2 A Mordell–Weil sieve

In this section, we present a Mordell–Weil sieve and apply it to prove Theorem 1.1.

We first describe how to obtain a suitable model for $X_0(N)$ for $N \in \mathcal {N}$ . Let g denote the genus of the modular curve $X_0(N)$ . We start by computing a basis $f_1, \ldots , f_g$ of cusp forms for $S_2(\Gamma _0(N))$ with integer Fourier coefficients such that the Atkin–Lehner involution $w_N$ satisfies $w_N(f_1) = f_1$ and $w_N(f_i) = -f_i$ for $2 \leq i \leq g$ (we refer to such a basis as a diagonalised basis). For each $N \in \mathcal {N}$ , the curve $X_0(N)$ is nonhyperelliptic of genus $>2$ and we may obtain a nonsingular model for $X_0(N)$ over $\mathbb {Q}$ in $\mathbb {P}^{g-1}_{x_1, \ldots , x_g}$ as the image of the canonical embedding on the cusp forms $f_1, \ldots , f_g$ . The details of this (standard) procedure are described in [Reference Galbraith9, pages 17–38], and the Magma code we used to do this is adapted from [Reference Ozman and Siksek14].

With this procedure, the Atkin–Lehner involution $w_N$ on this model is given by the map $(x_1 : x_2 : \cdots : x_g) \mapsto (-x_1 : x_2 : \cdots : x_g)$ . We denote by $ \psi : X_0(N) \rightarrow X_0^+(N) $ the degree $2$ map induced by quotienting by $w_N$ . In each case, we found that the projection map onto the coordinates $x_2, \ldots , x_g$ had degree $2$ and image $X_0^+(N)$ (and not some quotient of $X_0^+(N)$ ), so that the map $\psi $ is given by

$$ \begin{align*} \psi : X_0(N) & \longrightarrow X_0^+(N) \\ (x_1 : x_2 : \cdots : x_g) & \longmapsto (x_2 : \cdots : x_g). \end{align*} $$

In fact, we then obtained a Weierstrass model for $X_0^+(N)$ and composed $\psi $ with this transformation (see the example in Section 3). The reason for using a diagonalised model for $X_0(N)$ is twofold. First, it forces the coordinates of a quadratic point to be of a certain shape, as we see below. Second, it greatly speeds up the computations we perform in the sieving step.

Let $K = \mathbb {Q}(\sqrt {d})$ be a quadratic field and write $\sigma $ for the generator of $\mathrm {Gal}(K/\mathbb {Q})$ . Suppose that $P \in X_0(N)(K) \backslash X_0(N)(\mathbb {Q})$ (equivalently, P is a noncuspidal quadratic point). In projective coordinates, we may write

$$ \begin{align*} P & = (a_1 + b_1\sqrt{d} : a_2 + b_2 \sqrt{d} : \cdots : a_g + b_g \sqrt{d}), \\ w_N(P) & = (-a_1 - b_1\sqrt{d} : a_2 + b_2\sqrt{d} : \cdots : a_g + b_g \sqrt{d}), \quad\text{and} \\ P^\sigma & = (a_1 - b_1\sqrt{d} : a_2 - b_2 \sqrt{d} : \cdots : a_g - b_g \sqrt{d}), \end{align*} $$

where $a_i, b_i \in \mathbb {Z}$ for $1 \leq i \leq g$ .

As discussed in the introduction, thanks to the work of Box, and Najman and Vukorepa in [Reference Box5, Reference Najman and Vukorepa12], we know that $\psi (P) = \psi (P^\sigma ) \in X_0^+(N)(\mathbb {Q})$ , or equivalently, ${w_N(P) = P^\sigma }$ . It follows that

$$ \begin{align*} P = (b_1 \sqrt{d} : a_2 : \cdots : a_g), \end{align*} $$

with $b_1 \neq 0$ , and we may assume that $\gcd (b_1,a_2, \ldots , a_g) = 1$ by rescaling if necessary.

