$\ell $-adic images of Galois for elliptic curves over $\mathbb {Q}$ (and an appendix with John Voight)

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 $ .

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 2.2. 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 $[\operatorname {GL}_2({\widehat {\mathbb {Z}}})\!:\!H]=[\operatorname {GL}_2(N)\!:\!H(N)]$ .

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.

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.

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.

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 $p\ne \ell \le 37$ .