Elliptic Curves over Totally Real Cubic Fields are Modular

TL;DR

This paper proves that all elliptic curves over totally real cubic fields are modular, extending previous results from quadratic fields and recent breakthroughs in the field.

Contribution

It establishes the modularity of elliptic curves over totally real cubic fields, advancing the understanding of elliptic curves in higher degree number fields.

Findings

All elliptic curves over totally real cubic fields are modular.

Builds on prior work for quadratic fields and recent breakthroughs.

Extends modularity results to cubic fields.

Abstract

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

Equations

$$X(K)=X(\mathbb{Q})\cong \mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}.$$

$$X:y^2=(x^2+x-1)(x^6-5x^5-9x^3-5x-1).$$

$$J(\mathbb{Q})=\frac{\mathbb{Z}}{24\mathbb{Z}}\cdot \left[{\infty }_{-}-{\infty }_{+}\right]+\frac{\mathbb{Z}}{2\mathbb{Z}}\cdot \left[3(0,-1)-3{\infty }_{+}\right].$$

$$J\sim A_1\times A_2\times A_3$$

$$F_1(x_1,x_2)=(x_1^2+10x_1{x}_2+5x_2^2{)}^3,F_2:=x_1{x}_2^5,$$

$$G_1(y_1,y_2)=64\cdot {\left(y_1\cdot (y_1^2+7y_2^2)\cdot (y_1^2-7y_1{y}_2+14y_2^2)\cdot (5y_1^2-14y_1{y}_2-7y_2^2)\right)}^3,$$

$$G_2(y_1,y_2)=(y_1^3-7y_1^2{y}_2+7y_1{y}_2^2+7y_2^3{)}^7.$$

$$j:X(\mathrm{b}5)\to X(1),(x_1:x_2)\mapsto (F_1(x_1,x_2):F_2(x_1,x_2)),$$

$$j:X(\mathrm{ns}7)\to X(1),(y_1,y_2)\mapsto (G_1(y_1,y_2):G_2(y_1,y_2)).$$

$$C:F_1(x_1,x_2)G_2(y_1,y_2)=F_2(x_1,x_2)G_1(y_1,y_2).$$

$$D:5u^6-50u^{5}v+206u^4{v}^2-408u^3{v}^3+321u^2{v}^4+10u{v}^5-100v^6+9u^4{w}^2-$$

$$60u^{3}v{w}^2+80u^2{v}^2{w}^2+48u{v}^3{w}^2+15v^4{w}^2+3u^2{w}^4-10uv{w}^4+6v^2{w}^4-w^6=0.$$

$$p_1=(i:0:1),p_2=(-i:0:1),p_3=(0:\frac{1}{\sqrt{5}}:1),p_4=(0:-\frac{1}{\sqrt{5}}:1).$$

$$(-4{\eta }^2+21\eta +7:-{\eta }^2+7\eta :14),(4{\eta }^2-21\eta -7:{\eta }^2-7\eta :14).$$

$$J({\mathbb{F}}_3)\cong \mathbb{Z}/7\mathbb{Z}\times \mathbb{Z}/(7\cdot 23)\mathbb{Z},$$

$$J({\mathbb{F}}_{17})\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/(2^2\cdot 7^3\cdot 31\cdot 271)\mathbb{Z}.$$

$$(w_5-1)([3c_0-3c_{\infty }])=6[c_{\infty }-c_0]=[c_0-c_{\infty }].$$

$$(x-w_5(x))\sim k\cdot (c_0-c_{\infty })$$

$$\left(y-w_5(y)\right)\sim k\cdot \left(c_{0,{\mathbb{F}}_3}-c_{\infty ,{\mathbb{F}}_3}\right)$$

$$M={\left(({\omega }_i/d{u}_j)(P_j)\right)}_{i=1,2,3,4;j=1,2,3}.$$

Full text

Derickx is supported by Simons Foundation grant 550033. Najman is supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). Siksek is supported by EPSRC LMF: L-Functions and Modular Forms Programme Grant EP/K034383/1.