We now present the sieve in the case that $N \neq 65$ . In the case where $N = 65$ , we will need to adapt the sieve slightly, and we discuss this case in the proof of Theorem 1.1. For each $N \neq 65$ , we have $X_0^+(N)(\mathbb {Q}) \cong \mathbb {Z}$ , and we let R denote a generator of the Mordell–Weil group, so that

$$ \begin{align*} \psi(P) = m \cdot R \quad \text{for some } m \in \mathbb{Z}.\end{align*} $$

Let $\ell $ be a prime of good reduction for our models of $X_0(N)$ and $X_0^+(N)$ , and consider the following commutative diagram, where $\sim $ denotes reduction mod $\ell $ , or a prime of K above $\ell $ :

By commutativity, $\psi _\ell (\widetilde {P}) = m \cdot \widetilde {R}$ , so that $\widetilde {P} \in \psi _\ell ^{-1}(m \cdot \widetilde {R})$ . Write $G_\ell $ for the order of $\widetilde {R}$ in the group $\widetilde {X}_0^+(N)(\mathbb {F}_\ell )$ . Then,

$$ \begin{align*}\psi_l(\widetilde{P}) = m \cdot \widetilde{R} = m_0 \cdot \widetilde{R}\end{align*} $$

for some integer $m_0$ satisfying $0 \leq m_0 < G_\ell $ and $m \equiv m_0 \pmod {G_\ell }$ . We note that $\psi _\ell ^{-1}(m \cdot \widetilde {R}) = \psi _\ell ^{-1}(m_0 \cdot \widetilde {R})$ . For each integer $m_1$ with $0 \leq m_1 < G_\ell $ , we explicitly compute the set $\psi _\ell ^{-1}(m_1 \cdot \widetilde {R}) \subset \widetilde {X}_0(N)(\mathbb {F}_{\ell ^2})$ . There are three cases.

1. (i) The set $\psi _\ell ^{-1}(m_1 \cdot \widetilde {R})$ consists of a pair of distinct points defined over $\mathbb {F}_\ell $ .

   If $\ell $ is inert or ramifies in K, then $\psi _\ell ^{-1}(m \cdot \widetilde {R})$ will not consist of a pair of distinct points defined over $\mathbb {F}_{\ell }$ , and so $ m \not \equiv m_1 \pmod {G_\ell }$ .

2. (ii) The set $\psi _\ell ^{-1}(m_1 \cdot \widetilde {R})$ consists of a pair of points defined over $\mathbb {F}_{\ell ^2}$ (with each point not defined over $\mathbb {F}_\ell $ ).

   If $\ell $ splits or ramifies in K, then $\psi _\ell ^{-1}(m \cdot \widetilde {R})$ will not consist of a pair of points defined over $\mathbb {F}_{\ell ^2}$ , and so $ m \not \equiv m_1 \pmod {G_\ell }$ .

3. (iii) The set $\psi _\ell ^{-1}(m_1 \cdot \widetilde {R})$ consists of a single point defined over $\mathbb {F}_\ell $ .

Verifying the splitting behaviour of the prime $\ell $ in cases (i) and (ii) leaves us with a list of possible values for $m \pmod {G_\ell }$ .

We may then repeat this process with a list of primes $\ell _1, \ldots , \ell _s$ . For each i with $1 \leq i \leq s$ , we obtain a list of possibilities for $m \pmod {G_{\ell _i}}$ . This gives a system of congruences that we may solve using the Chinese remainder theorem to obtain a list of possibilities for $m \pmod {\mathrm {lcm}(G_{\ell _i})_{1 \leq i \leq s)}}$ . If no solution exists to this system of congruences, then we obtain a contradiction and conclude that $X_0(N)(K) = X_0(N)(\mathbb {Q})$ .

Proof of Theorem 1.1 for $N \ne 65$ .

Let $N \in \mathcal {N} \backslash \{65\}$ . We start by proving that if $d \in \mathcal {D}_N$ , then $X_0(N)(\mathbb {Q}(\sqrt {d})) \ne X_0(N)(\mathbb {Q})$ . We computed the preimages $\psi ^{-1}(t \cdot R)$ for $t \in \mathbb {Z}$ with $\lvert t \rvert \leq 5$ and verified the field of definition of the points we obtained. For each $d \in \mathcal {D}_N$ , we found a pair of quadratic points in $X_0(N)(\mathbb {Q}(\sqrt {d}))$ .

