2 The Modular curves $X_H$

2.1 Notation

In this article, $\ell $ and p denote primes, and N is a positive integer. We define

with the understanding that $\mathbb {Z}(1)$ is the zero ring and $\operatorname {GL}_2(1)$ and $\operatorname {SL}_2(1)$ are trivial groups. We then have ${\widehat {\mathbb {Z}}}=\varprojlim \mathbb {Z}(N)$ and $\operatorname {GL}_2({\widehat {\mathbb {Z}}})=\varprojlim \operatorname {GL}_2(N)$ and natural projection maps

$$\begin{align*}\pi_N\colon \operatorname{GL}_2({\widehat{\mathbb{Z}}})\to \operatorname{GL}_2(N). \end{align*}$$

We view $\mathbb {Z}(N)^2$ as a $\mathbb {Z}(N)$ -module of column vectors equipped with a left action of $\operatorname {GL}_2(N)$ via matrix-vector multiplication.

For $N>1$ , we use $B(N)$ to denote the Borel subgroup of $\operatorname {GL}_2(N)$ consisting of upper triangular matrices, $C_{\operatorname {\mathrm {sp}}}(N)$ to denote the split Cartan subgroup of diagonal matrices in $\operatorname {GL}_2(N)$ and $C_{\operatorname {\mathrm {ns}}}(N)$ to denote the nonsplit Cartan subgroup of $\operatorname {GL}_2(N)$ , which can be constructed by picking an imaginary quadratic order $\mathcal O$ in which every prime $\ell |N$ is inert and embedding $(\mathcal O/N\mathcal O)^{\times }$ in $\operatorname {GL}_2(N)$ via its action on a $\mathbb {Z}(N)$ -basis for $\mathcal O/N\mathcal O$ (see Section 12), which is unique up to conjugacy. We use $N_{\operatorname {\mathrm {sp}}}(N)$ to denote the subgroup of the normaliser of $C_{\operatorname {\mathrm {sp}}}(N)$ in $\operatorname {GL}_2(N)$ whose projection to $\operatorname {GL}_2(M)$ contains $C_{\operatorname {\mathrm {sp}}}(M)$ with index 2 for all $M>2$ dividing N, which is unique up to conjugacy and equal to the normaliser when $N>2$ is a prime power; we similarly define $N_{\operatorname {\mathrm {ns}}}(N)$ . If G is a subgroup of the projectivisation $\operatorname {PGL}_2(N)$ of $\operatorname {GL}_2(N)$ , then we denote by $G(N)$ the preimage of G in $\operatorname {GL}_2(N)$ .

2.2 Group theory and Galois representations

Let $N \ge 1$ be an integer, let k be a perfect field of characteristic coprime to N, and let

. A basis $(P_1,P_2)$ of $E[N]({\overline {k}})$ determines an isomorphism $E[N]({\overline {k}}) \overset \sim \rightarrow \mathbb {Z}(N)^2$ via the map $P_i \mapsto e_i$ . This gives rise to an isomorphism $\iota \colon \operatorname {\mathrm {Aut}} \left ( E[N]({\overline {k}})\right ) \simeq \operatorname {GL}_2(N)$ as follows: for $\phi \in \operatorname {\mathrm {Aut}} \left (E[N]({\overline {k}})\right )$ such that

$$ \begin{align*} \phi(P_1) = &\, aP_1 + cP_2 \\ \phi(P_2) = &\, bP_1 + dP_2, \end{align*} $$

we define

which yields a Galois representation $G_k \to \operatorname {GL}_2(N)$ given by $\sigma \mapsto \iota (\sigma _N)$ , where $\sigma _N$ is the automorphism of $E[N]({\overline {k}})$ induced by $\sigma $ .

Remark 2.1. Our choice of $\iota $ corresponds to the left action of $\operatorname {GL}_2(N)$ on $\mathbb {Z}(N)^2$ . Many sources are ambiguous about the choice of left vs. right action; this ambiguity often makes no difference (see [Reference Rouse and Zureick-BrownRZB15, Remark 2.2]), but it does when $H\le \operatorname {GL}_2(N)$ is not conjugate to its transpose. Our choice here is consistent with [Reference SutherlandSut16, Reference Sutherland and ZywinaSZ17], but differs from [Reference Rouse and Zureick-BrownRZB15], which uses right actions.

For each elliptic curve $E/k$ , we fix a system of compatible bases for $E[N]({\overline {k}})$ for all $N\ge 1$ coprime to the characteristic of k, meaning if $N=N_1N_2$ , then the bases for $E[N_1]({\overline {k}})$ and $E[N_2]({\overline {k}})$ are the images of the basis for $E[N]({\overline {k}})$ under multiplication by $N_2$ and $N_1$ , respectively. Let $\iota _E\colon \! \operatorname {\mathrm {Aut}}(E[N]({\overline {k}})) \overset \sim \longrightarrow \operatorname {GL}_2(N)$ denote the isomorphism determined by the basis for $E[N]({\overline {k}})$ , and define the Galois representation $\rho _{E,N}\colon G_k\to \operatorname {GL}_2(N)$ via . When k has characteristic zero, we shall also use $\iota _E$ to denote the isomorphism $\iota _E\colon \operatorname {\mathrm {Aut}}(E_{\mathrm {tor}}({\overline {k}})) \overset \sim \longrightarrow \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ determined by this system of bases and use it to define the Galois representation $\rho _E\colon G_k\to \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ via . If k is a number field and ${\mathfrak p}$ is a prime of good reduction for E, for each $N\ge 1$ coprime to $N({\mathfrak p})$ , we shall use the reduction of our chosen basis for $E[N]({\overline {k}})$ as a basis for $E_{\mathfrak p}[N](\overline {\mathbb F}_{\mathfrak p})$ .

For a subgroup $H \leq \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ , we use $H(N)$ to denote the image $\pi _N(H)$ of H under $\pi _N$ . An open subgroup $H \leq \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ contains the kernel of $\pi _N\colon \operatorname {GL}_2({\widehat {\mathbb {Z}}})\to \operatorname {GL}_2(N)$ for some $N\ge 1$ ; the least such N is the level of H, in which case $H = \pi _N^{-1}(H(N))$ is completely determined by $\pi _N(H)$ . We will typically work directly with subgroups of $\operatorname {GL}_2(N)$ instead of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ , with the understanding that every subgroup of $\operatorname {GL}_2(N)$ uniquely determines an open subgroup of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ as its inverse image under $\pi _N$ ; we may use the symbol H to denote both H and $\pi _N(H)$ , which we note have the same index $[\operatorname {GL}_2({\widehat {\mathbb {Z}}}):H]=[\operatorname {GL}_2(N):\pi _N(H)]$ . We define the level of a subgroup $H \leq \operatorname {GL}_2(N)$ to be the level of $\pi _N^{-1}(H)$ .

We may also identify open subgroups of $\operatorname {GL}_2(\mathbb {Z}_{\ell })$ with their preimages in $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ under the projection map $\operatorname {GL}_2({\widehat {\mathbb {Z}}})\to \operatorname {GL}_2(\mathbb {Z}_{\ell })$ . For any $e\ge 0$ , we may identify $H\le \operatorname {GL}_2(\ell ^e)$ with the corresponding open subgroups of $\operatorname {GL}_2(\mathbb {Z}_{\ell })$ or $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of the same index. In what follows, we shall only be interested in subgroups of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ up to conjugacy and view each subgroup H as a representative of its conjugacy class. All inclusions $H_1\le H_2$ of subgroups of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ should be understood to indicate that $H_1$ is conjugate to a subgroup of $H_2$ .

2.3 Modular curves

Let H be a subgroup of $\operatorname {GL}_2(N)$ of level N. We define the modular curve $Y_H$ (respectively, $X_H$ ) to be the coarse moduli space of the stack $\mathcal {M}^0_H$ (respectively, $\mathcal {M}_H$ ), over $\operatorname {\mathrm {Spec}} \mathbb {Z}[1/N]$ , which parametrises elliptic curves (respectively, generalised elliptic curves) with H-level structure. Equivalently, by [Reference Deligne and RapoportDR73, IV-3.1] (see also [Reference Rouse and Zureick-BrownRZB15, Lemma 2.1]), $X_H$ is isomorphic to the coarse space of the stack quotient $X(N)/H$ , where $X(N)$ is the classical modular curve parametrising full level structures.

More precisely, an H-level structure on an elliptic curve $E/{\overline {k}}$ is an equivalence class $[\iota ]_H$ of isomorphisms $\iota \colon E[N]({\overline {k}})\to \mathbb {Z}(N)^2$ , where $\iota \sim \iota '$ if $\iota = h\circ \iota '$ for some $h\in H$ . The set $Y_H({\overline {k}})$ consists of equivalence classes of pairs $(E,[\iota ]_H)$ , where $(E,[\iota ]_H)\sim (E',[\iota ']_H)$ if there is an isomorphism $\phi \colon E\to E'$ for which the induced isomorphism $\phi _N\colon E[N]\to E'[N]$ satisfies $\iota \sim \iota '\circ \phi _N$ . Equivalently, $Y_H({\overline {k}})$ consists of pairs $(j(E),\alpha )$ , where $\alpha =HgA_E$ is a double coset in $H\backslash \operatorname {GL}_2(N)/A_E$ , where with .

Each $\sigma \in G_k$ induces an isomorphism $E^{\sigma }[N]\to E[N]$ defined by $P \mapsto \sigma ^{-1}(P)$ , which we denote $\sigma ^{-1}$ . We define a right $G_k$ -action on $Y_H({\overline {k}})$ via $(E,[\iota ]_H)\mapsto (E^{\sigma },[\iota \circ \sigma ^{-1}]_H)$ . The set of k-rational points $Y_H(k)$ consists of the elements in $Y_H({\overline {k}})$ that are fixed by this $G_k$ -action. Each $P\in Y_H(k)$ is represented by a pair $(E,[\iota ]_H)\in Y_H(k)$ such that E is defined over k and for all $\sigma \in G_k$ there exist $\varphi \in \operatorname {\mathrm {Aut}}(E_{{\overline {k}}})$ and $h\in H$ such that

(2.1) $$ \begin{align} \iota\circ\sigma^{-1} = h\circ \iota\circ \varphi_N, \end{align} $$

as shown in [Reference ZywinaZyw15b, Lemma 3.1].Footnote ¹ Equivalently, a pair $(j(E),\alpha )$ with $\alpha =HgA_E$ lies in the set $Y_H(k)$ if and only if we have $j(E)\in k$ and $Hg\rho _{E,N}(\sigma ) A_E=HgA_E$ for all $\sigma \in G_k$ .