1. Introduction

Let $K$ be a totally real number field and let $E$ be an elliptic curve over $K$ with conductor $\mathcal{N}$. It is conjectured that such a curve $E$ is modular in the following sense: there is a level $\mathcal{N}$ Hilbert newform $\mathfrak{f}$ over $K$ of parallel weight $2$ and rational Hecke eigenvalues such that $L(E,s)=L(\mathfrak{f},s)$, where the $L$-function on the left is the Hasse–Weil $L$-function of $E$, and the $L$-function on the right is the Hecke $L$-function of $\mathfrak{f}$. This modularity conjecture is the natural generalization to totally real fields of the Shimura–Taniyama conjecture for elliptic curves over the rationals. The latter is a celebrated theorem due to Wiles [21], Breuil, Conrad, Diamond and Taylor [20]. The earliest results towards the modularity conjecture for elliptic curves going beyond the rationals were due to Jarvis and Manoharmayum [12], and established modularity of semistable elliptic curves over ${\mathbb{Q}}(\sqrt{2})$ and ${\mathbb{Q}}(\sqrt{17})$. In the last 10 years there has been a dramatic strengthening of modularity lifting theorems due to, for example, Breuil and Diamond [5], Kisin [15], Gee [9], and Barnet-Lamb, Gee and Geraghty [2], [3]. The following theorem is easily deduced from these aforementioned modularity lifting theorems and by now standard modularity switching arguments due to Wiles and to Manoharmayum [17]. (A proof is given by Freitas, Le Hung and Siksek [7, Theorems 3 and 4] but the arguments are well-known).

Theorem 1**.**

Let $p=3$, $5$ or $7$. Let $E$ be an elliptic curve over a totally real field $K$, and write $\overline{\rho}_{E,p}$ for its mod $p$ representation. Suppose that $\overline{\rho}_{E,p}(G_{K(\zeta_{p})})$ is absolutely irreducible. Then $E$ is modular.

A hypothetical non-modular $E/K$ would therefore necessarily have small mod $p$ image for $p=3$, $5$, $7$ and would give rise to a $K$-point on one of a number of modular curves—we make this precise later. In [7], the real quadratic points of these modular curves are shown to be either cuspidal, or to correspond to elliptic curves that have complex multiplication, or rational $j$-invariants, or that are ${\mathbb{Q}}$-curves. The authors deduce the following.

Theorem 2** (Freitas, Le Hung and Siksek).**

Elliptic curves over real quadratic fields are modular.

Whilst direct computation is used in [7] to study the real quadratic points on the relevant modular curves, Thorne [19] uses Iwasawa theory to control points on some of these modular curves over ${\mathbb{Z}}_{p}$-extensions of ${\mathbb{Q}}$, and deduces the following remarkable theorem.

Theorem 3** (Thorne).**

Let $p$ be a prime, and let $K$ be a number field which is contained in the cyclotomic ${\mathbb{Z}}_{p}$-extension of ${\mathbb{Q}}$. Let E be an elliptic curve over $K$. Then $E$ is modular.

Recently Theorem 1 was substantially strengthened in the cases $p=5$ and $p=7$, respectively by Thorne [18] and Kalyanswamy [13]. This means that several difficult steps in the proof of Theorem 2 can now be eliminated. In this paper we build on these theorems of Thorne and Kalyanswamy to prove the following.

Theorem 4**.**

Let $K$ be a totally real cubic number field. Let $E$ be an elliptic curve over $K$. Then $E$ is modular.

The computations in this paper were carried out in the computer algebra system Magma [4]. The reader can find the Magma scripts for verifying these computations at:

http://homepages.warwick.ac.uk/staff/S.Siksek/progs/cubicmodularity/

2. Images mod $3$, $5$, $7$ and modularity

Let $p\geq 3$ be a prime. Write $B(p)$ for a Borel subgroup of $\operatorname{GL}_{2}({\mathbb{F}}_{p})$, and $C_{\mathrm{s}}(p)$ and $C_{\mathrm{ns}}(p)$ respectively for a split and non-split Cartan subgroups. Let $C_{\mathrm{s}}^{+}(p)$ and $C_{\mathrm{ns}}^{+}(p)$ respectively be their normalizers. The three groups $B(p)$, $C_{\mathrm{s}}^{+}(p)$ and $C_{\mathrm{ns}}^{+}(p)$ correspond to modular curves defined over ${\mathbb{Q}}$ which are usually denoted by $X_{0}(p)$, $X_{\mathrm{split}}(p)$ and $X_{\mathrm{nonsplit}}(p)$. Instead we shall mostly follow the notation of [7] and denote these modular curves by: $X(\mathrm{b}p)$, $X(\mathrm{s}p)$ and $X(\mathrm{ns}p)$. The following is well-known. For a proof, see [7, Section 2.3].