For the converse, we suppose that $\lvert d \rvert < 100$ with $d \notin \mathcal {D}_N$ and aim to prove that $X_0(N)(\mathbb {Q}(\sqrt {d})) = X_0(N)(\mathbb {Q})$ . We note that this is immediate if $\mathbb {Q}(\sqrt {d}) = \mathbb {Q}$ , so we assume that $\mathbb {Q}(\sqrt {d})$ is a quadratic field. Suppose, for a contradiction, that ${P \in X_0(N)(\mathbb {Q}(\sqrt {d})) \backslash X_0(N)(\mathbb {Q})}$ . We applied the Mordell–Weil sieve described above with the following (ordered) choice of primes (we discuss this choice in Remark 2.1):

(2.1) $$ \begin{align} \mathcal{L} = \{ \ell \mid d : \ell \nmid 2N \} \cup \{ \ell < 1000 : \ell \nmid 2N \text{ and } q \nmid G_\ell \text{ for primes } q> 7 \}. \end{align} $$

In each case, this led to a contradiction.

The total computation time for the proof of Theorem 1.1 was 2500 seconds running on a 2200 MHz AMD Opteron.

Although we have presented this sieve for certain specific bielliptic modular curves $X_0(N)$ , the sieve could be suitably adapted to compute quadratic points on a wider range of curves. Indeed, it should even be possible to apply a similar sieve to compute quadratic points on any curve X with a degree $2$ quotient of genus $\geq 1$ , if there are finitely many quadratic points on X not arising as pullbacks of rational points on this quotient, and that these have all been computed. Although, as in the $X_0(43)$ case discussed above, there may be obstructions to the sieving process succeeding.

3 Example computations

In this section, we provide some details of computations in the case where $N = 53$ . We start by obtaining a model for the genus $4$ curve $X_0(53)$ on which the Atkin–Lehner involution acts diagonally. By searching for relations between a diagonalised basis of cusp forms, we obtain the following model in $\mathbb {P}^3_{x_1,x_2,x_3,x_4}$ :

$$ \begin{align*} & x_1^2 - x_2^2 + 2x_2x_3 - 6x_2x_4 + 11x_3^2 - 6x_3x_4 - x_4^2 = 0, \\ & x_1^3 - x_1x_2^2 + 2x_1x_2x_3 - 6x_1x_2x_4 + 11x_1x_3^2 - 6x_1x_3x_4 - x_1x_4^2 = 0, \\ & x_1^2x_2 - x_2^3 + 2x_2^2x_3 + 5x_2x_3^2 + 5x_2x_4^2 - 6x_3^2x_4 + 6x_4^3 = 0, \\ & x_1^2x_3 - x_2^2x_3 + 2x_2x_3^2 - 6x_2x_3x_4 + 11x_3^3 - 6x_3^2x_4 - x_3x_4^2 = 0, \\ & x_1^2x_4 - x_2x_3^2 + 3x_2x_3x_4 - 5x_2x_4^2 + 10x_3^2x_4 - 6x_3x_4^2 = 0, \\ & x_2^2x_4 - x_2x_3^2 + x_2x_3x_4 + x_2x_4^2 - x_3^2x_4 + x_4^3 = 0. \end{align*} $$

The equations for the Atkin–Lehner involution on this particular model are given by $w_{53} : (x_1 : x_2 : x_3 : x_4) \mapsto (-x_1 : x_2 : x_3 : x_4)$ . The degree $2$ map to the elliptic curve $X_0^+(53)$ is then simply given by the projection map $(x_1 : x_2 : x_3 : x_4) \mapsto (x_2 : x_3 : x_4)$ . We then apply a transformation to take the image of this projection map to the following Weierstrass form:

$$ \begin{align*} X_0^+(53): \; Y^2Z+XYZ + YZ^2 = X^3-X^2Z. \end{align*} $$

The map $\psi $ is given by

$$ \begin{align*} \psi: X_0(53) & \longrightarrow X_0^+(53) \\ (x_1 : x_2 : x_3 : x_4) & \longmapsto (x_2x_3 + x_3x_4 : x_2^2 + x_2x_4 - x_3x_4 + x_4^2 : x_3x_4). \end{align*} $$

We choose $R = (0 : -1 : 1) \in X_0^+(53)(\mathbb {Q})$ as a generator of the Mordell–Weil group.