Remark 2.1. For an elliptic curve $E/k$ , if $\rho _{E,N}(G_k)\le H$ , then there exists an isomorphism $\iota \colon E[N]\overset \sim \rightarrow \mathbb {Z}(N)^2$ for which $(E, [\iota ]_H)\in Y_H(k)$ . Conversely, assuming $\textrm {char}(k)\ne 2,3$ , if $(E, [\iota ]_H)\in Y_H(k)$ , then for every twist $E'$ of E, there is an isomorphism $\iota '\colon E'[N]\overset \sim \rightarrow \mathbb {Z}(N)^2$ with $(E', [\iota ']_H)\in Y_H(k)$ , and for at least one $E'$ , we have $\rho _{E',N}(G_k)\le H$ , as we now show.

If $E'$ is a twist of E, then there is an isomorphism $\phi \colon E^{\prime }_{\bar k}\overset {\sim }{\to }E_{\bar k}$ that induces an isomorphism $\phi _N\colon E'[N]({\overline {k}})\overset {\sim }{\to }E[N]({\overline {k}})$ , and for the pairs $(E,[\iota ]_H)$ and $(E',[\iota ']_H)$ represent the same point in $Y_H({\overline {k}})$ ; thus if $(E,[\iota ]_H)$ lies in $Y_H(k)$ , then so does $(E',[\iota ']_H)$ .

If the point $(E,[\iota ]_H)=(j(E),\alpha )$ lies in $Y_H(k)$ , say for $\alpha =HgA_E$ , then $\rho _{E,N}(G_k)$ must lie in $g^{-1}HgA_E$ . In other words, up to conjugacy in $\operatorname {GL}_2(N)$ , the subgroups $\rho _{E,N}(G_k)$ and H can differ only by elements of $A_E$ , which for $\textrm {char}(k)\ne 2,3$ is cyclic of order 2, 4, 6, the last two occurring only for $j(E)=0,1728$ . When $\#A_E=2$ , every twist of E is a quadratic twist, and [Reference SutherlandSut16, Lemma 5.24] implies that there is a quadratic twist $E'$ of E for which $\rho _{E',N}(G_k)\le H$ . Otherwise E has potential CM, and we may apply Proposition 12.1, which also addresses quartic and sextic twists. These results assume k is a number field, but the arguments apply whenever $A_E$ is cyclic (including $\textrm {char}(k)= 2,3$ provided $j(E)\ne 0,1728$ ).

Remark 2.2. The elliptic curves

$$\begin{align*}E\colon {y^2 = x^3 - 1}\qquad\text{and}\qquad E' \colon {y^2 = x^3 + 2} \end{align*}$$

with $j(E)=j(E') = 0$ are cubic twists that correspond to the same point in $X_0(2)(\mathbb {Q})$ , but E has a rational point of order 2 and $\rho _{E,2}(G_{\mathbb {Q}}) = B(2)$ , while $E'$ has no rational points of order 2 and $\rho _{E',2}(G_{\mathbb {Q}}) = \operatorname {GL}_2(2)$ . In particular, [Reference Rouse and Zureick-BrownRZB15, Lemma 2.1] is incorrectly stated: it holds only up to twist.

The complement is the set of cusps of $X_H$ and corresponds to generalised elliptic curves with H-level structure. The set of k-rational cusps $X_H^{\infty }(k)$ can be alternatively described as follows. Let . We define a right $G_k$ -action on $H\backslash \operatorname {GL}_2(N)/U(N)$ via $hgu\mapsto hg\chi _N(\sigma )u$ , where is defined by $\sigma (\zeta _N)=\zeta _N^e$ . By [Reference Deligne and RapoportDR73, IV-5.3], $X_H^{\infty }(k)$ is in bijection with the subset of $H\backslash \operatorname {GL}_2(N)/U(N)$ fixed by $\chi _N(G_k)$ .

We define the genus $g(H)$ of H and $\pi _N^{-1}(H)$ to be the genus of each of the geometric connected components of the modular curve $X_H$ , which can be directly computed from H via the usual formula

$$\begin{align*}g(H) = g(\Gamma_H) = 1+\frac{i(\Gamma_H)}{12} - \frac{\nu_2(\Gamma_H)}{4} - \frac{\nu_3(\Gamma_H)}{3} - \frac{\nu_{\infty}(\Gamma_H)}{2}, \end{align*}$$

if we take , let $i(\Gamma _H)=[\operatorname {SL}_2(N):\Gamma _H]$ , let $\nu _2$ (respectively, $\nu _3$ ) count the right cosets of $\Gamma _H$ in $\operatorname {SL}_2(N)$ that contain a conjugate of $\left [\begin {smallmatrix} 0 & 1 \\ -1 & 0 \end {smallmatrix}\right ]$ (respectively, $\left [\begin {smallmatrix} 0 & 1 \\ -1 & -1 \end {smallmatrix}\right ]$ ) and let $\nu _{\infty }(\Gamma _H)$ count the orbits of $\Gamma _H\backslash \operatorname {SL}_2(N)$ under the right action of $\left [\begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix}\right ]$ . The function GL2Genus in the file gl2.m in [Reference Rouse, Sutherland and Zureick-BrownRSZB21] implements this computation.

We define the index $i(H)$ of H to be the integer $[\operatorname {GL}_2({\widehat {\mathbb {Z}}})\!:\!H]=[\operatorname {GL}_2(N)\!:\!H(N)]$ .

Remark 2.3. For $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of level N, the inverse image $\Gamma $ of $\Gamma _H\leq \operatorname {SL}_2(N)$ in $\operatorname {SL}_2(\mathbb {Z})$ is a congruence subgroup of some level $M|N$ , but M may be strictly smaller than N; see [Reference Sutherland and ZywinaSZ17] for examples. In any case, the base change of $X_H$ to $\mathbb {Q}(\zeta _N)$ breaks into connected components that are geometrically connected and are each isomorphic over $\mathbb {C}$ to the modular curve $X_{\Gamma }$ obtained by taking the quotient of the extended upper half plane by $\Gamma $ .

Remark 2.4. For $H\le \operatorname {GL}_2(N)$ , if $\det H = \mathbb {Z}(N)^{\times }$ , then $X_H$ is geometrically connected. We mostly consider the case where $\det H = \mathbb {Z}(N)^{\times }$ in this paper, although in Appendix A, we make use of the case $H = \{I\}$ , where $X_H = X(N)$ . When $\det H < \mathbb {Z}(N)^{\times }$ is a proper subgroup, $X_H$ is still a curve defined over $\mathbb {Q}$ , but it is not geometrically connected; its connected components are defined over $\mathbb {Q}(\zeta _N)^H$ , indexed (via the Weil pairing on the level structures) by cosets of $\det H$ and isomorphic (over $\mathbb {Q}(\zeta _N)^H$ ) to $X_{\Gamma }$ . See Appendix A where this is worked out in more detail for $X(N)$ , and Example 6.3, where we discuss the isogeny decomposition of the ‘Jacobian’ of such an $X_H$ .

2.4 Subgroup labels

To help organise our work, we introduce the following labeling convention for uniquely identifying (conjugacy classes of) open subgroups H of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ , which will also serve as labels of the corresponding modular curves $X_H$ . All the subgroups we shall consider satisfy $\det (H)={\widehat {\mathbb {Z}}}^{\times }$ and have labels of the form

$$\begin{align*}\texttt{N.i.g.n} \end{align*}$$

where N, i, g, n are the decimal representations of integers $N,i,g,n$ defined as follows:

• ○ $N=N(H)$ is the level of H;

• ○ $i=i(H)$ is the index of H;

• ○ $g=g(H)$ is the genus of H;

• ○ $n\ge 1$ is an ordinal indicating the position of H among all subgroups of the same level, index and genus with $\det (H)={\widehat {\mathbb {Z}}}^{\times }$ under the ordering defined below.

To define the ordinal n, we must totally order the set $S(N,i,g)$ of open subgroups of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of level N, index i and genus g with $\det (H)={\widehat {\mathbb {Z}}}^{\times }$ , and we want this ordering to be efficiently computable. With this in mind, we define the following invariants for each $H\in S(N,i,g)$ :

• ○ the parent list of H is the lexicographically ordered list of labels of the subgroups $G\leq \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of which H is a maximal subgroup;

• ○ the orbit signature of H as the ordered list of triples $(e,s,m)$ , where m counts orbits of $\mathbb {Z}(N)^2$ under the left-action of H of size s and exponent e (the least integer that kills every element of the orbit);Footnote ²

• ○ the class signature of H is the ordered set of tuples $(o,d,t,s,m)$ , where m counts the conjugacy classes of elements of $H\le \operatorname {GL}_2(N)$ of size s whose elements have order o, determinant d and trace t;

• ○ the minimal conjugate of H is the least subgroup of $\operatorname {GL}_2(N)$ conjugate to H, where subgroups of $\operatorname {GL}_2(N)$ are ordered lexicographically as ordered lists of tuples of integers $(a,b,c,d)$ with $a,b,c,d\in [0,N-1]$ corresponding to the matrix $\left [\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ]\in H$ .

We now totally order the set $S(N,i,g,d)$ lexicographically by parent list, orbit signature, class signature and minimal conjugate; these invariants are intentionally ordered according to the difficulty of computing them, and in most cases the first two or three suffice to distinguish every element of $S(N,i,g,d)$ , meaning we rarely need to compute minimal conjugates, which is by far the most expensive invariant to compute.

Example 2.1. Let $H=N_{\operatorname {\mathrm {ns}}}(11^2)$ be the normaliser of the nonsplit Cartan subgroup of $\mathbb {Z}(11^2)$ , which has level $11^2=121$ and index $5\cdot 11^3 = 6655$ . There are just two subgroups $H\le \operatorname {GL}_2(11^2)$ of index 6655 (up to conjugacy), only one of which has $\det (H)={\widehat {\mathbb {Z}}}^{\times }$ , so computing $g(H)=511$ suffices to determine the label 121.6655.511.1 of $N_{\operatorname {\mathrm {ns}}}(11^2)$ .

If we instead take $H=N_{\operatorname {\mathrm {ns}}}(11)$ with level 11 and index 55, we find there are two subgroups of $\operatorname {GL}_2(11)$ of index 55; they have genus 1 and determinant index 1 and thus belong to the set $S(11,55,1,1)$ . These subgroups and have parent list $(\texttt {1.1.0.1})$ and orbit signature $(1,1,1), (11,120,1)$ . Their class signatures begin $(1,1,2,1,1), (2,1,9,1,1), (2,10,0,12,1)$ but differ in the fourth tuple, which is $(3,1,10,2,1)$ for $H_1$ but $(3,1,10,8,1)$ for $H_2$ , meaning $H_1$ contains a conjugacy class of size 2 whose elements have order 3, determinant 1, and trace 10, but $H_2$ does not. It follows that the labels of $H_1$ and $H_2$ are 11.55.1.1 and 11.55.1.2, and that $N_{\operatorname {\mathrm {ns}}}(11)=H_1$ .

