)$ | $\langle w_{247}\rangle$ | 8
251A21 | $X_{0}(251)$ | $\langle w_{251}\rangle$ | 4
257A21 | $X_{0}(257)$ | $\langle w_{257}\rangle$ | 7
Table 9. Atkin-Lehner quotients of modular curves of Shimura type of genus
$21$ indicating no smooth plane model

###  5.7. Proof of Theorem 1.1

We are ready to prove now our main theorem:

###### Proof.

The first claim is Theorem 2.1 and the second one is deduced from its proof.
The last claim is a consequence of the computations in this section. ∎

## References