# Shimura curves admitting a smooth plane model

Oana Padurariu Oana Padurariu
Max-Planck-Institut für Mathematik Bonn
Germany https://sites.google.com/view/oanapadurariu/home
<EMAIL_ADDRESS>
###### Abstract.

We prove that there are no Shimura curves $X_{0}^{D}(N)$ with $N$ squarefree
of genus $g\geq 2$ admitting a smooth plane model.

##  1\. Introduction

Modular curves and Shimura curves are central moduli spaces in number theory.
Modular curves parametrize elliptic curves equipped with level structure and
Shimura curves parametrize abelian surfaces with quaternionic multiplication.
Shimura curves are naturally compact, in contrast to modular curves that are
compactified by adding cusps. These cusps are extremely useful in the
computations of defining equations for modular curves, which lead to an
impressive database of such explicit equations. On the other hand, there is a
scarcity of defining equations for Shimura curves in the literature.

We briefly recall some results about the classification of Shimura curves.
González–Rotger [GR06] provided defining equations for non-elliptic Shimura
curves $X_{0}^{D}(N)$ of genus one. Voight [Voi09] fully classified Shimura
curves of genus at most two. Guo–Yang [GY17] gave a complete list of
geometrically hyperelliptic Shimura curves $X_{0}^{D}(N)$ with explicit
equations. The next step in the classification of Shimura curves is finding
equations for all Shimura curves admitting a smooth plane model, which is what
Anni–Assaf–Lorenzo-García [AALG23] achieved for modular curves. A natural
question, inspired by [AALG23], is whether there are any Shimura curves
$X_{0}^{D}(N)$ admitting a smooth plane model beyond those of genus at most
one. In this paper, we prove the following:

######  Theorem 1.1.

Let $X_{0}^{D}(N)$ be a Shimura curve for which the level $N$ is squarefree.
If $X_{0}^{D}(N)$ admits a smooth plane model, then the genus of
$X_{0}^{D}(N)$ is at most one.

The proof relies on the interplay between the genus and the gonality of
Shimura curves, and on computations of the number of CM points fixed by
Atkin–Lehner involutions. The paper is accompanied by code [Pad] written in
Magma [BCP97].

### Acknowledgements

It is a pleasure to thank Pete L. Clark, Ciaran Schembri, Frederick Saia, and
John Voight for many helpful conversations about Shimura curves. The author is
very grateful to Max-Planck-Institut für Mathematik Bonn for hospitality and
financial support.

##  2\. Some background on Shimura curves

Throughout the paper, $p\in\mathbb{Z}_{\geq 1}$ is a prime number. In this
section, we recall some definitions and results about Shimura curves.

###  2.1. Quaternion algebras

To define Shimura curves, one first needs to define quaternion algebras, which
are extensively covered in [Voi21]. A quaternion algebra $B$ defined over
$\mathbb{Q}$ is a central simple $\mathbb{Q}$-algebra of dimension $4$.
Alternatively, $B$ is a quaternion algebra defined over $\mathbb{Q}$ if and
only if there are elements $i,j\in B$ such that

$B\simeq\mathbb{Q}\cdot 1+\mathbb{Q}\cdot i+\mathbb{Q}\cdot j+\mathbb{Q}\cdot
ij$

where $i^{2},j^{2}\in\mathbb{Q}^{\times}$ with $ij=-ji$. An example of a
quaternion algebra is $M_{2}(\mathbb{Q})$. An order of $B$ is a full
$\mathbb{Z}$-lattice which is also a subring.

A very important invariant for a quaternion algebra is the discriminant. For
any place $v$ of $\mathbb{Q}$, we say that $B$ is ramified at $v$ if the
completion $B\otimes_{\mathbb{Q}}\mathbb{Q}_{v}$ is a division algebra. The
set of ramified places of $B$ is finite, of even cardinality, and uniquely
determines the isomorphism class of a quaternion algebra. Then the
discriminant $D$ of $B$ is defined as

$D\colonequals\prod_{\begin{subarray}{c}v\text{ finite}\\\
\text{ramified}\end{subarray}}v.$

We say that $B$ is indefinite if $B$ is unramified at the archimedean place.
Note that in this case $\omega(D)$ is even, where

$\omega(n)\colonequals\\#\\{p:p|n\\}.$

We say that $d$ is a Hall divisor of $n$ if $d|n$ and $\gcd(d,n/d)=1$. We
denote this by $d\parallel n$.