Proposition 2.1**.**

*Let $E$ be an elliptic curve over a totally real field $K$, and let $p\geq 3$ be a rational prime. If $\overline{\rho}(G_{K(\zeta_{p})})$ is absolutely reducible, then $\overline{\rho}(G_{K})$ is contained either in a Borel subgroup, or in the normalizer of a Cartan subgroup. In this case $E$ gives rise to a non-cuspidal $K$-point on $X(\mathrm{b}p)$, $X(\mathrm{s}p)$ or $X(\mathrm{ns}p)$. *

When the additional assumption $K\cap{\mathbb{Q}}(\zeta_{p})={\mathbb{Q}}$ is satisfied, we can substantially improve on Proposition 2.1. We shall only need this improvement for $p=3$; in this case, as $K$ is real, it is certainly satisfies the additional assumption. The following is part of Proposition 4.1 in [7].

Proposition 2.2**.**

Let $K$ be a totally real number field and let $E$ be an elliptic curve over $K$. Write $\overline{\rho}=\overline{\rho}_{E,3}$. Suppose $\overline{\rho}(G_{K(\zeta_{p})})$ is absolutely reducible. Then $\overline{\rho}(G_{K})$ is conjugate to a subgroup of $B(3)$ or $C_{\mathrm{s}}^{+}(3)$.

We shall need the following strengthening of Theorem 1 for $p=5$ due to Thorne [18].

Theorem 5** (Thorne).**

Let $K$ be a totally real field and $E$ an elliptic curve over $K$. Suppose $5$ is not a square in $K$, and $\overline{\rho}_{E,5}$ is irreducible. Then $E$ is modular.

We shall need the following theorem of Kalyanswamy [13, Proposition 4.3 and Theorem 4.4] which improves on Theorem 1 for $p=7$.

Theorem 6** (Kalyanswamy).**

Let $K$ be a totally real field and $E$ an elliptic curve over $K$. Suppose

• •

$K\cap{\mathbb{Q}}(\zeta_{7})={\mathbb{Q}}$.

• •

$\overline{\rho}_{E,7}$* is irreducible.*

• •

$\overline{\rho}_{E,7}(G_{K})$* is not conjugate to a subgroup of $C_{\mathrm{ns}}^{+}(7)$. *

Then $E$ is modular.

Kalyanswamy’s theorem is somewhat more precise, but we shall not need its full strength.

In this paper we shall deal with four modular curves, that we denote by $X(\mathrm{b}3,\mathrm{b}5)$, $X(\mathrm{s}3,\mathrm{b}5)$, $X(\mathrm{b}5,\mathrm{b}7)$, $X(\mathrm{b}5,\mathrm{ns}7)$. For the details of the notation, and the precise modular interpretation of these curves we refer to [7, Section 2.2.2]. These are often conveniently thought of as normalizations of fibre products: if $p$, $q$ are distinct primes and $\mathrm{u}$, $\mathrm{v}\in\{\mathrm{b},\mathrm{s},\mathrm{ns}\}$ then $X(\mathrm{u}p,\mathrm{v}q)$ is the normalization of $X(\mathrm{u}p)\times_{X(1)}X(\mathrm{v}q)$.

3. Modularity of Elliptic Curves over ${\mathbb{Q}}(\zeta_{7})^{+}$

In this section prove Theorem 4 for $K={\mathbb{Q}}(\zeta_{7})^{+}$.

Lemma 3.1**.**

Let $K={\mathbb{Q}}(\zeta_{7})^{+}$. Let $E$ be an elliptic curve defined over $K$. Then $E$ is modular.

Proof.