We now exhibit some steps in the sieving process for $d = -47$ . As in the previous section, we will assume that $P \in X_0(53)(\mathbb {Q}(\sqrt {d})) \backslash X_0(53)(\mathbb {Q})$ and write $\psi (P) = m \cdot R$ . We apply the sieve with the primes $ \ell = 5,7$ and $11$ . The prime $\ell = 5$ is inert in $\mathbb {Q}(\sqrt {d})$ and we find that $m \equiv 3$ or $5 \pmod {6}$ . Next, $\ell = 7$ splits and we find that $m \equiv 0, 3, 4, 7$ or $11 \pmod {12}$ . Combining this with the previous condition, we have $m \equiv 3$ or $11 \pmod {12}$ . Finally, the prime $\ell = 11$ is inert and we find that $m \equiv 1, 2, 5, 7$ or $10 \pmod {12}$ , which is a contradiction. We conclude that $X_0(53)(\mathbb {Q}(\sqrt {-47})) = X_0(53)(\mathbb {Q})$ .

We have in fact proven that $X_0(53)(K) = X_0(53)(\mathbb {Q})$ for any quadratic field K in which $5$ and $11$ are inert and $7$ splits. In a similar vein, when $d = 3$ , we achieved a contradiction using only the prime $\ell = 3$ , and this proves that $X_0(53)(K) = X_0(53)(\mathbb {Q})$ for any quadratic field K in which $3$ ramifies. This type of result is similar to those appearing in [Reference Najman and Trbović11], and we could seek to prove more results along these lines, but we do not pursue this here.

To verify that the sieve is working as expected, we can try applying it for a value ${d \in \mathcal {D}_{53}}$ . For example, applying the sieve with $d = -11 \in \mathcal {D}_{53}$ and the primes in $\mathcal {L}$ (defined as in (2.1)) outputs a list of possibilities for $m \pmod {\mathrm {lcm}(G_\ell )_{\ell \in \mathcal {L}}}$ , where $\mathrm {lcm}(G_\ell )_{\ell \in \mathcal {L}} = 63504000$ . We find that either $m = 1$ or that $m \ge 1905121$ . The fact that $m = 1$ remains as a possibility is because $\psi ^{-1}(1 \cdot R)$ consists of a pair of quadratic points defined over $\mathbb {Q}(\sqrt {-11})$ .

It is interesting to consider how the results of Theorem 1.1 overlap with the results one may obtain by applying the techniques of [Reference Ozman13], which give a criterion for testing whether $X_0(N)(\mathbb {Q}(\sqrt {d})) = X_0(N)(\mathbb {Q})$ by checking local points on the curve $X_0^{(d)}(N)$ . This curve is the quadratic twist of $X_0(N)$ by the Atkin–Lehner involution $w_N$ and the quadratic extension $\mathbb {Q}(\sqrt {d}) / \mathbb {Q}$ (see [Reference Clark7, page 628] for a precise definition). For $N = 53$ , in Theorem 1.1, we prove that $X_0(N)(\mathbb {Q}(\sqrt {d})) = X_0(N)(\mathbb {Q})$ for $117$ values of d, with d squarefree and $\lvert d \rvert < 100$ . Applying [Reference Ozman13, Theorem 1.1], we reproduced these results for $94$ of these values. (As discussed in [Reference Banwait1, page 39], some care is needed to interpret correctly parts (5) and (6) of [Reference Ozman13, Theorem 1.1].) For some values of d, where our sieving method works, but applying [Reference Ozman13, Theorem 1.1] fails, we can often obtain examples of curves that violate the Hasse principle. Continuing our example with $d = -47$ , we find that $X_0^{(-47)}(53)(\mathbb {Q})$ has points everywhere locally by applying [Reference Ozman13, Theorem 1.1]. However, $X_0^{(-47)}(53)(\mathbb {Q}) = \emptyset $ , as proven above. Similar examples are considered in [Reference Ozman13, pages 344–346], where a standard Mordell–Weil sieve is applied to twists of a hyperelliptic curve.