Remark 2.2. This labeling system can be extended to arbitrary open $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ using labels of the form M.d.m-N.i.g.n, where M.d.m uniquely identifies $\det (H)\le {\widehat {\mathbb {Z}}}^{\times }$ according to its level $M|N$ , index $d:=[{\widehat {\mathbb {Z}}}^{\times }\!\!:\!\det (H)]$ and an ordinal m that distinguishes $\det (H)$ among the open subgroups $D\le {\widehat {\mathbb {Z}}}^{\times }$ of level M and index d.Footnote ³ To define m, we order the index d subgroups $D\le (\mathbb {Z}/M\mathbb {Z})^{\times }$ lexicographically according to the lists of labels $\texttt {M.a}$ of Conrey characters $\chi _M(a,\cdot )$ of modulus M whose kernels contain D; see [Reference Best, Bober, Booker, Costa, Cremona, Maarten, Lee, Lowry-Duda, Roe, Sutherland and VoightBBB+21, Section 3.2] for the definition of $\chi _M(a,\cdot )$ . We then redefine N and i to be the relative level and index of H as a subgroup of the preimage G of $\det (H)$ in $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ : this means N is the least positive integer for which $H=\pi _N^{-1}(\pi _N(H))$ , where $\pi _N\colon G\to G(N)$ is the reduction map and $i=[G:H]$ . This convention ensures that if $E/K$ has Galois image $\rho _E(G_K)=H$ , the values of N, i, g in the label of $\rho _E(G_L)$ will be the same for every finite extension $L/K$ .

3 Arithmetically maximal subgroups

Properties of the Weil pairing imply that if $H = \rho _E(G_{\mathbb {Q}})$ , then $\det (H)= \mathbb {Z}^{\times }$ , and more generally, that if E is an elliptic curve over a number field K and $H=\rho _E(G_K)$ , then $[\mathbb {Z}^{\times }:\det (H)]=[K:K\cap \mathbb {Q}^{\mathrm {cyc}}]$ . In particular, condition (i) holds for every H that arises as $\rho _E(G_{\mathbb {Q}})$ for an elliptic curve $E/\mathbb {Q}$ . Condition (ii) is a necessary and sufficient condition for $X_{H}$ to have noncuspidal real points, by Proposition 3.5 of [Reference ZywinaZyw15b], and it follows from Proposition 3.1 of [Reference Sutherland and ZywinaSZ17] that every genus zero H of prime power level $N=\ell ^e$ that satisfies equations (i) and (ii) has infinitely many $\mathbb {Q}$ -points (the assumption $-I\in H$ is not used in the proof of Proposition 3.1 in [Reference Sutherland and ZywinaSZ17]). For odd $\ell $ , the matrices in condition (ii) are conjugate in $\operatorname {GL}_2(N)$ , and only the first is needed. This definition differs slightly from that used in [Reference Rouse and Zureick-BrownRZB15, Definition 3.1] for $\ell =2$ .

Let ${\mathcal S}_{\ell }(\mathbb {Q})$ be the set of arithmetically maximal subgroups of $\ell $ -power level, and let ${\mathcal S}_{\ell }^{\infty }(\mathbb {Q})$ be the set of open subgroups of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of $\ell $ -power level for which $j(X_H(\mathbb {Q}))$ is infinite.

The sets ${\mathcal S}_{\ell }^{\infty }(\mathbb {Q})$ are explicitly determined in [Reference Sutherland and ZywinaSZ17] (and in [Reference Rouse and Zureick-BrownRZB15] for $\ell =2$ ) and have cardinalities $1208,47,23,15,2,11$ for $\ell =2,3,5,7,11,13$ and cardinality $1$ for all $\ell>13$ .Footnote ⁴ To enumerate the set ${\mathcal S}_{\ell }(\mathbb {Q})$ , it would suffice to enumerate the maximal subgroups of all the groups in ${\mathcal S}_{\ell }^{\infty }(\mathbb {Q})$ , but we take a slightly different approach and use our knowledge of ${\mathcal S}_{\ell }^{\infty }(\mathbb {Q})$ to derive an upper bound on the level of the groups in ${\mathcal S}_{\ell }(\mathbb {Q})$ .

Lemma 3.2 reduces the problem of enumerating the sets ${\mathcal S}_{\ell }(\mathbb {Q})$ to a finite computation in $\operatorname {GL}_2(\ell ^{e_{\ell }+1})$ . This computation can be further constrained by computing the maximal index $i_{\ell }$ of any maximal subgroup of $H_0\in {\mathcal S}_{\ell }^{\infty }(\mathbb {Q})$ and then enumerating all (conjugacy classes of) subgroups of $\operatorname {GL}_2(\ell ^{e_{\ell }+1})$ of index at most $i_{\ell }$ , which can be efficiently accomplished using algorithms for subgroup enumeration that are implemented in Magma and other computer algebra systems. The advantages of this approach are that it enumerates subgroups up to conjugacy in $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ , rather than up to conjugacy in $H_0$ , and it includes all subgroups that have the same level and index of any $H\in {\mathcal S}_{\ell }(\mathbb {Q})$ , as well as all of the groups that contain them, which may be needed to compute the group labels defined in the previous section.

Table 4 summarises this computation. The ‘subgroups’ row counts the number of subgroups $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of $\ell $ -power level with $\det (H)={\widehat {\mathbb {Z}}}^{\times }$ that satisfy the level and index bounds in the first two rows, and the last three rows list the maximum level, index and genus that arise among the groups $H\in {\mathcal S}_{\ell }(\mathbb {Q})$ .

Table 4 Summary of arithmetically maximal $H\leq \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of $\ell $ -power level for $\ell \le 37$ . The level and index bounds are those implied by Lemma 3.2 and the paragraph following; the max level, index and genus rows are the maximum values realised by arithmetically maximal groups.

4 Bookkeeping

By [Reference MazurMaz78], [Reference SerreSer81, Section 8.4] and [Reference Bilu and ParentBP11], if $\ell \geq 17$ and $\rho _{E,\ell }$ is not surjective, then either $\ell = 17$ or $37$ and $j(E)$ is listed in Table 1, or the image $\rho _{E,\ell }(G_{\mathbb {Q}})$ is a subgroup of $N_{\operatorname {\mathrm {ns}}}(\ell )$ . In the latter case, for $\ell \equiv 1\pmod {3}$ , the image must be equal to $N_{\operatorname {\mathrm {ns}}}(\ell )$ (not a proper subgroup); and for $\ell \equiv 2\pmod {3}$ , the image is either $N_{\operatorname {\mathrm {ns}}}(\ell )$ or the index-3 subgroup

$$\begin{align*}G:=\big\{a^3: a \in C_{\operatorname{\mathrm{ns}}}(\ell)\big\} \cup \big\{ \left(\begin{smallmatrix}1 &0 \\0 & -1 \end{smallmatrix}\right) \cdot a^3: a \in C_{\operatorname{\mathrm{ns}}}(\ell) \big\} \end{align*}$$

by [Reference ZywinaZyw15a, Proposition 1.13]. Recent work by le Fourn and Lemos [Reference Le Fourn and LemosLFL21, Theorem 1.2] shows that if $\ell> 1.4\times 10^7$ , then $\rho _{E,\ell }(G_{\mathbb {Q}})$ cannot be a proper subgroup of $N_{\operatorname {\mathrm {ns}}}(\ell )$ .

Numerous individual cases remain. See Tables 12–17 in Appendix B for a list of all arithmetically maximal $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of $\ell $ -power level for $\ell \leq 37$ , and see Table 4 for a numerical summary of this data. Many of the corresponding modular curves $X_H$ are already handled in the literature, and these account for most of the exceptional points listed in Table 1. Below is a summary of prior results.

1. 1. Ligozat determined the rational points on $X_0(N)$ for $N = 27,49,11,17,19$ with labels 27.36.1.1, 49.56.1.1, 11.12.1.1, 17.18.1.1, 19.20.1.1 in [Reference LigozatLig75, 5.2.3.1]; there are exceptional points for $N = 11,17$ .

2. 2. Ligozat addressed $X_{S_4}(11)$ with label 11.55.1.2 in [Reference LigozatLig77, Proposition 4.4.8.1].

3. 3. Kenku determined the rational points on $X_0(13^2)$ (labeled 169.182.8.1) [Reference KenkuKen80] and on $X_0(125)$ (labeled 125.150.8.1) [Reference KenkuKen81]; the modular curve $X_0(5,25)$ with label 25.150.8.1 is isomorphic to $X_0(125)$ and thus also handled by Kenku.

4. 4. The curves $X_{\operatorname {\mathrm {sp}}}^+(25)$ and $X_{\operatorname {\mathrm {sp}}}^+(49)$ have labels 25.375.22.1 and 49.1372.94.1; they are isomorphic to $X_0^+(25^2)$ and $X_0^+(49^2)$ and are thus handled by Momose and Shimura [Reference Momose and ShimuraMS02, Theorems 0.1 and 3.14] and by Momose [Reference MomoseMom86, Theorem 3.6].

5. 5. Momose also handled $X_{\operatorname {\mathrm {sp}}}^+(11)$ with label 11.66.2.1 in [Reference MomoseMom84, Theorem 0.1].

6. 6. The curves $X_{\operatorname {\mathrm {ns}}}^+(13)$ and $X_{\operatorname {\mathrm {sp}}}^+(13)$ labeled 13.78.3.1 and 13.91.3.1 are handled via nonabelian Chabauty by Balakrishnan et al. in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM+19].

7. 7. The curves $X_{S_4}(13)$ and $X_{\operatorname {\mathrm {ns}}}^+(17)$ labeled 13.91.3.2 and 17.136.6.1 are handled via nonabelian Chabauty by Balakrishnan et al. in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM+21].

8. 8. Rouse and Zureick-Brown addressed all $X_H$ of $2$ -power level in [Reference Rouse and Zureick-BrownRZB15].

9. 9. The group with label 7.42.1.1 is the index 2 subgroup of $N_{\operatorname {\mathrm {sp}}}(7)$ containing the elements of $C_{\operatorname {\mathrm {sp}}}(7)$ with square determinant and the elements of $N_{\operatorname {\mathrm {sp}}}(7) - C_{\operatorname {\mathrm {sp}}}(7)$ with nonsquare determinant, handled by [Reference SutherlandSut12a, Section 3] and [Reference ZywinaZyw15a, Remark 4.3]. There is one exceptional j-invariant, which is interesting because elliptic curves with this mod 7 image admit a rational 7-isogeny locally everywhere but not globally; this is the unique example of this phenomenon over $\mathbb {Q}$ for any prime $\ell $ .

10. 10. Zywina addressed all remaining curves of prime level $\ell \leq 37$ except $N_{\operatorname {\mathrm {ns}}}(\ell )$ in [Reference ZywinaZyw15a].