By Theorem 5 we may suppose that $\overline{\rho}_{E,5}$ is reducible. By Theorem 1 and Proposition 2.2 we may suppose that the image of $\overline{\rho}_{E,3}$ is contained in $B(3)$ or $C_{\mathrm{s}}^{+}(3)$. Thus $E$ gives rise to a non-cuspidal $K$-point on one of the two modular curves $X(\mathrm{b}3,\mathrm{b}5)$, $X(\mathrm{s}3,\mathrm{b}5)$. It is shown in [7, Section 5.4.2] that these are in fact elliptic curves defined over ${\mathbb{Q}}$ with Cremona labels 15A1 and 15A3. We computed the Mordell–Weil groups $X(K)$ for $X=X(\mathrm{b}3,\mathrm{b}5)$, $X(\mathrm{s}3,\mathrm{b}5)$ using Magma. In both cases we found

[TABLE]

In particular $E$ gives rise to ${\mathbb{Q}}$-point on $X$ and so is a twist of an elliptic curve defined over ${\mathbb{Q}}$. It is therefore modular by [20]. ∎

4. An overview of the proof of Theorem 4

The remainder of the paper is devoted to the proof of the following two theorems.

Theorem 7**.**

*Let $K$ be a totally real cubic field. Then $X(\mathrm{b}5,\mathrm{b}7)(K)$ consists only of cusps. *

Theorem 8**.**

*Let $K$ be a cubic field. Then $X(\mathrm{b}5,\mathrm{ns}7)(K)$ consists only of cusps. *

In this section we complete the proof of Theorem 4 assuming Theorems 7 and 8. Let $K$ be a totally real cubic field and $E$ an elliptic curve over $K$. We would like to show that $E$ is modular. Suppose that it is not modular. By Theorem 5, the representation $\overline{\rho}_{E,5}$ is reducible. By Lemma 3.1 we know $K\neq{\mathbb{Q}}(\zeta_{7})^{+}$ and thus $K\cap{\mathbb{Q}}(\zeta_{7})={\mathbb{Q}}$. Thus by Theorem 6 the image of $\overline{\rho}_{E,7}$ is contained in $B(7)$ or in $C_{\mathrm{ns}}^{+}(7)$. Thus if $E$ is not modular then $E$ gives rise to a non-cuspidal $K$-point on either $X(\mathrm{b}5,\mathrm{b}7)$ or $X(\mathrm{b}5,\mathrm{ns}7)$. But by Theorems 7 and 8 there are no such points, giving a contradiction.

In summary, to prove Theorem 4 all we have to do is prove Theorems 7 and 8.

5. Proof of Theorem 7

Let $X=X(\mathrm{b}5,\mathrm{b}7)$ (in standard notation denoted by $X_{0}(35)$). It is known that $X$ has four ${\mathbb{Q}}$-points and that these are cusps. Let $K$ be a totally real cubic field. For the proof of Theorem 7 it will be sufficient to show that $X(K)=X({\mathbb{Q}})$. Suppose $P\in X(K)\backslash X({\mathbb{Q}})$. Let $P_{1}$, $P_{2}$, $P_{3}$ be the conjugates of $P$ given by the three embeddings of $K$ in $\overline{{\mathbb{Q}}}$, and write $D=P_{1}+P_{2}+P_{3}$. Then $D$ is an irreducible ${\mathbb{Q}}$-rational divisor on $X$ of degree $3$. We shall determine all the irreducible ${\mathbb{Q}}$-rational divisors of degree $3$ on $X$ and show that none of them arise from totally real cubic points, giving a contradiction.