###  2.2. Definition of Shimura curves

Let $B$ be an indefinite quaternion algebra defined over $\mathbb{Q}$ of
discriminant $D$ and let $O$ denote a maximal order. There are embeddings:

$\displaystyle\iota_{\infty}$ $\displaystyle:B\hookrightarrow
M_{2}(\mathbb{R})\simeq B\otimes_{\mathbb{Q}}\mathbb{R}$
$\displaystyle\iota_{p}$ $\displaystyle:O\hookrightarrow
M_{2}(\mathbb{Z}_{p})\simeq O\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$

for any prime $p$ not dividing $D$. We have a notion of level, similar to the
case of modular curves: for any $N\in\mathbb{Z}_{\geq 1}$ coprime to $D$, we
define the Eichler order of level $N$ to be

$O_{N}\colonequals\\{x\in O\ |\ \iota_{p}(x)\equiv\begin{pmatrix}\ast&\ast\\\
0&\ast\end{pmatrix}\ \text{mod}\ p^{e}\ \text{for all}\ p^{e}\parallel N\\}.$

Let $O_{N}^{1}$ be the set of norm one elements of $O_{N}$. Then $O_{N}^{1}$
acts on the upper half plane $\mathbb{H}=\\{x+iy\ |\ y>0\\}$ via
$\iota_{\infty}(O_{N}^{1})/\\{\pm 1\\}\leq\operatorname{PSL}_{2}(\mathbb{R})$
by

$\gamma\cdot z\colonequals\iota_{\infty}(\gamma)\cdot z=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot z=\frac{az+b}{cz+d}.$

Consider the Riemann surface

$\left(\iota_{\infty}(O_{N}^{1})/\\{\pm 1\\}\right)\backslash\mathbb{H}.$

If $B$ has discriminant $D=1$, then this Riemann surface is the classical
modular curve $Y_{0}(N).$

From now on we shall assume that $B$ is not isomorphic to $M_{2}(\mathbb{Q})$,
or equivalently that $D>1.$ By work of Shimura [Shi67, Main Theorem I],
[Shi70, Theorem 2.5], there is an algebraic curve

$X_{0}^{D}(N)$

such that there is an open immersion of Riemann surfaces

$\left(\iota_{\infty}(O_{N}^{1})/\\{\pm
1\\}\right)\backslash\mathbb{H}\hookrightarrow X_{0}^{D}(N)$

which is a biregular isomorphism. The curve $X_{0}^{D}(N)$ has a canonical
model defined over $\mathbb{Q}$ [Shi67, §3] and it is called a Shimura curve
of discriminant $D$ and level $N$.

As with modular curves, Shimura curves arise naturally as a moduli problem.
The curve $X_{0}^{D}(N)$ parameterizes pairs $(A,\iota)$, where

  * •

$A$ is an abelian surface, and

  * •

$\iota:O_{N}\hookrightarrow\operatorname{End}(A)$ is an embedding.

We call such a pair $(A,\iota)$ a QM-surface. It is well known that Shimura
curves have no real points [Shi75, Theorem 0], [Ogg85, §3], i.e.

$X_{0}^{D}(1)(\mathbb{R})=\emptyset.$

###  2.3. Atkin–Lehner involutions

As in the case of modular curves, Shimura curves also have Atkin–Lehner
involutions. Consider the normalizer group

$N_{B_{>0}^{\times}}(O_{N}^{1})\colonequals\\{\alpha\in B_{>0}^{\times}\ |\
\alpha^{-1}O_{N}^{1}\alpha=O_{N}^{1}\\},$

where $B_{>0}^{\times}$ are the units in $B$ of positive norm. Elements of
$N_{B_{>0}^{\times}}(O_{N}^{1})$ naturally define an automorphism of the
Shimura curve $X_{0}^{D}(N)$, with the scalars $\mathbb{Q}^{\times}$ acting
trivially. It is straightforward to check that two elements $\alpha,\beta$ of
$N_{B_{>0}^{\times}}(O_{N}^{1})$ define the same automorphism if and only if
$\alpha\beta^{-1}\in\mathbb{Q}^{\times}O_{N}^{1}.$ We define the group of
Atkin–Lehner involutions as follows:

$W(D,N)\colonequals
N_{B_{>0}^{\times}}(O_{N}^{1})/\mathbb{Q}^{\times}O_{N}^{1}.$

We have an Atkin–Lehner involution $w_{m}$ for every Hall divisor $m$ of $DN$.
There is an identification

$W(D,N)\simeq(\mathbb{Z}/2\mathbb{Z})^{\omega(DN)}$

and we write $W(D,N)=\\{w_{m}\ |\ m\parallel ND\\}$ [Voi21, Ch. 28].