11. 11. Elkies showed that the curve 9.81.1.1 has no rational points in [Reference ElkiesElk06, p. 5].

12. 12. Noncuspidal points on the modular curves labeled 49.168.12.1 and 49.168.12.2 are ‘ $7$ -exceptional’ in the sense of [Reference Greenberg, Rubin, Silverberg and StollGRSS14, Definition 1.1].Footnote ⁵ By [Reference Greenberg, Rubin, Silverberg and StollGRSS14, page 3], the only $7$ -exceptional elliptic curves are CM curves with $j = -15^3$ and $j = 255^3$ , which give rise to two points on the curve with label 49.168.12.2.

13. 13. The groups with labels 25.150.4.1, 25.150.4.2, 25.150.4.5 and 25.150.4.6 correspond to elliptic curves $E/\mathbb {Q}$ that admit a rational $5$ -isogeny with $5$ -adic image of index divisible by $25$ ; their existence is ruled out by [Reference GreenbergGre12, Theorem 2].

14. 14. The genus 2 rank 2 curves 25.50.2.1 and 25.75.2.1 are handled by Balakrishnan et al. in [Reference Balakrishnan, Dogra, Müller, Tuitman and VonkBDM+21, 4.3]. Each has a single exceptional j-invariant, and the minimal conductors for the two j-invariants are 396900 and 21175, respectively.

Of the 210 arithmetically maximal subgroups in Table 3, only 32 are not addressed above. Five are the groups $N_{\mathrm {ns}}(\ell )$ for $19\le \ell \le 37$ ; the remaining 27 are listed in Table 5.

Table 5 The arithmetically maximal subgroups of $\ell $ -power level for $\ell \le 17$ not addressed by previous results. Subgroups marked with asterisks are the subgroups H listed in Table 2 for which we are not able to determine $X_H(\mathbb {Q})$ .

5 Counting ${\mathbb F}_q$ -points on $X_H$

For any prime power $q=p^e$ coprime to N and $H\le \operatorname {GL}_2(N)$ , we may consider the modular curve $X_H$ over the finite field ${\mathbb F}_q$ . We can count ${\mathbb F}_q$ -rational points on $X_H$ via

$$\begin{align*}\#X_H({\mathbb F}_q) = \#X_H^{\infty}({\mathbb F}_q) + \#Y_H({\mathbb F}_q), \end{align*}$$

by counting fixed points of a right $G_{{\mathbb F}_q}$ -action on certain double coset spaces of $\operatorname {GL}_2(N)$ defined in Subsection 2.3. In this section, we explain how to do this explicitly and efficiently via a refinement of the method proposed in [Reference ZywinaZyw15b, Section 3]. Applications include:

• ○ checking for p-adic obstructions to rational points (by checking if $\#X_H({\mathbb F}_p)=0$ );

• ○ computing the zeta function of $X_H/{\mathbb F}_p$ (by computing $\#X_H({\mathbb F}_{p^r})$ for $1\le r\le g(H)$ );

• ○ determining the isogeny decomposition of $\operatorname {\mathrm {Jac}}(X_H)$ ;

• ○ determining the analytic rank of $\operatorname {\mathrm {Jac}}(X_H)$ .

The last two are described in Section 6. None require an explicit model for $X_H$ .

5.1 Counting points

If we put , we can use the permutation representation provided by the right $G_{{\mathbb F}_q}$ -action on the coset space $[\overline {H}\backslash \operatorname {GL}_2(N)]$ to compute

$$\begin{align*}\#X_{H}^{\infty}({\mathbb F}_q) = \#(H\backslash \operatorname{GL}_2(N)/U(N))^{G_{{\mathbb F}_q}}, \end{align*}$$

as the number of $\left [\begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix}\right ]$ -orbits of $[\overline {H}\backslash \operatorname {GL}_2(N)]$ stable under the action of $\left [\begin {smallmatrix} q & 0 \\ 0 & 1 \end {smallmatrix}\right ]$ , which we note depends only on $q\bmod N$ ; if one expects to compute $\#X_H^{\infty }({\mathbb F}_q)$ for many finite fields ${\mathbb F}_q$ (as when computing the L-function of $X_H/\mathbb {Q}$ , for example), one can simply precompute a table of rational cusp counts indexed by $(\mathbb {Z}/N\mathbb {Z})^{\times }$ .

To compute

(5.1) $$ \begin{align} \#Y_H({\mathbb F}_q) = \sum_{j(E)\in k} \#(H\backslash \operatorname{GL}_2(N)/A_E)^{G_{{\mathbb F}_q}}, \end{align} $$

we first note that , so we may replace H with $\overline {H}$ and work with the quotient $A_E/\pm I$ , which is trivial for $j(E)\ne 0,1728$ . It then suffices to count elements of $[\overline {H}\backslash \operatorname {GL}_2(N)]$ fixed by $G_{{\mathbb F}_q}$ using the action of the Frobenius endomorphism $\pi $ on $E[N]$ with respect to some basis; we are free to choose any basis we like, since the number of elements of $[\overline {H}\backslash \operatorname {GL}_2(N)]$ fixed by a matrix in $\operatorname {GL}_2(N)$ is invariant under conjugation. By [Reference Duke and TóthDT02, Theorem 2.1], we can use the reduction modulo N of the integer matrix

(5.2)

where $a,b,\Delta ,d\in \mathbb {Z}$ are defined as follows. Let , let , let , and let if $\mathbb {Z}[\pi ]\ne \mathbb {Z}$ , and let otherwise, so that we always have

(5.3) $$ \begin{align} 4q = a^2-b^2\Delta, \end{align} $$

and let . We then have $\operatorname {\mathrm {tr}} A_{\pi }\equiv \operatorname {\mathrm {tr}} \pi \bmod N$ and $\det A_{\pi }\equiv q\bmod N$ . We note that $A_{\pi }$ is uniquely determined by $\operatorname {\mathrm {tr}}\pi $ and $\Delta $ and represents an element of $\operatorname {GL}_2(N)$ for every integer N coprime to q.

Let $\chi _{\overline {H}}\colon \operatorname {GL}_2(N)\to \mathbb {Z}_{\ge 0}$ denote the character of the permutation representation given by the right $\operatorname {GL}_2(N)$ -set $[\overline {H}\backslash \operatorname {GL}_2(N)]$ . Each term in equation (5.1) for $j(E)\ne 0,1728$ can be computed as $\chi _{\overline {H}}(A_{\pi })$ . It follows from the theory of complex multiplication that for each imaginary quadratic discriminant $\Delta $ for which the norm equation (5.3) has a solution with $a>0$ coprime to q, there are exactly $h(\Delta )$ ordinary j-invariants of elliptic curves over ${\mathbb F}_q$ with $\operatorname {\mathrm {disc}}(R_{\pi })=\Delta $ , where $h(\Delta )$ is the class number; see [Reference SutherlandSut11, Proposition 1], for example. We thus compute the number of ${\mathbb F}_q$ -points on $X_H$ corresponding to ordinary $j(E)\ne 0,1728$ via

(5.4) $$ \begin{align} \#X_H^{\mathrm{{ord}'}}({\mathbb F}_q) = \sum_{\substack{0<a<2\sqrt{q}\\ \gcd(a,q)=1}}\ \sum_{\substack{4q=a^2-b^2\Delta\\ \Delta <-4}} h(\Delta)\chi_{\overline{H}}\bigl(A(a,b,\Delta)\bigr). \end{align} $$

To count supersingular points in $X_H({\mathbb F}_q)$ with $j\ne 0,1728$ , we rely on the following lemma.

Lemma 5.1. Let $q=p^e$ be a prime power, let $H\le \operatorname {GL}_2(N)$ with $\gcd (q,N)=1$ , let $h'=\lfloor h(-4p)/2\rfloor $ , and let $s_0=1$ for $p\equiv 2\bmod 3$ and $s_0=0$ otherwise. The number $\#X_H^{\mathrm {ss'}}({\mathbb F}_q)$ of ${\mathbb F}_q$ -points on $X_H$ corresponding to supersingular $j(E)\ne 0,1728$ can be computed as follows.

• ○ If $p\le 3$ , then $\#X_H^{\mathrm {ss'}}({\mathbb F}_q)=0$ .

• ○ If e is odd and $p\equiv 1\bmod 4$ , then

  $$\begin{align*}\#X_H^{\mathrm{ss'}}({\mathbb F}_q) = (h'-s_0)\chi_{\overline{H}}\bigl(A(0,p^{(e-1)/2},-4p)\bigr). \end{align*}$$

• ○ If e is odd and $p\equiv 3\bmod 4$ , then

  $$ \begin{align*} \#X_H^{\mathrm{ss'}}({\mathbb F}_q) = h'\chi_{\overline{H}}\bigl(A(0,2p^{(e-1)/2},-p)\bigr) + (h'-s_0)\chi_{\overline{H}}\bigl(A(0,p^{(e-1)/2},-4p)\bigr). \end{align*} $$

• ○ If e is even then

  $$\begin{align*}\#X_H^{\mathrm{ss'}}({\mathbb F}_q) = \Biggl(\frac{p-6+2\left(\frac{-3}{p}\right)+3\left(\frac{-4}{p}\right)}{12}\Biggr) \chi_{\overline{H}}\bigl(A(2p^{e/2},0,1)\bigr). \end{align*}$$

Proof. These formulas are derived from [Reference SchoofSch87, Theorem 4.6] by discarding ${\mathbb F}_q$ -isomorphism classes with $j(E)=0,1728$ and dividing by 2 to count j-invariants rather than counting ${\mathbb F}_q$ -isomorphism classes. For $p=2,3$ , there are no supersingular $j(E)\ne 0,1728$ , and for $p>3$ and $j(E)\ne 0,1728$ , we only need to consider supersingular ${\mathbb F}_q$ -isomorphism classes with trace $a=0$ and $\Delta =-0,-4p$ when e is odd, and with trace $a=2p^{e/2}$ and $\Delta =1$ when e is even. In each case, the number of supersingular ${\mathbb F}_q$ -isomorphism classes with j-invariant $0,1728$ that we need to discard can be determined using Table 6 below.

Table 6 ${\mathbb F}_{p^e}$ -isomorphism classes of elliptic curves with j-invariant $0$ or $1728$ .