The arithmetic of $X$ and its Jacobian are studied in [7, Section 5.1]. The curve $X$ is hyperelliptic of genus $3$. A model for $X$, derived by Galbraith [8, Section 4.4], is given by

[TABLE]

Write $\infty_{\pm}$ for the two points at infinity. Write $J$ for $J_{0}(35)$—the Jacobian of $X$. Then

[TABLE]

Let $D_{1},\dots,D_{48}$ be rational divisors of degree [math] on $X$ representing the $48$ classes in $J({\mathbb{Q}})$, and let $D_{i}^{\prime}=D_{i}+3\infty_{+}$. Recall that $D$ is an irreducible ${\mathbb{Q}}$-rational divisor of degree $3$. Then $D\sim D_{i}^{\prime}$ for some $i$. We shall write $\mathcal{L}(D_{i}^{\prime})$ for the Riemann–Roch space corresponding to $D_{i}^{\prime}$ and $\lvert D_{i}^{\prime}\rvert$ for the corresponding complete linear system. By Riemann–Roch and Clifford’s inequality, $\dim\mathcal{L}(D_{i}^{\prime})=1$ or $2$. Moreover, if $\dim\mathcal{L}(D_{i}^{\prime})=2$, then $\lvert D_{i}^{\prime}\rvert$ contains a base point (c.f. [1, Chapter I, Exercise D.9]), and therefore cannot contain an irreducible divisor. To sum up, $D\sim D_{i}^{\prime}$ for some $1\leq i\leq 48$ such that $\dim\mathcal{L}(D_{i}^{\prime})=1$. We computed these spaces using Magma; for this Magma uses an algorithm of Hess [10]. We found that $\dim\mathcal{L}(D_{i}^{\prime})=1$ for precisely $44$ of the $48$ divisors $D_{i}^{\prime}$. For these, letting $f_{i}$ be a ${\mathbb{Q}}$-basis for $\mathcal{L}(D_{i}^{\prime})$, gives $D=D_{i}^{\prime}+\operatorname{div}(f_{i})$ for some $i$. We found that precisely $28$ of the effective degree $3$ divisors $D_{i}^{\prime}+\operatorname{div}(f_{i})$ are irreducible. However, all of these split over a cubic field with a complex embedding giving the required contradiction.

6. The modular curve $X(\mathrm{b}5,\mathrm{ns}7)$

We shall henceforth restrict our attention to $X(\mathrm{b}5,\mathrm{ns}7)$. To simplify the notation we write $X=X(\mathrm{b}5,\mathrm{ns}7)$. We denote the Jacobian of $X$ by $J=J(\mathrm{b}5,\mathrm{ns}7)$. The curve $X$ and its Jacobian $J$ are studied in Le Hung’s thesis [11, Section 6.4] and we make extensive use of his results. In particular, this curve is non-hyperelliptic and has genus $6$.

6.1. The Jacobian $J=J(\mathrm{b}5,\mathrm{ns}7)$

Le Hung shows that

[TABLE]

where $\sim$ here denotes isogeny over ${\mathbb{Q}}$, and $A_{1}$, $A_{2}$, $A_{3}$ are modular abelian surfaces defined over ${\mathbb{Q}}$. Morever the $A_{i}$ are absolutely simple. The involution $w_{5}$ on $J$ is compatible with the isogeny and acts by multiplication by $1$, $-1$, $-1$ respectively on $A_{1}$, $A_{2}$, $A_{3}$. The analytic ranks of $A_{1}$, $A_{2}$, $A_{3}$ are respectively $2$, [math], [math]. In particular, by the work of Kolyvagin and Logachëv [16], the Mordell–Weil groups $A_{2}({\mathbb{Q}})$ and $A_{3}({\mathbb{Q}})$ are torsion. We immediately deduce the following.

Lemma 6.1**.**

Let $A/{\mathbb{Q}}$ be the abelian subvariety of $J$ that is the image of $w_{5}-1$. Then $A\sim A_{2}\times A_{3}$ has dimension $4$. Moreover, the Mordell–Weil group $A({\mathbb{Q}})$ is torsion.

6.2. Le Hung’s model for $X=X(\mathrm{b}5,\mathrm{ns}7)$

We need a good model for $X(\mathrm{b}5,\mathrm{ns}7)$. Le Hung [11, p. 47] gives a model which will be a good starting point for us. We briefly sketch Le Hung’s derivation of his model, but work with projective rather than affine coordinates. Later we explain how to derive a better model. The curves $X(\mathrm{b}5)$ and $X(\mathrm{ns}7)$ are both isomorphic to ${\mathbb{P}}^{1}$ over ${\mathbb{Q}}$. Let

[TABLE]

[TABLE]

and

[TABLE]

