On smooth plane models for modular curves of Shimura type

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.

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.

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).

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

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: $\phi_{K}(C)\simeq\mathbb{P}^{1}$.

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.

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 $\left(\begin{smallmatrix}1&h\mathbb{Z}\\ 0&1\end{smallmatrix}\right)=\Gamma\cap\left(\begin{smallmatrix}1&\mathbb{Z}\\ 0&1\end{smallmatrix}\right)$.

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

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.

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

Moreover, if we decompose the space by Dirichlet characters as

then we obtain

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 $6$.

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

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

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.

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.

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 $\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{g-1}$.

We deal next with the higher genus situations.

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 $C\rightarrow\mathbb{P}^{2}:\,P\mapsto(x(P):y(P):z(P))$.

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:

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:

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].

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.

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:

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

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 $MK^{2}\leq 500$.

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.

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