For $j(E)=0,1728$ , rather than computing double cosets fixed by $G_k$ , we instead compute a weighted sum of $\chi _{\overline {H}}(A_{\pi })$ over k-isomorphism classes of elliptic curves with $j(E)=0,1728$ via the table below. If we extend the permutation character $\chi _{\overline {H}}$ to the group ring $\mathbb {Q}[\operatorname {GL}_2(N)]$ , we can compute the number of k-rational points of $X_H$ above $j=0,1728\in X(1)$ via

$$\begin{align*}\# X_H^{j=0}({\mathbb F}_q)= \chi_{\overline{H}}\left(\sum_{\substack{j(E)=0\\p\ne 2}} \frac{A(a,b,\Delta)}{\#\operatorname{\mathrm{Aut}}(E)} \right), \qquad \#X_H^{j=1728}({\mathbb F}_q) = \chi_{\overline{H}}\left(\sum_{\substack{j(E)=1728\\p\ne 3}} \frac{A(a,b,\Delta)}{\#\operatorname{\mathrm{Aut}}(E)} \right), \end{align*}$$

where the sums are over ${\mathbb F}_q$ -isomorphism classes of elliptic curves $E/{\mathbb F}_q$ , and the values of $a,b,\Delta $ are listed in Table 6. These values are derived from [Reference WaterhouseWat69] and [Reference SchoofSch87], and we note that for each triple $(|a|,b,\Delta )$ with $|a|>0$ listed in the table with multiplicity m, the triples $(a,b,\Delta )$ and $(-a,b,\Delta )$ each occur with multiplicity $m/2$ .

To sum up, for any $H\le \operatorname {GL}_2(N)$ , we can compute $\#X_H({\mathbb F}_q)$ as

(5.5) $$ \begin{align} \#X_H({\mathbb F}_q) = \#X_H^{\infty}({\mathbb F}_q) + \#X_H^{\mathrm{ord'}}({\mathbb F}_q) +\#X_H^{\mathrm{ss'}}({\mathbb F}_q) +\#X_H^{j=0}({\mathbb F}_q) +\#X_H^{j=1728}({\mathbb F}_q), \end{align} $$

where each term in the sum is computed using the permutation representation $[\overline {H}\backslash \operatorname {GL}_2(N)]$ : the term $\#X_H^{\infty }({\mathbb F}_q)$ is computed by counting $\left [\begin {smallmatrix} q & 0 \\ 0 & q \end {smallmatrix}\right ]$ -stable $\left [\begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix}\right ]$ -orbits, and the remaining terms are weighted sums of values of the permutation character $\chi _{\overline {H}}$ on the matrices $A(a,b,\Delta )$ arising for elliptic curves $E/{\mathbb F}_q$ . A Magma implementation of the formula in equation (5.5) is provided by the function GL2PointCount in the file gl2.m in [Reference Rouse, Sutherland and Zureick-BrownRSZB21].

Example 5.2. Consider $H=\{\left [\begin {smallmatrix} 1 & * \\ 0 & * \end {smallmatrix}\right ]\}\le \operatorname {GL}_2(13)$ with label 13.168.2.1. The modular curve $X_H=X_1(13)$ parametrises elliptic curves with a rational point of order 13. Over ${\mathbb F}_{37}$ , there are four such elliptic curves, up to ${\mathbb F}_{37}$ -isomorphism, each with 12 points of order 13 with distinct j-invariants $0$ , $16$ , $26=1728$ , $35$ (none are supersingular) and automorphism groups of order 6, 2, 4, 2, respectively. We thus expect $\#Y_H({\mathbb F}_{37}) = 12/6 + 12/2 + 12/4 + 12/2 = 17$ , and one can check that 6 of the 12 cusps of $X_1(13)$ are ${\mathbb F}_{37}$ -rational (those of width of 13). This agrees with the fact that there are 23 rational points on the mod-37 reduction of the genus 2 curve $y^2+(x^3+x+1)y=x^5+x^4$ , which is a $\mathbb {Z}[1/13]$ -model for $X_H=X_1(13)$ .

To compute $\#X_H({\mathbb F}_{37})$ using the algorithm proposed above, we apply equation (5.5), which does not require us to enumerate elliptic curves with a point of order 13. We compute $\#X_H^{\infty }({\mathbb F}_{37})=6$ by noting that the coset space $[\overline {H}\backslash \operatorname {GL}_2(13)]$ has twelve $\left [\begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix}\right ]$ -orbits, six of size 1 and six of size 13, the latter of which are stable under the action of $\left [\begin {smallmatrix} 37 & 0 \\ 0 & 1 \end {smallmatrix}\right ]\equiv \left [\begin {smallmatrix} 11 & 0 \\ 0 & 1 \end {smallmatrix}\right ]$ . Enumerating integers $a\in [1,\lfloor 2\sqrt {37}\rfloor ]$ yields 17 triples $(a,b,\Delta )$ with $\Delta < -4$ that satisfy the norm equation (5.3). Among these, only $A(1,1,-147)$ fixes an element of $[\overline {H}\backslash \operatorname {GL}_2(13)]$ , and from equation (5.4), we obtain

$$\begin{align*}\#X_H^{\mathrm{{ord}'}}({\mathbb F}_q) = h(\Delta)\chi_{\overline{H}}\bigl(A(a,b,\Delta)\bigr) = h(-147)\chi_{\overline{H}}\bigl(A(1,1,-147)\bigr) = 2\cdot 6 = 12. \end{align*}$$

To compute $\#X_H^{\mathrm {ss'}}({\mathbb F}_{37})$ , we apply Lemma 5.1 and find that $\chi _{\overline {H}}(A(0,0,-4\cdot 37))=0$ , so $\#X_H^{\mathrm {ss'}}({\mathbb F}_q) =0$ . Using Table 6, we compute

$$ \begin{align*} \#X_H^{j=0}({\mathbb F}_{37}) &= \chi_{\overline{H}}(A(\pm 1,7,-3)/6 + A(\pm 10, 4,-3)/6 + A(\pm 11,3,-3)/6)=2+0+0=2\\ \#X_H^{j=1728}({\mathbb F}_{37}) &= \chi_{\overline{H}}(A(\pm 2,6,-4)/4 + A(\pm 12, 1,-4)/4)=0+3=3, \end{align*} $$

and equation (5.5) then yields

$$ \begin{align*} \#X_H({\mathbb F}_{37}) &= \#X_H^{\infty}({\mathbb F}_{37}) + \#X_H^{\mathrm{ord'}}({\mathbb F}_{37}) +\#X_H^{\mathrm{ss'}}({\mathbb F}_{37}) +\#X_H^{j=0}({\mathbb F}_{37}) +\#X_H^{j=1728}({\mathbb F}_{37})\\ &= 6 + 12 + 0 + 2 + 3 = 23. \end{align*} $$

In contrast to the method described in [Reference ZywinaZyw15b, Section 3], in our computation above, we did not enumerate elliptic curves or j-invariants over ${\mathbb F}_q$ , nor did we compute any Hilbert class polynomials (a commonly used but inefficient way to compute $A_{\pi }$ ). Instead, we enumerated possible Frobenius traces $a>0$ , solved norm equations, computed class numbers and computed values of the permutation character $\chi _{\overline {H}}$ . This can be viewed as an algorithmic generalisation of the Eichler–Selberg trace formula that works for any open $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ . It has roughly the same asymptotic complexity as the trace formula (the dependence on q is $\tilde O(q^{1/2})$ if we ignore the cost of computing class numbers, which in practice we simply look up in a table such as those available for $|D|\le 2^{40}$ in the LMFDB). But the constant factors are better, and even for groups that do not contain the $\operatorname {GL}_2$ -analog of $\Gamma _1(N)$ , we are always able to work at level N rather than level $N^2$ , as typically required by generalisations of the trace formula to handle congruence subgroups that do not contain $\Gamma _1(N)$ .

5.2 Local obstructions

An easy application of this point counting algorithm is checking for local obstructions to rational points on the arithmetically maximal modular curves $X_H$ . For $X_H$ of genus g, it suffices to compute $\#X({\mathbb F}_p)$ for $p < \lfloor 2g\sqrt {p}\rfloor $ , since the Weil bounds force $\#X({\mathbb F}_p)>0$ for all larger primes p. Doing this for the 210 arithmetically maximal groups summarised in Table 4 yields Theorem 5.1, whose results are summarised in Table 7.

Theorem 5.1. For $\ell \le 37$ , the arithmetically maximal $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of $\ell $ -power level for which $X_H$ has no ${\mathbb F}_{p}$ -points for some prime $p\ne \ell $ are precisely those listed in Table 7.

Table 7 Arithmetically maximal $H\le \operatorname {GL}_2({\widehat {\mathbb {Z}}})$ of $\ell $ -power level for which $X_H$ has no ${\mathbb F}_p$ -points for some prime $p\ne \ell \le 37$ .

6 Computing the isogeny decomposition and analytic rank of $J_H$

Let H be an open subgroup of $\operatorname {GL}_2({\widehat {\mathbb {Z}}})$ . In this section, we explain how the point-counting algorithm described in the previous section can be used to determine the isogeny factors of the Jacobian as modular abelian varieties; this decomposition can then be used to compute the analytic rank of $J_H$ . To simplify the exposition, we shall assume that $\det (H)={\widehat {\mathbb {Z}}}^{\times }$ , which holds in all the cases of interest to us here, but the methods we describe can also be applied when $\det (H)\ne {\widehat {\mathbb {Z}}}^{\times }$ ; see Example 6.3 for details.

The Jacobian $J_H$ is an abelian variety over $\mathbb {Q}$ that has good reduction at all primes p that do not divide the level N of H. It follows from Theorem A.7 that the simple factors of $J_H$ are each $\mathbb {Q}$ -isogenous to the modular abelian variety associated to the Galois orbit of a (not necessarily new, normalised) eigenform of weight 2 for $\Gamma _1(N)\cap \Gamma _0(N^2)$ . We may decompose this space of modular forms as

$$\begin{align*}S_2\bigl(\Gamma_1(N)\cap\Gamma_0(N^2)\bigr)=\bigoplus_{\operatorname{\mathrm{cond}}(\chi)|N}\!\!\!S_2\bigl(\Gamma_0(N^2),[\chi]\bigr), \end{align*}$$

where $[\chi ]$ denotes the Galois orbit of the Dirichlet character $\chi $ and the direct sum varies over Galois orbits of Dirichlet characters of modulus $N^2$ whose conductor divides N. The Galois orbit of a normalised eigenform $f\in S_2(\Gamma _0(N^2),[\chi ])$ has an associated trace form $\operatorname {\mathrm {Tr}}(f)$ (see [Reference Best, Bober, Booker, Costa, Cremona, Maarten, Lee, Lowry-Duda, Roe, Sutherland and VoightBBB+21, Section 4.5]), which has an integral q-expansion

with

; and for each prime p, the coefficient $a_p$ is the Frobenius trace of the abelian variety $A_f/\mathbb {Q}$ of dimension $\dim f$ associated to f via the Eichler–Shimura correspondence; equivalently, $a_p$ is the pth (arithmetically normalised) Dirichlet coefficient of the L-function $L(A_f,s)=\prod _{f\in [f]}L(f,s)$ .