###  2.4. Genus of Shimura curves

In this subsection we recall an explicit formula for the genus of
$X_{0}^{D}(N)$. For coprime positive integers $D$ and $N$ such that $D$ is
squarefree, define

$e_{k}(D,N)\colonequals\prod_{p|D}\left(1-\left(\frac{-k}{p}\right)\right)\prod_{q\parallel
N}\left(1+\left(\frac{-k}{q}\right)\right)\prod_{q^{2}|N}\nu_{p}(k),$

where

$\nu_{p}(k)=\begin{cases}2&\quad\text{if }\left(\frac{-k}{q}\right)=1,\\\
0&\quad\text{otherwise},\\\ \end{cases}$

and $\left(\frac{\cdot}{\cdot}\right)$ is the Kronecker quadratic symbol.

######  Remark 2.1.

The above quantities have geometric meaning, for example:

  * •

$e_{4}(D,N)$ is the number of $\mathbb{Z}[i]$-CM points on $X_{0}^{D}(N)$, and
also the number of order $2$ elliptic points on $X_{0}^{D}(N)$,

  * •

$e_{3}(D,N)$ is the number of
$\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$-CM points on $X_{0}^{D}(N)$,
and also the number of order $3$ elliptic points on $X_{0}^{D}(N)$.

Let $\varphi$ and $\psi$ be the multiplicative functions such that

$\varphi(p^{k})=p^{k}-p^{k-1},\quad\psi(p^{k})=p^{k}+p^{k-1}.$

######  Proposition 2.1.

[Ogg83, p.280, 301] For $D>1$, the genus of $X_{0}^{D}(N)$ is

$g(X_{0}^{D}(N))=1+\frac{\varphi(D)\psi(N)}{12}-\frac{e_{4}(D,N)}{4}-\frac{e_{3}(D,N)}{3}.$

######  Remark 2.2.

A Shimura curve $X_{0}^{D}(N)$ with $N$ squarefree is of genus zero if and
only if the pair $(D,N)$ is in the set:

$\\{(6,1),(10,1),(22,1)\\}.$

A Shimura curves $X_{0}^{D}(N)$ with $N$ squarefree is of genus one if and
only if the pair $(D,N)$ is in the set:

$\\{(14,1),(15,1),(21,1),(6,5),(10,3),(33,1),(34,1),(6,7),(46,1),(10,7),(6,13)\\}.$

Equations for these genus one curves and formulae for their Atkin–Lehner
involutions can be found in [GR06, Theorem 3.4].

A lower bound for the genus is given by the following:

######  Lemma 2.1.

[Sai22, Lemma 10.5] For $D>1$ an indefinite rational quaternion discriminant
and $N\in\mathbb{Z}_{\geq 1}$ relatively prime to $D$, we have

$g(X_{0}^{D}(N))>1+\frac{DN}{12}\left(\frac{1}{e^{\gamma}\log\log(DN)+\frac{3}{\log\log{6}}}\right)-\frac{7\sqrt{DN}}{3},$

where $\gamma$ is the Euler constant.

###  2.5. Gonality of a curve

For a field $K$ and a curve $C/K$, the $K$-gonality
$\operatorname{gon}_{K}(C)$ is defined to be the least degree of a non-
constant morphism $f:C\rightarrow\mathbb{P}^{1}$ defined over $K$. For
$\overline{K}$ an algebraic closure of $K$, the $\overline{K}$-gonality is
also called the geometric gonality. We will use the following result to obtain
an upper bound on the genus of $X_{0}^{D}(N)$:

######  Theorem 2.1.

[Abr96, Theorem 1.1]

$\frac{21}{200}\left(g(X_{0}^{D}(N))-1\right)\leq\operatorname{gon}_{\mathbb{C}}(X_{0}^{D}(N)).$

###  2.6. CM points

There is a special type of points on Shimura curve called CM points that play
a very important rule. As Shimura curves lack cusps, Guo–Yang [GY17] used CM
points to help in the computation of defining equations for all geometrically
hyperelliptic Shimura curves $X_{0}^{D}(N)$. Moreover, if a point is fixed by
some Atkin–Lehner involution, then it is a CM point. This observation is one
of the main ingredients in the proof of Theorem 1.1.