For appropriate choices of projective coordinates $(x_{1}:x_{2})$ for $X(\mathrm{b}5)$ and $(y_{1}:y_{2})$ on $X(\mathrm{ns}7)$, the $j$-maps are given by

[TABLE]

and

[TABLE]

As $X$ is the normalization of $X(\mathrm{b}5)\times_{X(1)}X(\mathrm{ns}7)$ we immediately deduce the following model for $X$ in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$:

[TABLE]

The curve $X$ is the normalization of this model. The parameterization $(x_{1}:x_{2})$ on $X(\mathrm{b}5)$ is chosen so that the [math] and $\infty$ cusps are $(x_{1}:x_{2})=(0:1)$ and $(x_{1}:x_{2})=(1:0)$ respectively. We shall denote these by $a_{0}$, $a_{\infty}$. Let $\zeta_{7}$ be a primitive $7$-th root of unity. Let $\eta=2(\zeta_{7}^{3}+\zeta_{7}^{-3})+3\in{\mathbb{Q}}(\zeta_{7})^{+}$. Then $F_{2}(\eta:1)=0$. The three cusps of $X(\mathrm{ns}7)$ are $(\eta:1)$ and its Galois conjugates. It follows that the cusps of $X$ are the points belonging to the normalization of $C$ lying above the points $(x_{1}:x_{2},y_{1}:y_{2})=(0:1,\eta:1)$, $(1:0,\eta:1)$ and their Galois conjugates. Although these points on $C$ are singular, it is easy to check (c.f. [7, Section 5.5.1]) that there is only one point on the normalization above each, and to deduce:

• •

$X$ has two Galois orbits of cusps, both of degree $3$ and defined over ${\mathbb{Q}}(\zeta_{7})^{+}$, which we denote by $c_{0}$, $c_{\infty}$;

• •

The three cusps in $c_{0}$ map to $a_{0}$, and the three cusps in $c_{\infty}$ map to $a_{\infty}$ on $X(\mathrm{b}5)$.

• •

The divisor of $x_{1}/x_{2}$ interpreted as a function on $X$ is $7\cdot\left(c_{0}-c_{\infty}\right)$. In particular, the class $\left[c_{0}-c_{\infty}\right]$ is an element of order $1$ or $7$. There are several ways to show that the divisor $c_{0}-c_{\infty}$ is not principal, and so its class has order $7$. One way is by direct computation using Magma, working with the model $D$ introduced below. Here is another way: we shall show below that $X$ has gonality $4$. As $c_{0}$, $c_{\infty}$ have degree $3$ they cannot be linearly equivalent.

6.3. A plane degree $6$ model for $X=X(\mathrm{b}5,\mathrm{ns}7)$

We used Magma to compute, starting with the model $C$, the canonical map and its image. The latter is indeed a smooth genus $6$ curve cut out in ${\mathbb{P}}^{5}$ by six homogeneous degree $2$ polynomials. By the Enriques–Babbage Theorem [1, p. 124], we know that $X$ is neither trigonal, nor isomorphic to a plane quintic. Moreover, as the factors $A_{i}$ of the Jacobian are $2$-dimensional and absolutely simple, we see that the curve is not bi-elliptic. It follows (c.f. [1, 209–210]) that $X$ has gonality $4$ and a degree $6$ planar model, with four ordinary double points as singularities. We used the inbuilt Magma implementation for writing down this model, and found that two of the four double points are defined over ${\mathbb{Q}}(i)$ and the other two over ${\mathbb{Q}}(\sqrt{5})$. After applying a ${\mathbb{Q}}$-rational automorphism of $\mathbb{P}^{2}$ to slightly simplify this degree $6$ model, it is given by the following equation:

[TABLE]

On this model $D$ the double points are

[TABLE]

It is clear that $D$ has an automorphism $(u:v:w)\mapsto(-u:-v:w)$. The curve $X$ has an obvious modular involution which is $w_{5}$. The following lemma proves that $w_{5}$ coincides with the automorphism $(u:v:w)\mapsto(-u:-v:w)$.