Let be the Galois orbits of eigenforms $f\in S_2(\Gamma _1(N)\cap \Gamma _0(N^2))$ with $\dim f\le \dim J_H=g(H)$ . Theorem A.7 implies that there is a sequence of nonnegative integers $e(H)=(e_1,\ldots ,e_m)$ for which we have

$$\begin{align*}L(J_H,s) = \prod_{i=1}^m L(A_{f_i},s)^{e_i}, \end{align*}$$

and this sequence also satisfies $\sum _i e_i\dim f_i =g(H)$ . Now let $T(B)$ be the $n\times m$ integer matrix whose ith column is $[a_1(\operatorname {\mathrm {Tr}}(f_i)),a_2(\operatorname {\mathrm {Tr}}(f_i)),a_3(\operatorname {\mathrm {Tr}}(f_i)),a_5(\operatorname {\mathrm {Tr}}(f_i)),\ldots ,a_p(\operatorname {\mathrm {Tr}}(f_i)),\ldots ]$ , where p varies over the $n-1$ primes bounded by B that do not divide the level of H, and let $a(H;B)$ be the vector of integers $[g(H),a_2(H),a_3(H),\ldots ,a_p(H),\ldots ]$ , where $a_p(H)$ denotes the trace of the Frobenius endomorphism of $J_H$ at a prime $p\le B$ not dividing the level of H, which we can compute as $a_p(H)= p+1-\#X_H({\mathbb F}_p)$ via equation (5.5). For $B\ge 1$ , we have

$$\begin{align*}T(B)e(H)=a(H;B), \end{align*}$$

and it follows from an effective form of multiplicity 1 for the Selberg class [Reference Kaczorowski and PerelliKP01, Reference SoundararajanSou04] that for all sufficiently large B, the matrix $T(B)$ and the vector $a(H,B)$ uniquely determine the vector $e(H)$ . Indeed, there are only finitely many sequences of nonnegative integers that sum to $g(H)$ , and the corresponding L-functions are all distinguished by their Dirichlet coefficients $a_p$ on the set of primes $p\nmid N$ by strong multiplicity 1; for each pair of these L-functions, there is some smallest $p\nmid N$ for which the $a_p$ differ, and we can take B to be the maximum of this p over all pairs. We note that L-functions of modular forms and all products of such L-functions not only lie in the Selberg class but also satisfy the polynomial Euler product condition used in [Reference Kaczorowski and PerelliKP01] and the weaker condition required by [Reference SoundararajanSou04].

This suggests the following approach to determining the exponents in the factorisation $L(J_H,s)=\prod _i L(A_{f_i},s)^{e_i}$ , from which both the analytic rank of $J_H$ and its isogeny-factor decomposition can be derived. We first determine the set $S(H)$ , which we note depends only on the genus and level of H. We then determine a value of B for which the $\#S(H)$ columns of $T(B)$ become linearly independent over $\mathbb {Q}$ : for example, by starting with a value of B that makes $T(B)$ a square matrix and successively increasing B by some constant factor $\alpha>1$ . Finally, we compute the vector $a(H;B)$ by computing $p+1-\#X_H({\mathbb F}_p)$ for primes $p\le B$ not dividing the level of H and determine $e(H)$ as the necessarily unique solution to the linear system $T(B)x=a(H;B)$ .

This approach is remarkably effective in practice. For the 210 arithmetically maximal H arising for $\ell \le 37$ , the value of B never exceeds 3000, and the matrix $T(B)$ with linear independent columns typically has fewer than 100 rows. To compute $T(B)$ , we use precomputed trace form data stored in the LMFDB along with additional data available in the file cmfdata.txt in [Reference Rouse, Sutherland and Zureick-BrownRSZB21] that we computed to handle cases where the set $S(H)$ contains Galois orbits of eigenforms that are not currently stored in the LMFDB, using the methods described in [Reference Best, Bober, Booker, Costa, Cremona, Maarten, Lee, Lowry-Duda, Roe, Sutherland and VoightBBB+21].

Having determined the vector $e(H)=[e_1,\ldots ,e_m]$ , we compute the rank of $J_H$ as $\sum _i e_ir_i$ , where each $r_i$ is the sum of the analytic ranks of the eigenforms in the ith Galois orbit $[f_i]$ (one expects them to have the same rank as $f_i$ ; this is true for all of the modular forms in the LMFDB and for all of the modular forms that we computed). The method used to rigorously compute the analytic ranks of these modular forms is described in [Reference Best, Bober, Booker, Costa, Cremona, Maarten, Lee, Lowry-Duda, Roe, Sutherland and VoightBBB+21, Section 9].

Example 6.2. Consider the group $H{\,=\,}\langle \left [\begin {smallmatrix} 4 & 1 \\ 21 & 2 \end {smallmatrix}\right ],\left [\begin {smallmatrix} 11 & 0 \\ 0 & 19 \end {smallmatrix}\right ]\rangle {\,\le \,} \operatorname {GL}_2(27)$ with label 27.36.3.1. Eleven Galois orbits of eigenforms $f\in S_2(\Gamma _1(27)\cap \Gamma _0(27^2))$ have $\dim f{\,\le \,} g(H){\,=\,} 3$ , with LMFDB labels 243.2.a.a, 27.2.a.a, 243.2.a.b, 243.2.c.a, 243.2.a.c, 81.2.c.a, 243.2.a.d, 81.2.a.a, 243.2.c.b, 243.2.a.e, and 243.2.a.f. For $B =73$ the $21\times 11$ matrix $T(B)$ has independent columns. Computing $\#X_H({\mathbb F}_p)$ for $p{\,=\,}2,5,7,11,\ldots , 73$ yields

, and the linear system

has the unique solution $e(H)=[1,0,0,0,0,0,1,0,0,0,0]^{\mathrm {T}}$ corresponding to the Galois orbits 243.2.a.a and 243.2.a.d of dimensions 1 and 2 with analytic ranks 1 and 0, respectively; thus $J_H$ has analytic rank 1.

We used the method described above to compute the analytic ranks and the decomposition of $J_H$ up to isogeny for all but one of the arithmetically maximal subgroups of $\ell $ -power level for $\ell \le 37$ . The sole exception is the group H with label 121.605.41.1 of level $121$ , where we were unable to compute all of the elements of $S(H)$ (we were able to do this for the group 121.6655.511.1 corresponding to $N_{\mathrm { ns}}(121)$ by restricting to eigenforms $f\in S_2(\Gamma _0(\ell ^{2n}))$ via the optimisation noted above). Computing all of the elements of $S(H)$ in this case would require decomposing the newspaces $S_2^{\mathrm {new}}(\Gamma _0(11^4),[\chi ])$ for all Dirichlet characters of modulus $11^4$ and conductor dividing $11^2$ , which we found impractical in the two cases where $\operatorname {\mathrm {cond}}(\chi )=11^2$ (the corresponding newspaces have $\mathbb {Q}$ -dimensions 10550 and 42200).

For the group 121.605.41.1, we apply the alternative strategy of verifying a candidate value for $e(H)$ by checking Fourier coefficients up to the Sturm bound. We first use the subset of $S(H)$ that we know to find a candidate $e(H)=[e_1,\ldots ,e_m]$ that works even for B much larger than necessary to ensure that the columns of $T(B)$ are linearly independent. To rigorously verify this candidate, it suffices to compare the Dirichlet coefficients $b_n,c_n\in \mathbb {Z}$ of the L-functions

$$\begin{align*}L(J_H,s)= \sum_{n=1}^{\infty} b_n n^{-s}\qquad\text{and}\qquad \prod_i L(A_{f_i},s)^{e_i}=\sum_{n=1}^{\infty} c_nn^{-s}, \end{align*}$$

for n up to the Sturm bound for the space $S_2\bigl (\Gamma _1(N)\cap \Gamma _0(N^2)\bigr )$ , which is . The integers $c_n$ uniquely determine the Fourier coefficients $a_n$ that appear in the q-expansion of the modular from for $n\le B$ , which uniquely determines F.Footnote ⁶

Both L-functions are defined by an Euler product, so it suffices to check $b_q=c_q$ for prime powers $q\le B$ . We compute the $c_q$ by computing Euler factors of $\prod _{f\in [f_i]}L(f,s)$ at all primes $p\le B$ using the modular symbols functionality in Magma. For $11\!\nmid \! q$ , we compute the $b_q$ by computing $\#X_H({\mathbb F}_{p^r})$ for $1\le r\le \log _p B$ via the point counting algorithm described in the previous section (these point counts determine enough coefficients of the degree-82 L-polynomial of $J_H$ at p to determine the $a_q$ for $q\le B$ ). To compute the Euler factor of $L(J_H,s)$ at 11 (which determines the $a_q$ for $11|q$ ), we use the $\operatorname {GL}_2$ -modular symbols functionality in the Magma package ModFrmGL2 recently developed by Assaf [Reference AssafAss21]. The most time-consuming part of this process is computing the $c_q$ for $q\le B=322\,102$ , which takes 15–20 hours. It takes about 15 minutes to compute the $b_q$ for $11\nmid q$ and another 15 or 20 minutes to compute the Euler factor of $L(J_H,s)$ at 11. Magma scripts that reproduce all three computations and verify that the results match are available at [Reference Rouse, Sutherland and Zureick-BrownRSZB21].

7 Computing equations for $X_H$ with $-I \in H$

We now discuss the computation of equations for $X_H$ for arithmetically maximal H containing $-I$ of $\ell $ -power level for $3\le \ell \le 13$ ; see Section 10 for the case $-I \not \in H$ .

In [Reference Rouse and Zureick-BrownRZB15], Rouse and Zureick-Brown construct the modular curves $X_{H}$ as covers of $X_{\widetilde {H}}$ , where $\widetilde {H}$ is a supergroup of H for which $[\widetilde {H} : H]$ is minimal. Because we are only recording equations for arithmetically maximal subgroups, in any case where we compute the covering $X_{H} \to X_{\widetilde {H}}$ , the genus of $X_{\widetilde {H}}$ is zero or one. In all cases, we consider the genus of $X_{\widetilde {H}}$ is zero, and we know a specific function $x \in \mathbb {Q}(X_{\widetilde {H}})$ so that $\mathbb {Q}(X_{\widetilde {H}}) = \mathbb {Q}(x)$ . The models are computed by constructing elements of the function field of $\mathbb {Q}(X_{H})$ built using products of weight $2$ Eisenstein series whose Fourier expansions are simple to compute and for which it is easy to describe the action of $\operatorname {GL}_2(N)$ on them. In this way, we find a generator y for $\mathbb {Q}(X_{H})$ and then compute the action of H-coset representatives of $\widetilde {H}$ on it. This allows us to compute the minimal polynomial for y over $\mathbb {Q}(x)$ , which gives a plane model for $X_{H}$ . We use the method of Novocin and van Hoeij [Reference van Hoeij and NovocinvHN05] to reduce the resulting algebraic function field and obtain a simple model.