Let $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field which admits an
embedding

$q:K\hookrightarrow B.$

Let $R\subset K$ be an order. Then $R$ is said to be optimally embedded in
$O_{N}$ if we have $q(K)\cap O_{N}=q(R).$

For an optimally embedded order $R$, there is exactly one fixed point $z$ of
the quotient $\left(\iota_{\infty}(O_{N}^{1})/\\{\pm
1\\}\right)\backslash\mathbb{H}$ under the action of $\iota_{\infty}(q(R)).$
We say that $z$ is a CM point for $R$. The set

$\operatorname{CM}(R):=\\{z\in\left(\iota_{\infty}(O_{N}^{1})/\\{\pm
1\\}\right)\backslash\mathbb{H}\ |\ \iota_{\infty}(q(R))\cdot z=z,\
q:R\hookrightarrow B\ \text{an optimal embedding}\\}$

is the set of CM points for $R$. We identify the CM points on
$\left(\iota_{\infty}(O_{N}^{1})/\\{\pm 1\\}\right)\backslash\mathbb{H}$ with
the corresponding points on $X_{0}^{D}(N)$.

For any $W\leq W(D,N)$, there is the natural quotient map

$\pi_{W}:X_{0}^{D}(N)\longrightarrow X_{0}^{D}(N)/W.$

We say a point $Q\in X_{0}^{D}(N)/W$ is a CM point of $X_{0}^{D}(N)/W$ if any
of its preimages on $X_{0}^{D}(N)$ is a CM point.

######  Remark 2.3.

CM points on $X_{0}^{D}(N)$ correspond to abelian surfaces with extra
endomorphisms by the CM field $K$:

$\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\simeq M_{2}(K).$

We state an alternative description of CM points. Recall the cover map

$X_{0}^{D}(N)\to X_{0}^{D}(1).$

A CM point for $R$ on $X_{0}^{D}(N)$ is the same as a point in the fibre over
$[(A,\iota)]\in X_{0}^{D}(1)$ such that

$\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\simeq M_{2}(K)$

and the ring of QM-equivariant endomorphisms of $A$ is isomorphic to $R$ (cf.
[Sai22, §2.2]).

######  Theorem 2.2.

[Ogg83, Section 1] Let $N$ be squarefree and $m|DN$. The set of fixed points
of $X_{0}^{D}(N)$ under the involution $w_{m}$ is

$X_{0}^{D}(N)^{\langle
w_{m}\rangle}=\begin{cases}\operatorname{CM}(\mathbb{Z}[\sqrt{-1}])\cup\operatorname{CM}(\mathbb{Z}[\sqrt{-2}])&\text{if}\
m=2,\\\
\operatorname{CM}(\mathbb{Z}[\sqrt{-m}])\cup\operatorname{CM}\left(\mathbb{Z}\left[\frac{1+\sqrt{-m}}{2}\right]\right)&\text{if}\
m\equiv 3\ \text{mod}\ 4,\\\ \operatorname{CM}(\mathbb{Z}[\sqrt{-m}])\
&\text{otherwise}.\end{cases}$

We would like to compute $\\#X_{0}^{D}(N)^{\langle w_{m}\rangle}$. To this
end, we need to first understand $\\#CM(R)$ for an order $R$.

Assume $N$ is squarefree. We will use the following notation:

  * •

$\Delta_{R}$ be the discriminant of $R$,

  * •

$f$ the conductor of $R$,

  * •

$h(R)$ the class number of $R$,

  * •

$\big{(}\frac{K}{p}\big{)}$ be the Kronecker symbol,

  * •

$\big{(}\frac{R}{p}\big{)}\colonequals\big{(}\frac{K}{p}\big{)}$ if $p\nmid f$
and $1$ otherwise.