Lemma 6.2**.**

The ${\mathbb{Q}}$-rational automorphism group of $X(b5,ns7)$ is generated by $w_{5}$, i.e. $\operatorname{Aut}_{\mathbb{Q}}(X)=\langle w_{5}\rangle\cong{\mathbb{Z}}/2{\mathbb{Z}}$.

Proof.

As described in [1, p 210–211.] a degree $6$ planar curve with four ordinary double points such as $D$ has exactly five different $g_{4}^{1}$. Namely, one given by the pencil of quadrics going through all four points, and the other four coming from the pencil of lines through each of the $p_{i}$. Since none of the $p_{i}$ are ${\mathbb{Q}}$-rational, only the first $g_{4}^{1}$ is defined over ${\mathbb{Q}}$. Now every $g_{6}^{2}$ on such a curve is residual to a $g_{4}^{1}$. This means that there is only one ${\mathbb{Q}}$-rational $g_{6}^{2}$, namely the one corresponding to the degree $6$ model given by $u$, $w$, $v$ above. In particular every ${\mathbb{Q}}$-rational automorphism has to come from an automorphism $h:\mathbb{P}^{2}_{\mathbb{Q}}\to\mathbb{P}^{2}_{\mathbb{Q}}$ in the degree $6$ model. Such an automorphism $h$ has to preserve the singular locus $\left\{p_{1},p_{2},p_{3},p_{4}\right\}$ and is in fact uniquely determined by what it does on this singular locus. Of the $24$ automorphisms of $\mathbb{P}^{2}_{\overline{{\mathbb{Q}}}}$ preserving $\left\{p_{1},p_{2},p_{3},p_{4}\right\}$, only the ones of the form $(u:v:w)\mapsto(\pm u:\pm v:w)$ are ${\mathbb{Q}}$-rational. One easily sees that of these four only the identity and $(u:v:w)\mapsto(-u:-v:w)$ are actually automorphisms of the curve. ∎

Transferring $c_{0}$ and $c_{\infty}$ to our new model $D$, we find that they respectively are the Galois orbits of the following two points defined over ${\mathbb{Q}}(\eta)={\mathbb{Q}}(\zeta_{7})^{+}$ by

[TABLE]

We note that these are interchanged by $w_{5}:(u:v:w)\mapsto(-u:-v:w)$ as expected.

6.4. The Mordell–Weil group $A({\mathbb{Q}})$

In Lemma 6.1 we defined the abelian subvariety $A$ of $J$ as the image of $w_{5}-1$ and observed that $A({\mathbb{Q}})$ is torsion. We can now pin down $A({\mathbb{Q}})$ precisely. In particular, applying the function field class group algorithm of Hess [10] (implemented in Magma) to our model $D$ obtain

[TABLE]

and

[TABLE]

Hence $J({\mathbb{Q}})_{\mathrm{tors}}$ is isomorphic to a subgroup of ${\mathbb{Z}}/7{\mathbb{Z}}$. Recall that the class $[c_{0}-c_{\infty}]$ has order $7$. Thus $[c_{0}-c_{\infty}]$ generates $J({\mathbb{Q}})_{\mathrm{tors}}$. Now since $w_{5}$ interchanges $c_{0}$ and $c_{\infty}$,

[TABLE]

Therefore $[c_{0}-c_{\infty}]\in A({\mathbb{Q}})$. We have now proved the following.

Lemma 6.3**.**

$A({\mathbb{Q}})=({\mathbb{Z}}/7{\mathbb{Z}})\cdot[c_{0}-c_{\infty}]$.

7. Proof of Theorem 8

In this section we prove Theorem 8 thereby completing the proof of Theorem 4. Recall $X=X(\mathrm{b}5,\mathrm{ns}7)$. Write $X^{(3)}$ for the $3$-rd symmetric power of $X$. We shall prove the following result which immediately implies Theorem 8.

Proposition 7.1**.**

$X^{(3)}({\mathbb{Q}})=\{c_{0},\,c_{\infty}\}$.

Proof.