We now describe another approach for computing the canonical model of a high genus curve, which we use to compute equations of $X_{\operatorname {\mathrm {ns}}}^{+}(27)$ (label 27.243.12.1) and $X_{H}$ , where H has label 27.729.43.1 or 25.625.36.1. Given $H \leq \operatorname {GL}_{2}(N)$ , there is a natural action of H on the field of modular functions for $\Gamma (N)$ with coefficients in $\mathbb {Q}(\zeta _{N})$ . In particular, $\left [\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ] \in \operatorname {SL}_{2}(N)$ acts via $f | \left [\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ] = f\left (\frac {az+b}{cz+d}\right )$ , while the matrix $h = \left [\begin {smallmatrix} 1 & 0 \\ 0 & d \end {smallmatrix}\right ]$ acts by sending $f = \sum a(n) q^{n}$ to $f | h = \sum \sigma _{d}(a(n)) q^{n}$ , where $\sigma _{d}$ is the automorphism of $\mathbb {Q}(\zeta _{N})$ sending $\sigma _{d}(\zeta _{N}) = \zeta _{N}^{d}$ . One can extend this map to a natural map on $M_{2k}(\Gamma (N), \mathbb {Q}(\zeta _{N}))$ of the space of weight $2k$ modular forms for $\Gamma (N)$ with Fourier coefficients in $\mathbb {Q}(\zeta _{N})$ . To do so, given a form $F \in M_{2k}(\Gamma (N), \mathbb {Q}(\zeta _{N}))$ , define $f = \frac {E_{4}(z)^{k} E_{6}(z)^{k}}{\Delta (z)^{k}} F$ , a modular function for $\Gamma (N)$ , and

Here

(with $q = e^{2 \pi i z}$ ) and

are the classical weight $4$ and $6$ Eisenstein series, respectively.

Given H, our goal is to find q-expansions for a basis for $S_{2}(\Gamma (N), \mathbb {Q}(\zeta _{N}))^{H}$ . This problem has been considered before in [Reference Banwait and CremonaBC14] and recently in [Reference ZywinaZyw21] and [Reference AssafAss21]. Instead, we continue our approach from [Reference Rouse and Zureick-BrownRZB15] of first computing the space of weight $2$ Eisenstein series for H. Compared to other methods, our method has the disadvantage of requiring a much larger number of Fourier coefficients. However, it only requires working with modular forms defined on H, and our method may be substantially more efficient if the index of $\Gamma (N)$ in H is large. The addition of modular forms functionality to PARI/GP in [Reference Belabas and CohenBC18] uses the strategy of multiplying Eisenstein series.

The next task is to construct the Fourier coefficients of weight $2$ cusp forms from those of the Eisenstein series. We do this by multiplying triples of weight $2$ Eisenstein series together to obtain a large collection of weight $6$ modular forms. Because we know how each weight $2$ Eisenstein series transforms under matrices in $\operatorname {GL}_{2}(N)$ , we can keep track of the values of these products of the Eisenstein series at each cusp of $X_{H}$ . There is no reason to expect that this is the entirety of the space of weight $6$ modular forms because weight $2$ modular forms must have a zero at each elliptic point of order $2$ and a double zero at each elliptic point of order $3$ . One can use the Riemann–Roch theorem to compute that the dimension of the subspace of weight $6$ modular forms with a zero of order $\geq 3$ at each elliptic point of order $2$ and zero of order $\geq 6$ at each elliptic point of order $3$ is $3 \nu _{\infty }(\Gamma _{H}) + 5(g-1)$ . This allows one to avoid computing products of all triples of weight $2$ Eisenstein series.

Next, one applies Hecke operators to compute the full space of weight $6$ modular forms. In [Reference KatoKat04], Kato defines a family of Hecke operators $T(n) \left [\begin {smallmatrix} n & 0 \\ 0 & 1 \end {smallmatrix}\right ]^{*}$ for modular forms on $\Gamma (N)$ and gives their action on Fourier expansions for $\gcd (n,N) = 1$ . As noted by Kato, this normalisation commutes with the action of $\operatorname {GL}_{2}(N)$ , but it does not preserve the connected components of $X(N)$ .

Let w be the width of the cusp at $\infty $ for $\Gamma _H$ , and suppose that $f(z) = \sum _{n=0}^{\infty } a(n) q^{n/w}$ is a modular form of weight k for $\Gamma _{H}$ , $q = e^{2 \pi i z}$ . Then Kato’s formulas show that if p is a prime with $p \equiv \pm 1 \pmod {N}$

$$\begin{align*}\left(\sum_{n=0}^{\infty} a(n) q^{n/w}\right) | T(p) = \sum_{n=0}^{\infty} a(pn) q^{n/w} + p^{k-1} \sum_{n \equiv 0 \pmod{p}}^{\infty} a(n/p) q^{n/w}. \end{align*}$$

The case $p \equiv -1 \pmod {N}$ requires , so we conjugate H to contain M.

We choose the smallest prime $p \equiv \pm 1 \pmod {N}$ and apply the Hecke operator $T(p)$ to all the products of three weight $2$ Eisenstein series. While we are not guaranteed to obtain the entire space $M_{6}(\Gamma (N), \mathbb {Q}(\zeta _{N}))^{H}$ , in the cases under consideration we do. It is not difficult to see how the Hecke operator affects values at the cusps, and in this way, we obtain a basis for the space $S_{6}(\Gamma (N), \mathbb {Q}(\zeta _{N}))^{H}$ . We apply the same tools to obtain a basis for the space $S_{8}(\Gamma (N), \mathbb {Q}(\zeta _{N}))^{H}$ .

Let h be a meromorphic modular form on $\Gamma (N)$ . We use the elementary observation that since the zero sets of $E_{4}(z)$ and $E_{6}(z)$ are disjoint, $h\in S_{2}(\Gamma (N), \mathbb {Q}(\zeta _{N}))^{H}$ if and only if $h E_{4} \in S_{6}(\Gamma (N), \mathbb {Q}(\zeta _{N}))^{H}$ and $h E_{6} \in S_{8}(\Gamma (N), \mathbb {Q}(\zeta _N))^{H}$ . In this way, one obtains q-expansions for the weight $2$ cusps forms for H. From these, we can easily obtain the canonical map for $X_{H}$ and its image in $\mathbb {P}^{g-1}$ and then check whether the curve is hyperelliptic. Let $\Omega $ be the sheaf of holomorphic $1$ -forms on the curve $X_{H}$ . If $g(H) \geq 3$ and $X_{H}$ is not hyperelliptic, the canonical ring $R = \oplus _{d \geq 0} H^{0}(X_{H}, \Omega ^{\otimes d})$ is generated in degree $1$ [Reference Voight and Zureick-BrownVZB22, Chapter 2].

Finally, we seek to identify $j(z)$ as a ratio of two elements of the canonical ring. From a computational standpoint, it is easier to write

$$\begin{align*}j(z) = \frac{1728 E_{4}(z)^{3}}{E_{4}(z)^{3} - E_{6}(z)^{2}} \end{align*}$$

and find representations of $E_{4}(z)$ and $E_{6}(z)$ as ratios of elements of the canonical ring. Let $h(z)$ be an element of the canonical ring of weight $k-4$ . Then $E_{4}(z) = \frac {f(z)}{h(z)}$ for some $f(z)$ in the canonical ring if and only if $h(z) E_{4}(z) dz^{k/2}$ is a holomorphic $k/2$ -form. Diamond and Shurman [Reference Diamond and ShurmanDS05, p. 86] show that

$$\begin{align*}\mathrm{div}(f(z) dz^{k/2}) = \mathrm{div}(f) - \sum \frac{k}{4} x_{2,i} - \sum \frac{k}{3} x_{3,i} - \sum \frac{k}{2} x_{i}, \end{align*}$$

where $\{x_{2,i} \}$ is the set of elliptic points of order $2$ , $\{ x_{3,i} \}$ is the set of elliptic points of order $3$ and $\{ x_{i} \}$ is the set of cusps on $X_{H}$ . Let $D = -\sum x_{2,i} - \sum x_{3,i} - \sum 2x_{i}$ . Then $h(z) E_{4}(z) dz^{k/2}$ is holomorphic if and only if $h(z) dz^{(k-4)/2} \in L^{(k-4)/2}(D)$ , the space of meromorphic $(k-4)/2$ -forms with poles at most at D. We have (by Lemma VII.4.11 from Miranda [Reference MirandaMir95]) that $L^{(k-4)/2}(D) \simeq L(D + ((k-4)/2)K)$ , where K is the canonical divisor. Riemann–Roch now guarantees the existence of such a D, provided that

$$\begin{align*}k \geq \frac{2\nu_{\infty}(\Gamma_{H}) + \nu_{2}(\Gamma_{H}) + \nu_{3}(\Gamma_{H}) + 5g(H)-4}{g(H)-1}. \end{align*}$$

A similar calculation shows that $E_{6}$ is a ratio of a holomorphic $(k/2)$ -form and a holomorphic $((k-6)/2)$ -form, provided that

$$\begin{align*}k \geq \frac{3 \nu_{\infty}(\Gamma_{H}) + \nu_{2}(\Gamma_{H}) + 2 \nu_{3}(\Gamma_{H}) + 7g(H)-6}{g(H)-1}. \end{align*}$$

With these bounds, one can use the q-expansions of the weight $2$ cusp forms to compute the q-expansions of the modular forms of weight k that correspond to elements of the canonical ring and use linear algebra to compute a ratio that is equal to $E_{4}(z)$ and $E_{6}(z)$ .

This technique is used to compute the canonical models of $X_{\operatorname {\mathrm {ns}}}^{+}(25)$ , $X_{\operatorname {\mathrm {ns}}}^{+}(27)$ , a genus $43$ curve with label 27.729.43.1 and a genus $36$ curve with label 25.625.36.1. A variant of this technique was used to compute a model of a genus $3$ modular curve defined over $\mathbb {Q}(\zeta _{3})$ (see Subsection 9.1). We compute models for all of the arithmetically maximal modular curves with $3 \leq \ell \leq 13$ with the exceptions of $X_{\mathrm {s}}^{+}(49)$ with label 49.1372.94.1, which is known not to have any noncuspidal non-CM rational points, 121.605.41.1 which is shown not to have any rational points in Section 8.7 and $X_{\operatorname {\mathrm {ns}}}^{+}(121)$ with label 121.6655.511.1. For the genus $69$ curve $X_{\operatorname {\mathrm {ns}}}^{+}(49)$ , we obtain a singular plane model, but we are unable to apply the method of Novocin and van Hoeij to simplify the equation.

8 Analysis of rational points

In this section, we determine the rational points on $X_H(\mathbb {Q})$ for each of the 21 arithmetically maximal subgroups that appear in Table 5 without an asterisk.