The following three quantities are needed in the computation of
$\\#\operatorname{CM}(R)$:

$D(R)\colonequals\prod_{p|D,\ \big{(}\frac{R}{p}\big{)}=-1}p,\ \ \
N(R)\colonequals\prod_{p|N,\ \big{(}\frac{R}{p}\big{)}=1}p,\ \ \
N^{\ast}(R)\colonequals\prod_{p|N,\ p\nmid f,\big{(}\frac{R}{p}\big{)}=1}p.$

######  Theorem 2.3 ([Ogg83, Section 1], [Jor81], [BD96, Lemma 2.5]).

The set $\operatorname{CM}(R)$ is nonempty if and only if
$\frac{DN}{D(R)N^{\ast}(R)}|\Delta_{R}$ and in this case

$\\#\operatorname{CM}(R)=2^{\omega(D(R)N(R))}\cdot h(R).$

The code that computes $\\#\operatorname{CM}(R)$ and $\\#X_{0}^{D}(N)^{\langle
w_{m}\rangle}$ was previously implemented as part of the code accompanying
[PS23].

##  3\. Proof of Theorem 1.1

Assume $X_{0}^{D}(N)$ is a Shimura curve with squarefree level $N$ that admits
a degree $d$ smooth plane model over $\mathbb{Q}$. We know that the geometric
gonality of a smooth plane curve of degree $d$ is $d-1$, see [CK90, Theorem
A], while the genus-degree formula states that
$g(X_{0}^{D}(N))=\frac{(d-1)(d-2)}{2}$. Then using Theorem 2.1 we obtain:

$\frac{21}{200}\left(\frac{(d-1)(d-2)}{2}-1\right)=\frac{21}{200}(g(X_{0}^{D}(N))-1)\leq\operatorname{gon}_{\mathbb{C}}(X_{0}^{D}(N))=d-1,$

which implies that $d\leq 21$. Thus the genus of $X_{0}^{D}(N)$ is at most
$190$.

By Lemma 2.1, we find that for $DN>110011$, we have $g(X_{0}^{D}(N))>190$.
Therefore, we only need to consider the pairs $(D,N)$ where $DN\leq 110011$ is
squarefree. We may consider only those pairs for which $\omega(D)$ is even, as
the set of ramified places has even cardinality. We further need that

$g(X_{0}^{D}(N))\in\left\\{\frac{(d-1)(d-2)}{2}:1\leq d\leq 21\right\\}.$

Thus we are left to analyze $312$ pairs $(D,N)$. The discriminant $D$ is at
most $6990$, while the level $N$ is at most $1033$. Let $(D,N)$ be such a
pair, for which

$g(X_{0}^{D}(N))=\frac{(d-1)(d-2)}{2}$

for some $d\in\\{1,\cdots,21\\}$.

By Theorem 2.2, only CM points can be fixed by an Atkin–Lehner involution
$w_{m}$. Further, by Theorem 2.3, one is able to compute
$\\#X_{0}^{D}(N)^{\langle w_{m}\rangle}$. We will combine these results with
the following:

######  Theorem 3.1.

[HKKO10, Remark 2.1 (i) and Theorem 2.2 with $n=2$] Let $C$ be a smooth plane
curve of degree $d\geq 4$ and $\sigma$ an involution of $C$. Then the
involution $\sigma$ has $f=d+\frac{1-(-1)^{d}}{2}$ fixed points.

If $X_{0}^{D}(N)$ admits a plane model of degree $d\geq 4$, then by Theorem
3.1 we have that:

(1) $\\#X_{0}^{D}(N)^{\langle
w_{m}\rangle}=d+\frac{1-(-1)^{d}}{2}\quad\text{for all }\,1<m\parallel DN.$

For $298$ out of the $312$ candidate pairs $(D,N)$ we have that
$g(X_{0}^{D}(N))\geq 3$, while for the remaining $14$ we have
$g(X_{0}^{D}(N))\leq 1$. Running our code [Pad] on these $298$ pairs $(D,N)$
we find that Property (1) does not hold for any of the curves with
$g(X_{0}^{D}(N))\geq 3$. Thus there is no Shimura curve $X_{0}^{D}(N)$ with
$N$ squarefree admitting a smooth plane model of degree $d\geq 4$. In
conclusion, if $X_{0}^{D}(N)$ admits a smooth plane model for some $N$
squarefree, then $g(X_{0}^{D}(N))\leq 1$.

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