Let $x\in X^{(3)}({\mathbb{Q}})$. By Lemma 6.3 we have $(1-w_{5})[x-c_{\infty}]=\ell\cdot[c_{0}-c_{\infty}]$ for some $\ell\in{\mathbb{Z}}/7{\mathbb{Z}}$. As $w_{5}(c_{\infty})=c_{0}$ we may rewrite this as

[TABLE]

for some $k\in\{-3,\dotsc,3\}$. We write $x_{{\mathbb{F}}_{3}}$, $c_{0,{\mathbb{F}}_{3}}$, $c_{\infty,{\mathbb{F}}_{3}}\in X^{(3)}({\mathbb{F}}_{3})$ for the reductions of $x$, $c_{0}$, $c_{\infty}$ modulo $3$ respectively. It follows that

[TABLE]

where $y=x_{{\mathbb{F}}_{3}}$. Using our model $D$ we enumerated $X^{(3)}({\mathbb{F}}_{3})$; this has precisely $40$ elements. For each $y\in X^{(3)}({\mathbb{F}}_{3})$ and for each $k=-3,\dotsc,3$ we tested the relation (2) and found that it holds only for $y=c_{0,{\mathbb{F}}_{3}}$ and $k=1$ and for $y=c_{\infty,{\mathbb{F}}_{3}}$ and $k=-1$. We therefore deduce that $x_{{\mathbb{F}}_{3}}=c_{0,{\mathbb{F}}_{3}}$ or $c_{\infty,{\mathbb{F}}_{3}}$. We would like to conclude that $x=c_{0}$ or $c_{\infty}$. As $w_{5}$ swaps $c_{0}$ and $c_{\infty}$ and also their mod $3$ reductions, we may suppose that $x_{{\mathbb{F}}_{3}}=c_{\infty}$. Let $\mu:X^{(3)}\rightarrow J$ be given by $z\mapsto[z-c_{\infty}]$ and $t:J\rightarrow A$ be simply $t=w_{5}-1$. Since $x_{{\mathbb{F}}_{3}}=c_{\infty,{\mathbb{F}}_{3}}$, the point $(t\circ\mu)(x)\in A({\mathbb{Q}})$ belongs to the kernel of reduction $A({\mathbb{Q}})\rightarrow A({\mathbb{F}}_{3})$. However as $A({\mathbb{Q}})$ is torsion, this kernel of reduction is trivial [14, Appendix]. Thus $(t\circ\mu)(x)=0$. To conclude that $x=c_{\infty}$ it is now enough to check that $t\circ\mu$ is a formal immersion at $c_{\infty,{\mathbb{F}}_{3}}$, and for this we shall use the formal immersion criterion due to Derickx, Kamienny, Stein and Stoll [6, Proposition 3.7].

Write $\Omega_{X}\cong\Omega_{J}$ for the $6$-dimensional space of $1$-forms on $X/{\mathbb{F}}_{3}$. We would like to write down the $4$-dimensional subspace $t^{*}(\Omega_{A})$. We easily do this since it is precisely that $-1$-eigenspace of $w_{5}^{*}$ acting on $\Omega_{X}$, and we know the action of $w_{5}$ on our model $D$ from which can write down the corresponding action on the $1$-forms. Let $\omega_{1},\dotsc,\omega_{4}$ be an ${\mathbb{F}}_{3}$-basis for $t^{*}(\Omega_{A})$. To check the formal immersion criterion of Derickx et al. at $c_{\infty,{\mathbb{F}}_{3}}$ we need to check that a certain $4\times 3$ matrix defined in [6, Proposition 3.7], which we denote by $M$, has rank $3$. As $3$ is inert in ${\mathbb{Q}}(\zeta_{7})^{+}$, we have $c_{\infty,{\mathbb{F}}_{3}}=P_{1}+P_{2}+P_{3}$, where $P_{i}\in X({\mathbb{F}}_{27})$ are distinct. This slightly simplifies the description of the matrix $M$. Let $u_{j}\in{\mathbb{F}}_{27}(X)$ be a uniformizing element for $P_{j}$. Then $\omega_{i}/d{u_{j}}$ is a regular function at $P_{j}$ and we may evaluate $(\omega_{i}/d{u_{j}})(P_{j})\in{\mathbb{F}}_{27}$. The matrix is simply

[TABLE]

We computed $M$ and checked that it has rank $3$ as required. This completes the proof. ∎