8.1 Genus 1

The three curves of genus 1 listed in Table 5 have labels

$$\begin{align*}{\texttt{9.12.1.1}},\hspace{10pt} {\texttt{9.54.1.1}}, \hspace{10pt} {\texttt{27.36.1.2}}, \end{align*}$$

whose rational points are as follows:

• ○ 9.12.1.1 is a degree $3$ cover of $X_{0}(3)$ isomorphic to the elliptic curve $y^{2} + y = x^{3}$ . It has three rational points, two of which are cusps; the other is a CM point with $j = 0$ .

• ○ 9.54.1.1 also known as $X_{\operatorname {\mathrm {sp}}}^{+}(9)$ is also isomorphic to $y^{2} + y = x^{3}$ (different j-map). In this case, the curve has one cusp and two CM points: $j = 8000$ and $j = -32768$ .

• ○ 27.36.1.2 is isomorphic to the elliptic curve $y^2 + y = x^3 + 2$ ; it has three rational points, two of which are cusps; the other is a CM point with $j = 0$ .

The file check-for-sporadic-points.m in [Reference Rouse, Sutherland and Zureick-BrownRSZB21] verifies these computations.

8.2 Genus 2

The curves corresponding to the labels

$$\begin{align*}{\texttt{9.36.2.1}}, \hspace{10pt} {\texttt{27.36.2.1}}, \hspace{10pt} {\texttt{27.36.2.2}} \hspace{7pt} \text{and} \hspace{7pt} {\texttt{9.54.2.2}} \end{align*}$$

have genus 2 and rank 0. The first two curves are isomorphic and have minimal Weierstrass model $y^{2} + (x^{3} + 1)y = -5x^{3} - 7$ , and $y^{2} + (x^{3} + 1)y = -2x^{3} - 7$ is a minimal Weierstrass model for the third curve. Using Magma’s RankBound command, it is straightforward to see that the Jacobians of these curves have rank zero; and using the command Chabauty0, one finds that each has two rational points. In each case, the rational points are cusps.

The fourth curve, with minimal Weierstrass model

$$\begin{align*}y^{2} + (x^{2} + x + 1)y = 3x^{6} + 2x^{4} + 10x^{3} + 6x^{2} - 32x + 14, \end{align*}$$

has no $3$ -adic points. The file genus-2-analysis.m in [Reference Rouse, Sutherland and Zureick-BrownRSZB21] verifies these computations.

8.3 Genus 3

The curves corresponding to the labels

$$\begin{align*}{\texttt{9.108.3.1}}, \hspace{10pt} {\texttt{27.36.3.1}} \hspace{7pt} \text{and} \hspace{7pt} {\texttt{27.108.3.1}} \end{align*}$$

have genus 3 and are not hyperelliptic. The first has canonical model

$$\begin{align*}X \colon x^{3} z - 6x^{2} z^{2} + 3xy^{3} + 3xz^{3} + z^{4} = 0. \end{align*}$$

Since X is a Picard curve (which is clear from the y coordinate), we can use Magma’s RankBound command to compute that the rank of $J_X(\mathbb {Q})$ is zero. In principle, one can now simply compute $J_X(\mathbb {Q}) = J_X(\mathbb {Q})_{{\operatorname {tors}}}$ and then compute $X(\mathbb {Q})$ as the preimage of an Abel–Jacobi map $X(\mathbb {Q}) \hookrightarrow J_X(\mathbb {Q})$ ; see [Reference Derickx, Etropolski, van Hoeij, Morrow and Zureick-BrownDEvH+, Subsection 5.2] for a longer discussion of such computations.

In practice, we were not able to compute $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ exactly. By [Reference KatzKat81, Appendix], reduction modulo an odd prime p gives an injection $J_X(\mathbb {Q})_{{\operatorname {tors}}} \hookrightarrow J_X({\mathbb F}_p)$ . We compute (using Magma’s ClassGroup command) that $J_X({\mathbb F}_5) \simeq \mathbb {Z}/198\mathbb {Z}$ and $J_X({\mathbb F}_{13}) \simeq \mathbb {Z}/45\mathbb {Z} \times \mathbb {Z}/45\mathbb {Z}$ . (See [Reference Derickx, Etropolski, van Hoeij, Morrow and Zureick-BrownDEvH+, Subsection 4.1] for a longer discussion of computing torsion on modular Jacobians). On the other hand, X has two visible rational points with coordinates $[0 : 1 : 0]$ , $[1 : 0 : 0]$ whose difference is a point of order 3 in $J_X(\mathbb {Q})$ . Thus the only two possibilities for $J_X(\mathbb {Q})$ are $\mathbb {Z}/3\mathbb {Z}$ and $\mathbb {Z}/9\mathbb {Z}$ . A long search for low-degree points on X does not produce a point of order 9 on $J_X(\mathbb {Q})$ , nor do further local computations (including refined techniques such as [Reference Ozman and SiksekOS19, Section 4]) give a better local bound than $\mathbb {Z}/9\mathbb {Z}$ . This suggests a local to global failure for $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ .

We circumvent this as follows. We enumerate $X({\mathbb F}_2)$ and find that for any $P,Q \in X({\mathbb F}_2)$ , the order of $P - Q$ is not equal to 9. Since $J_X(\mathbb {Q})$ has rank 0 and odd order, the reduction map $J_X(\mathbb {Q})_{{\operatorname {tors}}} \hookrightarrow J_X({\mathbb F}_2)$ is injective by [Reference KatzKat81, Appendix]. Thus, there also do not exist points $P,Q \in X(\mathbb {Q})$ such that the order of $P - Q$ is equal to 9 (but note that this does not exclude the possibility that $J_X(\mathbb {Q}) \simeq \mathbb {Z}/9\mathbb {Z}$ ).

Finally, we compute the preimage of the remaining points under an Abel–Jacobi map. Fix $P = [0 : 1 : 0]$ , $P' = [1 : 0 : 0]$ and $D = P'-P$ . Let $\iota _P \colon X \hookrightarrow J_X$ be the map $Q \mapsto Q-P$ . By the previous paragraph, for any $Q \in X(\mathbb {Q})$ , $\iota _P(Q)$ has order 1 or 3. By definition, $\iota _P(P) = 0\cdot D$ , and $\iota _P(Q) = 1\cdot D$ . Using Magma’s RiemannRoch command, we find that the linear system $|2\cdot D + P|$ is empty, so there is no $Q \in X(\mathbb {Q})$ such that $\iota _P(Q) = 2\cdot D$ . Since this exhausts $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ , we have computed $X(\mathbb {Q})$ . Using our equations for the j-map, we determine that the two known points are both CM points with $j = 0$ .

The second curve has canonical model

$$\begin{align*}X \colon -x^{3} y + x^{2} y^{2} - xy^{3} + 3xz^{3} + 3yz^{3} = 0 \end{align*}$$

and is also a Picard curve (which is clear from the z coordinate). It has an automorphism $\iota \colon X \to X$ that swaps x and y. The quotient is the elliptic curve $E\colon y^{2} + y = x^{3} + 20$ , which has rank one and corresponding newform 243.2.a.a. Applying Magma’s RankBound command reveals that the rank of $J_X(\mathbb {Q})$ is also one. In particular, $J_X$ is isogenous to $E \times A$ , where A is the abelian surface A corresponding to the newform 243.2.a.d; see Example 6.2. The analytic rank of $A(\mathbb {Q})$ is zero; by Corollary A.8, the algebraic rank is also zero.

One could in principle apply Chabauty’s method, but this is not directly implemented in Magma for nonhyperelliptic curves. We present an alternative argument. For any point $P \in X(\mathbb {Q})$ , P and $\iota (P)$ have the same image in $E(\mathbb {Q})$ , so $P - \iota (P)$ has trivial image in $E(\mathbb {Q})$ . Thus, since $A(\mathbb {Q})$ is torsion, the image of $P - \iota (P)$ in $E(\mathbb {Q}) \times A(\mathbb {Q})$ is also torsion; since the map $J(\mathbb {Q}) \to E(\mathbb {Q}) \times A(\mathbb {Q})$ is finite, we conclude that $P - \iota (P)$ is an element of $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ .

It thus suffices to compute $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ and the preimage of the map

$$ \begin{align*} \tau\colon X(\mathbb{Q}) &\to J_X(\mathbb{Q})_{{\operatorname{tors}}}\\ P &\mapsto P - \iota(P). \end{align*} $$

Local computations show that $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ is isomorphic to a subgroup of $\left (\mathbb {Z}/3\mathbb {Z}\right )^2$ , and differences of rational points generate a group of order 9.

To compute the preimage of $\tau $ , we first note that the $\tau $ is injective away from the fixed points of $\iota $ . Indeed, if $P - \iota (P) = Q - \iota (Q)$ as divisors, then either $P = Q$ , or $P = \iota (P)$ and $Q = \iota (Q)$ . Now suppose that $P - \iota (P) \neq Q - \iota (Q)$ as divisors, but $P - \iota (P) \sim Q - \iota (Q)$ . If $P = \iota (P)$ but $Q \neq \iota (Q)$ , this equivalence gives a $g^1_1$ , which is a contradiction since X is not rational (and similarly if $Q = \iota (Q)$ but $P \neq \iota (P)$ ). If $Q \neq \iota (Q)$ and $P \neq \iota (P)$ , then instead this gives a $g^1_2$ on a nonhyperelliptic curve, which is also a contradiction. Thus $\tau $ is injective away from the fixed points of $\iota $ .

Next, differences of rational points give an explicit basis $D_1, D_2$ for $J_X(\mathbb {Q})_{{\operatorname {tors}}}$ . For each $a,b \in \mathbb {Z}/3\mathbb {Z}$ such that $(a,b) \neq (0,0)$ , we either find an explicit point P for which we have $P - \iota (P) = aD_1 + bD_2$ , or we find a prime p such that there is no $P \in X({\mathbb F}_p)$ such that $P - \iota (P) = aD_1 + bD_2$ in $J_X({\mathbb F}_p)$ (essentially, ‘sieving’) and thus no such $P \in X(\mathbb {Q})$ . If $(a,b) = (0,0)$ , then $P - \iota (P) = aD_1 + bD_2$ if and only if $P = \iota (P)$ : that is, if and only if P is a fixed point of the involution $\iota $ ; there are finitely many such points, and they are straightforward to compute. Using our equations for the j-map, we find that two of the known points are cusps and one is a CM point with $j = 0$ .

The third curve is isomorphic to the first, which is easy to verify with the explicit models. The 2 rational points are not exceptional; they are both CM points with $j = 0$ .

The files 9.108.3.1.m, 27.36.3.1.m, 27.108.3.1.m in [Reference Rouse, Sutherland and Zureick-BrownRSZB21] verify these computations.